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UBC Theses and Dissertations

New mechanisms for external field control of microscopic interactions in ultracold gases Li, Zhiying 2009

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New mechanisms for external field control of microscopic interactions in ultracold gases by Zhiying Li A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in The Faculty of Graduate Studies (Chemistry) THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) November, 2009 c© Zhiying Li 2009 Abstract This Thesis describes new mechanisms for controlling elastic and inelastic collisions of ultracold atoms and molecules with static electromagnetic and laser fields. The dynamical properties of ultracold atoms are usually tuned in experiments by apply- ing an external magnetic field to induce a Feshbach resonance. The work presented in this Thesis demonstrates the possibility of inducing and manipulating Feshbach resonances with electric fields. We discuss in detail the mechanisms of electric-field- induced resonances in ultracold mixtures of alkali metal atoms and demonstrate that electric fields may shift and split the magnetic resonances. We show that electric fields may spin up the collision complex of ultracold atoms and induce anisotropic scattering which may be exploited in experiments on many-body dynamics of ul- tracold gaseous mixtures. The mechanisms of electric-field-induced resonances de- scribed in this Thesis allow for two-dimensional control of inter-particle interactions, leading to total control over ultracold gases. To guide future experiments, we gener- ate accurate interaction potentials for ultracold Li–Rb mixtures by fitting positions and widths of experimentally measured Feshbach resonances. Ultracold atomic and molecular gases can be confined by laser fields in one or two dimensions which pro- duces an optical lattice of ultracold particles. We develop a multichannel scattering theory for collisions of atoms and molecules in two dimensions and explore the ef- fects of the confining laser potential on inelastic and reactive collisions of ultracold atoms and molecules in a 1D optical lattice. We show that ultracold collisions can be controlled in a quasi-2D geometry by varying the orientation of a magnetic field with respect to the confinement plane normal and demonstrate that the threshold energy dependence of cross sections for inelastic collisions in an optical lattice can be tuned by varying the confining potential and the magnetic field. Our results show that applying laser confinement in one dimension may stabilize ultracold sys- tems with large scattering lengths, which may open up interesting opportunities for studies of ultracold controlled chemistry and might lead to a new research direction of ultracold chemistry in restricted geometries. ii Table of Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviii Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xx 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Ultracold atoms and molecules−properties and applications . . . . . 1 1.2 Cooling techniques for atoms and molecules . . . . . . . . . . . . . . 5 1.3 Feshbach resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.4 Ultracold gases in restricted geometries . . . . . . . . . . . . . . . . 12 1.5 Thesis outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2 Background material . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.1 The adiabatic approximation . . . . . . . . . . . . . . . . . . . . . . 17 2.2 Scattering amplitude and cross section . . . . . . . . . . . . . . . . 20 2.2.1 Typical scattering experiment . . . . . . . . . . . . . . . . . 20 2.2.2 Time-independent Schrödinger equation . . . . . . . . . . . . 23 2.2.3 Differential cross section . . . . . . . . . . . . . . . . . . . . 25 2.3 Single-channel scattering theory . . . . . . . . . . . . . . . . . . . . 26 2.3.1 Free-particle solutions . . . . . . . . . . . . . . . . . . . . . . 26 2.3.2 Scattering wave function . . . . . . . . . . . . . . . . . . . . 28 2.3.3 Differential and integral cross sections . . . . . . . . . . . . . 30 2.3.4 Numerical calculation of the phase shift . . . . . . . . . . . . 31 2.4 Multi-channel scattering theory . . . . . . . . . . . . . . . . . . . . 33 2.4.1 Multi-channel theory . . . . . . . . . . . . . . . . . . . . . . 34 iii Table of Contents 2.4.2 Numerical integration of multi-channel equations . . . . . . 39 3 Accurate interatomic potentials from interplay of ultracold exper- iment and theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.1 Ultracold mixtures of 6Li and 87Rb . . . . . . . . . . . . . . . . . . 42 3.2 Fitting procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.3 Asymptotic bound state model . . . . . . . . . . . . . . . . . . . . . 44 3.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 4 Electric-field-induced Feshbach resonances in alkali metal mixtures 58 4.1 Why electric fields? . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 4.2 Atomic collisions in combined electric and magnetic fields . . . . . . 60 4.3 Li–Cs system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 4.3.1 Electric-field-induced Feshbach resonances . . . . . . . . . . 64 4.3.2 Anisotropy of ultracold scattering . . . . . . . . . . . . . . . 69 4.4 Li–Rb system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 4.4.1 Li–Rb collisions in combined electric and magnetic fields . . 76 4.4.2 Mechanism of electric-field-induced shifts of magnetic Fesh- bach resonances . . . . . . . . . . . . . . . . . . . . . . . . . 78 4.4.3 Splitting of Feshbach resonances in an electric field . . . . . 81 4.4.4 Collision dynamics in non-parallel electric and magnetic fields 85 4.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 5 Ultracold inelastic collisions in two dimensions . . . . . . . . . . . 91 5.1 Why 2D? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 5.2 Close coupling theory of collisions in two dimensions . . . . . . . . . 93 5.2.1 Scattering amplitude and cross section . . . . . . . . . . . . 93 5.2.2 Elastic collisions in two dimensions . . . . . . . . . . . . . . 95 5.2.3 Numerical calculation of phase shift in 2D geometry . . . . . 98 5.2.4 Inelastic collisions in two dimensions . . . . . . . . . . . . . 99 5.2.5 Magnetic dipole-dipole interaction in 2D . . . . . . . . . . . 105 5.3 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 5.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 6 Inelastic collisions in a quasi-2D trapped gas . . . . . . . . . . . . 118 6.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 6.2 Ultracold quasi-2D gas . . . . . . . . . . . . . . . . . . . . . . . . . 119 iv Table of Contents 6.3 Elastic collisions in quasi-2D geometry . . . . . . . . . . . . . . . . 121 6.4 Inelastic collisions in quasi-2D geometry . . . . . . . . . . . . . . . . 124 6.5 Threshold laws for inelastic collisions in quasi-2D . . . . . . . . . . 127 6.6 Molecular scattering in quasi-2D . . . . . . . . . . . . . . . . . . . . 128 6.7 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 6.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 7 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 Appendices A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 D List of publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 v List of Tables 3.1 Definition of quantum numbers used in this Thesis. . . . . . . . . . . 46 3.2 Positions and widths of 6Li–87Rb Feshbach resonances for magnetic fields below 2 kG determined from the coupled-channel calculations. The experimentally measured Feshbach resonances (and absence of resonances below 1.2 kG) are also included for comparison. The ex- perimentally determined width 4Bexpt is the full width at half maxi- mum of the trap loss feature and, although related, it is not equivalent to 4B (defined only for s-wave resonances). Several resonances were found that exhibited a suppressed oscillation due to comparable cou- pling to inelastic channels and could not be assigned a width in the usual way. In these cases the maximum and minimum elastic scatter- ing lengths of the oscillation were identified and the distance between them is indicated in parentheses. . . . . . . . . . . . . . . . . . . . . 57 4.1 The positions (B0) and widths (∆B) of s-wave magnetic Feshbach resonances for Li–Cs at magnetic fields below 500 G. The notation |FaMFa〉 for the atomic states is the same as in Chapter 3. . . . . . . 65 4.2 The positions (B0) of p-wave magnetic Feshbach resonances for Li–Cs at magnetic fields below 1 kG. . . . . . . . . . . . . . . . . . . . . . 69 4.3 The positions (B0) and widths (∆B) of s-wave resonances induced by an external electric field of 100 kV/cm for 6Li–87Rb at magnetic fields below 2 kG. (d) denotes an s-wave electric-field-induced Feshbach res- onance arising from a high order coupling through the p-wave channel to a d-wave closed channel state. As a consequence, these resonances are exceedingly narrow. . . . . . . . . . . . . . . . . . . . . . . . . . 77 5.1 The energy dependence of the elastic and inelastic cross sections in 2D and 3D. k is the collision wave number, and l (ml) and l′ (m′l) are the orbital angular momenta (projections) before and after the collision in 3D (2D). . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 vi List of Figures 1.1 The schematic diagram of the effective interaction potentials for col- lisions in s-wave (upper panel) and p-wave (lower panel) collision channels. The labels “s” and “p” refer to the angular momentum describing the rotational motion of the collision complex. The an- gular momentum is zero for s-wave collisions and 1 bohr for p-wave collisions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 The scheme of a laser cooling experiment, in particular, an absorption- emission cycle: A - a photon in a laser beam interacts with an atom in the ground state; B - the atom absorbs the photon and is promoted to an excited state; C - the atom spontaneously emits a photon in a random direction and returns to the ground state. . . . . . . . . . . 6 1.3 The energy levels of a 133Cs atom in the presence of an external magnetic field: A – a low-field-seeking state, i.e., the potential energy of Cs increases with the increase of the field strength; B – a high- field-seeking state, i.e., the potential energy of Cs decreases as the field strength increases. . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.4 The scheme of evaporative cooling. The most energetic atoms and molecules are expunged from the trap by lowering the trap depth. The temperature of the remaining particles decreases after thermal re-equilibration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.5 The schematic diagram of a Feshbach resonance. A – a quasi-molecular state of a weakly bound pair of atoms in a closed collision channel; B – the collision energy of the colliding atomic pair in an open channel. A Feshbach resonance occurs when the energy of the quasi-molecular state is degenerate with the collision energy of the colliding atomic pair. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 vii List of Figures 1.6 The scattering length a varies as a function of the magnetic field near a Feshbach resonance. abg is the background scattering length associated with the interaction potential of an open channel. ∆B and B0 represent the width and the position of the resonance, respectively. The value of the scattering length diverges at the position of the resonance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.7 Optical lattices with different geometries. (a) 2D optical lattice formed by overlapping two orthogonal optical standing waves – particles can only move along a cigar-shaped potential; (b) 3D optical lattice cre- ated by three orthogonal optical standing waves – particles confined in each trapping site are only allowed to oscillate in a tightly 3D harmonic potential. Adapted with permission from Macmillan Pub- lishers Ltd.: I. Bloch, Nature Physics 2005, 1, 23. . . . . . . . . . . 13 2.1 The typical configuration of a conventional scattering experiment. A uniform incident beam α of particles with a certain collision energy and current density Jinc is incoming on a target containing collision centers. Particles can then be scattered into different directions and the number of outgoing particles in a solid angle dΩ is detected by a scattering detector. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.2 The coordinate system describing a scattering experiment. The inci- dent beam is directed along z-axis, the distance between the detector and the target is r, the angle between ~r and z-axis is θ, and the an- gle between the projection of ~r on the (x, y) plane and x-axis is ϕ. The surface element dS subtending the scattering solid angle dΩ is dS = r2 sin θdθdϕ = r2dΩ. . . . . . . . . . . . . . . . . . . . . . . . . 21 2.3 Space-fixed spherical polar coordinates for two-body collisions. Col- lisions between particles A and B can be treated as a problem of a virtual particle C with a reduced mass µ interacting with a fixed scat- tering center through the centrally symmetric potential V (r). The fixed scattering center is located at the coordinate origin. The dis- tance between the particle C and the scattering center is r and the orientation of the vector ~r is specified by angles θ and ϕ. . . . . . . . 23 viii List of Figures 3.1 The s- and p-wave molecular bound state energies as functions of magnetic fields for all the states with MF = 3/2 computed within the asymptotic bound state model. The solid line shows the threshold for the |12 , 12〉6Li ⊗ |1, 1〉87Rb collision channel (see text) while the dashed (dotted) lines indicate the s-wave (p-wave) states. These molecular state energies were computed given the least bound states ESl of the optimal singlet and triplet potentials E0l=0 = −0.106 cm−1, E0l=1 = −0.0870 cm−1, and E1l=0 = −0.137 cm−1, E1l=1 = −0.116 cm−1. The predicted resonance positions are close to the actual positions determined by the full coupled-channel calculation and are indicated by the solid dots (A, B, and C). . . . . . . . . . . . . . . . . . . . . . 49 3.2 The procedure of fitting the interactions potentials for ultracold 6Li– 87Rb collisions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3.3 The singlet scattering length varies periodically as a function of the fitting parameter bsinglet. This fitting parameter determines the slope of the repulsive wall of the 1Σ interaction potential. . . . . . . . . . 52 3.4 The triplet scattering length varies periodically as a function of the fitting parameter btriplet. This fitting parameter determines the slope of the repulsive wall of the 3Σ interaction potential. . . . . . . . . . 53 3.5 Locus of points in the (Esinglet, Etriplet) parameter space where an s-wave resonance occurs at one of the two experimentally determined locations 882.02 G [gray (green)] or 1066.92 G [dark (red)] for atoms in the |12 , 12〉6Li⊗|1, 1〉87Rb state. The dotted lines indicate the approx- imate values for Esinglet and Etriplet beyond which a new bound state enters the potential at zero energy. There are four regions (I-IV) indi- cated on the plot where an s-wave resonance occures simultaneously at 882.02 G and at 1066.92 G. Region V indicates a range of values for which an s-wave resonance occurs at 1066.92 G while a p-wave resonance (not presented in this plot) occurs at 882.00 G. . . . . . . 54 3.6 Magnetic field dependence of the s-wave (upper panel) and p-wave (lower panel) elastic scattering cross sections for atoms in the |12 , 12〉6Li⊗ |1, 1〉87Rb state. These results are from the coupled-channel calcula- tions for a collision energy of 144 nK and using the optimal singlet and triplet potentials. Only the ml = 0 contribution of the p-wave elastic scattering cross section is shown. Two s-wave resonances oc- cur at 1065 and 1278 G, while two p-wave resonances occur at 882 and 1066 G. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 ix List of Figures 4.1 The interaction potentials and dipole moment functions (inset) of the LiCs molecule in the 1Σ (solid lines) and 3Σ (dashed lines) states. The interaction potentials were taken from Ref.[219] and the dipole moment functions approximate the data of Ref. [206]. . . . . . . . . 61 4.2 The interaction potentials and dipole moment functions (inset) of the LiRb molecule in the 1Σ (solid lines) and 3Σ (dashed lines) states. The interaction potentials were taken from Ref.[170] and the dipole moment functions approximate the data of Ref. [206]. . . . . . . . . 62 4.3 The coordinate system in our calculations. ~ζ and ~d represent the vector of the external electric field and the dipole moment vector, re- spectively; γ specifies the orientation of the electric field with respect to the quantization axis; θ is the angle between the dipole moment vector and the z-axis; χ is the angle between ~ζ and ~d, and ϕγ and ϕθ are the angles between the x-axis and the projections of the vectors êζ and êd on the (x, y) plane, respectively. . . . . . . . . . . . . . . . 63 4.4 Cross sections for elastic s-wave (solid curves) and p-wave (broken curves) scattering of Li and Cs atoms in the states |1, 1〉7Li⊗|3, 3〉133Cs computed at different electric fields: ζ = 0 kV/cm (upper panel) and ζ = 100 kV/cm (lower panel). The collision energy is 10−7 cm−1. . . 66 4.5 Cross sections for elastic s-wave (solid curves) and p-wave (broken curves) scattering of Li and Cs atoms in the states |1, 0〉7Li⊗|3, 3〉133Cs computed at different electric fields: ζ = 0 kV/cm (upper panel) and ζ = 100 kV/cm (middle panel). The lower panel presents the cross section for the s→ p transition. The collision energy is 10−7 cm−1. . 67 4.6 Cross sections for elastic s-wave (solid curves) and p-wave (broken curves) scattering of Li and Cs atoms in the states |1,−1〉7Li⊗|3, 3〉133Cs computed at different electric fields: ζ = 0 kV/cm (upper panel) and ζ = 100 kV/cm (lower panel). The collision energy is 10−7 cm−1. . . 68 4.7 Cross sections for elastic s-wave collisions of Li and Cs atoms in the |1, 0〉7Li⊗ |3, 3〉133Cs states computed at different electric fields: curve labeled a – ζ = 0 kV/cm; curve labeled b – ζ = 30 kV/cm; curve labeled c – ζ = 50 kV/cm; curve labeled d – ζ = 70 kV/cm; broken curve labeled e – ζ = 100 kV/cm. The collision energy is 10−7 cm−1. 70 4.8 Electric-field dependence of the s-wave scattering cross section for collisions of Li and Cs atom in the |1, 0〉7Li ⊗ |3, 3〉133Cs states at the magnetic field strength 1071 G. The variation of the cross sections is due to shifts of the s-wave resonances shown in Fig. 4.7. . . . . . . . 71 x List of Figures 4.9 Variation of the cross sections for s-wave collisions of Li and Cs atoms in the states |1, 0〉7Li⊗ |3, 3〉133Cs with the electric field strength. The magnetic field is fixed at 1024 G (full curve) and 1026 G (broken curve). The collision energy is 10−7 cm−1. . . . . . . . . . . . . . . . 72 4.10 Differential scattering cross sections for ultracold collisions of Li and Cs atoms in the |1,−1〉7Li ⊗ |3, 3〉133Cs states computed at an electric field strength of 100 kV/cm. The collision energy is 10−5 cm−1 (full curve) , 10−6 cm−1 (broken curve) and 10−7 cm−1 (dotted-dashed curve). The magnetic field is fixed at 1162 G. . . . . . . . . . . . . . 73 4.11 Magnetic field dependence of the elastic cross section for collisions be- tween Li and Rb in the atomic spin state |12 , 12〉6Li⊗ |1, 1〉87Rb. These results were obtained for a collision energy of 10−7cm−1 and two dif- ferent electric fields. The solid and dash-dotted curves show the s- and p-wave cross sections with ζ = 0, while the dotted and dashed curves show the s- and p-wave cross sections when ζ = 100 kV/cm. Here, only the cross section for the ml = 0 state is shown for p-wave scattering. At A an s-wave resonance is induced by an intrinsic p- wave resonance. Figure 4.12 shows this feature in more detail. At B and at C an intrinsic s-wave resonance is shifted to higher magnetic fields (corresponding to a shift of the associated bound state to lower energy) due to the electric field coupling between bound states. The observation that the shift of higher field resonances (e.g. C) is typi- cally larger than that of lower field resonances (e.g. B) is discussed in the text. At D an intrinsic p-wave resonance is shifted to lower mag- netic fields (corresponding to a shift of the associated bound state to higher energy). At E an induced p-wave resonance appears (invisible on this scale) due to the intrinsic s-wave resonance at C. . . . . . . . 74 xi List of Figures 4.12 Magnetic field dependence of s- and p-wave elastic cross sections for atoms in the atomic spin state |12 , 12〉6Li ⊗ |1, 1〉87Rb computed at dif- ferent electric fields. This is the same feature at A in Fig. 4.11. The solid and dotted curves show the s-wave cross sections at ζ = 0 and ζ = 100 kV/cm, respectively. The dot-dashed and dashed curves show the p-wave cross sections at ζ = 0 and ζ = 100 kV/cm, respec- tively. This intrinsic p-wave resonance shifts to lower magnetic field (corresponding to the shift of the associated bound state to higher energy) as the electric field magnitude is increased. The s-wave in- duced resonance appears at the same location as the intrinsic p-wave resonance, and its width grows with the strength of the electric field (see Fig. 4.13). Here only the cross section of the ml = 0 compo- nent is shown for the p-wave state is shown (Fig. 4.17 shows the cross sections for all three components). The collision energy is 10−7cm−1. 75 4.13 The width (∆B) of the s-wave electric-field-induced Feshbach res- onance arising from the intrinsic p-wave resonance at 882 G as a function of the electric field magnitude. Here γ = 0 and the collision energy is 10−7cm−1. The width appears to scale quadratically with ζ, at least for the electric fields below 200 kV/cm, and suggests that this induced resonance arises from an indirect coupling [226]. The solid line is the fit ∆B = 1.76 × 10−4 ζ2 G, where ζ is in units of kV/cm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 4.14 Magnetic field dependence of the s-wave elastic cross section for atoms in the atomic spin state |12 ,−12〉6Li ⊗ |1,−1〉87Rb computed at dif- ferent electric fields: ζ = 0 kV/cm (solid curve), ζ = 30 kV/cm (dotted curve), ζ = 70 kV/cm (dashed curve) and ζ = 100 kV/cm (dot-dashed curve). An intrinsic s-wave resonance (whose position is 1611 G in the absence of an electric field) is observed to shift to lower magnetic fields as the electric field strength is increased. Note: the shift direction is in the opposite sense to that of the intrinsic s-wave resonances in Fig. 4.11. These results were obtained with a collision energy of 10−7cm−1. . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 xii List of Figures 4.15 A schematic illustrating the mechanism of the shifts and splitting of p- and d-wave bound states resulting in the shifts and splitting of the corresponding Feshbach resonances. For simplicity, only three adjacent bound state levels are shown. The different partial wave potentials of each state are on this scale almost indistinguishable and are drawn here as a single potential. The inset shows the energy levels associated with these three states. The dotted lines indicate their energies in the absence of an electric field. The coupling induced by the electric field is represented as double-ended arrows and shown for the case when the electric field is aligned along the magnetic field, i.e. when γ = 0, states with the same ml value are coupled. The coupling results in level repulsion and the new position of the states is indicated by the solid lines. The degeneracy of the p- and d-wave bound states is broken and the associated Feshbach resonance splits into a multiplet with l+ 1 distinct resonances as shown in Figs. 4.17 and 4.18. This simple picture predicts that the s-wave resonance should shift to higher magnetic fields (given that the energy of the threshold moves down with increasing magnetic fields) and that the ml = 0 partial wave component should produce a new resonance at a magnetic field below the |ml| = 1 component – consistent with the motion of the resonances in Fig. 4.11 and Fig. 4.17. Of course, each state is coupled to all other bound states within the same spin manifold and with an orbital angular momenta differing by ∆l = ±1, resulting in splittings and shifts (e.g. Fig. 4.14) which may not follow the predictions of this simple picture. . . . . . . . . . . . . . . . . . 80 4.16 Variation of the cross section for s-wave collisions as a function of the electric field strength with the magnetic field fixed at 1066 G (solid line) and 878 G (dotted line) for atoms in the spin state |12 , 12〉6Li ⊗ |1, 1〉87Rb. The large resonance feature shown in the solid curve is due to the shift of the intrinsic magnetic Feshbach resonance just below 1066 G to higher magnetic fields, while the small resonance feature at 16 kV/cm arises from the shift of an intrinsic p-wave reso- nance just above 1066 G to lower magnetic fields as the electric field increases. The dotted curve shows a resonance feature associated with an electric-field-induced resonance (shown in Fig. 4.12) which moves from 882 G at ζ = 0 down to a magnetic field below 877 G at ζ = 120 kV/cm. The collision energy is 10−7cm−1. . . . . . . . . . . 81 xiii List of Figures 4.17 Magnetic field dependence of p-wave elastic cross section (averaged over all three orbital angular momentum components) for atoms in the atomic spin state |12 , 12〉6Li ⊗ |1, 1〉87Rb computed at zero electric field (solid curve) and at ζ = 100 kV/cm (dot-dashed curve). The thin dotted curves show the magnetic field dependence of the cross section for the |ml| = 1 and the m = 0 components separately. The p-wave resonance splits into two distinct resonances, one occurring for the ml = 0 component and one for the |ml| = 1 components. When the electric and magnetic fields are not co-linear, this segregation of the resonance multiplet breaks down as seen in Fig. 4.20. The collision energy is 10−7 cm−1. . . . . . . . . . . . . . . . . . . . . . . 82 4.18 The upper panel shows the magnetic field dependence of the d-wave elastic cross section for atoms in the atomic spin state |12 , 12〉6Li ⊗ |1, 1〉87Rb computed at zero electric fields (solid curve). The lower panel shows the magnetic field dependence of d-wave elastic cross section (solid curve). The contributions to the cross section from the |ml| = 2, |ml| = 1 and the ml = 0 components are shown (dotted curves) at ζ = 100 kV/cm. The d-wave resonance splits into l+1 = 3 distinct resonances corresponding to the splitting of the d-wave bound state levels drawn schematically in the lower panel. The collision energy is 10−7cm−1. . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 4.19 Total elastic cross section for different components of p-wave scatter- ing versus the angle, γ, between the applied electric and magnetic fields. The cross sections are shown for collisions in the ml = 0 state (dashed curve), the |ml| = 1 states (dotted curve), and the average (solid curve) of the cross sections over all three components for the atomic state |12 , 12〉6Li ⊗ |1, 1〉87Rb and for ζ = 100 kV/cm . The up- per panel shows these cross sections at an applied magnetic field of 877.0 G which is near the resonance for the ml = 0 component while the lower panel is at a field of 881.9 G which is in between the reso- nances for the ml = 0 and |ml| = 1 components (see Fig. 4.17). We observe that the shape of this variation changes dramatically near a resonance. The collision energy is 10−7cm−1. . . . . . . . . . . . . . 86 xiv List of Figures 4.20 Magnetic field dependence of the elastic cross section for different components of p-wave scattering with an electric field, ζ = 100 kV/cm, tilted with respect to the magnetic field axis by γ = 45◦. The cross sections are shown for collisions in the ml = 0 state (dashed curve), the |ml| = 1 states (dotted curve), and the average (solid curve) of the cross sections over all three components for the atomic state |12 , 12〉6Li ⊗ |1, 1〉87Rb. The doublet structure of the p-wave resonance seen also in Fig. 4.17 now appears for each of the three angular mo- mentum projection components. The collision energy is 10−7cm−1. . 87 4.21 Magnetic field dependence of elastic cross sections for atoms in the atomic spin state |12 , 12〉6Li ⊗ |1, 1〉87Rb computed at ζ = 100 kV/cm with the orientation of the electric field at γ = 0◦ (solid curve), 45◦ (dotted curve), and 90◦ (dot-dashed curve). The collision energy is 10−7cm−1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 5.1 The threshold energy dependence of cross sections for elastic (upper panel) and inelastic (middle panel) s-wave collisions of Li and Cs atoms in 3D (diamonds) and 2D (circles). Symbols – numerical cal- culations; lines – analytical fits based on the analysis of the threshold laws (cf. Tab.5.1 and Eq. 5.81). The lower panel shows the ratio of cross sections for inelastic collisions in 2D and 3D. The initial states are |2,−2〉7Li ⊗ |3, 2〉133Cs. The calculations were carried out in a magnetic field of 100 G. . . . . . . . . . . . . . . . . . . . . . . . . . 109 5.2 The threshold energy dependence of cross sections for elastic (up- per panel) and inelastic (lower panel) p-wave collisions of Li and Cs atoms in 3D (diamonds) and 2D (circles). Symbols – numerical cal- culations; lines – analytical fits based on the analysis of the threshold laws (cf. Tab.5.1). The initial states are |2,−2〉7Li ⊗ |3, 2〉133Cs. The calculations were carried out in a magnetic field of 100 G. . . . . . . 110 5.3 The ratio of inelastic and elastic cross sections in 2D (red circles) and 3D (blue diamonds) for s- (upper panel) and p-wave (lower panel) col- lisions. The initial states are |2,−2〉7Li ⊗ |3, 2〉133Cs. The calculations were carried out in a magnetic field of 100 G. . . . . . . . . . . . . . 111 xv List of Figures 5.4 The modification of the threshold energy dependence of the cross sec- tions for s-to-d transitions induced by the magnetic dipole-dipole 1/r3 interaction in collisions of Li and Cs atoms in 3D. The graph shows a gradual convergence of the calculations to the line (circles) computed using the asymptotic form of the Bessel and Neumann functions. The s-to-d transitions are calculated at zero magnetic field for the max- imally stretched state |2, 2〉7Li ⊗ |4, 4〉133Cs. Rend specifies the prop- agation distance of the coupled differential equation (cf. Eq. 5.48). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 5.5 The modification of the threshold energy dependence of cross sections for s-to-d transitions induced by the magnetic dipole-dipole 1/ρ3 in- teraction in collisions of Li and Cs atoms in 2D. The graph shows a gradual convergence of the calculations to the line (circles) computed using the asymptotic form of the Bessel and Neumann functions. The s-to-d transitions are calculated at zero magnetic field for the max- imally stretched state |2, 2〉7Li ⊗ |4, 4〉133Cs. Rend specifies the prop- agation distance of the coupled differential equation (cf. Eq. 5.48). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 5.6 Collisional spin relaxation of ultracold atoms and molecules initially in a maximum spin state in 3D in the presence of a low (upper panel) and a high (lower panel) magnetic field. Solid curve – s-wave collision channel; dashed curve – collision channels with nonzero orbital an- gular momentum. Adapted with permission from R. V. Krems, Int. Rev. Phys. Chem. 24, 99 (2005). . . . . . . . . . . . . . . . . . . . 115 5.7 Collisional spin relaxation of ultracold atoms and molecules initially in a maximum spin state in 2D in the presence of a magnetic field. Left panel – the magnetic field is perpendicular to the plane of con- finement; right panel – the magnetic field axis is directed at a nonzero angle with respect to the confinement plane normal. . . . . . . . . . 116 6.1 The schematic diagram of a quasi-2D system. Particles are confined in the ground state of a harmonic potential with the oscillation length of the confining potential much larger than the characteristic radius re of inter-particle interaction potentials. . . . . . . . . . . . . . . . . 120 xvi List of Figures 6.2 The schematic diagram of an elastic collision in quasi-2D geometry: (i) at short interparticle separations r < re, the collision occurs in 3D; (ii) in the region of r between re and the characteristic de Broglie wavelength of the particles Λ̃ε, the wave function is proportional to the 3D s-wave scattering wave function [180]; (iii) in the asymptotic region, the wave function is the product of a circular wave function and the wave function for the ground state harmonic motion in the confining potential. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 6.3 The threshold energy dependence of cross sections for inelastic re- laxation in s-wave collisions of 6Li with 87Rb: filled circles−purely 2D geometry; filled squares−3D scattering cross section reduced by a factor of 4× 104; open circles−quasi-2D with |a|/l0 > 1 cross section reduced by a factor of 30; open squares−quasi-2D with |a|/l0  1. The initial states are |12 ,−12〉6Li ⊗ |1, 0〉87Rb. . . . . . . . . . . . . . . 131 6.4 Cross sections for s-wave inelastic collisions of 6Li and 87Rb atoms in 3D (solid curve) and quasi-2D scattering with a weak confinement (l0 = 104 bohr – dotted curve) and a strong confinement (l0 = 103 bohr – dot-dashed curve) as functions of the magnetic field. The inset shows the ratio of the cross sections for inelastic collisions in quasi-2D with l0 = 104 and 3D. The collision energy is 10−8 cm−1. The initial states are |12 ,−12〉6Li ⊗ |1, 0〉87Rb. . . . . . . . . . . . . . . . . . . . . 132 6.5 The ratios of inelastic and elastic cross sections for s-wave collisions of 6Li and 87Rb atoms as functions of l0 for |a| = 13.58 bohr (B = 200G) (circles) and |a| = 1704.43 bohr (B = 1104.9G) (triangles). The initial states are |12 ,−12〉6Li ⊗ |1, 0〉87Rb. The collision energy is 10−8cm−1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 6.6 The ratio of cross sections for elastic (circles) and inelastic (diamonds) collisions in quasi-2D and 3D as functions of l0 for the H2–H2 system. The collision energy is 10−8cm−1. The initial states are v1 = 0, N1 = 2; v2 = 0, N2 = 2 . . . . . . . . . . . . . . . . . . . . . . . . . . 134 6.7 The ratios of cross sections for elastic scattering (circles) and chemical reaction (triangles) in quasi-2D and 3D as functions of the confine- ment strength for 7Li + 6Li2(v = 0,N = 1) collisions. The collision energy is 10−8cm−1. . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 xvii Acknowledgements In deepest gratitude, I would like to thank my supervisor, Prof. Roman Krems, for his excellent guidance and generous help in my study and work at UBC. He inspired me with interesting discussions and motivated me by his enthusiasm, persistence, and hardworking. I appreciate his intelligent advice and unique angle to approach a research problem. He also cares about his students. He brought us to mountains and lighted us up by his positive thinking. Here, I want to borrow Dr. Albert Schweitzer’s words to express my thanks to him: “At times our own light goes out and is rekindled by a spark from another person. Each of us has cause to think with deep gratitude of those who have lighted the flame within us.” I would like to thank Prof. Kirk Madison for his suggestions and inspiring ques- tions. He looked at my project from different perspectives and the interaction with him helped me understand the insight of the research problems and related physics better. He also showed me a good example of being a well-organized researcher. My Ph.D studies would have been less colorful if I did not meet Prof. Sture Nordholm. His deep thinking has broadened my view of the world of science and let me find a new research direction for the future. I also want to thank my thesis committee members, especially Prof. Mark Thachuk. His careful reading and helpful comments made the Thesis more elu- cidative and demonstrative. I truly thank my colleagues in our group: Timur Tscherbul, Erik Abrahams- son, Sergey Alyabyshev, Chris Hemming, and Felipe Herrera, for their kindness and help. I have learned a lot from interacting with them and I feel very lucky to work in this friendly and inspiring group. I also want to thank my friends in the chem- istry department at Göteborg University: Erik Wernersson, Huaqinq Li, Magnus Gustafsson, Sergey Antipov, and Soheil Sharifi. Their warm help and interesting discussions made my visit to Sweden wonderful. This Thesis would not have been possible without the support of my parents. Every time when I was frustrated and discouraged, their love showed me the light. No word can express my appreciation for their love, I just quote an ancient Chinese poem here: xviii Acknowledgements A Poem By A Leaving Son – By Jiao Meng A thread is in my fond mother’s hand moving. For her son to wear the clothes ere leaving. With her whole heart she’s sewing and sewing. For fear I’ll e’er be roving and roving. Who says the little soul of grass waving. Could for the warmth repay the sun of spring. At last, I would like to say: although only one name appears on the cover of this Thesis, the work presented here should be attributed to all individuals who have assisted me, encouraged me, and made efforts together with me. From the bottom of my heart, I thank you. xix Dedication To my family. xx Chapter 1 Introduction An entirely new research field of ultracold physics and chemistry has recently emerged due to technological breakthroughs in cooling and thermally isolating atomic and molecular ensembles [1–7]. At ultracold temperatures (T < 0.001 Kelvin), the de Broglie wavelength of atoms and molecules is very large and thermal fluctuations are nearly absent. This renders ultracold materials novel properties, not present in ther- mal gases. For example, the dynamics of ultracold systems are entirely determined by quantum mechanics and the microscopic interactions in ultracold gases can be controlled by external electromagnetic fields of moderate strength. Therefore, the creation of ultracold atoms and molecules has opened up new opportunities for the study of long-standing research problems in physics and chemistry. This Chapter describes the unique properties of ultracold quantum gases, presents an overview of experimental techniques for cooling atoms and molecules to ultracold temperatures, and discusses the prospects for external field control of microscopic interactions in ultracold gases. A detailed outline of the Thesis is presented in the last section of this Chapter. 1.1 Ultracold atoms and molecules−properties and applications The de Broglie hypothesis suggests that matter exhibits particle-wave duality. Mas- sive particles in an ideal gas can be considered as quantum-mechanical wavepackets with an extension on the order of a thermal de Broglie wavelength Λ = √ h2/2pimkBT , where h is the Planck constant, m is the mass of the particles in the gas, kB is the Boltzmann constant, and T is the temperature of the gas. At thermal temperatures, Λ is much smaller than the mean distance between particles in the gas. Atoms and molecules can therefore be described classically and the distribution of their veloc- ities obeys the Maxwell-Boltzmann law. When the gas is cooled to a sufficiently low temperature, Λ becomes comparable to the mean spacing between particles, which leads to quantum degeneracy. If the gas consists of bosons – particles with integer total spin – cooling leads to the appearance of a Bose-Einstein condensate, 1 1.1. Ultracold atoms and molecules−properties and applications a system with all the particles occupying the same quantum state. For fermions – particles with half-integer total spin – the decrease of the temperature gradually brings the gas to a “Fermi sea”, a state in which every accessible energy level is filled by only one fermion in a specific internal state. The many-body behavior of ultracold particles is described by quantum statistics, i.e., Bose and Fermi gases obey the Bose-Einstein and the Fermi-Dirac distribution laws, respectively [8]. Bose-Einstein condensates of weakly interacting atomic gases were first realized experimentally in 1995 for rubidium [9], sodium [10], and lithium [11]. Other atomic species have later been Bose-condensed in many experimental studies [12–16]. The properties of Bose-Einstein condensates are very different from those of thermal gases, liquids, and solids. For example, the density of a Bose gas is very low, normally smaller than the density of air by five orders of magnitudes. However, the system cannot be described as an ideal gas. The atoms in the condensate are indistinguishable and their wave packets overlap. This leads to the formation of a giant coherent matter wave, composed of a macroscopic number of atoms populating the quantum state of the lowest energy. Four years after the first realization of a Bose-Einstein condensate, researchers developed the technology for cooling atomic Fermi gases to quantum degeneracy [17–20]. The quantum behavior of fermions is very different from that of bosons due to the Pauli exclusion principle. Identical fermions cannot simultaneously occupy the same quantum state. As mentioned above, they stack up in the lowest energy levels with one particle in a specific quantum state per energy level. As a result, the average energy per fermionic atom is larger than the energy obtained from classical physics or in a Bose gas. The size of a quantum degenerate Fermi gas is therefore larger than that of a Bose-Einstein condensate. This so-called Fermi pressure was observed experimentally in 2001 [19]. Ultracold gases have several unique characteristics: (i) The inter-particle interactions in an ultracold gas can be described by a single parameter known as the scattering length (usually denoted by a). At ultracold temperatures, the de Broglie wavelength of particles is on the order of micrometer (1 µm ≈ 1.9 × 104 bohr) whereas atomic diameters are usually less than 10 bohr. The scattering length can be considered as the radius of hypothetical hard spheres which mimic scattering dynamics determined by microscopic interaction potentials. It reflects the net consequence of complex interatomic or intermolecular interactions. The value and sign of the scattering length is associated with the details of the inter-particle interactions and can be tuned using external fields. A Bose-Einstein condensate is stable when the interactions are effectively repulsive (a > 0) and it collapses when the interactions are effectively attractive (a < 0). In the presence 2 1.1. Ultracold atoms and molecules−properties and applications of inelastic collisions, the scattering length is a complex number a = α + iβ. The imaginary part is related to the probability of inelastic collisions [21]. (ii) Ultracold collisions are isotropic. Inter-pariticle interactions can be described by an effective interaction potential curve, which incorporates the rotational energy of a collision complex into the potential energy of the system. As shown in Fig. 1.1, the rotational energy of the collision complex in states with non-zero angular mo- menta (e.g., p-wave) gives rise to a long-range centrifugal barrier. If the kinetic energy of the colliding particles is smaller than the energy of the barrier, the cen- trifugal repulsion prevents the particles from reaching the short-range interaction region. At ultracold temperatures, the kinetic energy of atoms and molecules is extremely small. Therefore, collisions of particles in states with non-zero angular momenta are suppressed and isotropic collisions occurring in the s-wave collision channel are dominant. (iii) The statistical and microscopic behavior of ultracold atoms and molecules can be tuned by an external electromagnetic field, since the perturbations due to interactions with external fields are comparable with or larger than the collision energy at ultralow temperatures. s-wave p-wave Figure 1.1: The schematic diagram of the effective interaction potentials for col- lisions in s-wave (upper panel) and p-wave (lower panel) collision channels. The labels “s” and “p” refer to the angular momentum describing the rotational motion of the collision complex. The angular momentum is zero for s-wave collisions and 1 bohr for p-wave collisions. 3 1.1. Ultracold atoms and molecules−properties and applications The creation of ultracold atoms has revolutionized the field of atomic physics. Ultracold atoms can be used to produce atom lasers – wave packets released coher- ently from a Bose-Einstein condensate [22–26] – and allow for the design of precise atomic clocks [27–34]. The experiments with ultracold atoms may also lead to the realization of quantum simulators [35–38] and new advances in quantum compu- tation research [39–43]. Experiments with ultracold tunable atomic gases provide a direct observation of many-body quantum phenomena [44–48] and can be used for the study of a wide range of important problems in condensed matter physics [19, 49–59]. Ultracold atoms can also be linked together to form ultracold molecules and produce molecular Bose-Einstein condensates [57, 60–65]. Inspired by the success of the experiments with ultracold atoms, many research groups have recently focused on the creation of ultracold molecules [3–7]. Molecules provide new properties such as vibrational and rotational degrees of freedom and electric dipole moments. The electric dipole moments of molecules give rise to long-range dipole-dipole interactions, leading to interesting quantum effects [66, 67]. Possible applications of ultracold molecules go far beyond what is feasible in the experiments with atoms [3–7]. Here I give two representative examples: (i) Ultracold molecules allow for high resolution spectroscopy measurements in a wide range of photon frequencies. The resolution of a molecular spectroscopy experiment is determined by the time molecules spend in an apparatus. At ther- mal temperatures, high velocities of gas-phase molecules limit our understanding of molecular structure, whereas the low translational energy of ultracold molecules provides opportunities to improve the measurement resolution by many orders of magnitude. Performing high precision spectroscopy measurements with ultracold molecules may offer sensitive tests of fundamental symmetries of nature [68–72] and help in the search for the time variation of fundamental constants [73–76]. (ii) Collisions of ultracold molecules can be exploited to extend the research of molecular dynamics and chemistry in different aspects. Scattering properties of ultracold molecules are determined by quantum phenomena that enhance the prob- abilities for both elastic and inelastic collisions. Large de Broglie wavelengths make tunneling under reaction barriers the dominant mechanism of chemical reactions. Rate constants for inelastic collisions and chemical reactions are temperature in- dependent and can be very large at zero Kelvin [77–95]. Moreover, the quantum statistical behavior of ultracold molecules may greatly enhance the reaction rates [96, 97]. The study of ultracold chemistry may thus elucidate the role of quantum effects and quantum statistics in the dynamics of chemical reactions. Interactions between molecules may form resonant structures by transferring the translational 4 1.2. Cooling techniques for atoms and molecules energy of molecules into vibrational or rotational energy. These interactions, not present in atomic collisions, can be used to probe fine details of intermolecular po- tential energy surfaces [98]. Collisions of ultracold molecules are extremely sensitive to intermolecular interaction potentials and relative energies of the initial and final scattering states. A slight variation of the molecular structure due to an applied electromagnetic field may dramatically change the outcome of an inelastic colli- sion or chemical reaction of ultracold molecules. Interactions of cold and ultracold molecules may consequently be controlled with external electromagnetic fields and studies of collision dynamics of ultracold molecules may lead to the development of a new research field of ultracold controlled chemistry [97, 99–104]. 1.2 Cooling techniques for atoms and molecules The experimental techniques for the production of ultracold gases have developed rapidly since the award of the 1997 Nobel prize for “development of methods to cool and trap atoms with laser light” [105–107]. In general, cooling atoms and molecules to ultracold temperatures involves two steps. The first step is to pre-cool the particles to the temperature of about 1 mK and capture them in an external field trap; the second step involves evaporative or sympathetic cooling to ultracold temperatures (T ∼ 10−9 − 10−3 K). One can also create ultracold molecules by linking ultracold atoms. The following discussion focusses on cooling techniques relevant to the work presented in this Thesis. Atomic ensembles are often pre-cooled using laser cooling techniques. The idea of laser cooling is based on the Doppler shift as well as the conservation of energy and momentum during the process of atom-photon interaction. Fig. 1.2 shows a cartoon picture of an absorption-emission cycle for an atom in a laser cooling experiment. When an atom initially prepared in its ground state (A) is exposed to laser light of appropriate frequency, it can be promoted to an excited state (B) by absorbing a photon. It may then return to the ground state by emitting a photon spontaneously (C). The emitted photons scatter in random directions (dotted arrows in C) so that the average velocity of the atom does not change during this process. As a result, the net effect of the laser light is to reduce the mean speed of the atom in a particular direction, opposite to the propagation of the laser beam. The Doppler shift makes a traveling atom absorb the light it moves towards shifted from resonance to the blue. The energy of the photon absorbed by the atom is thus lower than that of emitted photons. After an absorption-emission cycle, the total energy of the atom decreases. Eventually, the spontaneous emission damps the motion of atoms and reduces their 5 1.2. Cooling techniques for atoms and molecules temperature after consecutively transferring momentum and energy from atoms to photons. The lowest temperature that can be achieved by laser cooling is limited by the so-called recoil energy, i.e., the energy acquired by an atom at rest when it interacts with a photon. Conventional laser cooling produces atoms at temperatures of several hundreds of µK. Extending laser cooling techniques to molecules is generally difficult. Simple absorption-emission cycles cannot be performed with molecules due to their complex multi-level structure. In order to pre-cool molecules, researchers have developed several alternative methods, such as buffer-gas loading [108–111], Stark deceleration [112–116], skimming [117, 118], mechanical slowing [119] and crossed-beam collision experiments [120]. AB C Figure 1.2: The scheme of a laser cooling experiment, in particular, an absorption- emission cycle: A - a photon in a laser beam interacts with an atom in the ground state; B - the atom absorbs the photon and is promoted to an excited state; C - the atom spontaneously emits a photon in a random direction and returns to the ground state. Once the atoms and molecules are pre-cooled to cold temperatures (0.001 K < T < 1 K), they can be captured in a magneto-optical [121, 122] or static magnetic [1, 9, 107, 111, 123] and electric [124, 125] traps. A magnetic trap is created by superimposing magnetic fields to generate a harmonic potential energy well with the minimum at the center. If paramagnetic atoms or molecules are exposed to an external magnetic field, their energy levels split into Zeeman states, as shown in Fig. 1.3 for the 133Cs atom as an example. The potential energy of the particles in some states increases with the increase of the field strength (e.g., “A” in Fig. 1.3). 6 1.2. Cooling techniques for atoms and molecules These states are called “low-field-seeking states”. Due to the magnetic field gradi- ents of the magnetic trap, when the atoms and molecules in low-field-seeking states move away from the center, their potential energy increases thereby reducing the kinetic energy. As a result, the particles are confined to the middle of the experi- mental cell. It is critical for a magnetic trapping experiment that the confined atoms and molecules remain in the low-field-seeking states. If inelastic collisions between particles change the orientation of their magnetic moments with respect to the mag- netic field axis, the particles relax to high-field-seeking states, where the potential energy of the atoms and molecules decreases as the field strength increases (e.g., “B” in Fig. 1.3). These states are not trappable and transitions from low-field-seeking states to high-field-seeking states generally lead to trap loss. 0 1000 2000 3000 4000 5000 Magnetic field (G) -2×10-6 -1×10-6 -5×10-7 0 5×10-7 1×10-6 2×10-6 En er gy  (a .u. ) A B Figure 1.3: The energy levels of a 133Cs atom in the presence of an external magnetic field: A – a low-field-seeking state, i.e., the potential energy of Cs increases with the increase of the field strength; B – a high-field-seeking state, i.e., the potential energy of Cs decreases as the field strength increases. In order to cool trapped atoms and molecules to ultracold temperatures, re- searchers use evaporative cooling, of which the scheme is presented in Fig. 1.4. Nor- mally, the atoms at the trap edge have higher kinetic energy than the atoms at the center of the trap. By using radio-frequency lasers to induce low-field-seeking to high-field-seeking transitions or by lowering the trap depth, one can expunge the most energetic atoms and molecules from the trap. The kinetic energy of the re- maining particles is then re-equilibrated by two-body elastic collisions and the overall 7 1.2. Cooling techniques for atoms and molecules temperature of the trapped gas decreases. The evaporative cooling can be applied only if the atomic or molecular gas originally confined in the trap is dense enough and elastic collisions are dominant during the cooling process. Typical temperatures achieved in evaporative cooling experiments are ∼ 10 − 100 nK. The evaporative cooling technique is difficult to apply to Fermi gases due to the strange collision properties of fermions. The Pauli exclusion principle forbids s-wave elastic collisions between fermions in the same quantum state. The lack of elastic collisions impedes the conventional evaporative cooling of fermions. Fortunately, fermions in different internal states are allowed to interact through s-wave elastic collisions and therefore they can be cooled by what is known as sympathetic cooling technique, which involves a distinct gas of atoms as a refrigerant in a cooling process [14, 126]. As first demonstrated by DeMarco and Jin [17], a Fermi degenerate gas can be obtained through simultaneous cooling with two different spin states of fermionic alkali metal atoms. Fermi gases of several atomic species have recently been cooled to quantum degeneracy using the sympathetic cooling technique [18, 19, 127].More delicate methods: evaporative cooling Elastic collisions TextLength Trapping potential Figure 1.4: The scheme of evaporative cooling. The most energetic atoms and molecules are expunged from the trap by lowering the trap depth. The temperature of the remaining particles decreases after thermal re-equilibration. The creation of Bose-Einstein condensates and the study of many-body quan- tum phenomena require not only low temperatures but also high enough densities of atoms and molecules. Phase-space density (D) is usually defined as D = nΛ3 to describe a gaseous ensemble in terms of both particle density and temperature. Here, n is the number density (space) and Λ = (2pi~2/mkBT )1/2 is the thermal de Broglie wavelength - the position uncertainty associated with the distribution of 8 1.2. Cooling techniques for atoms and molecules velocities (phase). For a classical gas, the most occupied states have energies on the order of kBT or less. The number of these states per unit volume is approximately (mkBT/~2)3/2. So the phase-space density D can be considered as the ratio between the number density of particles and the density of the states that are significantly occupied. For a thermal gas D  1, whereas for a uniform Bose gas in a three- dimensional box, the phase transition forming a Bose-Einstein condensate occurs when D ≈ 2.61. Therefore, increasing the phase-space density while lowering the temperature is a general goal of cooling experiments. Most methods for pre-cooling molecules lower the temperature at the expense of the particle density. They lead to low phase-space densities of cold molecules, making evaporative cooling to ultra- cold temperatures difficult. An alternative way to create dense ultracold molecular gases is to start with ultracold atomic gases with high phase-space density and then convert atomic pairs into molecules. Photoassociation [128–131] and Feshbach res- onance linking (also known as magneto-association) [129, 132, 133] are two of the most widely used approaches for this purpose. These methods produce molecules with the translational temperature as low as that of the precursor atoms. In a photoassociation experiment, two atoms resonantly absorb one photon to form an electronically excited molecule. Such molecules are often unstable and dissociate back into the atoms or a bound molecule in the ground electronic state by spontaneous emission. Bound molecules can thus be obtained by spontaneous radiative decay from the excited state. The first stable molecules generated us- ing photoassociation were homonuclear alkali dimers, with temperatures of several hundreds of µK [134–138]. Later on, heteronuclear molecules were produced and state-identified as well [139–143]. The disadvantage of this method is that the ul- tracold molecules are usually produced in vibrationally excited states, i.e., they are vibrationally hot. Feshbach resonances offer an alternative approach to create molecules from ultracold atoms [57, 61–65, 133, 144, 145]. The concept of a Fes- hbach resonance is very important for this Thesis and it is described in detail in the next section. Molecules thus created are in weakly bound states near the dis- sociation threshold [57, 61–65, 133, 144, 145] and decay quickly due to unfavorable inelastic collisions [146, 147]. In order to produce ultracold molecules in deeply bound vibrational states of the ground electronic state, new schemes which combine photoassociation and Feshbach resonances, have recently been developed [148–153]. The formation rate of ultracold molecules by photoassociation increases dramatically in the vicinity of a Feshbach resonance [151–153]. Therefore, instead of perform- ing photoassociation with two colliding atoms, researchers developed techniques to transfer weakly bound Feshbach molecules coherently to the ground rovibrational 9 1.3. Feshbach resonances state by lasers. This leads to the formation of strongly-bound and stable ultracold molecules [154–159]. 1.3 Feshbach resonances The discovery of magnetic field tunable Feshbach resonances has led to many ground- breaking experiments in the field of ultracold atomic and molecular physics [57, 61– 65, 133, 144, 145, 152, 160–162]. Magnetic Feshbach resonances provide a powerful tool to control microscopic interactions in ultracold quantum gases [152, 161]. These resonances allow for tuning the s-wave scattering length with external fields, which may be used to improve the efficiency of cooling and study fundamental problems and explore novel physical phenomena in many-body systems [58, 163–167]. Mag- netic Feshbach resonances also offer an extremely sensitive probe of interatomic interaction potentials for collisions at ultracold temperatures [168–170]. For ex- ample, in Chapter 3 of this Thesis, we generate accurate interaction potentials for ultracold collisions in Li – Rb gaseous mixtures by fitting the experimentally mea- sured Feshbach resonances [170]. As mentioned in the previous section, ultracold molecules can be created by tuning external magnetic fields near a magnetic Fes- hbach resonance, leading to the formation of molecular Bose-Einstein condensates [57, 60–65] and quantum gases of polar molecules [156]. The large electric dipole moment of molecules in quantum dipolar gases leads to aniosotropic intermolecular interactions, which gives rise to exotic many-body phenomena [171, 172] and may find applications in quantum computation research [173]. Feshbach resonances arise due to couplings between a quasi-molecular bound state in a closed collision channel and the scattering wave function of the colliding atoms in an open channel. A collision channel is defined by a set of quantum num- bers which describes internal states of two initially separated atoms. The channel energy is the sum of the internal energies of the separated atoms. If the total energy of the colliding atoms is bigger than the channel energy, the channel is considered to be open; if the total energy is less than the channel energy, the channel is said to be closed. When an atom is exposed to an external magnetic field, the energy levels of different spin states are split and the energy splitting increases with the magnetic field strength (cf. Fig. 1.3). Correspondingly, the channel energies vary with the change of the external magnetic field and the magnitude of the varia- tion depends on individual atomic states. The splitting between different channel energies can thus be tuned by an external magnetic field. In terms of interatomic interaction potentials, this splitting corresponds to the separation between the disso- 10 1.3. Feshbach resonances A B Closed channel Open channel Figure 1.5: The schematic diagram of a Feshbach resonance. A – a quasi-molecular state of a weakly bound pair of atoms in a closed collision channel; B – the collision energy of the colliding atomic pair in an open channel. A Feshbach resonance occurs when the energy of the quasi-molecular state is degenerate with the collision energy of the colliding atomic pair. ciation thresholds of the interaction potential curves for different collision channels, as shown in Fig. 1.5. The position of the quasi-bound state in a closed channel, can thus be tuned by changing the energy splitting between the thresholds of the potential curves. When the energy of the quasi-bound state in a closed channel (“A” in Fig. 1.5) is degenerate with the energy of the colliding atoms in an open channel (“B” in Fig. 1.5), the colliding atoms form resonant dimers and a resonant scatter- ing process occurs. By ramping the magnetic field across a Feshbach resonance, the energy of the resonant state can be shifted from above to below the energy of the col- lision threshold, forcing the dimers to form molecules by rearranging their electronic and nuclear spins. In principle, as the magnetic field varies through a Feshbach res- onance, the scattering length of ultracold atomic and molecular gases changes from −∞ to +∞ (See Fig. 1.6 for an illustrative example). A simple expression for the s-wave scattering length a(B) = abg(1− ∆B B −B0 ) (1.1) is often used to describe magnetically tunable Feshbach resonances. Here, abg is the background scattering length associated with the interaction potential of an open channel, and ∆B and B0 represent the width and the position of the resonance, 11 1.4. Ultracold gases in restricted geometries 1060 1062 1064 1066 1068 1070 Magnetic field (G) -104 -104 -103 0 103 104 Sc at te rin g le ng th  (B oh r) abg B0 ∆Β Figure 1.6: The scattering length a varies as a function of the magnetic field near a Feshbach resonance. abg is the background scattering length associated with the interaction potential of an open channel. ∆B and B0 represent the width and the position of the resonance, respectively. The value of the scattering length diverges at the position of the resonance. respectively. As illustrated in Fig. 1.6, the value of the scattering length diverges at the position of the resonance. The width of the resonance ∆B is the difference between the magnetic field at the position of the resonance B0 and the magnetic field B where the scattering length is equal to zero. All of the experimental studies to date have used magnetic fields and the Zeeman splitting diagrams similar to Fig. 1.3 to induce Feshbach resonances in ultracold atomic gases. In Chapter 4 of this Thesis, we demonstrate the possibility of inducing and manipulating Feshbach resonances in heteronuclear mixtures of atomic gases by applying an external dc electric field [174, 175]. 1.4 Ultracold gases in restricted geometries Atomic ensembles cooled to ultracold temperatures can be confined by optical forces of counterpropagating laser beams to form a periodic lattice structure. Allowing for the variation of the depth and the geometry of confining potentials, optical lattices of ultracold atoms and molecules can be used as a powerful instrument to study fundamental problems and explore new phenomena in several areas of 12 1.4. Ultracold gases in restricted geometries physics [35, 47, 48, 59]. For example, they may allow for quantum simulations of novel condensed matter systems [38, 51] and the development of new schemes for quantum computation [39, 43]. Optical lattices can also be used to produce low- dimensional quantum gases by confining the motion of ultracold particles in one or two dimensions. In low-dimensional systems, the confinement modifies the in- teraction potentials of ultracold particles [176–180], which may lead to new states of matter and dynamical behavior not observable in three dimensional (3D) gases [53, 54, 171, 179, 181–188]. For instance, metastable alkaline earth atoms or polar molecules confined in two dimensions (2D) may repel each other at long range, which leads to the formation of repulsive bound pairs [55] and self-organizing crystals at ultracold temperatures [171]. The realization of ultracold low-dimensional quan- tum gases also suggests new possibilities to study ultracold chemistry in restricted geometries. Figure 1.7: Optical lattices with different geometries. (a) 2D optical lattice formed by overlapping two orthogonal optical standing waves – particles can only move along a cigar-shaped potential; (b) 3D optical lattice created by three orthogonal optical standing waves – particles confined in each trapping site are only allowed to oscillate in a tightly 3D harmonic potential. Adapted with permission from Macmillan Publishers Ltd.: I. Bloch, Nature Physics 2005, 1, 23. The confining optical forces originate from the interaction between an induced dipole moment in a neutral atom or molecule and an external electric field. A 13 1.5. Thesis outline laser beam offers an oscillating electric field which induces an oscillating dipole moment. At the same time, the electric field interacts with the induced dipole moment resulting in the confining force. The magnitude of this force is proportional to the intensity of the laser beam, which normally is a Gaussian beam, i.e. the electric field intensity profile in a plane perpendicular to the beam axis is a Gaussian function. This spatial dependence of the electric field intensity in the laser beam provides a confining potential in an optical trap. The confining potential can also be created by interfering two laser beams propagating in the opposite directions. The interference of these two beams forms a periodic standing optical wave, which can be used to create 1D optical lattices. It is a periodic system of ultracold atoms and molecules with the motion in one direction confined by a harmonic potential to zero point oscillations. The particles are only allowed to move in a pancake- shaped geometry, leading to the formation of quasi-2D quantum gases [181, 185, 189– 192]. In an optical lattice, the overall trapping configuration is the superposition of two confining potentials: the spatially dependent potential and the standing wave potential. By introducing more laser beams propagating in other directions, one can create 2D and 3D optical lattices. As shown in Fig. 1.7 (a), the 2D lattice is created by overlapping two orthogonal optical standing waves, so that the particles can only move along a cigar-shaped potential, giving rise to quasi-1D ultracold gases [53, 54]. Three orthogonal optical standing waves form a 3D cubic lattice, in which particles confined in each trapping site are only allowed to oscillate in a tightly confining harmonic potential, leading to the formation of an optical 3D lattice [51]. In the presence of a confinement, collision properties of atoms and molecules are different from those in three dimensions [180]. For example, the energy dependence of cross sections for elastic and inelastic collisions in the limit of vanishing collision energy depends on the dimensionality of the system [193]. The confining potential may therefore modify chemical reactions and inelastic collisions of ultracold atoms and molecules. In this Thesis, we first study the collision dynamics in a purely 2D geometry (Chapter 5) and then develop a formalism for rigorous calculations of probabilities for inelastic and chemically reactive collisions in quasi-2D atomic and molecular gases (Chapter 6). 1.5 Thesis outline Chapter 2 presents a scattering theory for two-body elastic and inelastic collisions in the presence of external fields. The theory is based on solutions to the time- independent Schrödinger equation in a space-fixed coordinate frame. We describe 14 1.5. Thesis outline typical experimental setup for the study of atomic and molecular collisions and present a derivation of the expressions for cross sections for elastic and inelastic collisions. A numerical approach to obtaining cross sections by solving close-coupling differential equations is also described in this Chapter. Chapter 3 describes a method of generating accurate interatomic interaction po- tentials using positions and widths of experimentally measured Feshbach resonances. We illustrate the approach by describing the procedure for fitting the experimental data and create model interaction potential curves for the 6Li–87Rb molecule in the singlet (1Σ) and triplet (3Σ) electronic states. We employ an asymptotic bound state model which yields several combinations of the least bound energies of the singlet and triplet interaction potentials giving rise to the experimentally measured Feshbach resonances. We then generate approximate singlet and triplet potential curves reproducing these bound energies. In the last step of our fitting procedure, we fine tune the potential curves to reproduce the positions and widths of the res- onances using full quantum scattering calculations. Based on these potentials, we predict the positions and widths of several experimentally relevant resonances in ultracold Li–Rb mixtures. The ultracold mixture of Li and Rb atoms is currently studied experimentally in the physics department at UBC. Our results may guide the experimental work. Chapter 4 explores the possibility of inducing Feshbach resonances in heteronu- clear mixtures of atomic gases by an external dc electric field. The mechanism is based on the interaction of the instantaneous dipole moment of the collision complex with the external electric field. This interaction couples collision states of different angular momenta and the coupling becomes very significant near a Feshbach reso- nance. We first present a theory for ultracold atomic collisions in the presence of superimposed magnetic and electric fields and then demonstrate the effects of an external electric field on collision dynamics in ultracold 6Li – 87Rb and 7Li – 133Cs mixtures. We have found that the external electric field induces s-wave resonances in the presence of p-wave resonances which we refer to as electric-field-induced Fes- hbach resonances. The electric field may also shift the positions of resonances and induce the anisotropy of ultracold scattering by rotating and spinning up the col- lision complex of ultracold atoms. We also discovered that electric fields may split Feshbach resonances of non-zero partial waves into separated peaks associated with different orbital angular momentum projections. Chapter 5 presents a multi-channel collision theory to describe ultracold atomic collisions in the presence of magnetic fields in a purely 2D geometry. Based on this theory, we carry out rigorous quantum calculations to study collisions in a 15 1.5. Thesis outline binary mixture of ultracold Li and Cs atoms in 2D. Our results present the first numerical test of the threshold collision laws for elastic and inelastic scattering in two dimensions. We show that the magnetic dipole-dipole interaction in atomic collisions may modify the energy dependence of cross sections for elastic scattering accompanied with changes of angular momentum near threshold. We also find that collisional spin relaxation of ultracold molecules initially in a maximum spin state must be strongly suppressed in the presence of weak magnetic fields. Our numerical calculations demonstrate a dramatic difference between scattering dynamics in 2D and 3D ultracold atomic gases, and indicate that ultracold chemical reactions and inelastic collisions may be suppressed in ultracold collisions in a purely 2D geometry. Chapter 6 presents a formalism for rigorous calculations of cross sections for in- elastic and reactive collisions of ultracold atoms and molecules in quasi-2D geometry. Our calculations show that the modification of the geometry changes the threshold laws for ultracold collisions and that the threshold behavior of ultracold atoms and molecules in quasi-2D can be tuned by varying the confinement potential and an external magnetic field. We elucidate the general features of inelastic collisions and chemical reactions in ultracold atomic and molecular gases in quasi-2D. Our results suggest that applying laser confinement in one dimension may stabilize ultracold systems with large scattering lengths, which may open up interesting possibilities for the research of ultracold controlled chemistry. 16 Chapter 2 Background material The study of collision dynamics in ultracold quantum gases is very important for understanding physics and chemistry at ultralow temperatures. Elastic collisions ensure that atoms and molecules remain in thermal equilibrium during a cooling process, whereas inelastic collisions and chemical reactions normally lead to heating and trap loss. Due to the extremely low density of ultracold quantum gases, binary collisions are usually dominant. By studying two-body ultracold collisions, one can obtain parameters such as the s-wave scattering length used to model many-body properties of ultracold atomic and molecular ensembles. This Chapter introduces a time-independent scattering theory of two-body elastic and inelastic collisions in the space-fixed coordinate frame and describes an approach for numerical calculations of atomic and molecular collision properties. The theory described in this Chapter is based on the work of Arthurs and Dalgarno [194], Lester [195], Marković [196], Mott and Massey [197], Rodberg and Thaler [198] and Krems and Dalgarno [199]. 2.1 The adiabatic approximation The calculations presented in this Thesis are based on the adiabatic approximation (or Born-Oppenheimer approximation), which separates electronic and nuclear de- grees of freedom. This approximation makes the computations of energy and wave function of molecules feasible [200]. Here, I illustrate it by the example of diatomic molecules. Let’s start from the non-relativistic time-independent Hamiltonian for a diatomic molecule, which can be written as Ĥ =− 2∑ α=1 ~2 2mα ∆α − N∑ i=1 ~2 2me ∆i + ∑ α ∑ β>α ZαZβe 2 Rαβ − ∑ α ∑ i Zαe 2 Riα + ∑ j ∑ i>j e2 Rij , (2.1) where the nuclei are labeled by α and β and the electrons are labeled by i and j, and Zα and Zβ are atomic numbers. Rαβ = |Rα − Rβ| denotes the distance 17 2.1. The adiabatic approximation between the nuclei of the molecules, Riα = |Ri−Rα| denotes the distances between electrons and nuclei, and Rij = |Ri − Rj | denotes the distances between electrons. The Schrödinger equation is then given by Ĥψ(Rα, Ri) = Eψ(Rα, Ri). (2.2) Here, ψ(Rα, Ri) is the total wave function of the molecular system, which depends on the nuclear and electronic coordinates, Rα and Ri, respectively. Due to the fact that nuclei are much heavier than electrons, i.e., mα  me, the total Hamiltonian can be decomposed into the nuclear and electronic components Ĥ = T̂N + Ĥel. (2.3) and the wave function of the diatomic system can be written as a product of the nuclear wave function φ(Rα) and electronic wave function un(Ri|Rα) ψ(Rα, Ri) = φ(Rα)un(Ri|Rα) (2.4) The electronic Hamiltonian is Ĥel = − N∑ i=1 ~2 2me ∆i + ∑ α ∑ β>α ZαZβe 2 Rαβ − ∑ α ∑ i Zαe 2 Riα + ∑ j ∑ i>j e2 Rij . (2.5) One can solve the electronic Schrödinger equation Ĥelun(Ri|Rα) = Enun(Ri|Rα) (2.6) to obtain the wave functions un(Ri|Rα) and energies En of the molecular system for a particular nuclear configuration. The results are determined by the electronic coordinates and only depend on the nuclear coordinates parametrically. One can thus obtain a set of electronic energies En(Rα) as a function of the nuclear coordinate by repeating this calculation for different nuclear configurations. The operator for the nuclear kinetic energy is then reintroduced to the total Hamiltonian. The electronic energy function En(Rα) is now used as a potential in the Schrödinger equation describing the nuclear motion[ T̂N + En(Rα) ] φ(Rα) = Eφ(Rα). (2.7) One can solve this equation to obtain the total energy of the molecule and the wave function for the translational, vibrational, and rotational motions of the diatomic 18 2.2. Scattering amplitude and cross section molecular system. In general, a collision problem can also be described by the relative motion part of Eq. 2.7. This step can be reduced to a set of coupled differential equations which include vibronic couplings. The total wave function can be expanded in terms of electronic eigenfunctions and nuclear wave functions as following ψ = ∑ n φn(Rα)un(Ri) (2.8) where 〈un|un′〉 = δnn′ . (2.9) The Schrödinger equation for the molecular system becomes (T̂N + Ĥel) ∑ n φn(Rα)un(Ri) = E ∑ n φn(Rα)un(Ri). (2.10) Multiplying Eq. 2.10 from the left by un′(Ri), integrating over the electronic co- ordinates, and using Eq. 2.9, we obtain the following set of coupled differential equations:∑ n′ 〈un(Ri)|T̂N|un′(Ri)〉φn′(Rα) + T̂ φn(Rα) + En(Rα)φn(Rα) = Eφn(Rα). (2.11) The wave functions for different electronic states are coupled by the nuclear kinetic energy terms. Normally the splitting between electronic energy levels is very large, so this coupling is negligibly small. The adiabatic approximation is not valid if the coupling is significant, for example, in the case when two electronic energy levels are degenerate. In this Thesis, we study the collision dynamics of alkali atoms at ultracold temperatures. The collision energy of atoms in our calculations is very low and the atoms remain in the electronic ground state. In the presence of electric fields, s-wave and p-wave collision channels are coupled. However, the coupling between partial waves of different electronic states is very small due to the large splitting between electronic energy levels. The interaction between atoms and external fields is treated as a perturbation to the electronic ground state. Therefore, our calculations do not require non-adiabatic corrections. 19 2.2. Scattering amplitude and cross section z where VS(R) denotes the adiabatic interaction potential of the molecule in the spin state S. To evaluate the matrix elements of the interaction potential V̂ (R), we write the product states |ILiMILi〉|SLiMSLi〉|IRbMIRb〉|SRbMSRb〉 as |ILiMILi〉|SLiMSLi〉|IRbMIRb〉|SRbMSRb〉 =∑ S ∑ MS (−1)MS(2S + 1)1/2  SLi SRb S MSLi MSRb −MS  |ILiMILi〉|IRbMIRb〉|SMS〉 (6) and note that 〈SMS|V̂ (R)|S ′M ′S〉 = VS(R) δSS′δMSM ′S . (7) The parentheses in Eq. (6) denote a 3j-symbol. The operator V̂ (R) is diagonal in the nuclear spin states and l and ml quantum numbers. The operator V̂E(R) can be written in the form V̂E(R) = −E( "E · "d) ∑ S ∑ MS |SMS〉dS(R)〈SMS| (8) where "E · "d represents the dot product of the dipole moment vector "d of the LiRb dimer and the vector of the external electric field "E, dS denotes the dipole moment functions of LiRb in the different spin states nd E is the electric field magnitude. If the electric field is oriented at a certain angle γ with respect to the quantization axis, "d · "E has the form "d · "E = cos(χ) with χ the angle between θ and γ. It can be written in terms of the first-degree Legendre polynomial cos(χ) = P1(cos(χ)) (9) = 4pi 3 [ Y ∗1−1(γ,φγ)Y1−1(θ,φθ) + Y ∗ 10(γ,φγ)Y10(θ,φθ) + Y ∗ 11(γ,φγ)Y11(θ,φθ) ] where φγ and φθ are the projections of angles γ and θ on the (x, y) plane and Yxx are spherical harmonics. The dipole moment functions are represented by the expressions dS(R) = D exp [−α(R−Re)2] (10) with the parameters Re = 7.2 bohr, α = 0.06 bohr−2 and D = 4.57 Debye for the singlet state, and Re = 5.0 bohr, α = 0.045 bohr−2 and D = 1.02 Debye for the triplet state. These 5 Incident beam (α) Scattering detector Chapter 2 Background Material The study of collision dynamics in ultracold quantum gases is very important in understanding physics and chemistry at ultralow temperatures. For example, elastic collisions play roles in keeping atoms and molecules in thermal equilibrium during a cooling process, whereas inelastic collisions normally lead to chemical reactions and trap loss. Due to the extremely low density of ultracold quantum gases, the binary collisions ar dominant in de ermining the collision dynamics in the gas s. By studying ultracold collisions between two particles, one can extract parameters such as the s-wave scattering length to explore many-body phenomena. This chapter introduces a time-independent scattering theory for two-body elastic and inelastic collisions in a space-fixed frame and provide an approach to implement the theory in numerical calculations. 2.1 Scattering amplitude and cross sections FigThe typical set-up of a scattering experiment dΩ 16 Scattered beam (β) Transmitted beam Target Figure 2.1: The typical configuration of a conventional scattering experiment. A uniform incident beam α of particles with a certain collision energy and current density Jinc is incoming on a target containing collision centers. Particles can then be scattered into different directions and the number of outgoing particles in a solid angle dΩ is detected by a scattering detector. 2.2 Scattering amplitude and cross section 2.2.1 Typical scattering experiment A conventional scattering experiment is typically carried out as shown in Fig. 2.1 (see Ref. [201] for more details). A uniform incident beam α of particles with a certain collision energy and current density Jinc is incoming on a target containing scattering centers. Particles can then be scattered into different directions and the number of outgoing particles in a solid angle dΩ per unit time can be detected by a scattering detector. The current density of the scattered beam β is denoted by Jsc. Assuming that the incident beam is directed along a space-fixed z-axis, the distance between the detector and the target is r, and the angle between r and the z-axis is θ, the configuration of the scattering experiment can be projected on a space- fixed spherical polar coordinate system, as shown in Fig. 2.2. We are interested in the number of particles (denoted by dN) per unit time passing through a surface element dS which subtends the vector solid angle dΩ. dS is then given by dS = r2 sin θdθdϕ = r2dΩ (2.12) 20 2.2. Scattering amplitude and cross section and dN = JscdS = r2JscdΩ. (2.13) The number of scattered particles increases with the current density of the incoming beam Jinc. A proportionality coefficient dσ can thus be defined as dN = r2JscdΩ = Jincdσ, (2.14) which is known as the differential scattering cross section. It has the dimension of length×length and may be considered as an effective area of the incident beam where particles are scattered into the solid angle dΩ. Integrating dσ over all the angles, one obtains the integral scattering cross section σ = ∫ dσ = ∫ 4pi ( dσ dΩ )dΩ. (2.15) It can be interpreted as an effective area of the incident beam corresponding to scattering in all directions. x y z where VS(R) denotes the adiabatic interaction potential of the molecule in the spin state S. To evaluate the matrix elements of the interaction potential V̂ (R), we write the product states |ILiMILi〉|SLiMSLi〉|IRbMIRb〉|SRbMSRb〉 as |ILiMILi〉|SLiMSLi〉|IRbMIRb〉|SRbMSRb〉 =∑ S ∑ MS (−1)MS(2S + 1)1/2  SLi SRb S MSLi MSRb −MS  |ILiMILi〉|IRbMIRb〉|SMS〉 (6) and note that 〈SMS|V̂ (R)|S ′M ′S〉 = VS(R) δSS′δMSM ′S . (7) The parentheses in Eq. (6) denote a 3j-symbol. The operator V̂ (R) is diagonal in the nuclear spin states and l and ml quantum numbers. The operato V̂E(R) can be written in the form V̂E(R) = −E( "E · "d) ∑ S ∑ MS |SMS〉dS(R)〈SMS| (8) where "E · "d repres nts h dot product of the dipole moment vec or "d of the LiRb dimer and the vector of the external electric field "E, dS denotes the dipole moment functions of LiRb in the different spin states and E is the electric field magnitude. If the electric field is oriented at a certain angle γ with respect to the quantization axis, "d · "E has the form "d · "E = cos(χ) with χ the angle between θ and γ. It can be written in terms of the first-degree Legendre polynomial cos(χ) = P1(cos(χ)) (9) = 4pi 3 [ Y ∗1−1(γ,φγ)Y1−1(θ,φθ) + Y ∗ 10(γ,φγ)Y10(θ,φθ) + Y ∗ 11(γ,φγ)Y11(θ,φθ) ] where φγ and φθ are the projections of angles γ and θ on the (x, y) plane and Yxx are spherical harmonics. The dipole moment functions are represented by the expressions dS(R) = D exp [−α(R−Re)2] (10) with the parameters Re = 7.2 bohr, α = 0.06 bohr−2 and D = 4.57 Debye for the singlet state, and Re = 5.0 bohr, α = 0.045 bohr−2 and D = 1.02 Debye for the triplet state. These 5 Chapter 2 Background Material The study of collision dynamics in ultracold quantum gases is very important in understanding physics and chemistry at ultralow temperatures. For example, elastic collisions play roles in keeping atoms and molecules in thermal equilibrium during a cooling process, wherea i elastic collisions normally lead to chemical r actions and trap loss. Due to the extremely low density of ultracold quantum gases, the binary collisions are dominant in de ermining the colli ion dynamics in the gases. By studying ultracold collisions between two particles, one can extract parameters such as the s-wave scattering length to explore many-body phenomena. This chapter introduces a time-independent scattering theory for two-body elastic and inelastic collisions in a space-fixed frame and provide an approach to implement the theory in numerical calculations. 2.1 Scattering amplitude and cross sections In a space-fixed frame, collisions between two particles A and B can be treated as a problem of a particle C interacting with a fixed scattering center through a central potential V (r), where C has a reduced mass φ µ = mAmB mA +mB (2.1) 16 r Axis of incident beam Scattered beam 2.1. Scattering amplitude and cross sections dS = r2 sin θdθdφ = r2dΩ dθ Let with the scattered current density Jsc. A Figure 2.1: z where VS(R) denotes the adiabatic interaction potential of the molecule in the spin state S. To evaluate the matrix elements of the interaction potential V̂ (R), we write the product states |ILiMILi〉|SLiMSLi〉|IRbMIRb〉|SRbMSRb〉 as |ILiMILi〉|SLiMSLi〉|IRbMIRb〉|SRbMSRb〉 =∑ S ∑ MS (−1)MS(2S + 1)1/2  SLi SRb S MSLi MSRb −MS  |ILiMILi〉|IRbMIRb〉|SMS〉 (6) and note that 〈SMS|V̂ (R)|S ′M ′S〉 = VS(R) δSS′δMSM ′S . (7) The parentheses in Eq. (6) denote a 3j-symbol. The operator V̂ (R) is diagonal in the nuclear spin states and l and ml quantum numbers. The operator V̂E(R) can be written in the form V̂E(R) = −E( "E · "d) ∑ S ∑ MS |SMS〉dS(R)〈SMS| (8) where "E · "d represents the dot product of the dipole moment vector "d of the LiRb dimer and the vector of the external electric field "E, dS denotes the dipole moment functions of LiRb in the differe t spin states and E is the ele tric field magnitude. If the electric field is oriented at a certain angle γ with respect to the quantization axis, "d · "E has the form "d · "E = cos(χ) with χ the angle between θ and γ. It can be written in terms of the first-degree Legendre polynomial cos(χ) = P1(cos(χ)) (9) = 4pi 3 [ Y ∗1−1(γ,φγ)Y1−1(θ,φθ) + Y ∗ 10(γ,φγ)Y10(θ,φθ) + Y ∗ 11(γ,φγ)Y11(θ,φθ) ] where φγ and φθ are the projections of angles γ and θ on the (x, y) plane and Yxx are spherical harmonics. The dipole moment functions are represented by the expressions dS(R) = D exp [−α(R−Re)2] (10) with the parameters Re = 7.2 bohr, α = 0.06 bohr−2 and D = 4.57 Debye for the singlet state, and Re = 5.0 bohr, α = 0.045 bohr−2 and D = 1.02 Debye for the triplet state. These 5 Incident beam (!) Scattering detector Chapter 2 Background Material The study of collision dynamics in ultracold quantum gases is very important in understanding physics and chemistry at ultralow temperatures. For example, elastic collisions play roles in keeping atoms and molecules in thermal equilibrium during a cooling process, whereas inelastic collisions normally lead to chemical reactions and trap loss. Due to the extremely low density of ultracold quantum gases, the binary collisions ar dominant in de ermining the collision dynamics in the gas s. By studying ultracold collisions between two particles, one can extract parameters such as the s-wave scattering length to explore many-body phenomena. This chapter introduces time-independe t scattering theory for two-body elastic a d inelastic collisions in a space-fixed frame and provide an approach to implement the theory in numerical calculations. 2.1 Scattering amplitude and cross sections FigThe typical set-up of a scattering experiment dΩ 16 Scattering beam (") Transmitted beam Target Figure 2.2: x y z where VS(R) denotes the adiabatic interaction potential of the molecule in the spin state S. To evaluate the matrix elements of the interaction potential V̂ (R), we write the product states |ILiMILi〉|SLiMSLi〉|IRbMIRb〉|SRbMSRb〉 as |ILiMILi〉|SLiMSLi〉|IRbMIRb〉|SRbMSRb〉 =∑ S ∑ MS (−1)MS(2S + 1)1/2  SLi SRb S MSLi MSRb −MS  |ILiMILi〉|IRbMIRb〉|SMS〉 (6) and note that 〈SMS|V̂ (R)|S ′M ′S〉 = VS(R) δSS′δMSM ′S . (7) The parentheses in Eq. (6) denote a 3j-symbol. The operator V̂ (R) is diagonal in the ucl r spin states and l and ml quantum numbers. The operator V̂E(R) can be written in the form V̂E(R) = −E( "E · "d) ∑ S ∑ MS |SMS〉dS(R)〈SMS| (8) where "E · "d represents the dot product of the dipole moment vector "d of the LiRb dimer and the vector of the external electric field "E, dS denotes the dipole moment functions of LiRb in the different spin states and E is the electric field magnitude. If the electric field is oriented at a certain angle γ with respect to the quantization axis, "d · "E has the form "d · "E = cos(χ) with χ the angle between θ and γ. It can be written in terms of the first-degree Legendre polynomial cos(χ) = P1(cos(χ)) (9) = 4pi 3 [ Y ∗1−1(γ,φγ)Y1−1(θ,φθ) + Y ∗ 10(γ,φγ)Y10(θ,φθ) + Y ∗ 11(γ,φγ)Y11(θ,φθ) ] where φγ and φθ are the projections of angles γ and θ on the (x, y) plane and Yxx are spherical harmonics. The dipole moment functions are represented by the expressions dS(R) = D exp [−α(R−Re)2] (10) with the parameters Re = 7.2 bohr, α = 0.06 bohr−2 and D = 4.57 Debye for the singlet state, and Re = 5.0 bohr, α = 0.045 bohr−2 and D = 1.02 Debye for the triplet state. These 5 Chapter 2 Backg ound Material The study of collision dynamics in ultracold quantum gases is very important in understanding physics and chemistry at ultralow temperatures. For example, elastic collisions play roles in keeping atoms and molecules in ther al equilibrium during a cooling process, whereas inelastic collisions normally lead to chemical reactions and trap loss. Due to the extremely low density of ultracold quantum gas s, th binary collisions are dominant in determining the collision dynamics in the gases. By studying ultracold collisions between two particles, one can extract parameters such as the s-wave scattering length to explore many-body phenomena. This chapter introduces a time-independent scattering theory for two-body elastic and inelastic collisions in a space-fixed frame and provide an approach to implement the theory in numerical calculations. 2.1 Scattering amplitude and cross sections In a space-fixed frame, collisions between two particles A and B can be treated as a problem of a particle C interacting with a fixed scattering center through a central potential V (r), where C has a reduced mass φ µ = mAmB mA +mB (2.1) 16 r Axis of incident beam Scattered beam Chapter 2 Backgr und Material The study of collision dynamics in ultracold quantum gases is very important in un- derstanding physics and chemistry at ultralow temperatures. For example, elastic collisions play roles in keeping atoms and molecules in thermal equilibrium during a cooling process, whereas inelastic collisions normally lead to chemical reactions and trap loss. Due to the extremely low density of ultracold quantum gases, the binary collisions are domin nt in determining the collision dyn mics in the gases. By studying ultracold collisions between two particles, one can extract parame- ters such as the s-wave scattering length to explore many-body phenomena. This chapter introduces a time-independent scattering theory for two-body elastic and inelastic collisions in a space-fixed frame and an approach to implement the theory in numerical calculations. 2.1 Scattering amplitude and cross sections A scattering experiment is typically carried out in the way shown in Fig. 2.1. A uniform incident beam α of particles with a cer ain collision energy and current density Jinc is incoming o a target containing collision cen ers. Particl s can th n be scattered into different directions and the number of outgoing particles in a solid angle dΩ is detected by a scattering detector. dS = r2 sin θdθdφ = r2dΩ Let with the scattered current density Jsc. A 16 17 2.1. Scattering amplitude and cross sections dS = r2 sin θdθdφ = r2dΩ dφ Let with the scattered curr nt density Jsc. A Figure 2.1: z where VS(R) denotes the adiabatic interaction potential of the molecule in the spin state S. To evaluate the matrix elements of the interaction potential V̂ (R), we write the product states |ILiMILi〉|SLiMSLi〉|IRbMIRb〉|SRbMSRb〉 as |ILiMILi〉|SLiMSLi〉|IRbMIRb〉|SRbMSRb〉 =∑ S ∑ MS (−1)MS(2S + 1)1/2  SLi SRb S MSLi MSRb −MS  |ILiMILi〉|IRbMIRb〉|SMS〉 (6) and note that 〈SMS|V̂ (R)|S ′M ′S〉 = VS(R) δSS′δMSM ′S . (7) The parentheses in Eq. (6) denote a 3j-symbol. The operator V̂ (R) is diagonal in the nuclear spin states and l and ml quantu umbers. The operator V̂E(R) can be written in the form V̂E(R) = −E( "E · "d) ∑ S ∑ MS |SMS〉dS(R)〈SMS| (8) where "E · "d represents the dot product of the dipole moment vector "d of the LiRb dimer and the vector of the external electric field "E, dS denotes the dipole moment functions of LiRb in the different spin states and E is the electric field magnitude. If the electric field is oriented at a certain angle γ with respect to the q antization axis, "d · "E has the f rm "d · "E = cos(χ) with χ the angle between θ and γ. It can be writte in terms of the first-degree Legendre polynomial cos(χ) = P1(cos(χ)) (9) = 4pi 3 [ Y ∗1−1(γ,φγ)Y1−1(θ,φθ) + Y ∗ 10(γ,φγ)Y10(θ,φθ) + Y ∗ 11(γ,φγ)Y11(θ,φθ) ] where φγ a d φθ are the projectio s of angles γ and θ on the (x, y) plane and Yxx are sph rical harmonics. The dipole moment functions are represented by the expressions dS(R) = D exp [−α(R−Re)2] (10) with the parameters Re = 7.2 bohr, α = 0.06 bohr−2 and D = 4.57 Debye for the singlet state, a d Re = 5.0 bohr, α = 0.045 bohr−2 a d D = 1.02 Debye for the triplet state. These 5 Incident beam (!) Scattering detector Chapter 2 Back round Materi The study of collision dynamics in ultracold quantum gases is very important in underst nding physics and chem stry at ultralow temperatures. For example, elastic collisions play roles in keeping atoms and mole ul s in thermal equilibrium during a cooling process, whereas inelastic collisions normally lead to chemical reactions and trap loss. Due to the extremely low density of ultracold quantum gases, the binary collisions ar dominant in de ermining the collision dynamics in the gas s. By studying ultracold collisions between two particles, one can extract parameters such as the s-wave scattering length to explo e ma y-body phenomena. This chapt r introduces a time-independent scattering theory for two-body elastic and inelastic collisions in a space-fixed frame and provide an approach to implement the theory in numerical calculations. 2.1 Scattering amplitude and cross sections FigThe typical s t-up of a scattering exp riment dΩ 16 Scattering beam (") Transmitted beam Target Figure 2.2: x y z where VS(R) denotes the adiabatic interaction potential of the molecule in the spin state S. To evaluate the m trix elements of the int r ction potential V̂ (R), we write the product states |ILiMILi〉|SLiMSLi〉|IRbMIRb〉|SRbMSRb〉 as |ILiMILi〉|SLiMSLi〉|IRbMIRb〉|SRbMSRb〉 =∑ S ∑ MS (−1)MS(2S + 1)1/2  SLi SRb S MSLi MSRb −MS  |ILiMILi〉|IRbMIRb〉|SMS〉 (6) and note that 〈SMS|V̂ (R)|S ′M ′S〉 = VS(R) δSS′δMSM ′S . (7) The parentheses in Eq. (6) denote a 3j-symbol. The operator V̂ (R) is diagonal in the nuclear spin states and l and ml quantum numbers. The operator V̂E(R) can be written in the form V̂E(R) = −E( "E · "d) ∑ S ∑ MS |SMS〉dS(R)〈SMS| (8) where "E · "d represents the dot product of the dipole moment vector "d of the LiRb dimer and the vector of the external electric field "E, dS d no s the dipole moment functions of LiRb in the different spin states and E is the electric field magnitude. If the electric field is oriented at a certain angle γ with r spect to the quantiz tion axis, "d · "E h s the form "d · "E = cos(χ) with χ the angle between θ and γ. It can be written in terms of the first-degree Legendre polynomial cos(χ) = P1(cos(χ)) (9) = 4pi 3 [ Y ∗1−1(γ,φγ)Y1−1(θ,φθ) + Y ∗ 10(γ,φγ)Y10(θ,φθ) + Y ∗ 11(γ,φγ)Y11(θ,φθ) ] where φγ and φθ are the projections of angles γ and θ on the (x, y) plane and Yxx are spherical harmonics. The dipole moment functions are represented by the expressions dS(R) = D exp [−α(R−Re)2] (10) with the parameters Re = 7.2 bohr, α = 0.06 bohr−2 and D = 4.57 Debye for the singlet state, and Re = 5.0 bohr, α = 0.045 bohr−2 and D = 1.02 Debye for the triplet state. These 5 Chapter 2 Background Material The study of collision dynamics in ul racold quantum gases is very important in understanding physics and chemistry at ultralow temperatures. For example, el stic collisions play roles in keeping atoms and molecules in ther al equilibrium during a cooling process, whereas inelastic collisions normally lead to chemical reactions and trap loss. Due to the extremely low density of ultracold quantum gases, the binary collisions are dominant in determining the collision dynamics in the gases. By studying ultracold collisions between two particles, one can extract parameters such as the s-wave scattering length to explore many-body phenomena. This chapter introduces a time-independent scattering theory for two-body elastic and inelastic collisions in a space-fixed frame and provide an approach to implement the theory in numerical calculations. 2.1 Scattering amplitude and cross sections In a space-fixed frame, collisions between two particles A and B can be treated as a problem of a particle C interacting with a fixed scattering center through a central potential V (r), where C has a reduced mass φ µ = mAmB mA +mB (2.1) 16 r Axis of incident beam Scattered beam Ch pter 2 Backgr und Mate ial The study of collision dynamic in ultracold quantum gases is very important in un- derstanding physics and chemistry at ultral w te peratures. For example, elastic collisions play roles in keeping a oms and molecules in thermal equilibrium during a cooling process, whereas inelastic collisions normally lead to chemical reactions and trap loss. Due to the x remely low density of ultracold quantum gases, the binary c llisions are domin nt in dete mi ing t e collision dyn mics i the gases. By studying ultracold coll sions be ween two particles, one a xtract arame- ters such as the s-wave scattering length to explore any-body phenomena. This chapter introduces a time-independent scattering theory for two-body elastic and inelastic collisions i a space-fixe frame and an approach to implement the theory in numerical calculations. 2.1 Scattering amplitude and cross sections A scattering experiment is typically carried out in the way shown in Fig. 2.1. A uniform incident beam α of particles with a cer ain collision energy and current density Jinc is incoming on a target containing collision cen ers. Particl s can th n be scattered into different directions and the number of outgoing particles in a solid angle dΩ is detected by a scattering detector. dS = r2 sin θdθdφ = r2 Ω Let with the scattered current density Jsc. A 16 17 Chapter 2 Background material The study of collision dynamics in ultracold quantum gases is very important for understanding physics and chemistry at ultralow temperatures. Elastic collisions ensure that atoms and molecules remain in thermal equilibrium during a cooling process, whereas inelastic collisions normally lead to chemical reactions and trap loss. Due to the extremely low density of ultracold quantum gases, the binary collisions are usually dominant. By studying two-body ultracold collisions, one can extract parameters such as the s-wave scattering length used to model many-body properties of ultracold atomic and molecular ensembles. This Chapter introduces a time-independent scattering theory of two-body elastic and inelastic collisions in the space-fixed coor inate frame and describes an approach to realizing numerical calculations based on the theory. 2.1 The scattering amplitude and cross section 2.1.1 The scattering experiment A scattering experiment is typically carried out as shown in Fig. 2.1. A uniform incident beam α of particles with a certain collision energy and current density Jinc is incoming on a target containing scattering centers. Particles can then be scattered into different directions and the number of outgoing particles in a solid angle dΩ per unit time can be detected by a scattering detector. The current density of the scattered beam β is denoted as Jsc. Assuming that t incident beam is coming along a space-fixed z-axis, the distance between the detector and the target is r, and the angle between r and the z-axis is θ, the configuration of the scattering experiment can be projected on a space-fixed spherical polar coordinate system, as show in Fig. 2.2. We are interested in the number of particles (denoted by dN) per unit time passing through a surface element dS which subtends the vector solid angle dΩ. dS is then given by dS = r2 sin θdθdϕ = r2dΩ (2.1) 17 2.1. The scattering amplitude and cross section x y z where VS(R) denotes the adiabatic interaction potential of the molecule in the spin state S. To evaluate the matrix elements of the interaction potential V̂ (R), we write the product states |ILiMILi〉|SLiMSLi〉|IRbMIRb〉|SRbMSRb〉 as |ILiMILi〉|SLiMSLi〉|IRbMIRb〉|SRbMSRb〉 =∑ S ∑ MS (−1)MS(2S + 1)1/2  SLi SRb S MSLi MSRb −MS  |ILiMILi〉|IRbMIRb〉|SMS〉 (6) and note that 〈SMS|V̂ (R)|S ′M ′S〉 = VS(R) δSS′δMSM ′S . (7) The parentheses in Eq. (6) denote a 3j-symbol. The operator V̂ (R) is diagonal in the nuclear spin states and l and ml quantum numbers. The operator V̂E(R) can be written in the form V̂E(R) = −E( "E · "d) ∑ S ∑ MS |SMS〉dS(R)〈SMS| (8) where "E · "d represents the dot product of the dipole moment vector "d of the LiRb dimer and the vector of the external electric field "E, dS denotes the dipole moment functions of LiRb in the different spin states and E is the electric field magnitude. If the electric field is oriented at a certain angle γ with respect to the quantization axis, "d · "E has the form "d · "E = cos(χ) with χ the angle between θ and γ. It can be written in terms of the first-degree Legendre polynomial cos(χ) = P1(cos(χ)) (9) = 4pi 3 [ Y ∗1−1(γ,φγ)Y1−1(θ,φθ) + Y ∗ 10(γ,φγ)Y10(θ,φθ) + Y ∗ 11(γ,φγ)Y11(θ,φθ) ] where φγ and φθ are the projections of angles γ and θ on the (x, y) plane and Yxx ar spherical harmonics. The dipole moment functions are represented by the expressions dS(R) = D exp [−α(R−R )2] (10) with the parameters Re = 7.2 b hr, α = 0.06 bohr−2 and D = 4.57 Debye for the single state, and Re = 5.0 bohr, α = 0.045 bohr−2 and D = 1.02 Debye for the triplet state. These 5 Chapter 2 Background Material The study of collision dynamics in ultracold quantum gases is very important in understanding physics and chemistry at ultralow temperatures. For example, elastic collisions play roles in keeping atoms and molecules in thermal equilibrium during a cooling process, whereas inelastic collisions normally lead to chemical reactions and trap loss. Due to the extremely low density of ultracold quantum gases, the binary collisions are dominant in determining the collision dynamics in the gases. By studying ultracold collisions between two particles, one can extract parameters such as the s-wave scattering length to explore many-body phenomena. This chapter introduces a time-independent scattering theory for two-body elastic and inelastic collisions in a space-fixed frame and provide an approach to implement the theory in numerical calculations. 2.1 Scattering amplitude and cross sections In a space-fixed frame, collisions between two particles A and B can be treated as a problem of a particle C interacting with a fixed scattering center through a central potential V (r), where C has a reduced mass φ µ = mAmB mA +mB (2.1) 16 r Axis of incident beam Scattered beam Chapter 2 Background Material The study of collision dynamics in ultracold quantum gases is very important in un- derstanding physics and chemistry at ultralow temperatures. For example, elastic collisions play roles in keeping atoms and molecules in thermal equilibrium during a cooling process, whereas inelastic collisions normally lead to chemical reactions and trap loss. Due to the extremely low density of ultracold quantum gases, the binary collisions are dominant in determining the collision dynamics in the gases. By studying ultracold collisions between two particles, one can extract parame- ters such as the s-wave scattering length to explore many-body phenomena. This chapter introduces a time-independent scattering theory for two-body elastic and inelastic collisions in a space-fixed frame and an approach to implement the theory in numerical calculations. 2.1 Scattering amplitude and cross sections A scattering experiment is typically carried out in the way shown in Fig. 2.1. A uniform incident beam α of particles with a cer ain collision energy and current density Jinc is incoming on a target containing collision centers. Particles can then be scattered into different directions and the number of outgoing particles in a solid angle dΩ is detected by a scattering detector. dS = r2 sin θdθdφ = r2dΩ Let with the scattered current density Jsc. A 16 2.1. Scattering am litude and cross sections dS = r2 sin θdθdφ = r2dΩ dθ Let with the scattered current density Jsc. A Figure 2.1: z where VS(R) denotes the a iab tic interaction potential of the mol cul in e spin state S. To evaluate the matrix elements of the interaction potential V̂ (R), we write the product states |ILiMILi〉|SLiMSLi〉|IRbMIRb〉|SRbMSRb〉 as |ILiMILi〉|SLiMSLi〉|IRbMIRb〉|SRbMSRb〉 =∑ S ∑ MS (−1)MS(2S + 1)1/2  SLi SRb S MSLi MSRb −MS  |ILiMILi〉|IRbMIRb〉|SMS〉 (6) and note that 〈SMS|V̂ (R)|S ′M ′S〉 = VS(R) δSS′δMSM ′S . (7) The parentheses in Eq. (6) denote a 3j-symbol. The operator V̂ (R) is diagonal in the nuclear spin states and l and ml quantum numbers. The operator V̂E(R) can be written in the form V̂E(R) = −E( "E · "d) ∑ S ∑ MS |SMS〉dS(R)〈SMS| (8) where "E · "d represents the dot product of the dipole moment vector "d of the LiRb dimer and the vector of the ext rnal electri field "E, dS denotes the dipole moment functions of LiRb in the differe t spin states and E is the ele tric field magnitude. If the electric field is oriented at a certain angle γ with respect to the quantization axis, "d · "E has the form "d · "E = cos(χ) with χ he angle between θ and γ. It can be written in terms of the first-degree Legendre polynomial cos(χ) = P1(cos(χ)) (9) = 4pi 3 [ Y ∗1−1(γ,φγ)Y1−1(θ,φθ) + Y ∗ 10(γ,φγ)Y10(θ,φθ) + Y ∗ 11(γ,φγ)Y11(θ,φθ) ] where φγ and φθ are the projections of angles γ and θ on the (x, y) plane and Yxx are spherical h rmonics. The dipole mome t f nctions are represented by the expre sions dS(R) = D exp [−α(R−Re)2] (10) with the parameters Re = 7.2 bohr, α = 0.06 bohr−2 and D = 4.57 Debye for the singlet state, and Re = 5.0 bohr, α = 0.045 bohr−2 and D = 1.02 Debye for the triplet state. These 5 Incident beam (!) Scattering detector Chapter 2 Background Material The study of collision dynamics in ultracold quantum gases is very important in understanding physics and chemistry at ultralow temperatures. For example, elastic collisions play roles in keeping atoms and molecules in thermal equilibrium during a cooling process, whereas inelastic collisions normally lead to chemical reactions and trap loss. Due to the extremely low density of ultracold quantum gases, the binary collisions ar dominant in de ermining the collision dynamics in the gas s. By studying ultracold collisions between two particles, one can extract parameters such as the s-wave scattering length to explore many-body phenomena. This chapter introduces time-independe t scattering theory for two-body elastic a d inelastic collisions in a space-fixed frame and provide an approach to implement the theory in numerical calculations. 2.1 Scattering amplitude and cross sections FigThe typical set-up of a scattering experiment dΩ 16 Scattering beam (") Transmitted beam Target Figure 2.2: x y z where VS(R) denotes the adiabatic interaction potential of the molecule in the spin state S. To evaluate the mat ix elements of the interaction potential V̂ (R), we write the product states |ILiMILi〉|SLiMSLi〉|IRbMIRb〉|SRbMSRb〉 as |ILiMILi〉|SLiMSLi〉|IRbMIRb〉|SRbMSRb〉 =∑ S MS (−1)MS(2S + 1)1/2  SLi SRb S MSLi MSRb −MS  |ILiMILi〉|IRbMIRb〉|SMS〉 (6) and note that 〈SMS|V̂ (R)|S ′M ′S〉 = VS(R) δSS′δMSM ′S . (7) The parentheses in Eq. (6) denote a 3j-symbol. The operator V̂ (R) is diagonal in the ucl r spin states and l and ml quantum numbers. The operator V̂E(R) can be written in the form V̂E(R) = −E( "E · "d) ∑ S ∑ MS |SMS〉dS(R)〈SMS| (8) where "E · "d represents the dot product of the dipole moment vector "d of the LiRb dimer and the vector of the external electric field "E, dS denotes the dipole moment functions of LiRb in the different spin states and E is the electric field magnitude. If the electric field is oriented at a certain angle γ with respect to the quantization axis, "d · "E has the form "d · "E = cos(χ) with χ the angle between θ and γ. It can be written in terms of the first-degree Legendre polynomial cos(χ) = P1(cos(χ)) (9) = 4pi 3 [ Y ∗1−1(γ,φγ)Y1−1(θ,φθ) + Y ∗ 10(γ,φγ)Y10(θ,φθ) + Y ∗ 11(γ,φγ)Y11(θ,φθ) ] where φγ and φθ are the projections of angles γ and θ on the (x, y) plane and Yxx are spherical harmonics. T e dipole moment functions are represe ted by the expressions dS(R) = D exp [−α(R−Re)2] (10) with the parameters Re = 7.2 bohr, α = 0.06 bohr−2 and D = 4.57 Deby for the single state, and Re = 5.0 bohr, α = 0.045 bohr−2 and D = 1.02 Debye for the triplet state. These 5 Chapter 2 Backg ound Material The study of collision dynamics in ultracold quantum gases is very important in understanding physics and chemistry at ultralow temperatur s. For example, elastic collisions play roles in keeping atoms and molecules in ther al equilibrium during a cooling process, whereas inelastic collisions normally lead to chemical reactions and trap loss. Due to the extre ely low density of ultracold quantu gas s, th binary collisions are dominant in determining the collision dynamics in the gases. By studying ultracold collisions between two particles, one can extract parameters such as the s-wave scattering length to explore many-body phe omena. This chapter introduces a time-independent scatteri g theory for two-body lastic and inelastic collisions in a space-fixed frame and provide a approach to implement the theory in numerical calculations. 2.1 Scattering amplitude and cross sections In a space-fixed frame, collisions betwee two p rticles A and B can be treated a a problem of a particle C interacting with a fixed scattering center through a central potential V (r), where C has a reduced mass φ µ = mAmB mA +mB (2.1) 16 r Axis of incident beam Scattered beam Chapter 2 Backgr und Material The study of collision dynamics in ultracold quantum gases is very important in un- derstanding physics and chemistry at ultralow temperatures. For example, elastic collisions play roles in keeping atoms and molecules in thermal equilibrium during a cooling process, whereas inelastic collisions normally lead to chemical reactions and trap loss. Due o the extr mely low density of ultracold quantum gases, th binary collisions are domin nt in determining the collision dyn mics in the gases. By studying ultracold collisions between two particles, one can extract parame- ters such as the s-wave scattering length to explore many-body phenomena. This chapter introduces a time-independent scattering theory for two-body elastic and in lastic collisions in a space-fixed frame and an approach to implement the theory in numerical calculations. 2.1 Scatte ing amplitude and cross sections A scattering experiment is typically carried out in the way shown in Fig. 2.1. A uniform incident beam α of particles with a cer ain collision energy and current density Jinc is incoming o a target containing collision cen ers. Particl s can th n be scattered into different directio s and the number of outg ing particles in a solid angle dΩ is detected by a scattering detecto . dS = r2 sin θdθdφ = r2dΩ Let with the scattered current density Jsc. A 16 17 2.1. Scattering amplitude and cross sections dS = r2 sin θdθdφ = r2dΩ dφ Let with the scattered curr nt density Jsc. A Figure 2.1: z where VS(R) denotes the adiabatic interaction potential of the molecule in the spin state S. To evaluate the matrix elements of the interaction potential V̂ (R), we write the product states |ILiMILi〉|SLiMSLi〉|IRbMIRb〉|SRbMSRb〉 as |ILiMILi〉|SLiMSLi〉|IRbMIRb〉|SRbMSRb〉 =∑ S ∑ MS (−1)MS(2S + 1)1/2  SLi SRb S MSLi MSRb −MS  |ILiMILi〉|IRbMIRb〉|SMS〉 (6) and note that 〈SMS|V̂ (R)|S ′M ′S〉 = VS(R) δSS′δMSM ′S . (7) The parentheses in Eq. (6) denote a 3j-symbol. The operator V̂ (R) is diagonal in the nuclear spin states and l and ml quantu umbers. The operator V̂E(R) can be written in the form V̂E(R) = −E( "E · "d) ∑ S ∑ MS |SMS〉dS(R)〈SMS| (8) where "E · "d represents the dot product of the dipole moment vector "d of the LiRb dimer and the vector of the external electric field "E, dS denotes the dipole mom n functions of LiRb in the different spin states and E is the electric field magnitude. If the electric field is oriented at a certain angl γ with resp ct to the q a tization xis, "d · "E has the f rm "d · "E = cos(χ) with χ the angle between θ and γ. It can be writte in terms of the first-degree Legendre polynomial cos(χ) = P1(cos(χ)) (9) = 4pi 3 [ Y ∗1−1(γ,φγ)Y1−1(θ,φθ) + Y ∗ 10(γ,φγ)Y10(θ,φθ) + Y ∗ 11(γ,φγ)Y11(θ,φθ) ] where φγ a d φθ are the projectio s of angles γ and θ on the (x, y) plane a d Yxx are sph rical harmonics. The dipole moment fu ctions are represented by the expressions dS(R) = D exp [−α(R−Re)2] (10) with the parameters Re = 7.2 bohr, α = 0.06 bohr−2 and D = 4.57 Debye for the si glet state, a d Re = 5.0 bohr, α = 0.045 bohr−2 a d D = 1.02 Debye for the triplet st te. These 5 Incident beam (!) Scattering detector Chapter 2 Back round ateri The study of collision dynamics in ultracold quantum gases is very important in underst nding physics and chemistry at ultralow temperatures. For example, elastic collisions play roles in keeping atoms and mole ul s in thermal equilibrium during a cooling process, whereas inelastic collisions normally lead to chemical reactions and trap loss. Due to the extremely low density of ultracold quantum gases, the binary collisions ar dominant in de ermining the collision dynamics in the gas s. By studying ultracold collisions between two particles, one can extract parameters such as the s-wave scattering length to xplo e ma y-bo y phenomena. This chapt r introduces a time-independent scattering theory for two-body elastic and inelastic collisions in a space-fixed frame and provide an approach to implement the theory in numerical calculations. 2.1 Scattering amplitude and cross sections FigThe typical s t-up of a scattering exp riment dΩ 16 Scattering beam (") Transmitted beam Target Figure 2.2: x y z where VS(R) denotes the adiabatic interaction potential of the molecule in the spin state S. To evaluate the m trix elements of the int r ction potential V̂ (R), we write the product states |ILiMILi〉|SLiMSLi〉|IRbMIRb〉|SRbMSRb〉 as |ILiMILi〉|SLiMSLi〉|IRbMIRb〉|SRbMSRb〉 =∑ S ∑ MS (−1)MS(2S + 1)1/2  SLi SRb S MSLi MSRb −MS  |ILiMILi〉|IRbMIRb〉|SMS〉 (6) and note that 〈SMS|V̂ (R)|S ′M ′S〉 = VS(R) δSS′δMSM ′S . (7) The parenth ses in Eq. (6) denote 3j-symbol. The operator V̂ (R) is diagonal in the nuclear spin states and l and ml quantum numbers. The operator V̂E(R) can be written in the form V̂E(R) = −E( "E · "d) ∑ S ∑ MS |SMS〉dS(R)〈SMS| (8) where "E · "d represents the dot product of the dipole moment vector "d of the LiRb dimer and the vector of the external electric field "E, dS d no s the dipole mom nt functions of Li b in the different spin states and E is the electric field magnitude. If the electric field is ori n ed at a certain angle γ with r spect to the quantiz tion axis, "d · "E h s the form "d · "E = cos(χ) with χ the angle between θ and γ. It can be written in terms of the first-degree Legendre polynomial cos(χ) = P1(cos(χ)) (9) = 4pi 3 [ Y ∗1−1(γ,φγ)Y1−1(θ,φθ) + Y ∗ 10(γ,φγ)Y10(θ,φθ) + Y ∗ 11(γ,φγ)Y11(θ,φθ) ] where φγ and φθ are the projections of angles γ and θ on the (x, y) plane and Yxx are spherical harmonics. The dipole moment functions are represented by the expressions dS(R) = D exp [−α(R−Re)2] (10) with the parameters Re = 7.2 bohr, α = 0.06 bohr−2 and D = 4.57 Debye for the single state, and Re = 5.0 bohr, α = 0.045 bohr−2 and D = 1.02 Debye for the triplet state. These 5 Chapter 2 Backgr und Material The study of collision dynamics in ul racold quantum gases is very important in understanding physics and chemistry at ultralow temperatures. For example, el stic collisions play roles in keeping atoms and molecules in ther al equilibrium during a cooling process, whereas inelastic collisions normally lead to chemical reactions and trap loss. Due to the extremely low density of ultracold quantum gases, the binary collisions are dominant in determining the collision dyn mics in the gases. By studying ultracold collisions between two particles, one can extract parameters such as the s-wave scattering length to explore many-body phenomena. This chapter introduces a time-independent scattering theory for two-body elastic and inelastic collisions in a space-fixed frame and provide an approach to implement the theory in numerical calculations. 2.1 S ttering amplitude and cross sections In a space-fixed frame, collisions between two particles A and B can be treated as a problem of a particle C interacting with a fixed scattering center through ce tral potential V (r), where C has a reduced mass φ µ = mAmB mA +mB (2.1) 16 r Axis of incident beam Scattered beam Ch pter 2 Backgr und Mate ial The study of collision dynamic in ultracold quantum gases is very important in un- derstanding physics and chemistry at ultral w emperatures. or example, elastic collisions play roles in keeping a oms and molecules in thermal equilibrium during a cooling process, whereas inelastic collisions normally lead to chemical reactions and trap loss. Due to the x remely low density of ultracold quantum gases, the binary c llisions are d min nt in dete mi ing t e collision dyn mics i the gases. By studying ultracold coll sions be ween two particles, one a xtract arame- ters such as the s-wave scattering length to explore any-body phenomena. This chapter introduces a time-independent scattering theory for two-body elastic and inelastic collisions i a space-fixe frame and an approach to implement the theory in numerical calculations. 2.1 Scattering amplitude and cross sections A scattering experiment is typically carried out in the way shown in Fig. 2.1. A uniform incident beam α of particles with a cer ain collision energy and current density Jinc is incoming on a target containing collision cen ers. Particl s c n th n be scattered into different directi ns and the num er of outgoing p rticles in a solid angle dΩ is detected by a scattering detector. dS = r2 sin θdθdφ = r2 Ω Let with the scattered current density Jsc. A 16 17 ! d! Figure 2.2: The coordinate system which desc i s a scat ering experi ent. T incident beam is coming along z- xis, the distance b twe n the ector n th target is r, the angle between r and z-axis θ, and the proje tion gl of "r on the (x, y) plane is ϕ. The surface el me t dS sub e ding the sca tering s lid angle dΩ is dS = r2 sin θdθdϕ = r2dΩ. be considered as a plane wave trave ing along the z-axis ψinc = Aeikz, (2.5) where A is a normalization factor. The inter ction of the wave with a scattering center gives rise to a scattered wave ψsc. If the scattering is isotropic, i.e. the scattering into all directions is equally probable, the scattered wave can be described by a spherically symmetric wavefunction ψsc,iso = A eikr r , (2.6) where the scattered wavefunction has the same normalization factor A for elastic collisions. However, the scatter d wave is normally a isotrop c. The anisotropy of the scattered wavefunction may be described by a modulation factor f(k, θ,ϕ). In 19 2.1. The sca tering amplitude and cross section x y z where VS(R) denotes the adiabatic interaction potential of the molecule in the spin state S. To evaluate the matrix elem n s of the interaction potential V̂ (R), we write the product states |ILiMILi〉|SLiMSLi〉|IRbMIRb〉|SRbMSRb〉 as |ILiMILi〉|SLiMSLi〉|IRbMIRb〉|SRbMSRb〉 =∑ S ∑ MS (−1)MS(2S + 1) /2  Li SRb S MSLi MSRb −MS  |ILiMILi〉|IRbMIRb〉|SMS〉 (6) and note that 〈SM |V̂ (R)|S ′M ′S〉 = VS(R) δSS′δMSM ′S . (7) Th parenthe es i Eq. (6) denote a 3j-symbol. The operator V̂ (R) is diagonal in the nuclear spin states and l and l quantum umbers. The oper tor V̂E(R) can be written in the form V̂E(R) = −E( "E · "d) ∑ S ∑ MS |SMS〉dS(R)〈SMS| (8) where "E · "d represents the dot product of the dipole moment vector "d of the LiRb dimer and the vector of the exter al lectric field "E, dS denotes the dipole moment functions of LiRb in the different spin states and E is the electric field magnitude. If the electric field is oriented at a cert in angle γ with respect to he quantization axis, "d · "E has the form "d · "E = cos(χ) with χ the angle b twe n θ nd γ. It can be written in terms of the first-degree Legendre po yno ial cos(χ) = P1(cos(χ)) (9) = 4pi 3 [ Y ∗1−1(γ,φγ)Y1−1(θ,φθ) + Y ∗ 10(γ,φγ)Y10(θ,φθ) + Y ∗ 11(γ,φγ)Y11(θ,φθ) ] where φγ and φθ are the projections of angles γ and θ on the x, y) plane and Yxx are spherical harmonics. T e dipole mom nt functions ar represented by the expressions dS(R) = D exp [ α( −Re)2 ] (10) with the parame ers Re = 7.2 bohr, α = 0.06 bohr−2 and D = 4.57 Debye for the singlet state, and Re = 5.0 bohr, α = 0.045 bohr−2 and D = 1.02 Debye for the triplet state. These 5 C apt 2 Ba k ro ate ial Th study of collisio dynamics in ul r cold quantu gase is v ry important in und rs anding physi s and chem s ry at ultralow temperat res. For example, elastic collisions play roles in keepi g atoms a d olecu s in therm l quilibrium during a cooling process, whe eas inel stic collisions ormally lead o chemical reactions and trap loss. Due to the extre ely low density of ultracold quantum gases, the binary collisions are domin nt in determining he collision dynamics in the gases. By studying ultracold collisions b twe two particl s, one can extract parameters such as the s-wave cattering le gth o explor many-b dy phenomena. This chapter introduces a time-independent scattering heory for two-body elastic and inelastic collisions in a s ac -fixed fram an provide an approach o implement the theory in num rical al ulations. 2.1 Scattering mplitude and cross sections In a space-fixed frame, collisions between two particles A and B can be treated as a problem of a par icl C n eracting with a fixed scattering center through a central potential V (r), where C has a reduced mass φ µ = mAmB mA +mB (2.1) 16 r Axis of incident beam Sca tered beam Chapter 2 Background Material The study of collision dynamics in ultracold quantum gases is very important in un- derstanding physics and chemistry at ultralow temperatures. For example, elastic collisions play roles in keeping atoms and molecules in thermal equilibrium during a cooling process, whereas inelastic collisions normally lead to chemical reactions and trap loss. Due to the extremely low density of ultracold quantum gases, the binary collisions are dominant in determining the collision dynamics in the gases. By studying ultracold collisions between two particles, one can extract parame- ters such as the s-wave scattering length to explore many-body phenomena. This chapter introduces a time-indep ndent scattering theory for two-body elastic and inelastic collisions i a space-fixed frame and an approach to implement the theory in numerical calculations. 2.1 Scatt ring amplitude and cross sections A sc t ing exp rimen is typically carried out in the way shown in Fig. 2.1. A uniform i cident beam α of p rticles with a cer ain collision energy and current density Jinc is incoming on a target containing collision centers. Particles can then be scattered into different directio s and the number of outgoing particles in a solid angle dΩ is detected by a scattering detector. dS = r2 sin θdθdφ = r2dΩ Let with the scattered current density Jsc. A 16 2.1. Scattering amplitude and cross sections dS = r2 sin θdθdφ = r2dΩ dθ Let with the scattered current density Jsc. A Figure 2.1: z where VS(R) denotes the a iab tic interaction potential of the mol cul in e spin state S. To valuate the matrix elements of the interaction potential V̂ (R), we write the product states |ILiMILi〉|SLiMSLi〉|IRbMIRb〉|SRbMSRb〉 as |ILiMILi〉|SLiMSLi〉|IRbMIRb〉|SRbMSRb〉 =∑ S ∑ MS (−1)MS(2S + 1)1/2  SLi SRb S MSLi MSRb −MS  |ILiMILi〉|IRbMIRb〉|SMS〉 (6) and note that 〈SMS|V̂ (R)|S ′M ′S〉 = VS(R) δSS′δMSM ′S . (7) The parentheses in Eq. (6) denote a 3j-symbol. The operator V̂ (R) is diagonal in the nuclear spin states and l and ml quantum numbers. The operator V̂E(R) can be written in the form V̂E(R) = −E( "E · "d) ∑ S ∑ MS |SMS〉dS(R)〈SMS| (8) where "E · "d represents the dot product of the dipole moment vector "d of the LiRb dimer and the vector of the external electric field "E, dS denotes the dipole moment functions of LiRb in the differe t spin states and E is the ele tric field magnitude. If the electric field is oriented at a certain angle γ with respect to the quantization axis, "d · "E has the form "d · "E = cos(χ) with χ the angle between θ and γ. It can be written in terms of the first-degree Legendre polynomial cos(χ) = P1(cos(χ)) (9) = 4pi 3 [ Y ∗1−1(γ,φγ)Y1−1(θ,φθ) + Y ∗ 10(γ,φγ)Y10(θ,φθ) + Y ∗ 11(γ,φγ)Y11(θ,φθ) ] where φγ and φθ are the projections of angles γ and θ on the (x, y) plane and Yxx are spherical harmonics. The dipole moment fu ctio s are represented by the expressions dS(R) = D exp [−α(R−Re)2] (10) with the parameters Re = 7.2 bohr, α = 0.06 bohr−2 and D = 4.57 Debye for the singlet state, and Re = 5.0 bohr, α = 0.045 bohr−2 and D = 1.02 Debye for the triplet state. These 5 Incident beam (!) Scattering detector Chapt r 2 Background Material The study of collision dynamics in ultracold quantum gases is very important in understanding physics and c emistry at ultralow temperatures. For example, elastic collisions play roles in keeping atoms and molecules in thermal equilibrium during a cooling process, whereas inelastic collisions normally lead to chemical reactions and trap loss. Due to the extremely low density of ultracold quantum gases, the binary collisions ar dominant in de ermining the collision dynamics in the gas s. By studying ultracold collisions between two particles, one can extract parameters such as the s-wave scattering lengt to explo e many-body phenomena. This chapter introduces time-independe t scattering theory for two-body elastic a d inelastic collisions in a space-fixed frame and provide an approach to implement the theory in numerical calculations. 2.1 Scattering amplitude and cross sections FigThe typical set-up of a scattering experiment dΩ 16 Scattering beam (") Transmitted beam Target Figure 2.2: x y z where VS(R) denotes the adiabatic interaction potential of the molecule in the spin state S. To evaluate the mat ix elements of the interaction potential V̂ (R), we write the product states |ILiMILi〉|SLiMSLi〉|IRbMIRb〉|SRbMSRb〉 as |ILiMILi〉|SLiMSLi〉|IRbMIRb〉|SRbMSRb〉 =∑ S ∑ MS (−1)MS(2 + 1)1/2  SLi SRb S MSLi MSRb −MS  |ILiMILi〉|IRbMIRb〉|SMS〉 (6) and note that 〈SMS|V̂ (R)|S ′M ′S〉 = VS(R) δSS′δMSM ′S . (7) The parentheses in Eq. (6) denote a 3j-symbol. The operator V̂ (R) is diagonal in the ucl r spin states and l and ml quantum numbers. The operator V̂E(R) can be written in the form V̂E(R) = −E( "E · "d) ∑ S ∑ MS |SMS〉dS(R)〈SMS| (8) where "E · "d represents the dot product of the dipole moment vector "d of the LiRb dimer and the vector of th external elec ric field "E, dS denotes the dipole moment functions of LiRb in the different spin states and E is the electric field magnitude. If the electric field is oriented at a certain angle γ with respect to the quantization axis, "d · "E has the form "d · "E = cos(χ) with χ the angle between θ d γ. It can be writt n in terms of the first-degree Legen polynomial cos(χ) = P1(cos(χ)) (9) = 4pi 3 [ Y ∗1−1(γ,φγ)Y1−1(θ,φθ) + Y ∗ 10(γ,φγ)Y10(θ,φθ) + Y ∗ 11(γ,φγ)Y11(θ,φθ) ] where φγ and φθ are the roje tions of angl s γ and θ on the (x, y) plane and Yxx are spherical harmonics. The dipole moment functions are represented by the expressions dS(R) = D exp [−α(R−Re)2] (10) with the parameters Re = 7.2 bohr, α = 0.06 bohr−2 and D = 4.57 Debye for the singlet state, and Re = 5.0 bohr, α = 0.045 bohr−2 and D = 1.02 Debye for the triplet state. These 5 Chapter 2 B ckg ound erial The study of colli ion dynamics in ultracold quantum gases is very important in understanding physics and chemistry at ultralow temperatures. For example, elastic collisions play roles in keeping atoms a d molecules in ther al equilibrium during a cooling process, whereas in lastic collisi ns n rmally lead to chemi al reactions and trap loss. Due to th extremely low density of ultracold quantum gas s, th binary collisions are dom nant in determining the collision dynamics in the gases. By studying ultracold collisions betwee two particles, one can extract parameters such as the s-wave scattering length to explore many-body phenomena. This chapter introduces a time-independent scattering theory for two-body elastic and inelastic collisions in a space-fixed frame and provide an approach to implement the theory in nu erical calculations. 2.1 Scat ing amplitude an cross s cti ns In a space-fixed frame, collisions between two particles A and B can be treated as a problem of a particle C interacting with a fixed scattering center through a central potential V (r), wh re C has a redu ed m ss φ µ = mAmB mA +mB (2.1) 16 r Axis of incident beam Scattered beam Chapter 2 Backgr und Material The study of collision dynamics in ultracold quantum gases is very important in un- derstanding physics and chemistry at ultralow temperatures. For example, elastic collisions play roles in keeping atoms and molecules in thermal equilibrium during a cooling process, whereas inelastic collisions normally lead to chemical reactions and trap loss. Due to the extremely low density of ultracold quantum gases, the binary collisions are domin nt in determining the collision dyn mics in the gases. By studying ltracold collisions be ween two particles, one can extract parame- ters such as the s-wave scattering length to explore many-body phenomena. This chapter introduces a time-independent scattering theory for two-body elastic and inelastic collisions in a space-fixed frame and an approach to implement the theory in numerical calculations. 2.1 Scattering amplitude and cross sections A scatte ing experiment is typically c rried out in the way shown in Fig. 2.1. A uniform incident beam α of particles with a cer ain collision energy and current density Jinc is incoming o a target containing collision cen ers. Particl s can th n be scattered into different directions and the number of outgoing particles in a solid angle dΩ is detected by a scattering detector. dS = r2 sin θdθdφ = r2dΩ Let with the sc ttered current density Jsc. A 16 17 2.1. Scattering amplitude and cross sections dS = r2 sin θdθdφ = r2dΩ dφ Let wi h the scattered curr nt density Jsc. A Figure 2.1: z where VS(R) denotes the adiabatic interaction potential of the molecule in the spin state S. To evaluate the matrix elements of the interaction potential V̂ (R), we write the product states |ILiMILi〉|SLiMSLi〉|IRbMIRb〉|SRbMSRb〉 as |ILiMILi〉|SLiMSLi〉|IRbMIRb〉|SRbMSRb〉 =∑ S ∑ MS (−1)MS(2S + 1)1/2  SLi SRb S MSLi SRb −MS  |ILiMILi〉|IRbMIRb〉|SMS〉 (6) and note that 〈SMS|V̂ ( )|S ′M ′S〉 = VS(R) δSS′δMSM ′S . (7) The parentheses in Eq. (6) denote a 3j-symbol. The operator V̂ (R) is diagonal in the nuclear spin states and l and ml quantu umbers. The operator V̂E(R) can be written in the form V̂E(R) = −E( "E · "d) ∑ S ∑ MS |SMS〉dS(R)〈SMS| (8) where "E · "d r presents the dot product of the dipole moment vector "d of the LiRb dimer and the vector of the external electric field "E, dS denotes the dipole moment functions of LiRb in the different spin states and E is the electric field magnitude. If the electric field is oriented at a certain angle γ with respect to the q antization axis, "d · "E has the f rm "d · "E = cos(χ) wi χ th angle between θ and γ. It can be writte in t rms of the first-degree Legendre polynomial cos(χ) = P1(cos(χ)) (9) = 4pi 3 [ Y ∗1−1(γ,φγ)Y1−1(θ,φθ) + Y ∗ 10(γ,φγ)Y10(θ,φθ) + Y ∗ 11(γ,φγ)Y11(θ,φθ) ] where φγ a d φθ are the projectio s of angles γ and θ on the (x, y) plane and Yxx are sph rical harmonics. The dipole moment functions are represented by the expressions S(R) = D exp [−α(R−Re)2] (10) with the parameters Re = 7.2 bohr, α = 0.06 bohr−2 and D = 4.57 Debye for he singlet state, a d e = 5.0 bohr, α = 0.045 bohr−2 a d D = 1.02 Debye for the triplet state. These 5 Incident beam (!) Scattering detector Chap er 2 Back rou Materi The study of collision dynamics in ultracold quantum gases is very important in underst nding physics and chemistry at ultralow temperatures. For example, elastic collisions play roles in keeping atoms and mole ul s in thermal equilibrium during a cooling process, whereas inelastic collisions normally lead to chemical reactions and trap loss. Due to the extre ely low density of ultracold quantum gases, the binary collisions ar dominan n de ermining he collision dyna ics in the gas s. By studying ultracold collisions between two particles, one can extract parameters such as the s-wave scattering length to explo e ma y-body phenomena. This chapt r introduces a time-independent scattering theory for two-body elastic and inelastic collisions in a space-fixed frame and provide an approach to implement the theory in numeric l calculations. 2.1 Scattering amplitude and cross sections FigThe typical s t-up of a scattering exp riment dΩ 16 Scattering beam (") Transmitted beam Target Figure 2.2: x y z where VS(R) denotes the adiabatic interaction potential of the molecule in the spin state S. To evaluate the m trix elements of the int r ction potential V̂ (R), we write the product states |ILiMILi〉|SLiMSLi〉|IRbMIRb〉|SRbMSRb〉 as |ILiMILi〉|SLiMSLi〉|IRbMIRb〉|SRbMSRb〉 =∑ S ∑ MS (−1)MS(2S + 1)1/2  SLi SRb S MSLi MSRb −MS  |ILiMILi〉|IRbMIRb〉|SMS〉 (6) and note that 〈SMS|V̂ (R)|S ′M ′S〉 = VS( ) δSS′δMSM ′S . (7) The parentheses in Eq. (6) denote a 3j-symbol. The operator V̂ (R) is diagonal in the nuclear spin states and l and ml quantum numbers. The ope tor V̂E(R) can be written in he form V̂E(R) = −E( "E · "d) ∑ S ∑ MS |SMS〉dS(R)〈SMS| (8) where "E · "d rep esents the dot product of the dipole moment vector "d of the LiRb dimer and the vector of the external electric field "E, dS d no s the dipole moment functions of LiRb in the different spin states and E is the electric field magnitude. If the electric field is oriented at a certain angle γ with r spect to the quantiz tion axis, "d · "E h s the form "d · "E = cos(χ) with χ the angle between θ and γ. It can b written in terms of the first-degree Legendre polynomial cos(χ) = P1(cos(χ)) (9) = 4pi 3 [ Y ∗1−1(γ,φγ)Y1−1(θ,φθ) + Y ∗ 10(γ,φγ)Y10(θ,φθ) + Y ∗ 11(γ,φγ)Y11(θ,φθ) ] where φγ and φθ are the projections of angles γ and θ on the (x, y) plane and Yxx are spherical harmonics. The dipol moment functio s are represented by the expressions dS(R) = D exp [−α(R−Re)2] (10) with the parameters Re = 7.2 bohr, α = 0.06 bohr−2 and D = 4.57 Deby for the singlet state, and Re = 5.0 bohr, α = 0.045 bohr−2 and D = 1.02 Debye for the triplet state. These 5 Chapter 2 Background Material The study f collisio dynamics in ul racold quantum gases is very important in understanding physics and chemistry at ultralow temperatures. For example, el stic collisions play roles in keeping atoms and molecules in ther al equilibrium during a cooling process, whereas inelastic collisions normally lead to chemical reactions and trap loss. Due to the extremely low density of ultracold quantum gases, the binary collisions are dominant in determining the c llision dynamics in the gases. By studying ul racold collisions between two particles, one can extract paramet rs such as the s-wave scattering length to explore many-body phenomena. This chapter introduces a time-independent sc tteri theory for two-body elastic and inelastic collisions in a space-fixed frame and provide an approach to implement the theory in numerical calculations. 2.1 Scattering amplitude nd c oss sections In a space-fixed frame, collisions between two particles A and B can be treated as a problem of a particle C inte acting with a fixed scattering center through a central potential V (r), where C has a reduced mass φ µ = mAmB mA +mB (2.1) 16 r Axis of incident beam Scattered beam Ch pter 2 Backgr u d Mate ial The study of collision dynamic in ultracold quantum gases is very important in un- d rstanding physics and chemistry at ultral w emperatures. For example, elastic collisions play roles in keeping a oms and molecules in thermal equilibrium during a co ling process, wh reas inelastic collisions normally lead t chemical reactions nd trap loss. Due to the x remely low density of ultracold quantum gases, the binary c llisions are d min nt in dete mi ing t e collision dyn mics i the gases. By studying ultracold coll sions be ween two particles, one a xtract arame- ters such as the s-wave scattering length to explore any-body phenomena. This chapter introduces a time-independent scattering theory for two-body elastic and inelastic collisions i a space-fixe frame and an approach to implement the theory in numerical calculations. 2.1 Scatte ing amplitude and c oss sections A scattering experime t is typically arried out in the way shown in Fig. 2.1. A unifo m incident beam α of particles with a cer ain collision energy and current density Jinc is incoming on a target containing collision cen ers. Particl s can th n be scattered into different directions and the number of outgoing particles in a solid angle dΩ is detected by a scattering detector. dS = r2 sin θdθdφ = r2 Ω Let with the scattered current d nsity Jsc. A 16 17 ! d! Figure 2.2: The coordinate system which desc ibes a scattering experi ent. T incident beam is coming along z-axis, the dist nce b ween the ector an th target is r, t e ngle between r and z-axis θ, and the proje tion gl of "r on the (x, y) plane is ϕ. The surface el me t dS s b e ding the sca tering s lid angle dΩ is dS = r2 sin θdθdϕ = r2dΩ. be considered as a plane wave traveling long the z-axis ψinc = Aeikz, (2.5) where A is a normalization facto . The inter c ion f the wave with a scat r g center gives rise to a scattered wave ψsc. If the scattering is isotropic, i.e. the scattering into all di ections is equally probable, the scattered wave can be described by a spherically symmetric wavefunction ψsc,iso = A eikr r , (2.6) where the scattered wavefunction has the same normalizatio factor A for lastic collisions. However, the scattered wave is normally niso ropic. The nisotropy f the scattered wavefunction may be described by a modulation factor f(k, θ,ϕ). In 19 Figure 2.2: The coordinate system describing a scatt ri g experiment. The incident beam is directed along z-axis, the distance between he detector and the target r, the angle between ~r and z-axis is θ, and the angle between the projection of ~r on the (x, y) plane and x-axis is ϕ. The surface element dS sub ending he scatte ing solid angle dΩ is dS = r2 sin θdθdϕ = r2 Ω. 21 2.2. Scattering amplitude and cross section A quantum mechanical particle with momentum ~~k in the incident beam can be considered as a plane wave traveling along the z-axis ψinc = Aeikz, (2.16) where A is a normalization factor. The interaction of the wave with a scattering center gives rise to a scattered wave ψsc. If the scattering is isotropic, i.e. scattering into all directions is equally probable, the scattered wave can be described by a spherically symmetric wave function ψsc,iso = A eikr r , (2.17) where the scattered wave function has the same normalization factor A for elastic collisions. However, the scattered wave is normally anisotropic. The anisotropy of the scattered wave function can be described by a modulation factor f(k, θ, ϕ). In general, the scattered wave function is written as ψsc,aniso = Af(k, θ, ϕ) eikr r , (2.18) where f(k, θ, ϕ) is called the scattering amplitude. We will assume that the scattering target is very thin and the particles in the incident beam are free particles, i.e. they do not interact with each other. So a valid description of the scattering event can be obtained in terms of two-body collisions. If two particles A and B interact with each other through a spherically symmetric potential V (r), the collision problem can be separated into the center-of-mass motion and the relative motion of the particles. The center-of-mass motion drops out of the scattering problem. The relative motion can be considered as a dynamical problem of a hypothetical particle C with the reduced mass µ = mAmB/(mA+mB) interacting with a fixed scattering center through the centrally symmetric potential V (r). Here, mA and mB are the masses of particles A and B, respectively. It is thus convenient to solve the collision problem in a space-fixed spherical polar coordinate system. The fixed scattering center is located at the coordinate origin and the distance between the particle C and the scattering center is r. The orientation of the vector ~r is specified by angles θ and ϕ, as shown in Fig. 2.3. 22 2.2. Scattering amplitude and cross section x y z where VS(R) denotes the adiabatic interaction potential of the molecule in the spin state S. To evaluate the matrix elements of the interaction potential V̂ (R), we write the product states |ILiMILi〉|SLiMSLi〉|IRbMIRb〉|SRbMSRb〉 as |ILiMILi〉|SLiMSLi〉|IRbMIRb〉|SRbMSRb〉 =∑ S ∑ MS (−1)MS(2S + 1)1/2  SLi SRb S MSLi MSRb −MS  |ILiMILi〉|IRbMIRb〉|SMS〉 (6) and note that 〈SMS|V̂ (R)|S ′M ′S〉 = VS(R) δSS′δMSM ′S . (7) The parentheses in Eq. (6) denote a 3j-symbol. The operator V̂ (R) is diagonal in the nuclear spin states and l and ml quantum numbers. The operator V̂E(R) can be written in the form V̂E(R) = −E( "E · "d) ∑ S ∑ MS |SMS〉dS(R)〈SMS| (8) where "E · "d represents the dot product of the dipole moment vector "d of the LiRb dimer and the vector of the external electric field "E, dS denotes the dipole moment functions of LiRb in the different spin states and E is the electric field magnitude. If the electric field is oriented at a certain angle γ with respect to the quantizatio axis, "d · "E has the form "d · "E = cos(χ) with χ the angle between θ and γ. It can be written in terms of the first-degree Legendre polynomial cos(χ) = P1(cos(χ)) (9) = 4pi 3 [ Y ∗1−1(γ,φγ)Y1−1(θ,φθ) + Y ∗ 10(γ,φγ)Y10(θ,φθ) + Y ∗ 11(γ,φγ)Y11(θ,φθ) ] where φγ and φθ are the projections of angles γ and θ on the (x, y) plane and Yxx are spherical harmonics. The dipole moment functions are represented by the expressions dS(R) = D exp [−α(R−Re)2] (10) with the parameters Re = 7.2 bohr, α = 0.06 bohr−2 and D = 4.57 Debye for the singlet state, and Re = 5.0 bohr, α = 0.045 bohr−2 and D = 1.02 Debye for the triplet state. These 5 Chapter 2 Background Material The study of collision dynamics in ultracold quantum gas s is very important in understanding physics and chemistry at ultralow temperatures. For example, elastic collisions play roles in keeping atoms and molecules in thermal equilibrium during a cooling process, whereas inelastic collisions normally lead to chemical reactions and trap loss. Due to the extrem ly low ensity of ultracol quantu gas s, the binary collisions are dominant in d termining the collision ynamics in the gases. By studying ultracold collisions between two particles, one can extract parameters such as the s-wave scattering length to explore many-body phenomena. This chapter introduces a time-independent scattering theory for two-body elastic and inelastic collisions in a space-fixed frame and provide an approach to implement the theory in numerical calculations. 2.1 Scattering amplitude and cross sections In a space-fixed frame, collisions between two particles A and B can be treated as a problem of a particle C interacting with a fixed scattering center through a central potential V (r), where C has a reduced mass φ µ = mAmB mA +mB (2.1) 16 C r 2.3. Multi-cha nel scattering theory numerical calculation. By examining those matrices, one can extract the probability of collisional energy transfer and thereby obtain the cross sections for both elastic and inelastic collisions. µ = mAmB mA +mB (2.74) 29 Figure 2.3: Spac -fixed spherical polar coordinates for tw -body collisions. Colli- sions between particles A and B can be treated as a problem of a virtual particle C with a reduced mass µ interacting with a fixed scattering center through the centrally symmetric potential V (r). The fixed scattering center is located at the coordinate origin. The distance between the particle C and the scattering center is r and the orientation of the vector ~r is specified by angles θ and ϕ. 2.2.2 Time-independent Schrödinger equation The relative motion of particles A and B is described by the time-independent Schrödinger equation Ĥψ = Eψ, (2.19) where ψ is the total wave function and E is the total energy of the system. Here, Ĥ = T̂ + V̂ with T̂ describing the free motion of the colliding particles and V̂ modeling the inter-particle interaction potentials. In the Cartesian coordinate system, T̂ has a form T̂ = − 1 2µ ∇2 = − 1 2µ ( ∂2 ∂x2 + ∂2 ∂y2 + ∂2 ∂z2 ) . (2.20) 23 2.2. Scattering amplitude and cross section Here and throughout this Thesis, we will use the atomic units with ~ = 1. Trans- forming Eq. 2.20 into the spherical polar coordinates, we obtain T̂ =− 1 2µ 1 r2 ∂ ∂r r2 ∂ ∂r (2.21) − 1 2µr2 [ 1 sin θ ∂ ∂θ (sin θ ∂ ∂θ ) + 1 sin2 θ ∂2 ∂ϕ2 ] , (2.22) with Eq. 2.21 representing the radial part and Eq. 2.22 the angular part of the kinetic energy operator. The rotational angular momentum operator l̂2(θ, ϕ) describing the angular motion of the collision complex is defined as l̂2(θ, ϕ) = − [ 1 sin θ ∂ ∂θ (sin θ ∂ ∂θ ) + 1 sin2 θ ∂2 ∂ϕ2 ] , (2.23) The time-independent Schrödinger equation then becomes[ − 1 2µ 1 r2 ∂ ∂r r2 ∂ ∂r + l̂2(θ, ϕ) 2µr2 + V̂ ] ψ(r, θ, ϕ) = Eψ(r, θ, ϕ). (2.24) Multiplying Eq. 2.24 by −2µ we obtain[ 1 r2 ∂ ∂r r2 ∂ ∂r − l̂ 2(θ, ϕ) r2 + k2 − 2µV̂ ] ψ(r, θ, ϕ) = 0, (2.25) where k2 = 2µE. If the potential V̂ approaches zero more rapidly than 1/r2 as r → ∞, it can be negelected at large r and Eq. 2.25 reduces to the free-particle Schrödinger equation[ 1 r2 ∂ ∂r r2 ∂ ∂r − l̂ 2(θ, ϕ) r2 + k2 ] ψ(r, θ, ϕ) = 0. (2.26) By solving Eq. 2.26, one can obtain the total wave function ψ(r, θ, ϕ) in the asymp- totic scattering region. At the same time, ψ(r, θ, ϕ) at r =∞ can be represented as a superposition of an incident and a scattered wave function, ψ(r) r→∞−→ ψinc(r) + ψsc(r) = A [ eikz + f(k, θ, ϕ) eikr r ] . (2.27) The scattering amplitude f(k, θ, ϕ) can therefore be obtained by matching Eq. 2.27 and the solutions of Eq. 2.26. 24 2.2. Scattering amplitude and cross section 2.2.3 Differential cross section From the wave function ψ(r) and its gradient, we can obtain the current density J(r), that is J(r) = 1 2iµ [ψ(r)∗(∇ψ(r))− (∇ψ(r))∗ψ(r)] = 1 µ Im [ψ(r)∗(∇ψ(r))] . (2.28) In spherical polar coordinates, the gradient operator is given by ∇ = ∂ ∂r r̂ + 1 r ∂ ∂θ θ̂ + 1 r sin θ ∂ ∂ϕ ϕ̂. (2.29) Asymptotically it becomes ∇ = ∂ ∂r r̂, (2.30) which means that the current in the asymptotic region is only in the radial direction. Substituting the incident wave function (Eq. 2.16) into Eq. 2.28, we obtain the current density of the incoming flux Jinc = |A|2 k µ = |A|2ν, (2.31) where ν is the velocity of the incoming flux. By normalizing the current density of the incoming flux to 1, we obtain the normalization factor |A| = ν−1/2. This normalization factor is not important for elastic collisions since it cancels out in the later derivation, whereas it plays a role in the derivation of the scattering amplitudes for inelastic collisions in the multi-channel collision theory. We will discuss it in more detail in Section 2.3. The current density of the outgoing flux is obtained by acting with the operator in Eq. 2.30 on the outgoing wave function given by Eq. 2.18 Jsc = |A|2 k µr2 |f(k, θ, ϕ)|2. (2.32) The substitution of Eqs. 2.31 and 2.32 into Eq. 2.14 gives us the relation between the differential cross section and the scattering amplitude dσ dΩ = |f(k, θ, ϕ)|2. (2.33) 25 2.3. Single-channel scattering theory 2.3 Single-channel scattering theory In order to obtain the scattering amplitude f(k, θ, ϕ), it is necessary to solve the time-independent free-particle Schödinger equation 2.26 and compare the solution with the form of the total wave function given by Eq. 2.27. 2.3.1 Free-particle solutions The eigenfunctions of the angular momentum operator l̂2(θ, ϕ) are spherical har- monics Ylml(θ, ϕ) l̂2Ylml(θ, ϕ) = l(l + 1)Ylml(θ, ϕ), (2.34) where l is the quantum number for the orbital angular momentum and ml is the projection of l̂ on the space-fixed quantization axis. The solutions of Eq. 2.26 with particular l and ml can therefore be written as a product of a radial function and the spherical harmonics ψlml(r, θ, ϕ) = Flml(k, r)Ylml(θ, ϕ). (2.35) The total wave function can be expanded as ψ(r, θ, ϕ) = ∑ l ∑ ml ψlml(r, θ, ϕ) = ∑ l ∑ ml Flml(k, r)Ylml(θ, ϕ). (2.36) For the problems we consider in this Thesis, the scattered wave is symmetric with respect to the z-axis. Therefore, the scattered wave function and the scattering amplitude are independent of the azimuthal angle ϕ, which means that the scattered wave function can be expanded in Legendre polynomials Pl, ψ(r, θ) = ∞∑ l=0 Fl(k, r)Pl(cos θ). (2.37) In this expansion, the components of the rotational motion with different orbital angular momenta l are known as partial waves. The substitution of this wave function into Eq. 2.26 and the use of the relation 1 r2 d dr r2 d dr = d2 dr2 + 2 r d dr (2.38) 26 2.3. Single-channel scattering theory lead to the radial equation[ d2 dr2 + 2 r d dr + k2 − l(l + 1) r2 ] Fl(k, r) = 0. (2.39) With the substitution ρ = kr, Eq. 2.39 is transformed into the spherical Bessel differential equation [ d2 dρ2 + 2 ρ d dρ + 1− l(l + 1) ρ2 ] Fl(ρ) = 0. (2.40) The solutions of Eq. 2.40 are Fl(k, r) = Bljl(kr) + Clnl(kr), (2.41) where jl(kr) and nl(kr) are the spherical Bessel and Neumann functions, respec- tively. In the asymptotic region, this radial wave function becomes Fl(k, r) r→∞−→ 1 kr [ Bl sin(kr − lpi2 )− Cl cos(kr − lpi 2 ) ] . (2.42) Since the operator in Eq. 2.26 is linear and real, we can always find a real regular solution of this equation and the ratio Cl/Bl must be real. We introduce two parameters Al and δl, which gives us Bl = Al cos δl (2.43) and Cl = −Al sin δl. (2.44) The radial wave function (Eq. 2.42) thus has a form Fl(k, r) r→∞−→ Al 1 kr [ sin(kr − lpi 2 ) cos δl + cos(kr − lpi2 ) sin δl ] (2.45) = Al 1 kr sin(kr − lpi 2 + δl), (2.46) where δl = arctan(−Cl/Bl) (2.47) is called the phase shift of the lth partial wave. 27 2.3. Single-channel scattering theory 2.3.2 Scattering wave function The next step is to transform the asymptotic form of the total wave function (Eq. 2.27) into a form in which the radial wave function is similar to Eq. 2.45. Since for elastic collisions the normalization constant A cancels out in later derivations, we rewrite Eq. 2.27 as ψ(r) r→∞−→ eikz + f(k, θ)e ikr r , (2.48) where the independence of the scattering amplitude on ϕ is taken into account. The incoming plane wave eikz and the scattering amplitude f(k, θ) can be ex- panded in Legendre polynomials as [8] eikz = ∞∑ l=0 (2l + 1)iljl(kr)Pl(cos θ), (2.49) f(k, θ) = ∞∑ l=0 fl(k)Pl(cos θ). (2.50) Asymptotically, Eq. 2.49 becomes, eikz = ∞∑ l=0 (2l + 1)il sin(kr − lpi2 ) kr Pl(cos θ). (2.51) Substituting Eq. 2.51 into 2.48, using the Legendre expansion of the scattering amplitude (Eq. 2.50), and comparing the result with Eq. 2.37, we find the expression for the radial part of the lth partial wave function Fl(k, r) r→∞−→ (2l + 1)il 1 kr sin(kr − lpi 2 ) + eikrfl(k) r (2.52) r→∞−→ (2l + 1)il 1 kr × [ sin(kr − lpi 2 ) ( 1 + i kfl(k) 2l + 1 ) + cos(kr − lpi 2 ) kfl(k) 2l + 1 ] . (2.53) Comparing this expression with Eq. 2.45, we get (2l + 1)il ( 1 + i kfl(k) 2l + 1 ) = Al cos δl, (2.54) 28 2.3. Single-channel scattering theory and (2l + 1)il kfl(k) 2l + 1 = Al sin δl. (2.55) The constant Al and the expansion coefficient fl are thus given by Al = (2l + 1)ileiδl(k) (2.56) and fl(k) = 2l + 1 k eiδl(k) sin δl(k) (2.57) = 2l + 1 2ik ( e2iδl(k) − 1 ) . (2.58) We then obtain the expression for the scattering amplitude f(k, θ) = 1 k ∞∑ l=0 (2l + 1)eiδl sin δlPl(cos θ). (2.59) Substituting the expression for the constant Al from Eq. 2.56 into Eq. 2.46, we obtain the final expression for Fl in the asymptotic region Fl(k, r) r→∞−→ (2l + 1)ileiδl 1 kr sin(kr − lpi 2 + δl), (2.60) which gives rise to the asymptotic form of the scattering wave function ψ(r, θ) r→∞−→ ∞∑ l=0 (2l + 1)ileiδl sin(kr − lpi2 + δl) kr Pl(cos θ). (2.61) Comparing this equation with the plane wave expansion (Eq. 2.51), we find that the elastic scattering process merely modifies the phase of each partial wave in elastic collisions and the absolute value of the amplitude remains unchanged. 29 2.3. Single-channel scattering theory 2.3.3 Differential and integral cross sections The differential cross section can be expressed in terms of the partial wave terms using Eq. 2.33, namely dσ dΩ =|f(θ)|2 = ∞∑ l=0 ∞∑ l′=0 f∗l (k)Pl(cos θ)fl′(k)Pl′(cos θ) = 1 k2 ∞∑ l=0 ∞∑ l′=0 (2l + 1)(2l′ + 1)ei(δl−δl′ ) sin δl sin δl′Pl(cos θ)Pl′(cos θ). (2.62) Taking into account the orthogonality property of the Legendre polynomials [202]∫ pi 0 dθ sin θPl(cos θ)Pl′(cos θ) = 2 2l + 1 δll′ , (2.63) we obtain the integral cross section by integrating the above expression over all angles to give σ(k) = ∫ dΩ|f(θ)|2 = ∫ 2pi 0 dϕ ∫ pi 0 dθ sin θ|f(θ)|2 = 4pi k2 ∞∑ l=0 (2l + 1) sin2 δl(k). (2.64) The scattering amplitude f(k, θ) can also be expressed in terms of the single channel analogues of the S, T , or K matrices. The reaction or reactance matrix element Kl(k) is defined as Kl(k) = tan δl(k) = −Cl/Bl, (2.65) the transition matrix element, Tl(k) is usually defined as Tl(k) = eiδl(k) sin δl(k), (2.66) and the scattering matrix element Sl(k) is related to the phase shift as Sl(k) = e2iδl(k). (2.67) 30 2.3. Single-channel scattering theory The relations between theK and S matrix elements and the S and T matrix elements are Sl(k) = [1 + iKl(k)] [1− iKl(k)]−1 (2.68) and Sl(k) = 1 + 2iTl(k). (2.69) Note that the K matrix elements are real while the S and T matrix elements are complex. The scattering amplitude f(k, θ) is then given by f(k, θ) = 1 k ∞∑ l=0 (2l + 1)Tl(k)Pl(cos θ) (2.70) = 1 2ik ∞∑ l=0 (2l + 1) [Sl(k)− 1]Pl(cos θ) (2.71) and the integral cross section may be expressed as σ(k) = 4pi k2 ∞∑ l=0 (2l + 1)|Tl(k)|2, (2.72) or in terms of the S-matrix element as σ(k) = pi k2 ∞∑ l=0 (2l + 1)|Sl(k)− 1|2. (2.73) 2.3.4 Numerical calculation of the phase shift In this section, we will describe a numerical approach to calculate the K matrix element, i.e. the phase shift, and construct the S-matrix using Eq. 2.68. From the S-matrix, we can obtain the integral scattering cross section as described above (cf. Eq. 2.73). Let’s go back to consider the radial Schördinger equation (Eq. 2.39). Now we want to transform it into a second-order ordinary differential equation and employ the log-derivative method [203] to solve it numerically. If we define Fl(r) = φl(r)/kr and use the relation 1 r d2 dr2 r = d2 dr2 + 2 r d dr , (2.74) Eq. 2.25 becomes [ d2 dr2 +W (r) ] φl(r) = 0 (2.75) 31 2.3. Single-channel scattering theory where W (r) = k2 − 2µVeff(r, l), (2.76) Veff(r, l) = V (r) + l(l + 1) 2µr2 , (2.77) and φl(r) = krFl(r). (2.78) The logarithmic derivative yl is defined as yl = φ′l φl . (2.79) Equation 2.75 can then be re-written in terms of yl as y′l(r) +W (r) + y 2 l (r) = 0. (2.80) The phase shift can thus be calculated by integrating Eq. 2.80 numerically with the boundary conditions: φl(r → 0) = 0 and φl(r →∞) = {the asymptotic form of the transformed wave function φl}. In practice, the integration is started at r deep into the classically forbidden region where yl is set equal to a large but finite number, e.g., 1030 in our calculations. The asymptotic form of the wave function is given by (cf. Eq. 2.41) φl(r) = krFl(k, r) r→∞−→ krBl [jl(kr) + Cl/Blnl(kr)] r→∞−→ krBl [jl(kr)−Klnl(kr)] r→∞−→ Bl [ ĵl(kr)−Kln̂l(kr) ] , (2.81) where the functions ĵl and n̂1 are the Ricatti-Bessel functions ĵl(kr) = krjl(kr), (2.82) n̂l(kr) = krnl(kr). (2.83) Differentiating Eq. 2.81, we get φ′l(r) = Bl [ ĵ′l(kr)−Kln̂′l(kr) ] , (2.84) where the prime indicates the derivative of the functions φl(r), ĵl(kr), and n̂l(kr) 32 2.4. Multi-channel scattering theory with respect to r. The definition of the logarithmic derivative yl = φ′lφ −1 l leads to: φ′l(r) = ylφl = Bl [ ĵl(kr)−Kln̂l(kr) ] yl. (2.85) Equating Eqs. 2.84 and 2.85 and rearranging the terms in the equation, we can obtain the expression for the K matrix in terms of the log-derivative yl, that is Kl = (yln̂l − n̂′l)−1(ylĵl − ĵ′l). (2.86) The S-matrix can then be constructed using Eq. 2.68 and the total scattering cross section is obtained using Eq. 2.73. 2.4 Multi-channel scattering theory In the single-channel scattering theory, particles are treated as structureless objects and their relative translational energy is conserved. More often, however, colliding atoms and molecules have internal structures. If the internal state of a particle changes during the scattering process, the translational energy is not conserved and the collision process is referred to as inelastic. To calculate the probability of inelastic scattering, one needs to employ a multi-channel scattering theory. This section introduces the essential idea of the theory and describes how scattering matrices are constructed based on numerical calculations. By examining those matrices, one can calculate the probability of collisional energy transfer and thereby obtain the cross sections for both elastic and inelastic collisions. The time-independent multichannel scattering theory is based on representing the total wave function of the collision complex as a basis set expansion. As a result, the relative motion of the colliding particles is described by a set of coupled differen- tial equations which can be solved in matrix form. In the absence of external fields, the total angular momentum of the collision system is conserved and it is convenient to break the collision problem into a subset of smaller problems corresponding to different total angular momenta. Therefore, most researchers to date have used a representation (i.e. basis functions) consisting of eigenfunctions of the total angular momentum operator, which is called the total angular momentum coupled represen- tation [194, 195]. In this Thesis, however, we are solving collision problems in the presence of external fields. The interaction with external fields breaks the isotropy of space which leads to couplings between states of different angular momenta. There- fore the total angular momentum representation is not advantageous for dynamical problems in external fields. Here, we use a formulation based on the fully uncoupled 33 2.4. Multi-channel scattering theory space-fixed representation introduced by Krems and Dalgarno in 2004 [199]. The solutions of the coupled differential equations in this basis yield the scattering S- matrix which describes the probabilities of state-resolved transitions in the presence of external fields. 2.4.1 Multi-channel theory Consider collisions between two atoms or molecules in the presence of an external electric or magnetic field. The interaction of the colliding particles with the external field leads to splitting of internal energy levels into manifolds of Zeeman or Stark states. The Schrödinger equation for two separated particles (asymptotic region) is Ĥasφα = αφα, (2.87) where φα and α are the wave function and the energy of the atoms or molecules for a particular collision channel α, respectively. The total Hamiltonian for the system is the sum of the Hamiltonian describing the relative motion of the colliding particles and the Hamiltonian accounting for the asymptotic states, that is Ĥ = − 1 2µ 1 r2 ∂ ∂r r2 ∂ ∂r + l̂2(θ, ϕ) 2µr2 + V̂ + Ĥas. (2.88) So the total wave function can be expanded in terms of the products of the wave functions describing the asymptotic states φα, the radial wave functions Fαlml(r), and the rotational wave functions Ylml(r̂) with r̂ denoting the orientation of the vector ~r (e.g. (θ, ϕ)) ψ = 1 r ∑ α′ ∑ l′ ∑ m′l Fα′l′m′l(r)φα′Yl′m′l(r̂). (2.89) Substituting this expansion into the Schrödinger equation Ĥψ = Eψ (2.90) with the Hamiltonian given by Eq. 2.88, multiplying the resultant from the left by φ∗αY ∗lml(r̂), integrating over θ and ϕ, and using the orthonormality of φα and Ylml(r̂), we obtain a set of coupled-channel differential equations (the derivation is presented in Appendix A),[ ∂2 ∂r2 − l(l + 1) r2 + k2α ] Fαlml(r) = 2µ ∑ α′ 〈φα|V̂ |φα′〉Fα′lml(r), (2.91) 34 2.4. Multi-channel scattering theory where k2α = 2µ(E − α). When the distance between the colliding particles is large enough, the interaction between them can be neglected, i.e. V = 0. The coupled differential equations (Eq. 2.91) become uncoupled[ ∂2 ∂r2 − l(l + 1) r2 + k2α ] Fαlml(r) = 0, (2.92) and the boundary condition for each equation becomes a combination of the spherical Bessel and Neumann functions (cf. Eq. 2.41) Fαlml(r)→ kαr [aαlmljl(kαr) + bαlmlnl(kαr)] . (2.93) The multi-channel scattering problem reduces to a single-channel problem and the wave function for a particular collision channel α at sufficiently large r is ψαlml(r)→ Aαkα [aαlmljl(kαr) + bαlmlnl(kαr)]φαYlml(r̂), (2.94) where Aα = ν −1/2 α is a normalization coefficient obtained by normalizing to unity the incoming flux of the atoms in the state α (cf. Eq. 2.31). Using the asymptotic forms of jl(kαr) and nl(kαr) aαlmljl(kαr) + bαlmlnl(kαr) r→∞−→ 1 kαr [ aαlml sin(kαr − lpi 2 ) + bαlml cos(kαr − lpi 2 ) ] , (2.95) we obtain the asymptotic form of the channel wave function ψαlml r→∞−→ Aα 1 r [ aαlml sin(kαr − lpi 2 ) + bαlml cos(kαr − lpi 2 ) ] φαYlml(r̂), (2.96) which can be re-written in terms of exponential functions as ψαlml r→∞−→ Aα 1 r [ Aαlmle−i(kαr− lpi 2 ) − Bαlmlei(kαr− lpi 2 ) ] φαYlml(r̂), (2.97) where e−i(kαr− lpi 2 ) and ei(kαr− lpi 2 ) describe the incoming and outgoing waves, respec- tively, and Aαlml = −(aαlml + ibαlml)/2iBαlml = (−aαlml + ibαlml)/2i. (2.98) 35 2.4. Multi-channel scattering theory The relationship between A and B defines the scattering matrix Bαlml = ∑ α′ ∑ l′ ∑ m′l Sαlml←α′l′m′lAα′l′m′l , (2.99) where Sαlml←α′l′m′l can be considered as the probability amplitude of atoms or molecules to go from one incoming channel α′l′m′l to the outgoing channel αlml and the sum is over all the possible incoming collision channels. The total asymptotic wave function for a particular channel αlml is then given by ψαlml r→∞−→ Aα 1 r φαYlml(r̂) × Aαlmle−i(kαr− lpi2 ) −∑ α′ ∑ l′ ∑ m′l Sαlml←α′l′m′lAα′l′m′le i(kαr− lpi2 )  . (2.100) Assuming that particles in the incoming channel α are scattered into all directions, the incident plane wave can be expanded as [8] ei ~k·~r = 4pi ∑ l ∑ ml iljl(kαr)Y ∗lml(k̂)Ylml(r̂) = i2pi kαr ∑ l ∑ ml il [ e−i(kαr− lpi 2 ) − ei(kαr− lpi2 ) ] Y ∗lml(k̂)Ylml(r̂). (2.101) The incident wave function thus has a form Aαφαe i~k·~r = Aαφα i2pi kαr ∑ l ∑ ml il [ e−i(kαr− lpi 2 ) − ei(kαr− lpi2 ) ] Y ∗lml(k̂)Ylml(r̂). (2.102) In the systems studied in this Thesis, the incoming particles are in a particular internal state. Therefore the amplitude of the incoming flux in channels which are different from α is zero. Comparing the coefficients in front of the term e−i(kαr− lpi 2 ) in Eqs. 2.100 and 2.102, we obtain Aαlml Aαlml = i2pi kα ilY ∗lml(k̂) (in channel α) (2.103) = 0 (in all other channels). 36 2.4. Multi-channel scattering theory Bαlml is then given by (cf. Eq. 2.99) Bαlml = ∑ l′ ∑ m′l Sαlml←αl′m′l i2pi kα il ′ Y ∗l′m′l(k̂). (2.104) Note that there is no sum over α′ now. The next step is to determine the scattering wave function and extract the scattering amplitude. Asymptotically, the total wave function can be written as ψ → ψinc + ψsc, (2.105) so the scattered part of the wave function is given by ψsc = ψ − ψinc = ψincoming + ψoutgoing − (ψincincoming + ψincoutgoing). (2.106) Because the incoming part of the total scattering wave function and the incoming part of the incident wave function must be identical, the scattering wave function is ψsc = ψoutgoing − ψincoutgoing. (2.107) The outgoing part of the total wave function is the sum over all energetically accessible collision channels ψoutgoing = ∑ α′ (ψα′)outgoing = ∑ α′ ∑ l′ ∑ m′l (ψα′l′m′l)outgoing = − ∑ α′ ∑ l′ ∑ m′l Aα′ 1 r Bα′l′m′le i(kα′r− lpi2 )φα′Yl′m′l(r̂). (2.108) Here, the normalization factor Aα′ is equal to Aα only for elastic collisions. Com- bining Eqs. 2.104 and 2.108 we obtain ψoutgoing ψoutgoing =− ∑ α′ ∑ l′ ∑ m′l Aα′ 1 r φα′Yl′m′l(r̂)e i(kα′r− l ′pi 2 ) × [∑ l ∑ ml Sα′l′m′l←αlml i2pi kα ilY ∗lml(k̂) ] . (2.109) 37 2.4. Multi-channel scattering theory The outgoing part of the incident wave function is given by ψincoutgoing →−Aαφα i2pi kαr ∑ l ∑ ml ilei(kαr− lpi 2 )Y ∗lml(k̂)Ylml(r̂) =− ∑ α′ ∑ l′ ∑ m′l ∑ l ∑ ml Aα′φα′ i2pi kαr ilei(kα′r− l′pi 2 ) Y ∗lml(k̂)Yl′m′l(r̂)δαα′δll′δmlm′l . (2.110) The scattering wave function is then obtained as ψsc =ψ − ψinc = ∑ α′ ∑ l′ ∑ m′l ∑ l ∑ ml Aα′φα′ i2pi kαr ilei(kα′r− l′pi 2 )Y ∗lml(k̂)Yl′m′l(r̂) × [ δαα′δll′δmlm′l − Sα′l′m′l←αlml ] . (2.111) The scattering wave function can also be written in terms of the scattering amplitude ψsc = ∑ α′ Aα′fα′←α eikα′r r φα′ . (2.112) Comparing Eqs. 2.111 and 2.112, we extract the expression for the scattering am- plitude fα′←α = ∑ l′ ∑ m′l ∑ l ∑ ml i2pi kα ilY ∗lml(k̂)Yl′m′l(r̂)e − il′pi 2 × [ δαα′δll′δmlm′l − Sα′l′m′l←αlml ] = ∑ l′ ∑ m′l ∑ l ∑ ml i2pi kα il−l ′ Y ∗lml(k̂)Yl′m′l(r̂)Tα′l′m′l←αlml , (2.113) where Tα′l′m′l←αlml = δαα′δll′δmlm′l − Sα′l′m′l←αlml . The differential cross section for α→ α′ transition is given by (cf. Eq. 2.33) dσα′←α dΩ = |fα′←α|2. (2.114) Integrating the above equation over all orientations and dividing the result by 4pi to account for the random orientation of the incoming flux, we obtain the total cross 38 2.4. Multi-channel scattering theory section [199] σα′←α = pi k2α ∑ l′ ∑ m′l ∑ l ∑ ml |δαα′δll′δmlm′l − Sα′l′m′l←αlml | 2. (2.115) 2.4.2 Numerical integration of multi-channel equations The coupled-channel equations of Eq. 2.91 can be written in the form of a matrix- vector equation[ ∂2 ∂r2 + k2n − Unn(r) ] Fn(r) = ∑ m6=n Unm(r)Fm(r), (2.116) where n corresponds to the incoming channel α whereas m 6= n corresponds to the outgoing channel α′ 6= α, and k2n = 2µ(E − n), (2.117) and Unm(r) = 2µ〈φn|V̂ |φm〉. (2.118) For each internal state of the colliding particles, we get one such matrix-vector problem. Solving Eq. 2.116 for all F-vectors simultaneously results in a matrix- matrix problem, where the columns of the new F-matrix are the old F-vectors. We can construct a log-derivative matrix y, which satisfies the equation y′(r) + W(r) + y2(r) = 0, (2.119) and use the method described in section 2.3.4 to solve the problem. Here, the coupled matrix elements are given by Wnm = k2nδmn − Unm(r). (2.120) The initial log-derivative matrix y(0) is diagonal with the matrix elements of infinite magnitude. In our calculation, we start the integration at r = a deep into the classically forbidden region (usually at r = 1.0 bohr) and set the matrix elements of the log-derivative matrix y(a) equal to very large numbers, e.g., y(a) = 1030I, where I is the identity matrix. When the integration is carried out to a sufficiently large r (usually > 800 bohr for our problems), the interaction potential can be neglected. 39 2.4. Multi-channel scattering theory The K-matrix is then constructed as (cf. Eq. 2.86) K = (yN−N′)−1(yJ− J′), (2.121) where [J]nm = δnmk − 1 2 n ĵl(knr) (2.122) [N]nm = δnmk − 1 2 n n̂l(knr). (2.123) As we include the normalization coefficient An = ν −1/2 n in the total wave function (cf. Eq. 2.94) for the boundary conditions in the derivation of the multi-channel scattering theory, we involve this coefficient in our numerical calculations by multi- plying the matrix elements in J and N by the factor k−1/2n ∝ ν−1/2n . The S-matrix can therefore be constructed using the relation given by Eq. 2.68 and from there we can calculate the integral cross sections for elastic and inelastic collisions (cf. Eq. 2.115). 40 Chapter 3 Accurate interatomic potentials from interplay of ultracold experiment and theory1 In order to describe collision dynamics in ultracold atomic gases, it is necessary to construct accurate interatomic interaction potentials. The collision properties of ultracold atoms are extremely sensitive to details of the interaction potentials, especially the long-range part of the interaction and the position of the least bound state supported by the interaction potential curve. The position of the least bound state determines the magnitude of the scattering length. Small variations of the interatomic potentials may lead to large variations of the scattering length and it is therefore very difficult to construct interatomic potentials that would describe accurately ultracold collision properties based on ab initio quantum chemistry calcu- lations alone. Magnetically tunable Feshbach resonances offer an extremely sensitive probe of interaction potentials. One can create model potential curves to describe collision dynamics in ultracold gases using numerical calculations reproducing po- sitions and widths of the resonances observed in experiments. In this Chapter, we describe a procedure of fitting the experimental data and generate accurate inter- action potentials for the 6Li–87Rb molecule in 1Σ and 3Σ electronic states. This allows us to predict quantitatively the positions of several experimentally relevant resonances in ultracold Li–Rb mixtures. The analysis is based on an asymptotic bound state model yielding the approximate energies of two least bound states of the triplet and singlet potentials and the positions of two experimentally measured Feshbach resonances. Guided by the results of the model analysis, we generate the corresponding potentials and fine-tune them to reproduce the measured positions and widths of the resonances using a full quantum scattering calculation. 1A part of this Chapter was presented in Ref. [1] of Appendix D. 41 3.1. Ultracold mixtures of 6Li and 87Rb 3.1 Ultracold mixtures of 6Li and 87Rb The Li–Rb system is very important for the study of both ultracold atomic and molecular gases. 6Li and 87Rb atoms are often used for studies of ultracold fermionic and bosonic atomic gases. At the same time, photo-association or magneto-association in ultracold Li–Rb mixtures can be used to make polar LiRb dimers. LiRb dimers have a relatively large dipole moment and are thus improtant for the study of ul- tracold polar molecules and for the experimental study of electric-field-induced Fes- hbach resonances [174]. Understanding the low temperature collision properties of atoms in the Li–Rb mixture is therefore of significant importance. In 2005, Silber et al. [204] created a quantum degenerate Bose-Fermi mixture of 6Li and 87Rb atoms in a magnetic trap with rubidium serving as the refrigerant. This experiment revealed the challenges of this approach to cooling lithium due to small magnitude of the interspecies scattering length at low magnetic fields. Subsequently, interspecies Feshbach resonances in this system were found by the same group. These resonances may provide a means to enhance cooling in this mixture by varying the scattering length [205] and a way to tune the interactions in the Bose-Fermi mixture [167]. Feshbach resonances may generally offer an efficient way of forming loosely bound LiRb dimers, which can then be transferred from the excited vibrational state near threshold to the ground vibrational state [143]. In deeply bound vibrational states, the LiRb dimer has a large electric dipole moment (up to 4.2 Debye) [206]. An ensemble of these molecules, polarized by an external electric field, will interact strongly via the long range and anisotropic dipole-dipole interaction, which is predicted to lead to a wide variety of novel phenomena [207]. It is therefore extremely important to understand and quantitatively characterize the properties and the number of Feshbach resonances in this system. The interaction between two alkali metal atoms in the ground electronic state gives rise to two molecular states of 1Σ and 3Σ symmetries. The molecular states are characterized by the corresponding scattering lengths. The interspecies triplet scattering lengths for the 6Li–87Rb mixtures have been evaluated through measure- ments of cross-thermalization: |a6,87triplet| = 20+9−6 bohr [204] and |a7,87triplet| = 59+19−19 bohr [208]. In addition, two heteronuclear Feshbach resonances were recently observed [205]. The signs of the triplet scattering lengths and the positions of Feshbach reso- nances in other atomic states, however, remain unknown. The sign of the scattering length is particularly important since it determines the global stability of the ultra- cold mixture. In this Chapter, we combine the experimental results from these two papers to produce a new set of high-precision LiRb interaction potentials which fully 42 3.2. Fitting procedure characterize the 6Li–87Rb scattering properties in any combination of spin states. We find that the sign of the 6Li–87Rb triplet scattering length must be negative. Using these potentials we also predict the positions and widths of all the Feshbach resonances below the magnetic field 2 kG for all 6Li–87Rb spin combinations where 6Li and 87Rb are in the lower hyperfine manifold. 3.2 Fitting procedure Our starting point for this work is to model the triplet a3Σ and singlet X1Σ interac- tion potentials of the Li–Rb molecule by an analytical function of the form originally proposed by Degli-Esposti and Werner [209] V = G(r)e−α(r−re) − T (r) 5∑ i=3 C2i r2i , (3.1) with G(r) = 8∑ l=0 glr l, (3.2) and T (r) = 1 2 [1 + tanh(1 + Tr)] . (3.3) The potential parameters were determined by varying this function to reproduce the overall shape and approximate number of bound states for the LiRb dimer predicted by the ab initio calculations reported in Ref. [206]. Here, we only tune the parameter g1 in G(r) to adjust the repulsive wall in the short range of the potential. In the Thesis, this fitting parameter is denoted as bsinglet and btriplet for singlet and triplet potentials, respectively. The long range behavior is adjusted to match the van der Waals coefficient C6 = 2545 Eh bohr6 (where Eh = 4.35974 × 10−18 J) calculated by Derevianko et al. [210]. The amplitude and sign of the pure triplet and singlet s-wave scattering lengths as well as the positions of the Feshbach resonances are almost completely deter- mined by the location of the least bound states of the potentials [129] if the bound state is close to the dissociation limit. When the bound state crosses the energy of the colliding atoms in the open channel at the dissociation threshold, Feshbach res- onances occur. Since the long range behavior of the potentials has been accurately determined [210], the potentials can only be refined by making small adjustments to the short range repulsive wall while keeping the long range part fixed. The full coupled-channel calculations are computationally intensive, so iteratively finding 43 3.3. Asymptotic bound state model the proper modification of the model potentials to reproduce the experimentally observed resonances can be a lengthy process. To simplify this search and to gain insight into the scattering properties of the Li–Rb system, we employ an asymptotic bound state model to first determine the energy of the last molecular bound states (closest to the threshold) as a function of the magnetic field and locate the cross- ings between the bound state energies and the threshold energy which give rise to Feshbach resonances. Once the crossings are located, we extract the energy of the last bound state of either the singlet or triplet potential from the calculations, and generate the corresponding potential energy curves. We then fine tune the potential curves by fitting the positions and widths of experimentally measured resonances using full quantum scattering calculations. 3.3 Asymptotic bound state model The asymptotic bound state model was proposed and described in Ref. [211]. The major simplifying assumption in the asymptotic bound state model is that the coupling between the channels (provided by the hyperfine interaction V̂hf) is small enough that the two-body bound states can be represented to first order by uncoupled orbital and spin states. In our calculations, we neglect the magnetic dipole-dipole interaction since it has no effect on the observables described in this Chapter. As a result, the projection MF of the total spin angular momentum on the magnetic field axis is conserved. We therefore only consider states with the same MF = MS +MIa +MIb values as the initial state. The atomic and molecular quantum numbers used in this Thesis are defined in Table I, where a and b represent different alkali metal atoms. The Hamiltonian for collisions between two alkali metal atoms is Ĥ = Ĥrel + V̂B + V̂hf , (3.4) where Ĥrel accounts for the relative motion of the atoms, V̂B models the interac- tion of the collision complex with external magnetic fields, and V̂hf represents the hyperfine interactions. Ĥrel can be written as Ĥrel = − 12µr ∂2 ∂r2 r + l̂2(θ, ϕ) 2µr2 + V̂ , (3.5) where µ is the reduced mass of the colliding atoms, r is the interatomic distance, l̂ is the operator describing the rotational motion of the collision complex and the angles 44 3.3. Asymptotic bound state model θ and ϕ specify the orientation of the interatomic axis in the space-fixed coordinate frame (cf., Chapter 2). V̂ can be written in the following form V̂ = ∑ S ∑ MS |SMS〉VS(r)〈SMS |, (3.6) where VS(r) denotes the adiabatic interaction potential of the molecule in either the pure singlet (S = 0) or the triplet (S = 1) states. If we expand the wave function of the last (least bound) two-body molecular bound states ψSMSl in the basis |RSMSl 〉|SMS〉, the matrix elements 〈ψSMSl |Ĥrel|ψ S′M ′S l 〉 are given by 〈ψSMSl |Ĥrel|ψ S′M ′S l 〉 = 〈SMS |〈RSMSl |Ĥrel|R S′M ′S l 〉|S′M ′S〉 = 〈SMS |〈RSMSl | − 1 2µr ∂2 ∂r2 r + l̂2(θ, ϕ) 2µr2 + V̂ |RS′M ′Sl 〉|S′M ′S〉 = 〈SMS |〈RSMSl | − 1 2µr ∂2 ∂r2 r + l̂2(θ, ϕ) 2µr2 + ∑ S′′ ∑ M ′′S |S′′M ′′S〉VS(r)〈S′′M ′′S ||RS ′M ′S l 〉|S′M ′S〉 = 〈RSMSl | − 1 2µr ∂2 ∂r2 r + l̂2(θ, ϕ) 2µr2 + VS(r)|RS ′M ′S l 〉δSS′δMSM ′S = ESl δSS′δMSM ′S , (3.7) where RSMSl and E S are the eigenfunctions and the energies of the last bound state of either the singlet or the triplet potentials, respectively. Therefore, the eigenfunctions of Hrel depend on the total electronic spin value implicitly and are parametrized by the spin value S. We assume that the overlap of the singlet and triplet wave functions of the same orbital angular momentum l is 〈R0l |R1l 〉 = 1. Here, R0l and R 1 l are the vibrational wave functions of the different potentials, however, they are both near the dissociation limit. So the character of the wave functions is mostly determined by the C6 coefficient, which is the same for both the singlet and triplet potentials. For deeply bound states, this assumption would not generally be valid. For the uncoupled representation, the uncoupled radial wave function of the last molecular bound state should be parametrized by the spin value Sa,MSa , Sb,MSb . We expand the wave function ψ Sa,MSa ,Sb,MSb l in the basisR Sa,MSa ,Sb,MSb l |SaMSa〉|SbMSb〉. 45 3.3. Asymptotic bound state model Table 3.1: Definition of quantum numbers used in this Thesis. l orbital angular momentum of a collision complex ml projection of l on the space-fixed quantization axis F total spin angular momentum of a two-atom system MF projection of F on the space-fixed quantization axis I total nuclear spin angular momentum of a two-atom system MI projection of I on the space-fixed quantization axis S total electronic spin angular momentum of a two-atom system MS projection of S on the space-fixed quantization axis Fa total spin angular momentum of atom a MFa projection of Fa on the space-fixed quantization axis Fb total spin angular momentum of atom b MFb projection of Fb on the space-fixed quantization axis Ia nuclear spin angular momentum of atom a MIa projection of Ia on the space-fixed quantization axis Ib nuclear spin angular momentum of atom b MIb projection of Ib on the space-fixed quantization axis Sa electronic spin angular momentum of atom a MSa projection of Sa on the space-fixed quantization axis Sb electronic spin angular momentum of atom b MSb projection of Sb on the space-fixed quantization axis The matrix elements of Ĥrel in the uncoupled representation thus have the form 〈ψSa,MSa ,Sb,MSbl |Ĥrel|ψ Sa,M ′Sa ,Sb,M ′ Sb l 〉 = 〈SaMSa |〈SbMSb |〈R Sa,MSa ,Sb,MSb l |Ĥrel|R Sa,M ′Sa ,Sb,M ′ Sb l |SaM ′Sa〉|SbM ′Sb〉 = ∑ SMS ∑ S′M ′S 〈SMS |SaMSaSbMSb〉〈S′M ′S |SaM ′SaSbM ′Sb〉 〈SMS |〈RSMSl |Ĥrel|R S′M ′S l 〉|S′M ′S〉 = ∑ SMS 〈SMS |SaMSaSbMSb〉〈SMS |SaM ′SaSbM ′Sb〉ESl , (3.8) where 〈SMS |SaMSaSbMSb〉 and 〈SMS |SaM ′SaSbM ′Sb〉 are the Clebsch-Gordan coef- ficients for the transformation between the coupled and uncoupled representations. Within the asymptotic bound model, the molecular bound state energies are found by diagonalizing the matrix of the total Hamiltonian given by Eq. 3.4. The operator V̂B describes the interaction between the atoms and an external magnetic 46 3.3. Asymptotic bound state model field V̂B = 2µ0B(ŜZa + ŜZb)−B ( µa Ia ÎZa + µb Ib ÎZb ) , (3.9) where B is the magnetic field strength, µ0 is the Bohr magneton and µa and µb denote the nuclear magnetic moments of two different alkali metal atoms. We assume that the magnetic field is directed along the z-axis. The hyperfine interaction V̂hf can be represented as V̂hf = γaÎa · Ŝa + γbÎb · Ŝb, (3.10) where γa and γb are the atomic hyperfine interaction constants. In the absence of an external magnetic field, the nuclear spin of the ground-state alkali atoms is coupled to the electronic spin by hyperfine interactions, yielding the energy splitting 4Ehf between states with different total angular momenta, i.e., hyperfine states. The operator for the total angular momentum is given by F̂a(b) = Îa(b) + Ŝa(b). (3.11) Squaring this expression yields Îa(b) · Ŝa(b) = 1 2 [ F̂ 2a(b) − Î2a(b) − Ŝ2a(b) ] , (3.12) The electronic spin of alkali metal atoms is Sa(b) = 1/2, giving rise to two Fa(b) values, Fa(b) = Ia(b) + 1/2 and Fa(b) = Ia(b) − 1/2. Therefore, the relation between the experimentally measured observable – the energy splitting – and the hyperfine interaction constant is 4Ehfa(b) = γa(b) ( Ia(b) + 1 2 ) . (3.13) In this calculation, we use γa = 152.14 MHz for the 6Li atom and γb = 3417.34 MHz for the 87Rb atom [212]. We expand the total wave function of the last bound state of the diatomic system in terms of the uncoupled radial function, and the eigenfunctions of Î2a , Î 2 b, ÎZa and ÎZb as follows: |Ψl〉 = |RSa,MSa ,Sb,MSbl 〉 ⊗ |α〉, (3.14) where |α〉 = |IaMIa〉|SaMSa〉|IbMIb〉|SbMSb〉. (3.15) 47 3.3. Asymptotic bound state model The operators for the interaction of the atoms with magnetic fields are diagonal in the representation |IaMIa〉|SaMSa〉|IbMIb〉|SbMSb〉. The matrix elements of the hyperfine interaction operators can be readily evaluated using the relations Îa · Ŝa = ÎZaŜZa + 1 2 (Îa+Ŝa− + Îa−Ŝa+) (3.16) and Îb · Ŝb = ÎZbŜZb + 1 2 (Îb+Ŝb− + Îb−Ŝb+), (3.17) where Î(Ŝ)a(b)+ and Î(Ŝ)a(b)− are the raising and lowing operators, respectively. The matrix elements of the raising and lowering operators can be evaluated as [202] 〈jmj |j±|j′m′j〉 = √ j(j + 1)−m′j(m′j ± 1)δjj′δmjm′j±1. (3.18) Since the long-range part of the potential is known, the energies ESl of the higher l > 0 states are uniquely determined by the l = 0 singlet and triplet energies. There- fore, in this model, Esinglet and Etriplet are the only two free parameters. They are adjusted until the threshold channel crosses the energy of the molecular states at the positions corresponding to the locations of the experimentally measured Feshbach resonances. Figure 3.1 shows both the s- and p-wave molecular bound state energies versus magnetic field strength for all the states with MF = 3/2 computed using the asymptotic bound state model. Although the asymptotic bound state model cannot predict the exact locations of the Feshbach resonances, it does predict reliably the energies of the molecular channels in regions far from the crossings. Therefore, in the limit that the effect of the interstate couplings on the energy is negligibly small, it provides an excellent estimate of the positions of the Feshbach resonances. After generating the approximate singlet and triplet potential curves using the asymptotic bound state model, we carry out full coupled-channel calculations with the total Hamiltonian (Eq. 3.4). The theory and method for numerical calcula- tions are presented in Section 2.3 of Chapter 2. We note that the matrix of the Hamiltonian in the basis |αlml〉 does not become diagonal as r →∞ (cf. Eq. 3.18). Therefore, the boundary conditions cannot be properly applied to the coupled chan- nel equations Eq. 2.91 in this representation. Before constructing the scattering S- matrix from the solutions of Eq. (2.91), we apply an additional transformation that diagonalizes the matrix of V̂B + V̂hf . This procedure was described in Ref. [199]. The scattering matrix thus obtained yields the probabilities of elastic and inelastic scattering of 6Li and 87Rb in the presence of magnetic fields. 48 3.4. Results 882 G. For each of the five candidate regions, the corre- sponding potentials were generated and the predicted elastic scattering cross sections as a function of magnetic field were computed using the full coupled-channel calculation. In ad- dition, the corresponding triplet scattering lengths were also computed. Each of the four purely s-wave cases were ruled out based on a variety of reasons. In region I, the lower resonance at 882 G is predicted to be a factor of ten larger in width than the upper resonance at 1067 G in violation of the experimentally measured widths of 1.27 and 10.62 G respec- tively !16". In region II, the relative widths of the resonances are #as in region I$ incorrect, and at these values for #Esinglet ,Etriplet$ there would have been three additional and wide #!5 G$ s-wave resonances below 200 G which were not observed in the experiment. While in regions III and IV the ordering of the resonances is consistent with the experi- mental measurements #the upper resonance is larger than the lower resonance$, in region III there is an additional wide #!10 G$ s-wave resonance below 200 G not observed in the experiment, and in region IV there is an additional s-wave resonance at approximately 960 G #!1 G$ in between the two observed resonances. In addition, the triplet scattering length for regions III and IV corresponding to Etriplet= −0.377 cm−1 is atriplet 6,87 =105aB. This value is in disagreement with the experimentally determined value from measure- ments of the cross thermalization in magnetically trapped 6Li-87Rb mixtures which indicate that the interspecies triplet scattering length is %atriplet 6,87 %=20 −6 +9aB !33". In order to verify the robustness of these findings, we constructed a pair of Lennard-Jones potentials !V#R$ =C12 /R12−C6 /R6" with the same C6 coefficient, roughly the same number of bound states, and the same least bound state energies #Esinglet ,Etriplet$ as the fitted potential in each of the regions considered. Using these potentials, we verified that the Feshbach resonance locations and scattering lengths are essentially the same as for the fitted potentials and are insen- sitive to the short-range details of the potentials. This check provides an important verification of our characterization of the four purely s-wave candidate regions. The conclusion is that the experimentally observed Feshbach resonances are inconsistent with pure s-wave resonances, and we must con- sider the possibility that at least one of the resonances origi- nates from a p-wave molecular state. Region V in Fig. 4 represents the only location in the #Esinglet ,Etriplet$ parameter space for which only one s-wave resonance occurs below 1.2 kG #at 1067 G$ and a p-wave resonance occurs at 882 G. All other branches displayed in Fig. 4 involve at least one additional s-wave resonance oc- curring in a location where none was observed experimen- tally. Along the locus of #Esinglet ,Etriplet$ values for which these two resonances occur at the correct locations, an addi- tional p-wave resonance was found to occur somewhere be- tween 1081 and 1024 G, while the width of the s-wave reso- nance at 1065 G was found to vary from 5 to 35 G. At the precise #Esinglet ,Etriplet$ values for which the second p-wave resonance was coincident with the s-wave resonance at 1065 G, the s-wave resonance width was "B=11.53 G, con- sistent with the experimentally measured value "Bexpt =10.62 G for the full width at half maximum for the trap loss feature. For these optimal singlet and triplet potentials, the bound state energies are Esinglet=−0.106 cm−1 and Etriplet=−0.137 cm−1. Figure 5 shows the result of the full coupled-channel cal- culation performed using the refined potentials. The elastic 0 500 1000 1500 2000 Magnetic Field (G) -0.4 -0.3 -0.2 -0.1 0 0.1 E ne rg y (c m -1 ) A B C FIG. 3. #Color online$ Molecular bound state energies versus magnetic field computed with the asymptotic bound state model. The threshold for the % 12 , 1 2 &6Li! %1,1&87Rb collision channel is shown by the solid line while the dashed #dotted$ lines indicate the s-wave #p-wave$ states. These molecular state energies were computed given the least bound states ES l of the optimal singlet and triplet potentials of E0 l=0 =−0.106 cm−1, E0 l=1 =−0.0870 cm−1, and E1 l=0 = −0.137 cm−1, E1 l=1 =−0.116 cm−1. The predicted resonance loca- tions are close to the actual locations determined by the full coupled-channel calculation and are indicated by the solid dots. Near 890 G #A$ the threshold crosses a p-wave molecular state and the corresponding p-wave elastic scattering cross section shown in Fig. 5 is observed to rapidly diverge and then return to the back- ground level. Likewise, near 1070 G #B$ the threshold crosses both a p-wave and an s-wave molecular state and both the s-wave and p-wave elastic scattering cross sections are affected. Finally, near 1300 G #C$ a second s-wave-induced Feshbach resonance occurs. -0.4 -0.3 -0.2 -0.1 0 Esinglet ( cm -1 ) -0.4 -0.3 -0.2 -0.1 0 E tr ip le t (c m -1 ) I II III IV V FIG. 4. #Color online$ Locus of points in the #Esinglet ,Etriplet$ parameter space where an s-wave resonance occurs at one of the two experimentally determined locations 882.02 !gray #green$" or 1066.92 G !dark #red$" for atoms in the % 12 , 1 2 &6Li! %1,1&87Rb state. The dotted lines indicate the approximate values for Esinglet and Etriplet beyond which a new bound state enters the potential at zero energy. There are four regions #I–IV$ indicated on the plot where an s-wave resonance occurs simultaneously at 882.02 and at 1066.92 G. Region V indicates a range of values for which an s-wave resonance occurs at 1066.92 while a p-wave resonance #not represented in this plot$ occurs at 882 G. For each of these five candidate regions, the character of the predicted elastic cross sec- tions as a function of magnetic field was studied, and the results of this analysis are discussed in the text. FESHBACH RESONANCES IN ULTRACOLD 85RB-… PHYSICAL REVIEW A 78, 022710 #2008$ 022710-5 Figure 3.1: The s- and p-wave molecular bound state energies as functions of mag- netic fields for all the states with MF = 3/2 computed within the asymptotic bound state model. The solid line shows the threshold for the |12 , 12〉6Li⊗ |1, 1〉87Rb collision channel (see text) while the dashe (dotted) lines indicate the s-wave (p-wave) states. These molecular state energies were computed given the least bound states ESl of the optimal singlet and triplet potentials E0l=0 = −0.106 cm−1, E0l=1 = −0.0870 cm−1, and E1l=0 = −0.137 cm−1, E1l=1 = −0.116 cm−1. The predicted resonance positions are close to the actual positions determined by the full coupled-channel calculation and are indicated by the solid dots (A, B, and C). 3.4 Results Deh et al. have found in their experiments two Feshbach resonances for atoms in the |12 , 12〉6Li ⊗ |1, 1〉87Rb state with positions (B0) of 882.02 G and 1066.92 G, and widths (∆B) of 1.27 G and 10.62 G, respectively [205]. Here, |Fa,MFa〉|Fb,MFb〉 is the usual notation for the atomic hyperfine states. Note that Fa and Fb are not good quantum numbers in the presence of a magnetic field. However, at low and moderate magnetic fields they can be used to specify atomic states correlating with particular hyperfine states at zero magnetic field. By fitting these two resonances, we want to generate accurate interaction potentials describing ultracold 6Li–87Rb collisions. At first, we thought fitting these two potential curves would be an easy problem and could be solved in two days. The original procedure we planned to follow is shown in Fig. 3.2. However, after obtaining one pair of the bound energies Esinglet and Etriplet 49 3.4. Results with the asymptotic bound state model, we found that this project was far more complicated than what we had expected. We discovered that there is an infinite number of combinations of singlet and triplet potential curves which can reproduce the positions of the experimentally measured Feshbach resonances, i.e., the results of step A are not unique. At the same time, there is also an infinite number of triplet potential energy curves yielding the same scattering length as the experimental measurement. As shown in Figs. 3.3 and 3.4, the scattering length varies periodically as a function of the fitting parameters of the potentials. Our task was to find the right and only combination of singlet and triplet potential energy curves which can reproduce all of the experimentally observed quantities consistently. We first used a try-and-error method to search for the right combination, however, this approach was like a near-sighted person looking for a weed in a big forest: time-consuming and inefficient. Eite Tiesinga – a researcher from NIST – gave us advice to calculate the positions of the resonances as functions of both Esinglet and Etriplet and plot them as shown in Fig. 3.5. With this graph, we could visualize the consequences of all the combinations and see the entire picture. Here, we truly thank Eite Tiesinga for his insight and advice on the use and utility of the asymptotic bound state model. The fitting approach has thus been modified. First, from Figs. 3.3 and 3.4, we selected an interval of fitting parameters which yield the scattering lengths from, say, -500 to 500 bohr, and found the corresponding interval of Esinglet and Etriplet. Then, we plotted in Fig. 3.5 the locus of points in the (Esinglet, Etriplet) parameter space, where an s-wave resonance occurs at one of the two experimentally determined positions 882.02 G or 1066.92 G for atoms in the |12 , 12〉6Li ⊗ |1, 1〉87Rb state. We identified four regions (I – IV) indicated on the plot where an s-wave resonance occures simultaneously at 882.02 G and at 1066.92 G. Region V indicates a range of values for which an s-wave resonance occurs at 1066.92 G while a p-wave resonance (not presented in this plot) occurs at 882 G. For each of the five candidate regions, the corresponding potentials are generated and the predicted elastic scattering cross sections as a function of the magnetic field are computed using the full coupled- channel calculations. In addition, the corresponding triplet scattering lengths are also computed. Each of the four purely s-wave cases are ruled out based on a variety of reasons. In region I, the lower resonance at 882 G is predicted to be a factor of ten larger in width than the upper resonance at 1067 G in violation of the experimentally measured widths of 1.27 and 10.62 G respectively [205]. In region II, the relative widths of the resonances are (as in region I) incorrect, and at these values for (Esinglet, Etriplet) there would have been three additional and wide (> 5 G) s-wave resonances below 200 G which were not observed in the experiment. While in 50 3.4. Results The positions of experimentally measured Feshbach resonances The optimal least bound energies for singlet and triplet potentials Calculate the experimentally measured scattering lengths Reproduce both the positions and width of the resonances Adjust the short-range repulsive wall to generate the singlet and triplet potential curves A: The asymptotic bound state model B: Discrete variable representation calculations C: Full coupled-channel calculations Make consistent Figure 3.2: The procedure of fitting the interactions potentials for ultracold 6Li– 87Rb collisions. regions III and IV the ordering of the resonances is consistent with the experimental measurements (the upper resonance is wider than the lower resonance), in region III there is an additional wide (>10 G) s-wave resonance below 200 G not observed in the experiment, and in region IV there is an additional s-wave resonance at approximately 960 G (>1 G) in between the two observed resonances. In addition, the triplet scattering length for regions III and IV corresponding to Etriplet = −0.377 cm−1 is a6,87triplet = 105 bohr. This value is in disagreement with the experimentally determined value from the measurements of the cross thermalization in magnetically trapped 6Li–87Rb mixtures which indicate that the interspecies triplet scattering length is |a6,87triplet| = 20+9−6 bohr [204]. In order to verify the robustness of these findings, we constructed a pair of Lennard-Jones potentials V = C12/r12−C6/r6 with the same C6 coefficient, roughly the same number of bound states, and the same least bound state energies (Esinglet, Etriplet) as the fitted potential in each of the regions considered. Using these po- tentials, we verified that the Feshbach resonance locations and scattering lengths 51 3.4. Results 1.641 1.644 1.647 1.65 1.653 1.656 1.659 b singlet -6 -4 -2 0 2 4 6 Sc at te rin g Le ng th  (x 10 3 B oh r) Figure 3.3: The singlet scattering length varies periodically as a function of the fitting parameter bsinglet. This fitting parameter determines the slope of the repulsive wall of the 1Σ interaction potential. are essentially the same as for the fitted potentials and are insensitive to the short- range details of the potentials. This check provides an important verification of our characterization of the four purely s-wave candidate regions. The conclusion is that the experimentally observed Feshbach resonances are inconsistent with pure s-wave resonances, and we must consider the possibility that at least one of the resonances originates from a p-wave molecular state. Region V in Fig. 3.5 represents the only location in the (Esinglet, Etriplet) param- eter space for which only one s-wave resonance occurs below 1.2 kG (at 1067 G) and a p-wave resonance occurs at 882 G. All other branches displayed in Fig. 3.5 involve at least one additional s-wave resonance occurring in a location where none was observed experimentally. Along the locus of (Esinglet, Etriplet) values for which these two resonances occur at the correct locations, an additional p-wave resonance was found to occur somewhere between 1081 and 1024 G, while the width of the s-wave resonance at 1065 G was found to vary from 5 to 35 G. At the precise (Esinglet, Etriplet) values for which the second p-wave resonance was coincident with the s-wave resonance at 1065 G, the s-wave resonance width is 4B = 11.53 G, consistent with the experimentally measured value 4B = 10.62 G for the full width 52 3.4. Results 2.06 2.065 2.07 2.075 2.08 btriplet -6 -4 -2 0 2 4 6 Sc at te rin g Le ng th  (x  10 3 B oh r) Figure 3.4: The triplet scattering length varies periodically as a function of the fitting parameter btriplet. This fitting parameter determines the slope of the repulsive wall of the 3Σ interaction potential. at half maximum for the trap loss feature in the experiment. For these optimal singlet and triplet potentials, the bound state energies are Esinglet = −0.106 cm−1 and Etriplet = −0.137 cm−1. Figure 3.6 shows the results of the full coupled-channel calculation performed using the refined potentials. The elastic scattering cross sections for the |12 , 12〉6Li ⊗ |1, 1〉87Rb state show divergences at magnetic fields of 1065 and 882 G, in excellent agreement with the experimentally determined Feshbach resonance positions. In ad- dition, the triplet scattering length from the fine-tuned triplet potential was found to be |a6,87triplet| = −19.8 bohr, also in excellent agreement with the experimentally determined value. For the reduced mass corresponding to the 7Li–87Rb complex, the optimal fine-tuned triplet potential predicts a7,87triplet = 448 bohr, in disagreement with the experimental measurement of |a7,87triplet| = 59+19−19 bohr [208]. This is because the slope of the short range repulsive wall of the interaction potentials is not well- defined. The potentials refined by the structure of the Feshbach resonances, which depends only on the positions of the least bound states, do not provide information about the short range shape of the interatomic interactions. Therefore, the interac- tion potentials thus generated are not unique. The reduced mass of the 7Li–87Rb 53 3.4. Results 882 G. For each of the five candidate regions, the corre- sponding potentials were generated and the predicted elastic scattering cross sections as a function of magnetic field were computed using the full coupled-channel calculation. In ad- dition, the corresponding triplet scattering lengths were also computed. Each of the four purely s-wave cases were ruled out based on a variety of reasons. In region I, the lower resonance at 882 G is predicted to be a factor of ten larger in width than the upper resonance at 1067 G in violation of the experimentally measured widths of 1.27 and 10.62 G respec- tively !16". In region II, the relative widths of the resonances are #as in region I$ incorrect, and at these values for #Esinglet ,Etriplet$ there would have been three additional and wide #!5 G$ s-wave resonances below 200 G which were not observed in the experiment. While in regions III and IV the ordering of the resonances is consistent with the experi- mental measurements #the upper resonance is larger than the lower resonance$, in region III there is an additional wide #!10 G$ s-wave resonance below 200 G not observed in the experiment, and in region IV there is an additional s-wave resonance at approximately 960 G #!1 G$ in between the two observed resonances. In addition, the triplet scattering length for regions III and IV corresponding to Etriplet= −0.377 cm−1 is atriplet 6,87 =105aB. This value is in disagreement with the experimentally determined value from measure- ments of the cross thermalization in magnetically trapped 6Li-87Rb mixtures which indicate that the interspecies triplet scattering length is %atriplet 6,87 %=20 −6 +9aB !33". In order to verify the robustness of these findings, we constructed a pair of Lennard-Jones potentials !V#R$ =C12 /R12−C6 /R6" with the same C6 coefficient, roughly the same number of bound states, and the same least bound state energies #Esinglet ,Etriplet$ as the fitted potential in each of the regions considered. Using these potentials, we verified that the Feshbach resonance locations and scattering lengths are essentially the same as for the fitted potentials and are insen- sitive to the short-range details of the potentials. This check provides an important verification of our characterization of the four purely s-wave candidate regions. The conclusion is that the experimentally observed Feshbach resonances are inconsistent with pure s-wave resonances, and we must con- sider the possibility that at least one of the resonances origi- nates from a p-wave molecular state. Region V in Fig. 4 represents the only location in the #Esinglet ,Etriplet$ parameter space for which only one s-wave resonance occurs below 1.2 kG #at 1067 G$ and a p-wave resonance occurs at 882 G. All other branches displayed in Fig. 4 involve at least one additional s-wave resonance oc- curring in a location where none was observed experimen- tally. Along the locus of #Esinglet ,Etriplet$ values for which these two resonances occur at the correct locations, an addi- tional p-wave resonance was found to occur somewhere be- tween 1081 and 1024 G, while the width of the s-wave reso- nance at 1065 G was found to vary from 5 to 35 G. At the precise #Esinglet ,Etriplet$ values for which the second p-wave resonance was coincident with the s-wave resonance at 1065 G, the s-wave resonance width was "B=11.53 G, con- sistent with the experimentally measured value "Bexpt =10.62 G for the full width at half maximum for the trap loss feature. For these optimal singlet and triplet potentials, the bound state energies are Esinglet=−0.106 cm−1 and Etriplet=−0.137 cm−1. Figure 5 shows the result of the full coupled-channel cal- culation performed using the refined potentials. The elastic 0 500 1000 1500 2000 Magnetic Field (G) -0.4 -0.3 -0.2 -0.1 0 0.1 E ne rg y (c m -1 ) A B C FIG. 3. #Color online$ Molecular bound state energies versus magnetic field computed with the asymptotic bound state model. The threshold for the % 12 , 1 2 &6Li! %1,1&87Rb collision channel is shown by the solid line while the dashed #dotted$ lines indicate the s-wave #p-wave$ states. These molecular state energies were computed given the least bound states ES l of the optimal singlet and triplet potentials of E0 l=0 =−0.106 cm−1, E0 l=1 =−0.0870 cm−1, and E1 l=0 = −0.137 cm−1, E1 l=1 =−0.116 cm−1. The predicted resonance loca- tions are close to the actual locations determined by the full coupled-channel calculation and are indicated by the solid dots. Near 890 G #A$ the threshold crosses a p-wave molecular state and the corresponding p-wave elastic scattering cross section shown in Fig. 5 is observed to rapidly diverge and then return to the back- ground level. Likewise, near 1070 G #B$ the threshold crosses both a p-wave and an s-wave molecular state and both the s-wave and p-wave elastic scattering cross sections are affected. Finally, near 1300 G #C$ a second s-wave-induced Feshbach resonance occurs. -0.4 -0.3 -0.2 -0.1 0 Esinglet ( cm -1 ) -0.4 -0.3 -0.2 -0.1 0 E tr ip le t (c m -1 ) I II III IV V FIG. 4. #Color online$ Locus of points in the #Esinglet ,Etriplet$ parameter space where an s-wave resonance occurs at one of the two experimentally determined locations 882.02 !gray #green$" or 1066.92 G !dark #red$" for atoms in the % 12 , 1 2 &6Li! %1,1&87Rb state. The dotted lines indicate the approximate values for Esinglet and Etriplet beyond which a new bound state enters the potential at zero energy. There are four regions #I–IV$ indicated on the plot where an s-wave resonance occurs simultaneously at 882.02 and at 1066.92 G. Region V indicates a range of values for which an s-wave resonance occurs at 1066.92 while a p-wave resonance #not represented in this plot$ occurs at 882 G. For each of these five candidate regions, the character of the predicted elastic cross sec- tions as a function of magnetic field was studied, and the results of this analysis are discussed in the text. FESHBACH RESONANCES IN ULTRACOLD 85RB-… PHYSICAL REVIEW A 78, 022710 #2008$ 022710-5 Figure 3.5: Locus of points in the (Esinglet, Etriplet) parameter space where an s-wave resonance occurs at one of the two experimentally determined locations 882.02 G [gray (green)] or 1066.92 G [dark (red)] for atoms in the |12 , 12〉6Li ⊗ |1, 1〉87Rb state. Th dotted lines indicate the approximate values for Esinglet and Etriplet beyond which a new bound state enters the potential at zero energy. There are four regions (I-IV) indicated on the plot where an s-wave resonance occures simultaneously at 882.02 G and at 1066.92 G. Region V indicates a range of values for which an s-wave resonance occurs at 1066.92 G while a p-wave resonance (not presented in this plot) occurs at 882.00 G. complex is about 16% different from that of the 6Li–87Rb dimer, so the interaction potentials for 6Li–87Rb may not be reliable to describe the collisions between atoms 7Li and 87Rb. Close inspection of the potential also reveals that there is a bound state very close to the dissociation threshold for the 7Li–87Rb system. Therefore, a small uncertainty in the exact location of this very weakly bound state translates into a very large uncertainty in the predicted triplet scattering length for 7Li–87Rb mixtures. Using the refined potentials we calculated the s- and p-wave scattering cross sections as a function of magnetic field for all spin combinations where 6Li and 87Rb are in the lower hyperfine manifold, and the location and widths of all resonances below 2 kG are summarized in Table 3.2. In experiments with 6Li–87Rb mixtures, no Feshbach resonances were observed below 1.2 kG for the |12 , 12〉6Li ⊗ |1, 0〉87Rb, 54 3.4. Results3.4. esults scattering cross sections for the ! 12 , 1 2 "6Li! !1,1"87Rb state show divergences at magnetic fields of 1065 and 882 G, in excellent agreement with the experimentally determined Fes- hbach resonance positions. In addition, the triplet scattering length from the fine-tuned triplet potential was found to be atriplet 6,87 =−19.8aB, also in excellent agreement with the experi- mentally determined value. For the reduced mass corre- sponding to a 7Li-87Rb complex, the optimal fine-tuned trip- let potential predicts atriplet 7,87 =448aB, in disagreement with the experimental measurement of !atriplet 7,87 !=59 −19 +19aB #41$. Close inspection of this potential reveals that there is a bound state very close to the dissociation threshold for the 7Li-87Rb trip- let state. In this case, a small uncertainty in the exact location of this very weakly bound state %arising from uncertainties in the exact shape of the potential& translates into a very large uncertainty in the predicted triplet scattering length for 7Li-87Rb mixtures. We verified the robustness of these results by changing the short-range part of the potentials so that the number of bound states was different from the optimal po- tentials by more than 20% while still producing the same energy of the least bound states. As a result, the Feshbach resonance locations and scattering lengths did not change significantly. In addition, we generated a set of Lennard- Jones potentials which reproduced the same least bound state energies and Feshbach resonance structure as the optimal fitted potentials. These potentials have a very different short- range shape than the fitted potentials and they resulted in triplet scattering lengths of atriplet 6,87 =−22.6aB and atriplet 7,87 = −333aB confirming that the determination for the 6Li-87Rb triplet scattering length is very reliable %independent of the details of the short-range part of the potential& while the 7Li-87Rb triplet scattering length cannot be reliably predicted given the proximity of a zero-energy resonance for this com- bination #43$. We note that, if the triplet scattering length for the 7Li-87Rb mixture is, in fact, negative, the experimental determination of its absolute magnitude can be complicated TABLE II. Position and width of 6Li-87Rb Feshbach resonances for magnetic fields below 2 kG determined from the coupled- channel calculations. The experimentally measured Feshbach reso- nances %and absence of resonances below 1.2 kG& are also included for comparison. The experimentally determined width !Bexpt is the full width at half maximum of the trap loss feature and, although related, it is not equivalent to !B %defined only for s-wave reso- nances&. Several resonances were found that exhibited a suppressed oscillation due to comparable coupling to inelastic channels #28$ and could not be assigned a width in the usual way #in accordance with Eq. %5&$. In these cases, the maximum and minimum elastic scattering lengths of the oscillation were identified and the distance between them is indicated in parentheses. Atomic states Theory Experiment #16$ !f ,mf"6 !f ,mf"87 B0 %G& !B %G& B0 %G& !Bexpt %G& ! 12 , 1 2 " !1,1" 882 p wave 882.02 1.27 1065 11.5 1066.92 10.62 1066 p wave 1278 0.07 ! 12 , 1 2 " !1,0" 889 p wave None below 1.2 kG 1064 17 1096 p wave 1308.5 %3& 1361.7 p wave ! 12 , 1 2 " !1,−1" 1348 %4& None below 1.2 kG ! 12 ,− 1 2 " !1,1" 773 p wave 923 "0.001 926 p wave 1108.6 11 1119.5 p wave 1331 0.08 ! 12 ,− 1 2 " !1,0" 923 p wave 1105 16.3 1150 p wave 1362 %3& ! 12 ,− 1 2 " !1,−1" 1408 %4& 1611 0.06 ! 32 , 3 2 " !1,1" None None below 1.2 kG ! 32 , 3 2 " !1,0" None ! 32 , 3 2 " !1,−1" 953 48.5 1236.6 p wave ! 32 ,− 3 2 " !1,1" 809 p wave 960 "0.001 971 p wave 1156 11.7 ! 32 ,− 3 2 " !1,0" 973 p wave 1149 16.7 ! 32 ,− 3 2 " !1,−1" 1609 0.07 10 0 10 2 10 4 10 6)2 B (u ni ts of a sσ 0 500 1000 1500 2000 Magnetic Field (G) 10 -8 10 -6 10 -4 10 -2 )2 B (u ni ts of a pσ FIG. 5. %Color online& Magnetic field dependence of the s-wave %upper panel& and p-wave %lower panel& elastic scattering cross sec- tions for atoms in the ! 12 , 1 2 "6Li! !1,1"87Rb state. These results are from the coupled-channel calculations for a collision energy of 144 nk and using the optimal singlet and triplet potentials. Only the ml=0 contribution to the p-wave elastic scattering cross section is shown. Two s-wave resonances occur at 1065 and 1278 G, while two p-wave resonances occur at 882 and 1066 G. LI et al. PHYSICAL REVIEW A 78, 022710 %2008& 022710-6 Figure 3.6: Magnetic field dependence of the s-wave (upper panel) and p-wave (lower panel) elastic scattering cross sections for atoms in the |12 , 12〉6Li ⊗ |1, 1〉87Rb state. These results are from the coupled-channel calculations for a collision energy of 144 nk and using the optimal singlet and triplet p tentials. Only theml = 0 contribution of the p-wave elastic scattering cross section is shown. Two s-wave resonances occur at 1065 and 1278 G, while two p-wave resonances occur at 882 and 1066 G. bound state arising from uncertainties in the exact shape of the potential translates into a very large uncertainty in the predicted triplet scattering length for 7Li–87Rb mixtures. Using the refined potentials we calculated the s- and p-wave scattering cross sections as a function of magnetic field for all spin combinations where 6Li and 87Rb are in the lower hyperfine manifold, and the location and widths of all resonances below 2 kG are summarized in Table 3.2. In experiments with 7Li–87Rb mixtures, no Feshbach resonances were obs rved below 1.2 kG for the |12 , 12〉6Li ⊗ |1, 0〉87Rb, |12 , 12〉6Li⊗ |1,−1〉87Rb, and |32 , 32〉6Li⊗ |1, 1〉87Rb stat s. The results pres nted in Table 3.2 are in agreement with the last two of these observations but not the fi st. It is possible that, because the resonances present in |12 , 12〉6Li ⊗ |1, 0〉87Rb combination are very similar in position and width to those of the |12 , 12〉6Li ⊗ |1, 1〉87Rb state, they may have been observed and erroneously concluded to arise from an impure state preparation. This was recently confirmed by B. Deh, the lead author of the experiment papers. 52 3. 4. R es u lt s 1 .6 4 1 1 .6 4 4 1 .6 4 7 1 .6 5 1 .6 5 3 1 .6 5 6 1 .6 5 9 b si n g le t -6-4-20246 Scattering Length (x10 3  Bohr) F ig ur e 3. 3: T he si ng le t sc at te ri ng le ng th va ri es pe ri od ic al ly as a fu nc ti on of th e fit ti ng pa ra m et er b s in g le t. T hi s fit ti ng pa ra m et er de te rm in es th e sl op e of th e re pu ls iv e w al l of th e 1 Σ in te ra ct io n po te nt ia l. G ) s- w av e re so na nc e be lo w 20 0 G no t ob se rv ed in th e ex pe ri m en t, an d in re gi on IV th er e is an ad di ti on al s- w av e re so na nc e at ap pr ox im at el y 96 0 G (> 1 G ) in be tw ee n th e tw o ob se rv ed re so na nc es . In ad di ti on ,t he tr ip le t sc at te ri ng le ng th fo r re gi on s II I an d IV co rr es po nd in g to E tr ip le t = −0 .3 77 cm −1 is a 6 ,8 7 tr ip le t = 10 5 B oh r. T hi s va lu e is in di sa gr ee m en t w it h th e ex pe ri m en ta lly de te rm in ed va lu e fr om th e m ea su re m en ts of th e cr os s th er m al iz at io n in m ag ne ti ca lly tr ap pe d 6 L i– 8 7 R b m ix tu re s w hi ch in di ca te th at th e in te rs pe ci es tr ip le t sc at te ri ng le ng th is |a6 ,8 7 tr ip le t| = 20 + 9 −6 B oh r [1 78 ]. σ s (B oh r2 ) In or de r to ve ri fy th e ro bu st ne ss of th es e fin di ng s, w e co ns tr uc te d a pa ir of L en na rd -J on es po te nt ia ls V (r ) = C 1 2 / r1 2 − C 6 / r6 w it h th e sa m e C 6 co effi ci en t, ro ug hl y th e sa m e nu m be r of bo un d st at es ,a nd th e sa m e le as t bo un d st at e en er gi es (E si n g le t, E tr ip le t) as th e fit te d po te nt ia l in ea ch of th e re gi on s co ns id er ed . U si ng th es e po te nt ia ls , w e ve ri fie d th at th e Fe sh ba ch re so na nc e lo ca ti on s an d sc at te ri ng le ng th s ar e es se nt ia lly th e sa m e as fo r th e fit te d po te nt ia ls an d ar e in se ns it iv e to th e sh or t- ra ng e de ta ils of th e po te nt ia ls . T hi s ch ec k pr ov id es an im po rt an t ve ri fic at io n of ou r ch ar ac te ri za ti on of th e fo ur pu re ly s- w av e ca nd id at e re gi on s. T he co nc lu si on is th at th e ex pe ri m en ta lly ob se rv ed Fe sh ba ch re so na nc es ar e in co ns is te nt w it h pu re 49 3 .4 . R es u lt s 1 .6 4 1 1 .6 4 4 1 .6 4 7 1 .6 5 1 .6 5 3 1 .6 5 6 1 .6 5 9 b si n g le t -6-4-20246 Scattering Length (x10 3  Bohr) F ig ur e 3. 3: T he si ng le t sc at te ri ng le ng th va ri es pe ri od ic al ly as a fu nc ti on of th e fit ti ng pa ra m et er b s in g le t. T hi s fit ti ng pa ra m et er de te rm in es th e sl op e of th e re pu ls iv e w al l of th e 1 Σ in te ra ct io n po te nt ia l. G ) s- w av e re so na nc e be lo w 20 0 G no t ob se rv ed in th e ex pe ri m en t, an d in re gi on IV th er e is an ad di ti on al s- w av e re so na nc e at ap pr ox im at el y 96 0 G (> 1 G ) in be tw ee n th e tw o ob se rv ed re so na nc es . In ad di ti on ,t he tr ip le t sc at te ri ng le ng th fo r re gi on s II I an d IV co rr es po nd in g to E tr ip le t = −0 .3 77 cm −1 is a 6 ,8 7 tr ip le t = 10 5 B oh r. T hi s va lu e is in di sa gr ee m en t w it h th e ex pe ri m en ta lly de te rm in ed va lu e fr om th e m ea su re m en ts of th e cr os s th er m al iz at io n in m ag ne ti ca lly tr ap pe d 6 L i– 8 7 R b m ix tu re s w hi ch in di ca te th at th e in te rs pe ci es tr ip le t sc at te ri ng le ng th is |a6 ,8 7 tr ip le t| = 20 + 9 −6 B oh r [1 78 ]. σ p (B oh r2 ) In or de r to ve ri fy th e ro bu st ne ss of th es e fin di ng s, w e co ns tr uc te d a pa ir of L en na rd -J on es po te nt ia ls V (r ) = C 1 2 / r1 2 − C 6 / r6 w it h th e sa m e C 6 co effi ci en t, ro ug hl y th e sa m e nu m be r of bo un d st at es ,a nd th e sa m e le as t bo un d st at e en er gi es (E si n g le t, E tr ip le t) as th e fit te d po te nt ia l in ea ch of th e re gi on s co ns id er ed . U si ng th es e po te nt ia ls , w e ve ri fie d th at th e Fe sh ba ch re so na nc e lo ca ti on s an d sc at te ri ng le ng th s ar e es se nt ia lly th e sa m e as fo r th e fit te d po te nt ia ls an d ar e in se ns it iv e to th e sh or t- ra ng e de ta ils of th e po te nt ia ls . T hi s ch ec k pr ov id es an im po rt an t ve ri fic at io n of ou r ch ar ac te ri za ti on of th e fo ur pu re ly s- w av e ca nd id at e re gi on s. T he co nc lu si on is th at th e ex pe ri m en ta lly ob se rv ed Fe sh ba ch re so na nc es ar e in co ns is te nt w it h pu re 49 Figure 3.6: Magnetic field dependence of the s-wave (upper panel) and p-wave (lower panel) elastic scattering cr ss sections for atoms in the |12 , 12〉6Li ⊗ |1, 1〉8 state. These results are from the coupled-channel calculations for a collision energy of 144 nK and using the optimal singlet and triplet potentials. Only the ml = 0 contribution of the p-wave elastic scattering cross section is shown. Two s-wave resonances occur at 1065 and 1278 G, while two p-wave resonances occur at 882 and 1066 G. |12 , 12〉6Li⊗|1,−1〉87Rb, and |32 , 32〉6Li⊗|1, 1〉87Rb states. The results presented in Table 3.2 are in agreement with the last two of these observations but not the first. It is possible that, because the resonances present in |12 , 12〉6Li ⊗ |1, 0〉87Rb combination are very similar in position and width to those of the |12 , 12〉6Li ⊗ |1, 1〉87Rb state, they may have been observed and erroneously concluded to arise from an impure state preparatio . This was recently confirmed by B. Deh, the le d author of the experiment papers [205]. 55 3.5. Conclusions 3.5 Conclusions In this Chapter, we have generated a set of accurate Li–Rb interaction potentials which reproduce the positions and widths of the measured resonances with high precision. We used the asymptotic bound state model to obtain the combinations of the least bound energies of the singlet and triplet interaction potentials that give rise to the positions of two experimentally measured Feshbach resonances. We then generated the approximate singlet and triplet potential curves supporting these bound energies and fine-tuned them to reproduce the positions and widths of the resonances using full quantum scattering calculations. Our potentials indicate that the 6Li–87Rb triplet scattering length is a6,87triplet = −19.8 bohr, which is consistent with cross-thermalization measurements. We have verified that the predictions of these fine-tuned potentials are robust in the sense that they only depend on the well- known long-range C6 coefficient and are independent of both the details of the short range shape and the exact number of bound states of the interaction potentials. Using these refined potentials, we predicted new experimentally relevant resonances for the 6Li–87Rb mixture to guide future experiments. The ultracold mixture of Li and Rb atoms is currently studied experimentally by Kirk Madison’s group in the physics department at UBC. Our results may help them to identify other Feshbach resonances in order to create bound Li–Rb dimers and control ultracold collisions between Rb and Li atoms. 56 3.5. Conclusions Table 3.2: Positions and widths of 6Li–87Rb Feshbach resonances for magnetic fields below 2 kG determined from the coupled-channel calculations. The experimentally measured Feshbach resonances (and absence of resonances below 1.2 kG) are also included for comparison. The experimentally determined width 4Bexpt is the full width at half maximum of the trap loss feature and, although related, it is not equivalent to 4B (defined only for s-wave resonances). Several resonances were found that exhibited a suppressed oscillation due to comparable coupling to inelastic channels and could not be assigned a width in the usual way. In these cases the maximum and minimum elastic scattering lengths of the oscillation were identified and the distance between them is indicated in parentheses. Atomic States Theory Experiment [205] |f,mf 〉6|f,mf 〉87 B0 (G) 4B (G) B0 (G) 4Bexpt (G) |12 , 12〉 |1, 1〉 882 p-wave 882.02 1.27 1065 11.5 1066.92 10.62 1066 p-wave 1278 0.07 |12 , 12〉 |1, 0〉 889 p-wave None below 1.2 kG 1064 17 1096 p-wave 1308.5 (3) 1361.7 p-wave |12 , 12〉 |1,−1〉 1348 (4) None below 1.2 kG |12 ,−12〉 |1, 1〉 773 p-wave 923 < 0.001 926 p-wave 1108.6 11 1119.5 p-wave |12 ,−12〉 |1, 0〉 923 p-wave 1105 16.3 1150 p-wave 1362 (3) |12 ,−12〉 |1,−1〉 1408 (4) 1611 0.06 |32 , 32〉 |1, 1〉 None None below 1.2 kG |32 , 32〉 |1, 0〉 None |32 , 32〉 |1,−1〉 953 48.5 1236.6 p-wave |32 ,−32〉 |1, 1〉 809 p-wave 960 < 0.001 971 p-wave 1156 11.7 |32 ,−32〉 |1, 0〉 973 p-wave 1149 16.7 |32 ,−32〉 |1,−1〉 1609 0.07 57 Chapter 4 Electric-field-induced Feshbach resonances in alkali metal mixtures 2 All experiments to date have focused on the study of magnetic field tunable Feshbach resonances. Recent theoretical work has demonstrated the possibility of inducing Feshbach resonances in heteronuclear mixtures of ultracold atomic gases by applying a dc electric field [213]. The mechanism of electric-field-induced resonances is based on the interaction of the instantaneous dipole moment of the collision complex with the external electric field. In this Chapter, we extend the work of Ref. [213] and present a theory for ultracold atomic collisions in the presence of superimposed magnetic and electric fields. We study in detail the effects of external electric fields on collision dynamics in ultracold Li–Cs and Li–Rb mixtures. Our calculations reveal several new phenomena and provide the physical insight into the dynamics of ultracold heteronuclear mixtures near Feshbach resonances. 4.1 Why electric fields? Marinescu and You [214] and Melezhik and Hu [215] proposed to control ultra- cold atomic gases by polarizing atoms with strong electric fields. The polarization changes the long-range form of the interatomic interaction potentials and modifies the scattering cross sections in the limit of zero collision energy. The interaction between an atom and an electric field is, however, extremely weak and fields of as much as 250 to 700 kV/cm were required to alter the elastic scattering cross sections of ultracold atoms in these calculations. The results obtained in our re- search group [213] suggest an alternative mechanism for electric field control of ultracold atomic interactions and demonstrate that collisions and interactions in binary mixtures of ultracold atoms can be effectively manipulated by electric fields below 100 kV/cm. The mechanism of electric field control is based on the interac- 2A part of this Chapter was presented in Refs. [2] and [3] of Appendix D. 58 4.1. Why electric fields? tion of the instantaneous dipole moment of the collision pair with external electric fields. This interaction couples collision states of different angular momenta and the coupling becomes very strong near scattering resonances, giving rise to electric- field-induced Feshbach resonances. The duration of an ultracold collision is so long that the interaction, while insignificant in thermal gases, may dramatically change the dynamics of atomic collisions at temperatures near absolute zero. Electric field control of microscopic interactions may offer several advantages to study fundamental problems and explore new phenomena in ultracold physics and chemistry. Electric fields can be tuned much faster than magnetic fields. Electric- field-induced Feshbach resonances may therefore be more versatile for quantum com- putation schemes than magnetic resonances. Magnetic field control of interatomic interactions is limited to para-magnetic species. The possibility of inducing scat- tering resonances with electric fields may expand the scope of studies of correlation phenomena in ultracold gases to systems without magnetic moments. Evaporative cooling of atoms and molecules is usually carried out in a magnetic trap, which may complicate the manipulation of ultracold collisions with magnetic fields. Electric fields may therefore provide an additional degree of control over atomic and molec- ular collisions, which may be important for experiments with gases in a trap with large magnetic field gradients or for systems where magnetic resonances cannot be tuned in the available magnetic field interval. Following our work [174, 213], other researchers demonstrated that the combination of electric and magnetic fields may be used to control the positions and widths of Feshbach resonances independently, leading to total control over ultracold collisions [216]. In this Chapter, we explore in detail the effects of external electric fields on ul- tracold atomic collisions in Li–Cs and Li–Rb binary mixtures. Ultracold Li–Cs and Li–Rb mixtures have recently been created in several experiments [204, 217, 218] and accurate interaction potentials have been generated to describe the collision dynamics of Li with Cs and Rb [170, 206, 219]. Using these potentials, we carry out full coupled-channel calculations to study the collision dynamics of atoms in ultracold Li–Cs and Li–Rb mixtures in the presence of superimposed magnetic and electric fields. Our results show that electric fields may induce new resonances by coupling different partial waves and may shift the positions of magnetic Feshbach resonances. Electric fields may also spin up the collision complex of ultracold atoms at substantial rates and induce the anisotropy of the differential scattering cross sections at ultracold temperatures. We demonstrate that electric fields may split Feshbach resonances in states of non-zero angular momenta into several peaks. This effect is more significant than the splitting of magnetic Feshbach resonances due to 59 4.2. Atomic collisions in combined electric and magnetic fields the magnetic dipole-dipole interaction discovered by other researchers [165, 220]. We also find that rotating an electric field with respect to the magnetic field has no significant effect on average cross sections for p-wave elastic scattering, but mod- ifies the magnitude of elastic cross sections in states of different orbital angular momentum projections. 4.2 Atomic collisions in combined electric and magnetic fields Collision dynamics of alkali metal atoms in the presence of superimposed electric and magnetic fields are described by the following Hamiltonian (cf. Eqs. 3.4 and 3.5) Ĥ = − 1 2µr ∂2 ∂r2 r + l̂2(θ, ϕ) 2µr2 + V̂ + V̂ζ + V̂B + V̂hf , (4.1) where V̂ζ represents the interaction of the collision complex with external electric fields, V̂B models the interaction of atoms with magnetic fields, and V̂hf describes the hyperfine interactions. The explicit expressions for V̂B and V̂hf are given in Chapter 3 (cf. Eqs 3.9 and 3.10). We expand the total wave function in the uncoupled representation (cf. Eqs. 3.14 and 3.15). The matrix elements of the operators V̂B and V̂hf are given in Chapter 3. The electronic interaction potential V̂ has the form V̂ = ∑ S ∑ MS |SMS〉VS(r)〈SMS |, (4.2) where VS(r) denotes the adiabatic interaction potential of the molecule in the spin state S. To evaluate the matrix elements of the interaction potential V̂ , we write the product states |IaMIa〉|SaMSa〉|IbMIb〉|SbMSb〉 as |IaMIa〉|SaMSa〉|IbMIb〉|SbMSb〉 = ∑ S ∑ MS (−1)MS (2S + 1)1/2 × ( Sa Sb S MSa MSb −MS ) |IaMIa〉|IbMIb〉|SMS〉 (4.3) and note that 〈SMS |V̂ |S′M ′S〉 = VS(r) δSS′δMSM ′S , (4.4) 60 4.2. Atomic collisions in combined electric and magnetic fields 8 12 16 20 r (Bohr) -5000 0 5000 E n e r g y  ( c m - 1 ) 6 9 12 15 r (Bohr) 0 1 2 3 4 5 6 D i p o l e m o m e n t ( D e b y e ) C6 = 2.9338x10 3 a.u. C8 = 3.1253x10 5 a.u. C10 = 1.7515x10 8 a.u. Figure 4.1: The interaction potentials and dipole moment functions (inset) of the LiCs molecule in the 1Σ (solid lines) and 3Σ (dashed lines) states. The interaction potentials were taken from Ref.[219] and the dipole moment functions approximate the data of Ref. [206]. and that the operator V̂ is diagonal in the nuclear spin states and l and ml quantum numbers. The parentheses in Eq. 4.3 denote a 3j-symbol. The operator V̂ζ can be written in the form V̂ζ = −ζ(êζ · êd) ∑ S ∑ MS |SMS〉dS(r)〈SMS |, (4.5) where êζ · êd represents the dot product of the unit vectors in the direction of the external electric field êζ and the dipole moment êd of the collision complex, dS denotes the dipole moment functions of the diatomic molecule in the different spin states (S = 0 and S = 1) and ζ is the electric field magnitude. The dipole moment functions are represented by the following expression dS(r) = D exp [−α(r − re)2]. (4.6) For LiCs, re = 7.7 bohr, α = 0.1 bohr−2 and D = 6 Debye for the singlet state; and re = 5.0 bohr, α = 0.1 bohr−2 and D = 0.5 Debye for the triplet state. For 61 4.2. Atomic collisions in combined electric and magnetic fields 5 10 15 20 r (Bohr) -3000 0 3000 6000 E n e r g y  ( c m - 1 ) 5 10 15 20 r (Bohr) 0 1 2 3 4 5 D i p o l e m o m e n t ( D e b y e ) C6 = 2.5457x10 3 a.u. C8 = 2.2825x10 5 a.u. C10 = 2.5647x10 7 a.u. Figure 4.2: The interaction potentials and dipole moment functions (inset) of the LiRb molecule in the 1Σ (solid lines) and 3Σ (dashed lines) states. The interaction potentials were taken from Ref.[170] and the dipole moment functions approximate the data of Ref. [206]. LiRb, re = 7.2 bohr, α = 0.06 bohr−2 and D = 4.57 Debye for the singlet state; and re = 5.0 bohr, α = 0.045 bohr−2 and D = 1.02 Debye for the triplet state. These analytical expressions approximate the numerical data for the dipole moment functions computed by Aymar and Dulieu [206]. Figures 4.1 and 4.2 show the interaction potentials VS(r) and the dipole moment functions dS(r) in the 1Σ and 3Σ states for the LiCs and LiRb molecules, respectively. If the electric field is oriented at a certain angle γ with respect to the quantization axis as depicted in Fig. 4.3, êζ · êd has the form êζ · êd = cos(χ) with χ = θ − γ. It can be written in terms of the first-degree Legendre polynomial [221] êζ · êd = cos(χ) = P1(cos(χ)) = 4pi3 [Y ∗ 1−1(γ, ϕγ)Y1−1(θ, ϕθ) +Y ∗10(γ, ϕγ)Y10(θ, ϕθ) + Y ∗ 11(γ, ϕγ)Y11(θ, ϕθ)], (4.7) where Yxx are spherical harmonics and ϕγ and ϕθ are the angles between the x-axis and the projections of the vectors êζ and êd on the (x, y) plane, respectively. The effect of an electric field on the collision dynamics in a system of alkali metal 62 4.2. Atomic collisions in combined electric and magnetic fields4.2. Theory Figure 4.3: The coordinate system in our calculations. !E and !d represent the vector of the external electric field and the dipole moment vector, respectively; γ specifies the orientation of the electric field with respect to the quantization axis; θ is the angle between the dipole moment vector and the z-axis; χ is the angle between !E and !d, and ϕγ and ϕθ are the projections of angles γ and θ on the (x, y) plane. magnetic field. Without loss of generality, we can assume that êE is in the x − z plane, i.e., ϕγ = 0. The matrix elements of V̂E(r) are therefore evaluated using the expressions 〈lml|êE · êd|l′m′l〉 = √ 2 2 sin γ(−1)m′l √ [(2l + 1)(2l′ + 1)] ( l 1 l′ 0 0 0 )( l 1 l′ ml −1 −m′l ) + cos γ(−1)m′l √ [(2l + 1)(2l′ + 1)] ( l 1 l′ 0 0 0 )( l 1 l′ ml 0 −m′l ) − √ 2 2 sin γ(−1)m′l √ [(2l + 1)(2l′ + 1)] ( l 1 l′ 0 0 0 )( l 1 l′ ml 1 −m′l ) (4.8) and 〈SMS | ∑ S′′ ∑ M ′′S |S′′M ′′S〉dS′′〈S′′M ′′S |  |S′M ′S〉 = dS δSS′δMSM ′S . (4.9) 60 4.2. Theory 5 10 15 20 r (Bohr) -3000 0 3000 6000 E n e r g y  ( c m - 1 ) 5 10 15 20 r (Bohr) 0 1 2 3 4 5 D i p o l e  m o m e n t ( D e b y e ) C 6  = 2.5457x10 3  a.u. C 8  = 2.2825x10 5  a.u. C 10  = 2.5647x10 7  a.u. Figure 4.2: The interaction potentials and dipole moment functions (inset) of the LiRb molecule in the ground 1 ∑ (solid lines) and excited 3 ∑ (dashed lines) states. The interaction potentials were taken from Ref.[4] and the dipole moment functions approximate the data of Ref. [3]. α = 0.045 bohr−2 and D = 1.02 Debye for the triplet state. These analytical expres- sions approximate the numerical data for the dipole moment functions computed by Aymar and Dulieu [3]. Figs. 4.1 and 4.2 show the interaction potentials VS(r) and the dipole moment functions dS(r) in the 1Σ and 3Σ states for the LiCs and LiRb molecules, respectively. If the electric field is oriented at a certain angle γ with respect to the quantization axis, êE · êd has the form êE · êd = cos(χ) with χ the angle between θ and γ. It can be written in terms of the first-degree Legendre polynomial êE · êd = cos(χ) = P1(cos(χ)) (4.7) = 4pi 3 [ Y ∗1−1(γ,ϕγ)Y1−1(θ,ϕθ) + Y ∗ 10(γ,ϕγ)Y10(θ,ϕθ) + Y ∗ 11(γ,ϕγ)Y11(θ,ϕθ) ] where ϕγ and ϕθ are the projections of angles γ and θ on the (x, y) plane and Yxx are spherical harmonics. Fig. 4.3 depicts the coordinate system in detail. The effect of a rotating electric field on the collision dynamics in a system of alkali metal atoms depends on the relative angle between the electric field and the 59 4.2. Theory 5 10 15 20 r (Bohr) -3000 0 3000 6000 E n e r g y  ( c m - 1 ) 5 10 15 20 r (Bohr) 0 1 2 3 4 5 D i p o l e  m o m e n t ( D e b y e ) C 6  = 2.5457x10 3  a.u. C 8  = 2.2825x10 5  a.u. C 10  = 2.5647x10 7  a.u. Figure 4.2: The interaction potentials and dipole moment functions (inset) of the LiRb molecule in the ground 1 ∑ (solid lines) and excited 3 ∑ (dashed lines) states. The interaction potentials were taken from Ref.[4] and the dipole moment functions approximate the data of Ref. [3]. α = 0.045 bohr−2 and D = 1.02 Debye for the triplet state. These analytical expres- sions approximate the numerical data for the dipole moment functions computed by Aymar and Dulieu [3]. Figs. 4.1 and 4.2 show the interaction potentials VS(r) and the dipole moment functions dS(r) in the 1Σ and 3Σ states for the LiCs and LiRb molecules, respectively. If the electric field is oriented at a certain angle γ with respect to the quantization axis, êE · êd has the form êE · êd = cos(χ) with χ the angle between θ and γ. It can be written in terms of the first-degree Legendre polynomial êE · êd = cos(χ) = P1(cos(χ)) (4.7) = 4pi 3 [ Y ∗1−1(γ,ϕγ)Y1−1(θ,ϕθ) + Y ∗ 10(γ,ϕγ)Y10(θ,ϕθ) + Y ∗ 11(γ,ϕγ)Y11(θ,ϕθ) ] where ϕγ and θ are the projections of angles γ and θ on the (x, y) plane and Yxx are spherical harmonics. Fig. 4.3 depicts the coordinate system in detail. The effect of a rotating electric field on the collision dynamics in a system of lkali metal atoms depends on the relative angle between the electric field and the 59 4.2. Theory4.2. Theory Figure 4.3: The coordinate system in our calculations. !E and !d represent the vector of the external electric field and the dipole moment vector, respectively; γ specifies th orientation of the electric field with respect to the quantization axis; θ is the angl be w en the dipole moment vector and the z-axis; χ is the angle between !E and !d, and ϕγ and ϕθ r he projections of angl s γ and θ on the (x, y) plane. magnetic field. Without loss of generality, we can assume that êE is in the x − z plane, i.e., ϕγ = 0. The matrix elements of V̂E(r) are therefore evaluated using the expressions 〈lml|êE · êd|l′m′l〉 = √ 2 2 sin γ(−1)m′l √ [(2l + 1)(2l′ + 1)] ( l 1 l′ 0 0 0 )( l 1 l′ ml −1 −m′l ) + cos γ(−1)m′l √ [(2l + 1)(2l′ + 1)] ( l 1 l′ 0 0 0 )( l 1 l′ ml 0 −m′l ) − √ 2 2 sin γ(−1)m′l √ [(2l + 1)(2l′ + 1)] ( l 1 l′ 0 0 0 )( l 1 l′ ml 1 −m′l ) (4.8) and 〈SMS | ∑ S′′ ∑ M ′′S |S′′M ′′S〉dS′′〈S′′M ′′S |  |S′M ′S〉 = dS δSS′δMSM ′S . (4.9) 60 4.2. Theory 5 10 15 20 r (Bohr) -3000 0 3000 6000 E n e r g y  ( c m - 1 ) 5 10 15 20 r (Bohr) 0 1 2 3 4 5 D i p o l e  m o m e n t ( D e b y e ) C 6  = 2.5457x10 3  a.u. C 8  = 2.2825x10 5  a.u. C 10  = 2.5647x10 7  a.u. Figure 4.2: The interaction potentials and dipole moment functions (inset) of the LiRb molecule in the ground 1 ∑ (solid lines) and excited 3 ∑ (dashed lines) states. The interaction potentials were taken from Ref.[4] and the dipole moment functions approximate the data of Ref. [3]. α = 0.045 bohr−2 and D = 1.02 Debye for the triplet state. These analytical expres- sions approximate the numerical data for the dipole moment functions computed by Aymar and Dulieu [3]. Figs. 4.1 and 4.2 show the interaction potentials VS(r) and the dipole moment functions dS(r) in the 1Σ and 3Σ states for the LiCs and LiRb molecules, respectively. If the electric field is oriented at a certain angle γ with r spect to the quantization axis, êE · êd has the form êE · êd = cos(χ) with χ the angle between θ and γ. It can be written in terms of the first-degree Legendre polynomial êE · êd = cos(χ) = P1(c s(χ)) (4.7) = 4pi 3 [ Y ∗1−1(γ,ϕγ)Y1−1(θ,ϕθ) + Y ∗ 10(γ,ϕγ)Y10(θ,ϕθ) + Y ∗ 11(γ,ϕγ)Y11(θ,ϕθ) ] where ϕγ and ϕθ are the projections of angles γ and θ on the (x, y) plane and Yxx are spherical harmonics. Fig. 4.3 depicts the coordinate system in detail. The effect of a rotating electric field on the collision dynamics in a system of alkali metal atoms depends on the relative ngle betwe n the electric field and the 59 4.2. Theory 5 10 15 20 r (Bohr) -3000 0 3000 6000 E n e r g y  ( c m - 1 ) 5 10 15 20 r (Bohr) 0 1 2 3 4 5 D i p o l e  m o m e n t ( D e b y e ) C 6  = 2.5457x10 3  a.u. C 8  = 2.2825x10 5  a.u. C 10  = 2.5647x10 7  a.u. Figure 4.2: The interaction potentials and dipole moment functions (inset) of the LiRb molecule in the ground 1 ∑ (solid lines) and excited 3 ∑ (dashed lines) states. The interaction potentials were taken from Ref.[4] and the dipole moment functions approximate the data of Ref. [3]. α = 0.045 bohr−2 and D = 1.02 Debye for the triplet state. These anal tical expres- sions approximate the numerical data for the dipole moment functions computed by Aymar and Dulieu [3]. Figs. 4.1 and 4.2 show the interaction potentials VS(r) and the dipole moment functions dS(r) in the 1Σ and 3Σ states for the LiCs and LiRb molecules, respectively. If the electric field is oriented at a certain angle γ with respect to the quantization axis, êE · êd has the form êE · êd = cos(χ) with χ the angle between θ and γ. It can be written in terms of the first-degree Legendre polynomial êE · êd = cos(χ) = P1(cos(χ)) (4.7) = 4pi 3 [ Y ∗1−1(γ,ϕγ)Y1−1(θ,ϕθ) + Y ∗ 10(γ,ϕγ)Y10(θ,ϕθ) + Y ∗ 11(γ,ϕγ)Y11(θ,ϕθ) ] where ϕγ and θ are the projections of angles γ and θ on the (x, y) plane and Yxx are spherical harmonics. Fig. 4.3 depicts e coordinate system in d tail. The effect of a rotating electric field on the collision dynamics in a system of lkali metal atoms depends on the relative angle between the electric field and the 59 Figure 4.3: The coordinate system in our calculations. !E and ! represent the vector of the external electric field and the dipole moment vector, respectively; γ specifies the orientation of the electric field with respect to the quantization axis; θ is the angle between the dipole moment vector and the z-axis; χ is the angle between !E and !d, and ϕγ and ϕθ are the projections of angles γ and θ on the (x, y) plane. magnetic field. Without loss of ge erality, we can assume that êE is in the x − z plane, i.e., ϕγ = 0. The matrix elements of V̂E(r) are therefore evaluated using the expressions 〈lml|êE · êd|l′m′l〉 = √ 2 2 sin γ(−1)m′l √ [(2l + 1)(2l′ + 1)] ( l 1 l′ 0 0 0 )( l 1 l′ ml −1 −m′l ) + cos γ(−1)m′l √ [(2l + 1)(2l′ + 1)] ( l 1 l′ 0 0 0 )( l 1 l′ ml 0 −m′l ) − √ 2 2 sin γ(−1)m′l √ [(2l + 1)(2l′ + 1)] ( l 1 l′ 0 0 0 )( l 1 l′ ml 1 −m′l ) (4.8) and 〈SMS | ∑ S′′ ∑ M ′′S |S′′M ′′S〉dS′′〈S′′M ′′S |  |S′M ′S〉 = dS δSS′δMSM ′S . (4.9) 60 4.2. Theory4.2. Theory Figure 4.3: The coordinate system in our calculations. !E and !d represent the vector of the external electric field and the dipole moment vector, respectively; γ specifies the orientation of the electric field wi h respect to the quantization axis; θ is th angle between the dipole moment vector and the z-axis; χ is the angle between !E and !d, and ϕγ and ϕθ are the projections of angles γ and θ on the (x, y) plane. magnetic field. Wit out loss of generality, we can ssum that êE is in the x − z lane i. ., ϕγ = 0. Th matrix element of V̂E(r) are therefore evaluated using the expressions 〈lml|êE · êd|l′m′l〉 = √ 2 2 sin γ(−1)m′l √ [(2l + 1)(2l′ + 1)] ( l 1 l′ 0 0 0 )( l 1 l′ ml −1 −m′l ) + cos γ(−1)m′l √ [(2l + 1)(2l′ + 1)] ( l 1 l′ 0 0 0 )( l 1 l′ ml 0 −m′l ) − √ 2 2 sin γ(−1)m′l √ [(2l + 1)(2l′ + 1)] ( l 1 l′ 0 0 0 )( l 1 l′ ml 1 −m′l ) (4.8) and 〈SMS | ∑ S′′ M ′′S |S′′ ′′ ′′ ′′S |  |S′M ′S〉 = dS δSS′δMS ′S . (4.9) 60 4.2. Theory 5 10 15 20 r (Bohr) -3000 0 3000 6000 E n e r g y  ( c m - 1 ) 5 10 15 20 r (Bohr) 0 1 2 3 4 5 D i p o l e  m o m e n t ( D e b y e ) C 6  = 2.5457x10 3  a.u. C 8  = 2.2825x10 5  a.u. C 10  = 2.5647x10 7  a.u. Figure 4.2: The interaction potentials and dipole moment functions (inset) of the LiRb molecule in the ground 1 ∑ (solid lines) and excited 3 ∑ (dashed lines) states. The interaction potentials were taken from Ref.[4] and the dipole moment functions approximate the data of R f. [3]. α = 0.045 bohr−2 and D = 1.02 Debye for the tripl t state. Thes analytical expres- sions approximate the numerical data for the dipole moment functions computed by Ayma and Dulieu [3]. Figs. 4.1 and 4.2 show the interaction potentials VS(r) and the dipole moment functions dS(r) in the 1Σ and 3Σ states for the LiCs a d LiRb molecules, respectively. If the electric field is oriented at a certain angle γ with respect to the quantization axis, êE · êd has the form êE · êd = cos(χ) with χ the angle between θ and γ. It can be written in terms of the first-degree Legendre polynomial êE · d̂ = cos(χ) = P1(cos(χ)) (4.7) = 4pi 3 [ Y ∗1−1(γ,ϕγ)Y1−1(θ,ϕθ) + Y ∗ 10(γ,ϕγ)Y10(θ,ϕθ) + Y ∗ 11(γ,ϕγ)Y11(θ,ϕθ) ] where ϕγ and ϕθ are the projections of angles γ and θ on the (x, y) plane and Yxx are spherical harmonics. Fig. 4.3 depicts the coordinate system in deta l. The effect of a rotating electric field on the collision dynamics in a system of alkali metal atoms depends on the relative angle be w n the e ctric field and the 59 4.2. Theory 5 10 15 20 r (Bohr) -3000 0 3000 6000 E n e r g y  ( c m - 1 ) 5 10 15 20 r (Bohr) 0 1 2 3 4 5 D i p o l e  m o m e n t ( D e b y e ) C 6  = 2.5457x10 3  a.u. C 8  = 2.2825x10 5  a.u. C 10  = 2.5647x10 7  a.u. Figure 4.2: The interaction potentials and dipole moment functions (inset) of the LiRb molecule in the ground 1 ∑ (solid lines) and excited 3 ∑ (dashed lines) states. The interaction potentials were taken from Ref.[4] and the dipole moment functions approximate the data of Ref. [3]. α = 0.045 bohr−2 and D = 1.02 Debye for the triplet state. These analytical expres- sions approximate the numerical data for the dipole moment functions computed by Aymar and Dulieu [3]. Figs. 4.1 and 4.2 show the interaction potentials VS(r) and the dipole moment functions dS(r) in the 1Σ and 3Σ states for the LiCs and LiRb molecules, respec ively. If the electric field is oriented at a certain angle γ ith respect to the quantization axis, êE · êd has the form êE · êd = cos(χ) with χ the angle between θ and γ. It can be written in terms of the first-degree Legendre polynomial êE · êd = cos(χ) = P1(cos(χ)) (4.7) = 4pi 3 [ Y ∗1−1(γ,ϕγ)Y1−1(θ,ϕθ) + Y ∗ 10(γ,ϕγ) 10(θ,ϕθ) + ∗ 11(γ,ϕγ)Y11(θ,ϕθ) ] where ϕγ and θ are the projections of a gle γ and θ on the (x, y) plane and Yxx are spherical harmonics. Fig. 4.3 depicts the coordinate system in detail. The effect of a rotating electric field on the collision dynamics in a system of lkali metal atoms depends on the relativ angle between the electric field and the 59 Figure 4.3: Th coordinate system in our calculati ns. !E and ! represent the vector of the external electric field and the dipole moment vector, respectively; γ specifies the orientation of the electric field with respect to the quantization axis; θ is the angle betwee the dipole moment vector and the z- xis; χ is the angle between !E and !d, and ϕγ and ϕθ are the projections of angles γ and θ on the (x, y) plane. magnetic field. Without loss of generality, we can ssume that êE is in the x − z lane, i.e., ϕγ = 0. The matrix elements of V̂E(r) are therefore evaluated using the expressions 〈lml|êE · êd|l′m′l〉 = √ 2 2 si γ(−1)m′l √ [(2l + 1)(2l′ + 1)] ( l 1 l′ 0 0 0 )( l 1 l′ ml −1 −m′l ) + cos γ(−1)m′l √ [(2l + 1)(2l′ + 1)] ( l 1 l′ 0 0 l 1 l′ l 0 −m′l ) − √ 2 2 sin γ(−1)m′l √ [(2l + 1)(2l′ + 1)] ( l 1 l′ 0 0 0 ( l 1 l′ ml 1 −m′l ) (4.8) and 〈SMS | ∑ S′′ M ′′S |S′′ ′′〉 ′′〈 ′′ ′′S |  |S′M ′S〉 = dS δSS′δMS ′S . (4.9) 60 4.2. Theory4.2. Theory Figure 4.3: The coordi ate system in ou calcula ion . !E nd ! represent he vector of the external e ectric field and the dipole oment vector, respectively; γ sp ifies the orient ion of the electric field with resp ct to the qu tization axis; θ is the angle betw en t dip l moment vector and the z-axis; χ is the angle between !E and !d, and ϕγ and ϕθ are the projections of angl s γ and θ o the (x, y) plane. magnetic field. Without loss of generality, we can assume that êE is in the z plane, i.e., ϕγ = 0. The matrix elements of V̂E(r) are therefore eval ate using the expressions 〈lml|êE · êd| ′m′l〉 = √ 2 2 sin γ(−1)m′l √ [(2l + 1)(2l′ + 1)] ( l 1 l′ 0 0 0 )( l 1 l′ l −1 −m′l ) + cos γ(−1)m′l √ [(2l + 1)(2l′ + 1)] ( l 1 l′ 0 0 0 ) l 1 l′ ml 0 −m′ ) − √ 2 2 sin γ(−1)m′l √ [(2l + 1)(2l′ + 1)] ( l 1 l′ 0 0 0 )( l 1 l′ ml 1 −m′l ) (4.8) and 〈SMS | ∑ S′′ ∑ M ′′S |S′′M ′′S〉dS′′〈S′′M ′′S |  |S′M ′S〉 = dS δSS′δMSM ′S . (4.9) 60 4.2. Theory 5 10 15 20 r (Bohr) -3000 0 3000 6000 E n e r g y  ( c m - 1 ) 5 10 15 20 r (Bohr) 0 1 2 3 4 5 D i p o l e  m o m e n t ( D e b y e ) C 6  = 2.5457x10 3  a.u. C 8  = 2.2825x10  a.u. C 10  = 2.5647x10 7  a.u. Figure 4.2: The interaction potentials and dipole moment functions (inset) of the LiRb molecule in the ground 1 ∑ (solid lines) and excited 3 ∑ (dashed lines) states. The interaction potentials were taken from Ref.[4] and the dipole moment functions approximate the data of Ref. [3]. α = 0.045 bohr−2 and D = 1.02 Debye for the t iplet state. These analytical expres- sions approximate the numerical data for the dipole moment functions computed by Aymar and Dulieu [3]. Figs. 4 1 a d 4.2 show the interaction pot ntials VS(r) and the dipole moment functions dS(r) in the 1Σ and 3Σ states for the LiCs and LiRb molecules, respectively. If the electric field is oriented at a certain angle γ with respect to the quantization axis, êE · êd has the form êE · êd = cos(χ) with χ the angl between θ and γ. It can be written in terms of the first-degree Legendre poly omial êE · êd = cos(χ) = P1(cos(χ)) (4.7) = 4pi 3 [ Y ∗1−1(γ,ϕγ)Y1−1(θ,ϕθ) + Y ∗ 10(γ,ϕγ)Y10(θ,ϕθ) + Y ∗ 11(γ,ϕγ)Y11(θ,ϕθ) ] where ϕγ and ϕθ are the projections of angles γ and θ on the (x, y) plane and Yxx are spherical harmonics. Fig. 4.3 de icts the coordinate syste in detail. The effect of a rot ting electric field on the collisi n dynamics in a system of alkali metal atoms de ends on the relative angle betw en el ric field and th 59 4.2. Theory 5 10 15 20 r (Bohr) -3000 0 3000 6000 E n e r g y  ( c m - 1 ) 5 10 15 20 r (Bohr) 0 1 2 3 4 5 D i p o l e  m o m e n t ( D e b y e ) C 6  = 2.5457x10 3  a.u. C 8  = 2.2825x10 5  a.u. C 10  = 2.5647x10 7  a.u. Figure 4.2: The interaction potentials and dipole moment functions (inset) of the LiRb molecule in the ground 1 ∑ (solid lines) and excited 3 ∑ (dashed lines) states. The interaction potentials were taken from Ref.[4] and the dipole moment functions approximate the data of Ref. [3]. α = 0.045 bohr−2 and D = 1.02 Debye for the triplet state. These analytical expres- sions approximate the numeric l data for the dipole moment functions computed by Aymar and Dulieu [3]. Figs. 4.1 and 4.2 show the interaction potentials VS(r) and the dipole moment fu ctions dS(r) in the 1Σ and 3Σ states for th LiCs and LiRb molecules, respectively. If the electric field is oriented at a certain angle γ with respect to the quantization axis, êE · êd has the form êE · êd = cos(χ) with χ the angle between θ and γ. It can be written in terms of the first-degree Legendre polynomial êE · êd = cos(χ) = P1(cos(χ)) (4.7) = 4pi 3 [ Y ∗1−1(γ,ϕγ)Y1−1(θ,ϕθ) + Y ∗ 10(γ,ϕγ)Y10(θ,ϕθ) + Y ∗ 11(γ,ϕγ)Y11(θ,ϕθ) ] where ϕγ a d θ are the projections of a gl s γ a d θ on the (x, y) plane and Yxx are spherical harmonics. Fig. 4.3 depicts the coordinate system in detail. The effect of a rotating electric field on the co lisio dynamic in a system of lkali metal atoms depends on the relative angle betwe n the electric field and the 59 Figur 4.3: The coordina e syst m in our calculations. !E and !d represent the vector of the external electric field and the dipole mome t vector, respectively; γ specifies the orienta ion of the lectric field wit respect t th quantization axis; θ is the angle be w en the dip le moment vector and th z-axis; χ is h angle b tween !E and !d, and ϕγ and ϕθ ar the projections f angl s γ and θ on th (x, y) plane. magnetic fiel . Without loss of generality, we can assume that êE is i the x − z plane, i.e., ϕγ = 0. The matrix elements of V̂E(r) are therefore evaluated using the expressions 〈lml|êE · êd|l′m′l〉 = √ 2 2 sin γ(−1)m′l √ [(2l + 1)(2l′ + 1)] ( l 1 l′ 0 0 0 )( l 1 l′ l −1 −m′l ) + cos γ(−1)m′l √ [(2l + 1)(2l′ + 1)] ( l 1 l′ 0 0 0 ) l 1 l′ ml 0 −m′l ) − √ 2 2 sin γ(−1)m′l √ [(2l + 1)(2l′ + 1)] ( l l′ 0 0 0 )( l l′ ml 1 −m′l ) (4.8) and 〈SMS | ∑ S′′ ∑ M ′′S |S′′M ′′S〉dS′′〈S′′M ′′S |  |S′M ′S〉 = dS δSS′δMSM ′S . (4.9) 60 4.2. Theory ϕθ ϕγ γ χ θ Figure 4.3: The coordinate system in our calculations. !ζ and !d represent th vector of the external electric field and e dipole m ment vector, respectively; γ sp cifies he orientation of the el ctric field with respect to the quantization axis; θ is the angle between the dipole moment vector and the z-axis; χ is the angle between !ζ and !d, and ϕγ and ϕθ are the projections of angles γ and θ on the (x, y) plane. magnetic field. Without loss of generality, we can assume that êE is in the x − z plane, i.e., ϕγ = 0. The matrix elements of V̂E(r) are ther fore evaluated using the expressions 〈lml|êE · êd|l′m′l〉 = √ 2 2 sin γ(−1)m′l √ [(2l + 1)(2l′ + 1)] ( l 1 l′ 0 0 0 )( l 1 l′ ml −1 −m′l ) + cos γ(−1)m′l √ [(2l + 1)(2l′ + 1)] ( l 1 l′ 0 0 0 )( l 1 l ml 0 −m′l ) − √ 2 2 sin γ(−1)m′l √ [(2l + 1)(2l′ + 1)] ( l 1 l′ 0 0 0 )( l 1 l′ ml 1 −m′l ) (4.8) and 〈SMS | ∑ S′′ ∑ ′′ S |S′′M ′′S〉dS′′〈S′′M ′′S |  |S′M ′S〉 = dS δSS′δMSM ′S . (4.9) 60 4.2. Theory ϕθ ϕγ γ θ Figure 4.3: The coordinate system in ur calculations. !ζ and !d repres nt the vector of the external electric field and the dipole mome t vector, respectively; γ specifies the orientation f the el ctric field with respect to the quantization axis; θ is the ngle be ween the dip le m ment vector and the z-axis; χ is the angle between !ζ and !d, and ϕγ and ϕθ ar the pr jections of angles γ and θ on the (x, y) plane. magnetic field. Without loss of generality, we can assume that êE is in the x − z plane, i.e., ϕγ = 0. The matrix elements of V̂E(r) are therefore evaluated using the expressions 〈lml|êE · êd|l′m′l〉 = √ 2 2 sin γ(−1)m′l √ [(2l + 1)(2l′ + 1)] ( l 1 l′ 0 0 0 )( 1 l′ ml −1 −m′l ) + cos γ(−1)m′l √ [(2l + 1)(2l′ + 1)] ( l 1 l′ 0 0 0 )( l 1 l′ ml 0 −m′l ) − √ 2 2 sin γ(−1)m′l √ [(2l + 1)(2l′ + 1)] ( l 1 l′ 0 0 0 )( l 1 l′ ml 1 −m′l ) (4.8) and 〈SMS | ∑ S′′ ∑ M ′′S |S′′M ′′〉dS′′〈S′′M ′′S |  |S′ ′S〉 = dS δS ′δ SM ′S . (4.9) 60 Figure 4.3: The coordinate system in our calculations. ~ζ and ~d represent the vector of the external electric field and the dipole moment vector, respectively; γ specifies the orientation of the electric field with respect to the quantization axis; θ is the angle between the dipole moment vector and the z-axis; χ is the angle between ~ζ and ~d, and ϕγ and ϕθ are the angles between the x-axis and the projections of the vectors êζ and êd on the (x, y) plane, respectively. atoms depends on the relative angle between the electric field and the magnetic field. Without loss of generality, we can assume that êζ is in the (x, z) plane, i.e., ϕγ = 0. The matrix elements of V̂ζ are therefore evaluated using the expressions 〈lml|êζ · êd|l′m′l〉 = √ 2 2 sin γ(−1)m′l √ [(2l + 1)(2l′ + 1)] ( l 1 l′ 0 0 0 )( l 1 l′ ml −1 −m′l ) + cos γ(−1)m′l √ [(2l + 1)(2l′ + 1)] ( l 1 l′ 0 0 0 )( l 1 l′ ml 0 −m′l ) − √ 2 2 sin γ(−1)m′l √ [(2l + 1)(2l′ + 1)] ( l 1 l′ 0 0 0 )( l 1 l′ ml 1 −m′l ) (4.8 63 4.3. Li–Cs system and 〈SMS | ∑ S′′ ∑ M ′′S |S′′M ′′S〉dS′′〈S′′M ′′S |  |S′M ′S〉 = dS δSS′δMSM ′S . (4.9) If the electric field is directed along the z-axis, i.e., γ = 0, Eq. 4.8 reduces to 〈lml| cos θ|l′m′l〉 = (−1)ml ( l 1 l′ −ml 0 ml )( l 1 l′ 0 0 0 )[ (2l + 1)(2l′ + 1) ] 1 2 δmlm′l . (4.10) The numerical approach for solving the coupled differential equations and con- structing the scattering S-matrix is described in Chapter 2. Here, we also apply an additional transformation that diagonalizes the matrix of V̂ζ + V̂B + V̂hf before constructing the S-matrix [199]. 4.3 Li–Cs system Ultracold mixtures of Li and Cs gases have recently been created in the labora- tory of Weidemüller in Freiburg [217, 218] for the formation of ultracold polar LiCs molecules through photoassociation [222]. An alternative method of producing ul- tracold molecules is based on linking ultracold atoms with magnetic-field-induced Feshbach resonances. As mentioned in Chapter 1, Feshbach resonances may also en- hance the probability for photoassociation [223] and provide detailed information for the analysis of interatomic interaction potentials [224]. Experimental measurements of the positions and widths of magnetic Feshbach resonances are therefore very im- portant for dynamical studies of ultracold gases. To guide future experiments in the search of Feshbach resonances, we present in Table 4.1 the positions and widths of purely magnetic s-wave resonances calculated with the spectroscopically determined potentials of Staanum et al. [219]. 4.3.1 Electric-field-induced Feshbach resonances In the absence of electric fields, different partial wave states |lml〉 of the collision complex are uncoupled and s-wave scattering entirely determines the collision cross sections at ultralow kinetic energies. The interaction of the atoms with electric fields (Eq. 4.5), however, induces couplings between different angular momentum states and may thus affect the scattering length. Figures 4.4, 4.5 and 4.6 display the 64 4.3. Li–Cs system Table 4.1: The positions (B0) and widths (∆B) of s-wave magnetic Feshbach res- onances for Li–Cs at magnetic fields below 500 G. The notation |FaMFa〉 for the atomic states is the same as in Chapter 3. Atomic States B0 (G) 4B (G) |1, 1〉7Li ⊗ |3,−2〉133Cs 2.03 > 2.00 |1, 1〉7Li ⊗ |3,−3〉133Cs 1.49 > 2.00 21.50 > 2.00 387.81 > 2.00 |1, 0〉7Li ⊗ |3,−3〉133Cs 4.92 > 2.00 |2,−2〉7Li ⊗ |4, 4〉133Cs 20.05 0.70 |2,−2〉7Li ⊗ |3, 3〉133Cs 0.86 0.03 2.27 0.16 7.06 1.68 |2,−2〉7Li ⊗ |3, 2〉133Cs 1.02 0.06 2.64 0.34 7.20 2.00 |2,−2〉7Li ⊗ |3, 1〉133Cs 0.47 0.02 1.25 0.12 3.15 0.42 6.65 1.50 |2,−2〉7Li ⊗ |3, 0〉133Cs 0.59 0.04 1.65 0.22 4.47 901.06 |2,−2〉7Li ⊗ |3,−1〉133Cs 0.63 0.10 2.49 0.56 magnetic field dependence of the s-wave (solid curve) and p-wave (dashed curve) scattering cross sections calculated for various Zeeman states of Li and Cs at zero electric field (upper panel) and at ζ = 100 kV/cm (lower panel). The examination of Figs. 4.4, 4.5 and 4.6 leads to two important observations: (i) The couplings between s- and p-wave scattering states induce new s- (p-) wave resonances at the location of magnetic p- (s-) wave Feshbach resonances, i.e., s- and p-wave resonances mirror each other in the presence of an electric field. We refer to these new resonances as electric-field-induced resonances. The scattering length is dramatically modified by the resonant enhancement. As shown in the third panel of Fig. 4.5, the s-to-p couplings induced by electric fields are significant both in the presence of p-wave resonances and near s-wave resonances. (ii) The interaction of Li and Cs atoms with electric fields shifts the positions of both the s-wave and p-wave resonances. Figure 4.7 is an expanded view of two s-wave resonances in Fig. 4.5, which shows that relatively weak electric fields (∼ 30 65 4.3. Li–Cs system 4.3. Li–Cs system 600 700 800 900 1000 1100 1200 10 -8 10 -4 10 0 10 4 10 8 C ro ss  s ec ti o n  ( a. u .) 600 700 800 900 1000 1100 1200 Magnetic field (G) 10 -8 10 -4 10 0 10 4 10 8 C ro ss  s ec ti o n  ( a. u .) E = 0 kV/cm E = 100 kV/cm Figure 4.4: Cross sections for elastic s-wave (solid curves) and p-wave (broken curves) scattering of Li and Cs atoms in the states |1, 1〉7Li⊗ |3, 3〉133Cs computed at zero electric field (upper panel) and an electric field strength of 100 kV/cm (lower panel). The collision energy is 10−7 cm−1. 63 4.3. Li–Cs system If the electric field is directed along the z-axis, i.e., γ = 0, Eq. 4.8 reduces to 〈lml| cos θ|l′m′l〉 (4.10) = (−1)ml ( l 1 l′ −ml 0 ml )( l 1 l′ 0 0 0 )[ (2l + 1)(2l′ + 1) ] 1 2 δmlm′l The approach of propagating the coupled differential equations and constructing the scattering S-matrix is the same as described in Chapter 2. Here, we also apply an additional transformation that diagonalizes the matrix of V̂ζ + V̂B + V̂hf before constructing the S-matrix [173]. 4.3 Li–Cs system Ultracold mixtures of Li and Cs gases have recently been created in the laboratory of Weidemüller [185, 186] for the formation of ultracold polar LiCs molecules through photoassociation [188]. An alternative method of producing ultracold molecules is based on linking ultracold atoms together with magnetic-field-induced Feshbach res- onances. Feshbach resonances may also enhance the probability for photoassociation [189] and provide detailed information for the analysis of interatomic interaction po- tentials [190]. Experimental measurements of the positions and widths of magnetic Feshbach resonances are therefore very important for dynamical studies of ultra- cold gases. To guide future experiments in the search of Feshbach resonances, we present in Table 4.1 the positions and widths of purely magnetic s-wave resonances calculated with the potentials of Staanum et al. [2]. 4.3.1 Electric-field-induced Feshbach resonances In the absence of electric fields, different partial wave states |lml〉 of collision com- plex are uncoupled and s-wave scattering entirely determines the collision cross sections at ultralow kinetic energies. The interaction of the atoms with electric fields (Eq. 4.5), however, induces couplings between different angular momentum states and may thus affect the scattering length. Figs. 4.4, 4.5 and 4.6 display the magnetic field dependence of the s-wave (solid curve) and p-wave (dashed curve) scattering cross sections calculated for various Zeeman states of Li and Cs at zero electric field (upper panel) and at ζ 00 kV/cm (lower panel). The examination of Figs. 4.4, 4.5 and 4.6 leads to two important observations: (i) The couplings between s- and p-wave scattering induce new s- (p-) wave resonances at the location of magnetic p- (s-) wave Feshbach resonances, i.e., s- and 61 4.4. Li–Rb system FIG. 8: Differential scattering cross sections for ultracold collisions of Li and Cs in the MLi = −1 and MCs = 3 states computed at an electric field strength of 100 kV/cm. The collision energy is 10−5 cm−1 (full curve) , 10−6 cm−1 (broken curve) and 10−7 cm−1 (dotted curve). The magnetic field is fixed at 1162 G. 0 50 100 150 Scattering angle (degrees) 10 0 10 2 D if fe re n ti al  s ca tt er in g  c ro ss  s ec ti o n  ( a. u .) 22 Figure 4.10: Differential scattering cross sections for ultracold collisions of Li and Cs atoms in the |1,−1〉7Li ⊗ |3, 3〉133Cs states computed at an electric field strength of 100 kV/cm. The collision energy is 10−5 cm−1 (full curve) , 10−6 cm−1 (broken curve) and 10−7 cm−1 (dotted-dashed curve). The magnetic field is fixed at 1162 G. We repor for the first time th observation that the coupling induced by electric fi lds splits Feshbach resonances int m ltiple resonances for states of non-zero an- gular momenta. This new phenomenon offers a complementary way to produce and tune an anisotropic interaction and to study its effect on the many-body physics of heteronucle r atomic gases. 4.4.1 Li–Rb collisions in combined electric and magnetic fields ζ = 0 kV/cm ζ = 100 kV/cm As discussed in Section 4.2, in the absence of electric fields, different partial wave states |lml〉 of the Li–Rb collision complex are uncoupled and s-wave scattering is dominant in ultracold collisions. The interaction with an electric field induces couplings between states of different orbital angular momenta with ∆l = ±1. As a result, a resonant enhancement of the s-wave cross section appears at magnetic fields near intrinsic p-wave resonances – as demonstrated in the previous Section for 70 Figure .4: Cross sections for el e (solid curves) and p-wave (broken curves) scattering of Li and Cs atoms in the states |1, 1〉7Li ⊗ |3, 3〉1 3Cs compute at different electric fields: ζ = 0 kV/cm (upper panel) and ζ = 100 kV/cm (lower panel). The collision energy is 10 . 66 4.3. Li–Cs system 700 800 900 1000 1100 1200 10 -8 10 -4 10 0 10 4 10 8 C ro ss  s ec ti o n  ( a. u .) 700 800 900 1000 1100 1200 10 -8 10 -4 10 0 10 4 10 8 C ro ss  s ec ti o n  ( a. u .) 700 800 900 1000 1100 1200 Magnetic field (G) 10 -3 10 0 10 3 C ro ss  s ec ti o n  ( a. u .) E = 0 kV/cm E = 100 kV/cm E = 100 kV/cm 4.3. Li–Cs system If the electric field is directed along the z-axis, i.e., γ = 0, Eq. 4.8 reduces to 〈lml| cos θ|l′m′l〉 (4.10) = (−1)ml ( l 1 l′ −ml 0 ml )( l 1 l′ 0 0 0 )[ (2l + 1)(2l′ + 1) ] 1 2 δmlm′l The approach of propagating the coupled differential equations and constructing the scattering S-matrix is the same as described in Chapter 2. Here, we also apply an additional transformation that diagonalizes the matrix of V̂ζ + V̂B + V̂hf before constructing the S-matrix [173]. 4.3 Li–Cs system Ultracold mixtures of Li and Cs gases have recently been created in the laboratory of Weidemüller [185, 186] for the formation of ultracold polar LiCs molecules through photoassociation [188]. An alternative method of producing ultracold molecules is based on linking ultracold atoms together with magnetic-field-induced Feshbach res- onances. Feshbach resonances may also enhance the probability for photoassociation [189] and provide detailed information for the analysis of interatomic interaction po- tentials [190]. Experimental measurements of the positions and widths of magnetic Feshbach resonances are therefore very important for dynamical studies of ultra- cold gases. To guide future experiments in the search of Feshbach resonances, we present in Table 4.1 the positions and widths of purely magnetic s-wave resonances calculated with the potentials of Staanum et al. [2]. 4.3.1 Electric-field-induced Feshbach resonances In the absence of electric fields, different partial wave states |lml〉 of collision com- plex are uncoupled and s-wave scattering entirely determines the collision cross sections at ultralow kinetic energies. The interaction of the atoms with electric fields (Eq. 4.5), however, induces couplings between different angular momentum states and may thus affect the scattering length. Figs. 4.4, 4.5 and 4.6 display the magnetic field dependence of the s-wave (solid curve) and p-wave (dashed curve) scattering cross sections calculated for various Zeeman states of Li and Cs at zero electric field (upper panel) and at ζ = 100 kV/cm (lower panel). The examination of Figs. 4.4, 4.5 and 4.6 leads to two important observations: (i) The couplings between s- and p-wave scattering induce new s- (p-) wave resonances at the location of magnetic p- (s-) wave Feshbach resonances, i.e., s- and 61 4.4. Li–Rb system FIG. 8: Differential scattering cross sections for ultracold collisions of Li and Cs in the MLi = −1 and MCs = 3 states computed at an electric field strength of 100 kV/cm. The collision energy is 10−5 cm−1 (full curve) , 10−6 cm−1 (broken curve) and 10−7 cm−1 (dotted curve). The magnetic field is fixed at 1162 G. 0 50 100 150 Scattering angle (degrees) 10 0 10 2 D if fe re n ti al  s ca tt er in g  c ro ss  s ec ti o n  ( a. u .) 22 Figure 4.10: Differential scattering cross sections for ultracold collisions of Li and Cs atoms in the |1,−1〉7Li ⊗ |3, 3〉133Cs states computed at an electric field strength of 100 kV/cm. The collision energy is 10−5 cm−1 (full curve) , 10−6 cm−1 (broken curve) and 10−7 cm−1 (dotted-dashed curve). The magnetic field is fixed at 1162 G. We report for the first time the observation that the coupling induced by electric fields splits Fe hbach resonances into multiple resonances for states of non-zero an- gular momenta. This new phenomenon offers a complementary way to produce and tune an anisotropic interaction and to study its effect on the many-body physics of heteronuclear atomic gases. 4.4.1 Li–Rb collisions in combined electric and magnetic fields ζ = 0 kV/cm ζ = 100 kV/cm As discussed in Section 4.2, in the absence of electric fields, different partial wave states |lml〉 of the Li–Rb collision complex are uncoupled and s-wave scattering is dominan in ultracold collisions. The interaction with an electric field induces couplings between states of different orbital angular momenta with ∆l = ±1. As a result, a resonant enhancement of the s-wave cross section appears at magnetic fields near intrinsic p-wave resonances – as demonstrated in the previous Section for 70 4.3. Li–Cs system If the electric field is directed along the z-axis, i.e., γ = 0, Eq. 4.8 reduces to 〈lml| cos θ|l′m′l〉 (4.10) = (−1)ml ( l 1 l′ −ml 0 l )( l 1 l′ 0 0 0 )[ (2l + 1)(2l′ + 1) ] 1 2 δmlm′l The approach of propagating the coupled differential equations and constructing the scattering S-matrix is the same as described in Chapter 2. Here, we also apply an additional transformation that diagonalizes the matrix of V̂ζ + V̂B + V̂hf before constructing the S-matrix [173]. 4.3 Li–Cs system Ultracold mixtures of Li and Cs gases have recently been created in the laboratory of Weidemüller [185, 186] for the formation of ultracold polar LiCs molecules through photoassociation [188]. An alternative method of producing ultracold molecules is based on linking ultracold atoms together with magnetic-field-induced Feshbach res- onances. Feshbach resonances may also enhance the probability for photoassociation [189] and provide detailed information for the analysis of interatomic interaction po- tentials [190]. Experimental measurements of the positions and widths of magnetic Feshbach resonances are therefore very important for dynamical studies of ultra- cold gases. To guide future experiments in the search of Feshbach resonances, we present in Table 4.1 the positions and width of purely magnetic s-wave resonances calculated with the potentials of Staanum et al. [2]. 4.3.1 Electric-field-induced Feshbach resonances In the absence of electric fields, different partial wave states |lml〉 of collision com- plex are uncoupled and s-wave scattering entirely determines the collision cross sections at ultralow kinetic energies. The interaction of the atoms with electric fields (Eq. 4.5), however, induces couplings between different angular momentum states and may thus affect the scattering length. Figs. 4.4, 4.5 and 4.6 display the magnetic field dependence of the s-wave (solid curve) and p-wave (dashed curve) scattering cross sections calculated for various Zeeman states of Li and Cs at zero electric field (upper panel) and at ζ = 100 kV/cm (lower panel). The examination of Figs. 4.4, 4.5 and 4.6 leads to two important observations: (i) The couplings between s- and p-wave scattering induce new s- (p-) wave resonances at the location of magnetic p- (s-) wave Feshb ch resonances, i.e., s- and 61Figure 4.5: Cross sections for elastic s-wave (solid curves) and p-wave (broken curves) scattering of Li and Cs atoms in the states |1, 0〉7Li ⊗ |3, 3〉133Cs computed at different electric fields: ζ = 0 kV/cm (upper panel) and ζ = 100 kV/cm (middle panel). The lower panel presents the cross section for the s → p transition. The collision energy is 10−7 cm−1. 67 4.3. Li–Cs system 800 900 1000 1100 1200 1300 10 -8 10 -4 10 0 10 4 10 8 10 12 C ro ss  s ec ti o n  ( a. u .) 800 900 1000 1100 1200 1300 Magnetic field (G) 10 -8 10 -4 10 0 10 4 10 8 10 12 C ro ss  s ec ti o n  ( a. u .) E = 0 kV/cm E = 100 kV/cm 4.3. Li–Cs system If the electric field is directed along the z-axis, i.e., γ = 0, Eq. 4.8 reduces to 〈lml| cos θ|l′m′l〉 (4.10) = (−1)ml ( l 1 l′ −ml 0 ml )( l 1 l′ 0 0 0 )[ (2l + 1)(2l′ + 1) ] 1 2 δmlm′l The approach of propagating the coupled differential equations and constructing the scattering S-matrix is the same as described in Chapter 2. Here, we also apply an additional transformation that diagonalizes the matrix of V̂ζ + V̂B + V̂hf before constructing the S-matrix [173]. 4.3 Li–Cs system Ultracold mixtures of Li and Cs gases have recently been created in the laboratory of Weidemüller [185, 186] for the formation of ultracold polar LiCs molecules through photoassociation [188]. An alternative method of producing ultracold molecules is based on linking ultracold atoms together with magnetic-field-induced Feshbach res- onances. Feshbach resonances may also enhance the probability for photoassociation [189] and provide detailed information for the analysis of interatomic interaction po- tentials [190]. Experimental measurements of the positions and widths of magnetic Feshbach resonances are therefore very important for dynamical studies of ultra- cold gases. To guide future experiments in the search of Feshbach resonances, we present in Table 4.1 the positions and widths of purely magnetic s-wave resonances calculated with the potentials of Staanum et al. [2]. 4.3.1 Electric-field-induced Feshbach resonances In the absence of electric fields, different partial wave states |lml〉 of collision com- plex are uncoupled and s-wave scattering entirely determines the collision cross sections at ultralow kinetic energies. The interaction of the atoms with electric fields (Eq. 4.5), however, induces couplings between different angular momentum states and may thus affect the scattering length. Figs. 4.4, 4.5 and 4.6 display the magnetic field dependence of the s-wave (solid curve) and p-wave (dashed curve) scattering cross sections calculated for various Zeeman states of Li and Cs at zero electric field (upper panel) and at ζ = 100 kV/cm (lower panel). The examination of Figs. 4.4, 4.5 and 4.6 leads to two important observations: (i) The couplings between s- and p-wave scattering induce new s- (p-) wave resonances at the location of magnetic p- (s-) wave Feshbach resonances, i.e., s- and 61 4.4. Li–Rb system FIG. 8: Differential scattering cross sections for ultracold collisions of Li and Cs in the MLi = −1 and MCs = 3 states computed at an electric field strength of 100 kV/cm. The collision energy is 10−5 cm−1 (full curve) , 10−6 cm−1 (broken curve) and 10−7 cm−1 (dotted curve). The magnetic field is fixed at 1162 G. 0 50 100 150 Scattering angle (degrees) 10 0 10 2 D if fe re n ti al  s ca tt er in g  c ro ss  s ec ti o n  ( a. u .) 22 Figure 4.10: Differential scattering cross sections for ultracold collisions of Li and Cs atoms in the |1,−1〉7Li ⊗ |3, 3〉133Cs states computed at an electric field strength of 100 kV/cm. The collision energy is 10−5 cm−1 (full curve) , 10−6 cm−1 (broken curve) and 10−7 cm−1 (dotted-dashed curve). The magnetic field is fixed at 1162 G. We report for the first time the observation that the coupling induced by electric fields splits Feshbach resonances into multiple resonances for states of non-zero an- gular momenta. This new phenomenon offers a complementary way to produce and tune an anisotropic interaction and to study its effect on the many-body physics of heteronuclear atomic gases. 4.4.1 Li–Rb collisions in combined electric and magnetic fields ζ = 0 kV/cm ζ = 100 kV/cm As discussed in Section 4.2, in the absence of electric fields, different partial wave states |lml〉 of the Li–Rb collision complex are uncoupled and s-wave scattering is dominant in ultracold collisions. The interaction with an electric field induces couplings between states of different orbital angular momenta with ∆l = ±1. As a result, a resonant enhancement of the s-wave cross section appears at magnetic fields near intrinsic p-wave resonances – as demonstrated in the previous Section for 70 Figure 4.6: Cross sections for elastic s-wave (solid curves) and p-wave (broken curves) scattering of Li and Cs atoms in the states |1,−1〉7Li ⊗ |3, 3〉133Cs computed at different electric fields: ζ = 0 kV/cm (upper panel) and ζ = 100 kV/cm (lower panel). The collision energy is 10−7 cm−1. 68 4.3. Li–Cs system kV/cm) may shift the resonances significantly. The shift of both magnetic Feshbach resonances and electric-field-induced resonances can lead to dramatic changes of the scattering length. Consider, for example, the variation of the scattering length with increasing electric field at the magnetic field value 1071 G (shown in Fig. 4.8), and 1024 G and 1026 G (shown in Fig. 4.9). At a fixed magnetic field, one can thus control microscopic interactions in ultracold atomic gases by varying external electric fields. However, the shift is not always sensitive to electric fields. For instance, we found that the resonances at low magnetic fields listed in Table 4.1 shift by less than 1 G in an electric field of 100 kV/cm so no variation of these resonances with electric fields should be expected. Generally, the shift of resonances at low magnetic fields is less sensitive than that at high magnetic fields. This phenomenon is explained in more detail in Sec. 4.4. Table 4.2: The positions (B0) of p-wave magnetic Feshbach resonances for Li–Cs at magnetic fields below 1 kG. Atomic States Bres (G) |1,−1〉7Li ⊗ |3, 3〉133Cs 953.54 |1, 0〉7Li ⊗ |3, 3〉133Cs 862.74, 907.55, 965.61 |1, 1〉7Li ⊗ |3, 2〉133Cs 998.79 |1, 1〉7Li ⊗ |3, 3〉133Cs 785.57, 862.47 P -wave magnetic resonances are essential for the electric-field-induced reso- nances in ultracold atomic collisions. To guide future experiments in the search of such resonances, we present in Table 4.2 the positions of p-wave magnetic Fesh- bach resonances calculated at zero electric field. 4.3.2 Anisotropy of ultracold scattering Ultracold s-wave scattering is isotropic: the probability to find the atoms after s- wave collisions does not depend on the scattering angle. The interaction with electric fields (Eq. 4.5), however, couples the spherically symmetric s-waves to anisotropic p-waves so electric fields may induce the anisotropy of ultracold scattering. The differential scattering cross section in the presence of external fields is defined as [199] 69 4.3. Li–Cs system FIG. 6: Cross sections for elastic s-wave collisions of Li and Cs in theMLi = 0 and MCs = 3 states computed at zero electric field (curve labeled a) and electric field strengths of 30 kV/cm (curve labeled b), 50 kV/cm (curve labeled c), 70 kV/cm (curve labeled d) and 100 kV/cm (broken curve labeled e). The collision energy is 10−7 cm−1. 1050 1100 1150 1200 Magnetic field (G) 10 0 10 4 10 8 C ro ss  s ec ti o n  ( a. u .) a b c d e ab c d e 20 Figure 4.7: Cross sections for elastic s-wave collisions of Li and Cs atoms in the |1, 0〉7Li ⊗ |3, 3〉133Cs states computed at different electric fields: curve labeled a – ζ = 0 kV/cm; curve labeled b – ζ = 30 kV/cm; curve labeled c – ζ = 50 kV/cm; curve labeled d – ζ = 70 kV/cm; broken curve labeled e – ζ = 100 kV/cm. The collision energy is 10−7 cm−1. dσβ→β′ dr̂idr̂ = 4pi2 kβ 2 ∑ l1 ∑ ml1 ∑ l2 ∑ ml2 ∑ l′1 ∑ m′l1 ∑ l′2 ∑ m′l2 il ′ 1−l1+l2−l′2Yl1ml1 (r̂i) × Y ∗l2ml2 (r̂i)Y ∗ l′1m ′ l1 (r̂)Yl′2m′l2 (r̂)T ∗βl1ml1→β′l′1m′l1 Tβl2ml2→β′l′2m′l2 (4.11) where β and β′ label the initial and final scattering states, kβ is the collision wave number and r̂i and r̂ specify the direction of the initial and final collision fluxes. Assuming that the initial collision flux is directed along the field axis, i.e., z-axis, we can write the differential cross section (Eq. 4.11) for elastic scattering in terms of the s-wave and s-to-p wave elements of the T -matrix in the form σd(θ) = dσ dr̂ = pi 4k2α {|Tl=0→l′=0|2 + 3 cos2 θ|Tl=0→l′=1|2 + 2 √ 3 cos θ × [Re(Tl=0→l′=0)Im(Tl=0→l′=1)− Im(Tl=0→l′=0)Re(Tl=0→l′=1)]}, (4.12) where θ is the angle between the initial and final collision fluxes. The first term is 70 4.3. Li–Cs system FIG. 7: Electric-field dependence of the s-wave scattering cross section for collisions of Li and Cs in the MLi = 0 and MCs = 3 states at the magnetic field strength 1024 G (full line), 1026 G (dashed line) and 1071 (dot-dashed line). The variation of the cross sections is due to shifts of the s-wave resonances shown in Fig. 6. 0 20 40 60 80 100 120 Electric field (kV/cm) 10 -3 10 0 10 3 10 6 10 9 C ro ss  s e c ti o n  ( a .u .) 21 Figure 4.8: Electric-field dependence of the s-wave scattering cross section for col- lisions of Li and Cs atom in the |1, 0〉7Li ⊗ |3, 3〉133Cs states at the magnetic field strength 1071 G. The variation of the cross sections is due to shifts of the s-wave resonances shown in Fig. 4.7. independent of the scattering angle and it usually dominates at ultralow collision energies. Figure 4.9, however, shows that at certain values of the magnetic and electric fields, the s-wave cross section becomes very small. At these points, the differential scattering may be determined by the third term in Eq. 4.12, which leads to angular dependence of the scattering cross sections. Figure 4.10 shows the angular dependence of differential cross sections for ultracold collisions of Li and Cs atoms. Without external electric fields, the contribution of elastic p-wave scattering at the collision energies below 10−5 cm−1 is normally negligible and the scattering dynamics is isotropic. In the presence of electric fields, however, the collision complex is rotated and spinned up leading to the anisotropy of the collisions. Figure 4.5 demonstrates that the probability of the s→ p transition near s-wave and p-wave threshold resonances is sensitive to the magnitude of the electric field. The s- and p-wave scattering channels are degenerate at infinite interatomic separation and the s → p transition must be suppressed by the centrifugal barrier in the p- state. The rate constant for this transition therefore vanishes in the limit of zero 71 4.4. Li–Rb system FIG. 5: Electric-field-induced resonances: Variation of the cross sections for s-wave collisions of Li and Cs in the states MLi = 0 and MCs = 3 with the electric field strength. The magnetic field is fixed at 1024 G (full curve) and 1026 G (broken curve). The collision energy is 10−7 cm−1. 0 20 40 60 80 100 120 140 Electric field (kV/cm) 10 -3 10 0 10 3 10 6 10 9 C ro ss  s e c ti o n  ( a .u .) 19 Figure 4.9: Variation of the cross sections for s-wave collisions of Li and Cs atoms in the states |1, 0〉7Li⊗ |3, 3〉133Cs with the electric field strength. The magnetic field is fixed at 1024 G (full curve) and 1026 G (broken curve). The collision energy is 10−7 cm−1. temperature and it varies with temperature as [225] Rs→p(T ) = A ( 8 piµ )1/2 (kBT )3/22!, (4.13) where kB is the Boltzmann constant and A is a proportionality constant given by the ratio of the cross section and the collision energy. Using the value of the cross section in Fig. 4.5 at B = 1027 G and E = 100 kV/cm, we estimate the rate constant for the s → p excitation due to the electric field to be about 3 × 10−14 cm3 s−1 at 10 nK, and at 1 µK it is on the order of 3× 10−11 cm3 s−1. 4.4 Li–Rb system As described in Chapter 3, the Li–Rb mixture is an important system for the study of both ultracold atomic and molecular gases. The LiRb dimers have a relatively large electric dipole moment (up to 4.2 Debye) [206], which makes the Li–Rb system a good candidate for the research on ultracold dipolar gases and the experimental study of electric-field-induced Feshbach resonances [174]. Motivated by these fea- 72 4.4. Li–Rb system FIG. 8: Differential scattering cross sections for ultracold collisions of Li and Cs in the MLi = −1 and MCs = 3 states computed at an electric field strength of 100 kV/cm. The collision energy is 10−5 cm−1 (full curve) , 10−6 cm−1 (broken curve) and 10−7 cm−1 (dotted curve). The magnetic field is fixed at 1162 G. 0 50 100 150 Scattering angle (degrees) 10 0 10 2 D if fe re n ti al  s ca tt er in g  c ro ss  s ec ti o n  ( a. u .) 22 Figure 4.10: Differential scattering cross sections for ultracold collisions of Li and Cs atoms in the |1,−1〉7Li ⊗ |3, 3〉133Cs states computed at an electric field strength of 100 kV/cm. The collision energy is 10−5 cm−1 (full curve) , 10−6 cm−1 (broken curve) and 10−7 cm−1 (dotted-dashed curve). The magnetic field is fixed at 1162 G. tures and the availability of the accurate interatomic interaction potentials gener- ated as described in Chapter 3, we explore in this section the effects of combined external electric and magnetic fields on elastic collisions in ultracold Li–Rb mix- tures. To guide future experimental studies, we predict the positions and widths of electric-field-induced Feshbach resonances for several spin states and explore the effect of the orientation of the electric field with respect to the magnetic field on ultracold elastic collisions. The work presented here represents the first quantitative analysis of electric-field-induced resonances based on precise inter-atomic potentials. In addition, our analysis provides insights into the detailed physical mechanism of electric-field-induced interactions in ultracold binary mixtures of alkali metal atoms. We report for the first time the observation that the coupling induced by electric fields splits Feshbach resonances into multiple resonances for states of non-zero an- gular momenta. 73 4.4. Li–Rb system 900 1000 1100 1200 1300 Magnetic field (G) 10-8 10-4 100 104 108 C r o s s  s e c t i o n  ( a . u . ) E A B D C Figure 4.11: Magnetic field dependence of the elastic cross section for collisions between Li and Rb in the atomic spin state |12 , 12〉6Li⊗ |1, 1〉87Rb. These results were obtained for a collision energy of 10−7cm−1 and two different electric fields. The solid and dash-dotted curves show the s- and p-wave cross sections with ζ = 0, while the dotted and dashed curves show the s- and p-wave cross sections when ζ = 100 kV/cm. Here, only the cross section for the ml = 0 state is shown for p-wave scattering. At A an s-wave resonance is induced by an intrinsic p-wave resonance. Figure 4.12 shows this feature in more detail. At B and at C an intrinsic s-wave resonance is shifted to higher magnetic fields (corresponding to a shift of the associated bound state to lower energy) due to the electric field coupling between bound states. The observation that the shift of higher field resonances (e.g. C) is typically larger than that of lower field resonances (e.g. B) is discussed in the text. At D an intrinsic p-wave resonance is shifted to lower magnetic fields (corresponding to a shift of the associated bound state to higher energy). At E an induced p-wave resonance appears (invisible on this scale) due to the intrinsic s-wave resonance at C. 74 4.4. Li–Rb system 874 876 878 880 882 Magnetic field (G) 10-8 10-4 100 104 108 C r o s s  s e c t i o n  ( a . u . ) Figure 4.12: Magnetic field dependence of s- and p-wave elastic cross sections for atoms in the atomic spin state |12 , 12〉6Li ⊗ |1, 1〉87Rb computed at different electric fields. This is the same feature at A in Fig. 4.11. The solid and dotted curves show the s-wave cross sections at ζ = 0 and ζ = 100 kV/cm, respectively. The dot-dashed and dashed curves show the p-wave cross sections at ζ = 0 and ζ = 100 kV/cm, respectively. This intrinsic p-wave resonance shifts to lower magnetic field (corresponding to the shift of the associated bound state to higher energy) as the electric field magnitude is increased. The s-wave induced resonance appears at the same location as the intrinsic p-wave resonance, and its width grows with the strength of the electric field (see Fig. 4.13). Here only the cross section of the ml = 0 component is shown for the p-wave state is shown (Fig. 4.17 shows the cross sections for all three components). The collision energy is 10−7cm−1. 75 4.4. Li–Rb system 0 50 100 150 200 250 Electric field (kV/cm) 0 2 4 6 8 W i d t h  ( G ) Figure 4.13: The width (∆B) of the s-wave electric-field-induced Feshbach resonance arising from the intrinsic p-wave resonance at 882 G as a function of the electric field magnitude. Here γ = 0 and the collision energy is 10−7cm−1. The width appears to scale quadratically with ζ, at least for the electric fields below 200 kV/cm, and suggests that this induced resonance arises from an indirect coupling [226]. The solid line is the fit ∆B = 1.76× 10−4 ζ2 G, where ζ is in units of kV/cm. 4.4.1 Li–Rb collisions in combined electric and magnetic fields As discussed in Section 4.2, in the absence of electric fields, different partial wave states |lml〉 of a two-atom collision complex are uncoupled and s-wave scattering is dominant in ultracold collisions. The interaction with an electric field induces couplings between states of different orbital angular momenta with ∆l = ±1. This leads to a resonant enhancement of the s-wave cross section at magnetic fields near intrinsic p-wave resonances – as demonstrated in the previous Section for Li–Cs scattering. The same occurs in collisions of Li and Rb atoms. Figure 4.11 shows the magnetic field dependence of s- and p-wave elastic cross sections for Li and Rb atoms in the spin state |12 , 12〉6Li ⊗ |1, 1〉87Rb computed at zero electric field and at ζ = 100 kV/cm. Here, the electric field is directed along the quantization axis (γ = 0). In the presence of 100 kV/cm electric field, an s-wave resonant peak appears (indicated at A) at the magnetic field of 877.50 G which arises from an intrinsic p-wave resonance. This is an electric-field-induced Feshbach resonance. 76 4.4. Li–Rb system Figure 4.12 shows this feature in more detail. Table 4.3 lists the positions and widths of electric-field-induced Feshbach resonances for several atomic spin states of the 6Li–87Rb system at magnetic fields below 2 kG. We extract the positions and widths from the magnetic field dependence of the scattering length (Eq. 1.1) for each resonance. In this calculation, we also observe new p-wave resonances induced by the coupling to a d-wave state, and we find that these p-wave resonances give rise to new s-wave electric-field-induced Feshbach resonances, denoted by (d) in Table 4.3. We emphasize that these calculations are based on precise interatomic inter- action potentials (obtained in Chapter 3) and rigorous quantum mechanical theory. The results presented in Table 4.3 should therefore be considered as quantitative predictions of the positions of the resonances. The width of the electric-field-induced Feshbach resonances is determined by the strength of the coupling, which is in turn determined by the magnitude of the electric field. In Fig. 4.13, we plot the width of the s-wave resonance (shown in Fig. 4.12) induced by the intrinsic p-wave resonance near 882 G as a function of the electric field magnitude. We find that the width can be well represented by a quadratic function of ζ, at least for the electric fields below 200 kV/cm, which suggests that this induced resonance arises from an indirect coupling [226]. Table 4.3: The positions (B0) and widths (∆B) of s-wave resonances induced by an external electric field of 100 kV/cm for 6Li–87Rb at magnetic fields below 2 kG. (d) denotes an s-wave electric-field-induced Feshbach resonance arising from a high order coupling through the p-wave channel to a d-wave closed channel state. As a consequence, these resonances are exceedingly narrow. Atomic States B0 4B |f,mf 〉6|f,mf 〉87 (G) (G) |12 , 12〉 |1, 1〉 536.65 (d) 0.01 877.5 2.3 654.52 < 0.01 |12 , 12〉 |1, 0〉 555.88 (d) < 0.01 885.8 2.6 |12 ,−12〉 |1, 1〉 578.58 (d) 0.01 707.70 < 0.01 770.50 < 0.01 |12 ,−12〉 |1, 0〉 596.01 (d) < 0.01 926.8 2.6 |32 , 32〉 |1,−1〉 1242.5 12.7 77 4.4. Li–Rb system 4.4.2 Mechanism of electric-field-induced shifts of magnetic Feshbach resonances As demonstrated in Sections 4.3.1 and 4.4.1, the electric field not only induces new resonances but also shifts the positions of intrinsic magnetic Feshbach resonances. Our calculations reported in Sections 4.3.1 and 4.4.1 provided the first observation of the electric-field-induced shifts of magnetic Feshbach resonances. Here, we use the accurate interaction potentials obtained for the Li–Rb collision system in Chapter 3 to explore this phenomenon in detail and provide an explanation for why some resonances shift significantly, while other resonances remain insensitive to electric fields. Figure 4.11 shows that the interaction of Li–Rb dipole moment with the electric field shifts the positions of both s- and p-wave resonances. Due to the interaction of the Li–Rb dimers with the external electric field, at B and at C, an intrinsic s-wave resonance is shifted to higher magnetic fields, which corresponds to a shift of the associated bound state to lower energy. At D an intrinsic p-wave resonance is shifted to lower magnetic fields corresponding to a shift of the associated bound state to higher energy. For the most part, the shift of the Feshbach resonance positions arises from the coupling between different bound states whereas the coupling of a given bound state to the scattering state results in a broadening of the associated electric- field-induced resonance. We note that the shift of magnetic Feshbach resonances at higher magnetic fields (e.g. C) is more significant than the shift at lower magnetic fields (e.g. B), the trend observed already in our first calculation of Li–Cs scattering (Section 4.41). This generic behavior results from the fact that resonances associated with higher magnetic fields are typically more deeply bound than those associated with lower magnetic fields. As a result, the wave function of the bound state giving rise to Feshbach resonances at higher fields samples smaller interatomic distances where the dipole moment function is much larger. Another example of the shift induced by the electric field couplings is shown in Fig. 4.14 for atoms in the atomic spin state |12 ,−12〉6Li ⊗ |1,−1〉87Rb. An electric field of 30 kV/cm is large enough to shift the position of this s-wave resonance by almost 2 G – much larger than its width – while a field of 100 kV/cm produces a shift of almost 9 G. It is important to note that this s-wave resonance shifts to lower magnetic fields as the electric field increases, and this is opposite to the shift of the s-wave resonances shown in Fig. 4.11. The shift of a resonance results from level repulsion between the closed channel bound states and therefore depends on the proximity, position, and coupling strengths of the nearby bound states. Therefore, the direction of the resonance shift and its dependence on the electric field magnitude 78 4.4. Li–Rb system 1602 1604 1606 1608 1610 1612 1614 Magnetic field (G) 100 103 106 109 C r o s s  s e c t i o n  ( a . u . ) Figure 4.14: Magnetic field dependence of the s-wave elastic cross section for atoms in the atomic spin state |12 ,−12〉6Li⊗|1,−1〉87Rb computed at different electric fields: ζ = 0 kV/cm (solid curve), ζ = 30 kV/cm (dotted curve), ζ = 70 kV/cm (dashed curve) and ζ = 100 kV/cm (dot-dashed curve). An intrinsic s-wave resonance (whose position is 1611 G in the absence of an electric field) is observed to shift to lower magnetic fields as the electric field strength is increased. Note: the shift direc- tion is in the opposite sense to that of the intrinsic s-wave resonances in Fig. 4.11. These results were obtained with a collision energy of 10−7cm−1. do not exhibit a generic behavior but depend on the particular environment of a given resonance. The interaction mechanism giving rise to the shifts is schematically depicted in Fig. 4.15. These shifts provide a way to dramatically and rapidly modify the s-wave scat- tering length by tuning into and out of an intrinsic magnetic field resonance. Fig- ure 4.16 presents two such tuned resonances arising from the variation of the electric field for atoms in the atomic spin state |12 , 12〉6Li ⊗ |1, 1〉87Rb. This figure shows the cross section for s-wave collisions as a function of the electric field strength with the magnetic field fixed at 1066 G (solid line) and 878 G (dotted line). The solid curve shows a large resonance feature due to the intrinsic magnetic Feshbach resonance at 1067 G which shifts to higher magnetic field as the electric field increases (see Fig. 4.11). The small resonance feature which appears at the electric field strength of approximately 16 kV/cm in the solid curve is due to an electric-field-induced res- 79 4.4. Li–Rb system Closed channel Open channel ml = 1 ml = 2 ml = −1 ml = −2 ml = 0 ml = 1ml = −1 ml = 0 ml = 0 s-wave p-wave d-wave l = 0 l = 1 l = 2 Energy Figure 4.15: A schematic illustrating the mechanism of the shifts and splitting of p- and d-wave bound states resulting in the shifts and splitting of the corresponding Feshbach resonances. For simplicity, only three adjacent bound state levels are shown. The different partial wave potentials of each state are on this scale almost indistinguishable and are drawn here as a single potential. The inset shows the energy levels associated with these three states. The dotted lines indicate their energies in the absence of an electric field. The coupling induced by the electric field is represented as double-ended arrows and shown for the case when the electric field is aligned along the magnetic field, i.e. when γ = 0, states with the same ml value are coupled. The coupling results in level repulsion and the new position of the states is indicated by the solid lines. The degeneracy of the p- and d-wave bound states is broken and the associated Feshbach resonance splits into a multiplet with l + 1 distinct resonances as shown in Figs. 4.17 and 4.18. This simple picture predicts that the s-wave resonance should shift to higher magnetic fields (given that the energy of the threshold moves down with increasing magnetic fields) and that the ml = 0 partial wave component should produce a new resonance at a magnetic field below the |ml| = 1 component – consistent with the motion of the resonances in Fig. 4.11 and Fig. 4.17. Of course, each state is coupled to all other bound states within the same spin manifold and with an orbital angular momenta differing by ∆l = ±1, resulting in splittings and shifts (e.g. Fig. 4.14) which may not follow the predictions of this simple picture. 80 4.4. Li–Rb system 0 20 40 60 80 100 120 Electric field (kV/cm) 10-3 100 103 106 109 C r o s s  S e c t i o n  ( a . u . ) Figure 4.16: Variation of the cross section for s-wave collisions as a function of the electric field strength with the magnetic field fixed at 1066 G (solid line) and 878 G (dotted line) for atoms in the spin state |12 , 12〉6Li ⊗ |1, 1〉87Rb. The large resonance feature shown in the solid curve is due to the shift of the intrinsic magnetic Feshbach resonance just below 1066 G to higher magnetic fields, while the small resonance feature at 16 kV/cm arises from the shift of an intrinsic p-wave resonance just above 1066 G to lower magnetic fields as the electric field increases. The dotted curve shows a resonance feature associated with an electric-field-induced resonance (shown in Fig. 4.12) which moves from 882 G at ζ = 0 down to a magnetic field below 877 G at ζ = 120 kV/cm. The collision energy is 10−7cm−1. onance arising from the intrinsic p-wave resonance just above 1066 G which shifts to lower magnetic field as the electric field increases. In the same plot, the dotted curve shows a resonance feature due to the shift of an electric-field-induced reso- nance arising from the intrinsic p-wave resonance at 882 G. Figure 4.12 shows that the p-wave state responsible for this resonance shifts to lower magnetic fields as the electric field increases. 4.4.3 Splitting of Feshbach resonances in an electric field In the presence of an electric field, the couplings between different partial wave states can push the bound states in the s-wave and p-wave interaction potentials apart. This level repulsion gives rise to the electric-field-induced shift of the intrinsic s- and 81 4.4. Li–Rb system 874 876 878 880 882 884 886 888 Magnetic field (G) 10-12 10-9 10-6 10-3 100 C r o s s  s e c t i o n  ( a . u . ) |ml| = 1ml = 0 Figure 4.17: Magnetic field dependence of p-wave elastic cross section (averaged over all three orbital angular momentum components) for atoms in the atomic spin state |12 , 12〉6Li ⊗ |1, 1〉87Rb computed at zero electric field (solid curve) and at ζ = 100 kV/cm (dot-dashed curve). The thin dotted curves show the magnetic field dependence of the cross section for the |ml| = 1 and the m = 0 components separately. The p-wave resonance splits into two distinct resonances, one occurring for the ml = 0 component and one for the |ml| = 1 components. When the electric and magnetic fields are not co-linear, this segregation of the resonance multiplet breaks down as seen in Fig. 4.20. The collision energy is 10−7 cm−1. p-wave magnetic Feshbach resonances – as described in the previous section. Since the couplings depend on the orbital angular momentum projection, ml, we also expect the electric-field-induced coupling to split Feshbach resonances for states of non-zero angular momenta. This mechanism is illustrated in Fig. 4.15 where we show three adjacent bound state levels as well as the coulping induced by an applied electric field with γ = 0. Without external electric fields, the bound states in the p-wave interaction potential are degenerate, whereas the electric field lifts this degeneracy. In the case where the electric field points along the quantization axis (γ = 0), the ml = 0 bound state in the p-wave potential is coupled to bound states in both the s- and d-wave potentials whereas the |ml| = 1 bound states are only 82 4.4. Li–Rb system 10-16 10-14 10-12 Cr os s S ec tio n (a. u.) 537 538 539 Magnetic field (G) 10-18 10-16 10-14 10-12 10-10 ml = 0 |ml| = 1 |ml| = 2 ml = -2 ml = -1 ml = 0 ml = 2 ml = 1 Figure 4.18: The upper panel shows the magnetic field dependence of the d-wave elastic cross section for atoms in the atomic spin state |12 , 12〉6Li ⊗ |1, 1〉87Rb com- puted at zero electric fields (solid curve). The lower panel shows the magnetic field dependence of d-wave elastic cross section (solid curve). The contributions to the cross section from the |ml| = 2, |ml| = 1 and the ml = 0 components are shown (dotted curves) at ζ = 100 kV/cm. The d-wave resonance splits into l + 1 = 3 distinct resonances corresponding to the splitting of the d-wave bound state levels drawn schematically in the lower panel. The collision energy is 10−7cm−1. 83 4.4. Li–Rb system coupled to bound states in the d-wave potential. This occurs because the system is cylindrically symmetric and therefore the total angular momentum projection is conserved. Also, the couplings between internal spin states and partial wave states are negligible. As a result, the coupled states repel and the ml = 0 state is shifted differently than the |ml| = 1 states splitting the p-wave resonance into a doublet. For the purposes of simplifying the discussion, we have neglected the coupling to yet higher order partial wave states and we have neglected the possible presence of other closed channel states in the near vicinity. This mechanism generally applies to all nonzero partial waves. For a state with an orbital angular momentum l, the number of peaks is l + 1 corresponding to the number of distinct values for |ml|. Figures 4.17 and 4.18 show the splitting of a p-wave and a d-wave Feshbach resonance, respectively. In the presence of a 100 kV/cm electric field, the p-wave resonance splits into two peaks (corresponding to the |ml| = 1 and ml = 0 com- ponents) with a separation of 4 G (dot-dashed line in Fig. 4.17). The shift of the |ml| = 1 peak in Fig. 4.17 to higher magnetic fields (lower energy) is consistent with coupling between the p-wave bound states and a d-wave state which resides at a higher energy (illustrated in Fig. 4.18). The splitting of a d-wave bound state gives rise to three separated resonances and is shown in Fig. 4.18. An interval of 1 G opens up between ml = 0 and |ml| = 1 and an interval of 2 G appears between |ml| = 1 and |ml| = 2. Since it is only very weakly coupled to higher partial-wave states, the |ml| = 2 component remains in essentially the same location as the resonance at zero electric field. Jin and coworkers [165] and Ticknor et al. [220] have previously observed that p-wave Feshbach resonances for collisions of 40K atoms split into a doublet due to the magnetic dipole-dipole interaction. In the work presented here, we neglect the magnetic dipole-dipole interaction since it produces a negligible effect compared to the electric field coupling and the splitting we predict for Feshbach resonances is entirely due to the effect of the electric field. As discussed in Ref. [165], the ability to introduce and tune an anisotropic interaction using high-partial-wave resonances may have far reaching consequences for the study of novel forms of superfluidity using cold atomic gases [227]. The splitting of the nonzero-partial-wave resonances arising from the magnetic dipole-dipole interaction is very small and will disappear as the resonance becomes broad with increasing temperature. In contrast, the splittings observed here, occurring for heteronuclear atomic mixtures, are more than an order of magnitude larger. In addition, the splitting can be used as a signature of Feshbach resonances in nonzero partial waves. For example, in the work of Deh and coworkers [205], the p-wave resonance measured in the Li–Rb system could be identified in the 84 4.4. Li–Rb system presence of external electric fields. 4.4.4 Collision dynamics in non-parallel electric and magnetic fields So far, we have discussed the modifications of Feshbach resonances induced by the application of an electric field parallel to the magnetic field (γ = 0). In this Section, we study the effect of non-parallel fields (γ 6= 0). Figure 4.19 shows the variation of the total elastic cross section for p-wave collisions given fixed electric (100 kV/cm) and magnetic fields as a function of the angle γ between them. In the upper panel of the Figure, the magnetic field is 877 G which is near a p-wave resonance for the ml = 0 component (see Fig. 4.17). In the lower panel, the magnetic field is 881.9 G and falls in between the ml = 0 and |ml| = 1 resonances in the p-wave doublet. In the latter case, the variation of the cross sections as a function of γ is only a factor of 10, while at a magnetic field near one of the resonances, the cross section varies by almost 4 orders of magnitude as γ changes by less than 30◦. Figure 4.20 presents the magnetic field dependence of the total elastic cross section for different components of p-wave scattering at γ = 45◦ near the intrinsic p-wave resonance at 882 G. In this case, because the electric field couples states of different ml values, the doublet structure of the p-wave resonance appears for each of the threeml components of the open channel. This is in contrast to the case with γ = 0 shown in Fig. 4.17 where the coupling is only between states with the sameml value and each component exhibits a single resonance. It should be clarified here that the electric-field-induced s-wave resonance arising from this p-wave resonance exhibits only the single resonance corresponding to the ml = 0 component of the p-wave bound state. This is because (neglecting the magnetic dipole-dipole interaction) the orbital angular momentum projection along the electric field axis is conserved by the Hamiltonian, and ml = 0 for s-wave collisions in all coordinate frames. On the other hand, a state with orbital angular momentum l and projection ml defined with respect to the magnetic field axis will be a linear combination of states with all possible values of ml when represented with respect to the electric field axis [221]. Figure 4.21 presents the magnetic field dependence of the average p-wave elastic scattering cross section (averaged over all three components) for atoms in the spin state |12 , 12〉6Li⊗ |1, 1〉87Rb at ζ = 100 kV/cm and with three orientations of the elec- tric field, γ = 0◦, 45◦, and 90◦. The main point of this plot is to illustrate that the position of the resonances remains unchanged for different values of γ. This is par- ticularly important for the experimental search for these effects since it means that any variation of the orientation of the electric and magnetic fields does not adversely 85 4.4. Li–Rb system 10-8 10-6 10-4 Cr os s S ec tio n ( a.u .) 0 30 60 90 120 150 180 ! (Degree) 10-6 10-5 Figure 4.19: Total elastic cross section for different components of p-wave scattering versus the angle, γ, between the applied electric and magnetic fields. The cross sections are shown for collisions in the ml = 0 state (dashed curve), the |ml| = 1 states (dotted curve), and the average (solid curve) of the cross sections over all three components for the atomic state |12 , 12〉6Li⊗ |1, 1〉87Rb and for ζ = 100 kV/cm . The upper panel shows these cross sections at an applied magnetic field of 877.0 G which is near the resonance for the ml = 0 component while the lower panel is at a field of 881.9 G which is in between the resonances for the ml = 0 and |ml| = 1 components (see Fig. 4.17). We observe that the shape of this variation changes dramatically near a resonance. The collision energy is 10−7cm−1. 86 4.5. Conclusions 874 876 878 880 882 884 886 888 Magnetic field (G) 10-9 10-6 10-3 100 C r o s s  S e c t i o n  ( a . u . ) Figure 4.20: Magnetic field dependence of the elastic cross section for different components of p-wave scattering with an electric field, ζ = 100 kV/cm, tilted with respect to the magnetic field axis by γ = 45◦. The cross sections are shown for collisions in the ml = 0 state (dashed curve), the |ml| = 1 states (dotted curve), and the average (solid curve) of the cross sections over all three components for the atomic state |12 , 12〉6Li⊗|1, 1〉87Rb. The doublet structure of the p-wave resonance seen also in Fig. 4.17 now appears for each of the three angular momentum projection components. The collision energy is 10−7cm−1. affect the visibility of these multiplet features. Consequently, any inhomogeneities in the direction of the electric field over the confinement size of the atomic ensem- ble would not affect their visibility either. Nevertheless, since the positions of the resonances do depend on the electric field strength, any inhomogeneities in the mag- nitude of the electric field would result in inhomogenous broadening of the observed resonances. 4.5 Conclusions In this Chapter, we have presented a detailed analysis of Feshbach resonances in ul- tracold collisions in Li–Cs and Li–Rb mixtures in the presence of superimposed elec- tric and magnetic fields. Our calculations show that electric fields below 100 kV/cm may significantly modify the collision dynamics in binary mixtures of ultracold gases 87 4.5. Conclusions 876 878 880 882 884 886 888 Magnetic field (G) 10-6 10-4 10-2 100 C r o s s  s e c t i o n  ( a . u . ) Figure 4.21: Magnetic field dependence of elastic cross sections for atoms in the atomic spin state |12 , 12〉6Li ⊗ |1, 1〉87Rb computed at ζ = 100 kV/cm with the ori- entation of the electric field at γ = 0◦ (solid curve), 45◦ (dotted curve), and 90◦ (dot-dashed curve). The collision energy is 10−7cm−1. by inducing couplings between collision channels with different partial waves. These couplings generate copies of intrinsic resonances previously restricted to a particular partial-wave collision to other partial wave channels, which we call electric-field- induced Feshbach resonances. We have shown that external electric fields can also shift the positions of s-wave resonances significantly. These shifts lead to the varia- tion of scattering lengths as functions of the electric field. Therefore, electric fields can be used for tuning the scattering length like in the experiments with magnetic Feshbach resonances [4]. We have also provided important insights into the detailed physical mechanism of electric-field-induced interactions in ultracold binary mix- tures of alkali metal atoms and have reported for the first time the observation that the coupling induced by electric fields splits Feshbach resonances into multiple peaks for states of non-zero angular momenta. It was recently observed that the magnetic dipole-dipole interaction can also lift the degeneracy of a p-wave state splitting the associated p-wave Feshbach resonance into two distinct resonances at different mag- netic fields [165, 220]. The primary differences with that work are that the splitting studied here is produced only in heteronuclear collisions, is continuously tunable 88 4.5. Conclusions using an applied electric field, and is more than an order of magnitude larger than the splitting induced by magnetic dipole-dipole interactions. Electric-field-induced resonances discussed here are three-state resonances in- volving the scattering s- and p-wave channels and a bound state of the molecules. The coupling between the s-wave or p-wave scattering channel and the molecular state can be induced by magnetic fields as in the experiments of Ticknor et al. [220]. The coupling between the s- and p-wave scattering channels can be induced by elec- tric fields as shown in this Chapter. Electric-field-induced Feshbach resonances may thus allow for two-dimensional control of interatomic interactions with both mag- netic and electric fields. Electric-field-induced resonances may also be used in the search for p-wave resonances at ultracold temperatures. In the absence of electric fields, the s- and p-wave channels are uncoupled and ultracold collisions are dom- inated by s-wave scattering. It is therefore difficult, if not impossible, to detect p-wave resonances in ultracold gases of binary mixtures directly. Applying an elec- tric field may thus be an important tool for spectroscopic studies of ultracold atoms. The measurements of p-wave resonances may provide important information for the analysis of interatomic interaction potentials, especially for systems with anisotropic long-range interactions [228]. We have also shown that electric fields may spin up the collision complex and induce anisotropic scattering. Controllable angle-dependent scattering may modify the properties of ultracold gases such as the expansion of Bose-Einstein conden- sates released from the trap [229]. Measurements of the differential scattering cross sections may probe the anisotropy of interatomic interactions [230] and provide de- tailed information on molecular structure. We believe that the additional degrees of control offered by electric field interactions will play an important role in future experiments on the many-body physics of heteronuclear atomic gases. Interactions between heteronuclear molecules will generally be characterized by significant dipole moment functions and the resonances described in this Chapter will similarly occur in ultracold collisions of molecules. Electric fields induce cou- plings between different total angular momenta of the collision complex of molecules. Feshbach resonances of higher total angular momenta may thus affect ultracold molecular collisions through electric-field-induced couplings. The density of Fesh- bach resonances in molecule – molecule collisions is quite large [231, 232] and we expect that the effects of electric fields on ultracold collisions of molecules will be even more pronounced than the effects observed here. Finally, we would like to point out that the mechanism of electric-field control described here does not perturb the separated atoms. The atoms interact with electric fields only when in a molecular 89 4.5. Conclusions collision complex so the wave function of the isolated atoms may be more immune to decoherence than in varying magnetic or optical fields. Coupled with the possibility of tuning the electric fields very fast, this makes electric-field-induced resonances a useful tool for the development of quantum computation with ultracold atoms and molecules. 90 Chapter 5 Ultracold inelastic collisions in two dimensions 3 The creation of low-dimensional quantum gases has opened up exciting possibilities for new research with ultracold atoms and molecules. The low-dimensional systems are generated by confining ultracold atoms in one or two dimensions by a harmonic optical potential, leading to the formation of quasi-2D and quasi-1D gases. In this Thesis, we develop a theory to explore inelastic and chemically reactive collisions of ultracold atoms and molecules in quasi-2D geometry. In order to understand the general features of ultracold collisions in restricted geometries, we first study the scattering properties of particles in a purely 2D geometry. In this Chapter, we present a multi-channel scattering theory of atomic and molecular collisions in 2D and report the results of rigorous quantum mechanical calculations elucidating the dynamics of inelastic collisions in 2D. In particular, we present the first numerical test of threshold collision laws in 2D. 5.1 Why 2D? Two-dimensional ultracold gases exhibit many interesting features. For example, the presence and characteristics of Bose-Einstein condensates and phase transitions in 2D are different from those in three dimensions [185, 233–235]. A “true” Bose- Einstein condensate with long range order is not stable in a homogeneous 2D gas and Bose-Einstein condensation is allowed to occur in 2D only at absolute zero. However, the 2D gas becomes superfluid and Bose-Einstein condensation can be observed in a local region when temperature is below a certain value. The transition from the disordered phase at high temperatures to this low temperature locally-ordered phase is a Kosterlitz-Thouless transition. The first experimental study of 2D systems was carried out by trapping atomic hydrogen at the surface of liquid helium [234–236], where evidence for the 2D phase transition was reported. With the development of laser cooling techniques, researchers demonstrated new 3The results of numerical calculations of this Chapter were presented in Ref. [4] of Appendix D. 91 5.1. Why 2D? possibilities to produce and manipulate quantum gases in quasi-confined geometries [50, 191, 192, 237]. A harmonic confining potential is used in these experiments to restrict the motion of atoms and molecules in a pancake-shaped trap. One degree of freedom of the confined particles is frozen to zero point oscillations leading to the formation of a quasi-2D gas [181, 185, 189–192, 237]. Ultracold quantum gases confined in 2D exhibit unique properties and can be used as model systems to explore new phenomena in several areas of physics [181, 184, 185, 187, 189, 190, 237–239]. For example, metastable alkaline earth atoms or polar molecules in 2D may repel each other at long range, which leads to the formation of self-organizing crystals at ultracold temperatures [55, 171] or the possibility to design spin lattice models [240]. The many-body behavior of quantum gases in low dimensions can be manipulated by adjusting the depth of the confining potential [127, 176, 181, 191]. Confining atoms in 2D may also result in interesting decoherence dynamics of quantum gases [241]. Collision properties are different for atoms and molecules in 2D and 3D geometries. Sadeghpour and coworkers [193] have shown that the energy dependence of cross sections for elastic and inelastic collisions near threshold depends on the dimensionality of the system. Confinement may therefore modify chemical reactions and inelastic collisions of molecules at ultralow temperatures. In this Thesis, we want to understand the collision dynamics and explore new control mechanisms in quasi-2D gases. For this purpose, we first study the scattering properties of ultracold atoms in a purely 2D geometry. The purely 2D geometry is the limit of a quasi-2D geometry with an extremely tight confinement. In the limit of zero collision energy, the threshold behavior of collision cross sections in a quasi- 2D gas must smoothly approach the threshold energy dependence for scattering in a purely 2D geometry. The analysis of elastic and inelastic scattering in a purely 2D geometry therefore provides a reference point for the study of laser field effects on atomic and molecular collisions in a quasi-2D gas. In this Chapter, we present a quantum-mechanical scattering theory of ultracold inelastic collisions in 2D. Based on this theory, we carry out rigorous quantum calculations to study collisions in a binary mixture of ultracold Li and Cs atoms in 2D. Our results provide a numerical test of the threshold laws in 2D and show that the magnetic dipole-dipole interaction in atomic collisions may modify the energy dependence of cross sections for elastic and inelastic collisions in 2D. Our numerical calculations demonstrate a dramatic difference of collision dynamics in 2D and 3D ultracold gases and show that inelastic scattering in 2D geometries is significantly suppressed. 92 5.2. Close coupling theory of collisions in two dimensions 5.2 Close coupling theory of collisions in two dimensions 5.2.1 Scattering amplitude and cross section The relative motion of two atoms confined in a plane is best described using cylin- drical polar coordinates referred to the quantization axis directed along the normal to the confinement plane. The time-independent Schrödinger equation in 2D is[ − 1 2µρ d dρ ρ d dρ + l̂2z(ϕ) 2µρ2 + V̂ (ρ) ] ψ(ρ, ϕ) = Eψ(ρ, ϕ), (5.1) where E is the total energy of the collision complex, µ is the reduced mass of the colliding particles, ρ is the interatomic distance, ϕ specifies the orientation of the interatomic axis in the confinement plane, l̂z is the operator describing the rotation of the collision complex about the quantization axis, and V̂ (ρ) is the electronic interaction potential. The incoming flux in our system is directed along the x-axis. In the asymptotic region where the interaction between particles can be ignored, the stationary wave function in 2D is a superposition of an incident and a scattered wave [242], which are a plane wave and a circular wave, respectively [242]: ψ(ρ) ρ→∞−→ A [ eikx + √ i k f(k, ϕ) eikρ√ ρ ] . (5.2) Here, A is the normalization factor and f(k, ϕ) is the scattering amplitude in purely 2D geometry. This expression is different from the regular asymptotic form of the wave function in 2D [243] ψ(ρ) ρ→∞−→ A [ eikx + f(k, ϕ) eikρ√ ρ ] (5.3) by the factor of (i/k) 1 2 . This factor ensures that the scattering amplitude has the correct analytic properties and that the optical theorem derived from Eq. 5.2 is of the same form as in 3D [242]. The probability current density can be obtained from the wave function and its gradient (cf. Eq. 2.28). Applying Eq. 2.28 to the incident and the scattered part of the total wave function in 2D (Eq. 5.2), we obtain the incident and scattered flux 93 5.2. Close coupling theory of collisions in two dimensions density, namely jinc = |A|2 k µ , (5.4) jsc = |A|2 1 µρ |f(k, ϕ)|2. (5.5) The differential cross section in 2D is defined as dσ(k, ϕ)dϕ = jscρdϕ jinc = 1 k |f(k, ϕ)|2dϕ, (5.6) which has the dimension of length. We thus obtain the relationship between the differential cross section and the scattering amplitude dσ(k, ϕ) = 1 k |f(k, ϕ)|2. (5.7) The scattering amplitude f(k, ϕ) can be expanded in terms of the eigenfunctions of the operator l̂z(ϕ), f(k, ϕ) = ∞∑ ml=−∞ fml(k)e imlϕ, (5.8) where the rotational motion is restricted to the (x, y) plane and ml is the analogue of the angular momentum in 3D. Substituting Eq. 5.8 into Eq. 5.7 and integrating over ϕ, we can obtain the relationship between the integral scattering cross section and the scattering amplitude, that is σ(k) = 1 k ∫ 2pi 0 ∞∑ ml=−∞ fml(k)e imlϕ ∞∑ m′l=−∞ f∗m′l(k)e −im′lφdϕ = 1 k ∞∑ ml=−∞ ∞∑ m′l=−∞ fml(k)f ∗ m′l (k) ∫ 2pi 0 ei(ml−m ′ l)φdϕ = 1 k ∞∑ ml=−∞ ∞∑ m′l=−∞ fml(k)f ∗ m′l (k)2piδmlm′l = 2pi k ∞∑ ml=−∞ |fml(k)|2. (5.9) 94 5.2. Close coupling theory of collisions in two dimensions 5.2.2 Elastic collisions in two dimensions The wave function ψ(ρ, ϕ) in Eq. 5.1 can be expanded in terms of the eigenfunctions of the operator l̂z ψ(ρ, ϕ) = ∞∑ ml=−∞ Fml(k, ρ)e imlϕ. (5.10) The components of the rotational motion with different ml can be considered as partial waves in 2D. At sufficiently large inter-particle distance ρ, we obtain the ra- dial part of the time-independent free-particle Schrödinger equation whose solution is a combination of the regular Bessel (Jml) and Neumann (Nml) functions, Fml(k, ρ) = BmlJml(kρ) + CmlNml(kρ). (5.11) In the asymptotic region, Fml(k, ρ) becomes: Fml(k, ρ) ρ→∞−→ Bml √ 2 pikρ cos ( kρ− mlpi 2 − pi 4 ) +Cml √ 2 pikρ sin ( kρ− mlpi 2 − pi 4 ) . (5.12) We define δml as the 2D phase shift of the mlth partial wave, that is δml = arctan(−Cml/Bml), (5.13) which leads to the definition of the coefficients Bml = Aml cos δml , (5.14) Cml = −Aml sin δml . (5.15) The radial function is then given by Fml(k, ρ) ρ→∞−→ Aml √ 2 pikρ cos ( kρ− mlpi 2 − pi 4 + δml ) . (5.16) In the asymptotic region, the total wave function can also be written in terms of the plane and circular waves as in Eq. 5.2. The plane wave eikx can be represented by the Jacobi-Anger expansion [244] as eikx = ∞∑ ml=−∞ imljml(kρ)e imlϕ. (5.17) 95 5.2. Close coupling theory of collisions in two dimensions At ρ→∞ it becomes eikx ρ→∞−→ ∞∑ ml=−∞ imleimlϕ √ 2 pikρ cos ( kρ− mlpi 2 − pi 4 ) . (5.18) The substitution of Eqs. 5.18 and 5.8 into Eq. 5.2 leads to the asymptotic form of the total wave function, namely ψ(ρ, ϕ) ρ→∞−→ ∞∑ ml=−∞ [ iml √ 2 pikρ cos ( kρ− mlpi 2 − pi 4 ) + √ i k fml(k) eikρ√ ρ ] eimlϕ.(5.19) From there we obtain the radial part of the wave function as Fml(k, ρ) = i ml √ 2 pikρ cos ( kρ− mlpi 2 − pi 4 ) + √ i k fml(k) eikρ√ ρ = iml √ 2 pikρ [ cos ( kρ− mlpi 2 − pi 4 ) + √ pi 2 i 1 2 −mlfml(k)e ikρ ] = iml √ 2 pikρ { cos ( kρ− mlpi 2 − pi 4 ) + √ pi 2 × [ cos ( 1 2 −ml ) pi 2 + i sin ( 1 2 −ml ) pi 2 ] fml(k) (cos kρ+ i sin kρ) } = iml √ 2 pikρ { cos ( kρ− mlpi 2 − pi 4 ) + √ pi 2 fml(k) [ − sin ( kρ− mlpi 2 − pi 4 ) + i cos ( kρ− mlpi 2 − pi 4 )]} = iml √ 2 pikρ { cos ( kρ− mlpi 2 − pi 4 )[ 1 + i √ pi 2 fml(k) ] − √ pi 2 fml(k) sin ( kρ− mlpi 2 − pi 4 )} . (5.20) From Eq. 5.16, we have Fml(k, ρ) ρ→∞−→ Aml √ 2 pikρ { cos ( kρ− mlpi 2 − pi 4 ) cos δml − sin ( kρ− mlpi 2 − pi 4 ) sin δml } . (5.21) 96 5.2. Close coupling theory of collisions in two dimensions Comparison of Eqs. 5.21 and 5.20 leads to iml [ 1 + i √ pi 2 fml(k) ] = Aml cos δml , (5.22) iml √ pi 2 fml(k) = Aml sin δml . (5.23) We thus obtain the coefficient Aml and the expansion coefficient fml(k) as Aml = i mleiδml , (5.24) fml(k) = √ 2 pi eiδml sin δml . (5.25) The asymptotic form of the scattering wave function is therefore given by the ex- pression (cf. Eq. 5.16) ψ(ρ, ϕ) = ∞∑ ml=−∞ imleiδml √ 2 pikρ cos ( kρ− mlpi 2 − pi 4 + δml ) . (5.26) Comparing the above equation with the asymptotic form of the plane wave expansion (cf. Eq. 5.18), we find that it is also valid in 2D that the elastic scattering merely modifies the phase of each partial wave. From Eqs. 5.9 and 5.25, we obtain the total scattering cross sections in 2D as σ(k) = ∞∑ ml=−∞ 2pi k |fml(k)|2 = 4 k ∞∑ ml=−∞ sin2 δml . (5.27) It can also be expressed in terms of the single channel analogues of the S, T , or K matrices. By analogy with 3D scattering theory (Chapter 2), we can define the S, T , and K matrices in 2D as: Tml = e iδml sin δml , (5.28) Sml = e 2iδml , (5.29) and Kml = tan δml = −Cml/Bml . (5.30) The usual relation between the K, S, and T matrices will therefore also hold in 2D 97 5.2. Close coupling theory of collisions in two dimensions so that Sml = 1 + 2iTml = (1 + iKml)(1− iKml)−1. (5.31) The expression for the integral scattering length thus becomes σ = 4 k ∞∑ ml=−∞ |Tml |2 = 1 k ∞∑ ml=−∞ |Sml − 1|2. (5.32) Now the calculation of the total scattering cross sections reduces to a problem of determining the phase shifts in 2D collisions. 5.2.3 Numerical calculation of phase shift in 2D geometry From Eq. 5.1 we obtain the radial part of the Schrödinger equation in 2D as 1 ρ d dρ ( ρ d dρ ) Fml(ρ) + ( k2 − Û(ρ)− m 2 l ρ2 ) Fml(ρ) = 0, (5.33) where k2 = 2µE and Û(ρ) = 2µV̂ (ρ). Noting that 1 ρ d dρ ( ρ d dρ ) 1 ρ 1 2 = ρ− 1 2 d2 dρ2 + 1 4 ρ− 5 2 , (5.34) we can write Eq. 5.33 as [ d2 dρ2 +W (ρ) ] φml(ρ) = 0, (5.35) where W (ρ) = k2 − 2µV̂eff(ρ,ml), (5.36) V̂eff(ρ,ml) = V̂ (ρ) + 4ml2 − 1 8µρ2 , (5.37) (5.38) and φml(ρ) = (kρ) 1 2Fml(ρ). (5.39) 98 5.2. Close coupling theory of collisions in two dimensions The logarithmic derivative y of φml is defined as yml = φ′ml φml , (5.40) which leads to a new form of Eq. 5.35 in terms of yml as y′ml(ρ) +W (ρ) + y 2 ml (ρ) = 0. (5.41) The phase shift can thus be calculated by integrating Eq. 5.41 numerically with the boundary conditions: φml(r → 0) = 0 and φml(r → ∞) = {the asymptotic form of the transformed wave function φml}. Here, the asymptotic form of φml is the combination of the regular Bessel Jml(ρ) and Neumann Nml(ρ) functions for a given ml, that is φml(ρ) = (kρ) 1 2 [BmlJml(kρ) + CmlNml(kρ)] = (kρ) 1 2Bml [Jml(kρ) + Cml/BmlNml(kρ)] = (kρ) 1 2Bml [Jml(kρ)−KmlNml(kρ)] = Bml [ Ĵml(kρ)−KmlN̂ml(kρ) ] , (5.42) where Ĵml(kρ) = (kρ) 1 2Jml(kρ), (5.43) N̂ml(kρ) = (kρ) 1 2Nml(kρ). (5.44) Following the same derivation as described in Chapter 2, we obtain the expression for the K matrix in terms of the log-derivative matrix in 2D, that is Kml = (ymlN̂ml − N̂ ′ml)−1(yml Ĵml − Ĵ ′ml). (5.45) 5.2.4 Inelastic collisions in two dimensions This section generalizes the results of Sections 5.21–5.23 to multi-channel inelastic collisions. Our formulation is based on the fully uncoupled-space-fixed representa- tion [199]. We consider collisions between ultracold atoms in the presence of an external field. However, our theory is general and can be applied to describe colli- sions between molecules. The relative motion of two atoms in 2D is described by 99 5.2. Close coupling theory of collisions in two dimensions the Hamiltonian Ĥ = − 1 2µρ ∂ ∂ρ ρ ∂ ∂ρ + l̂2z(ϕ) 2µρ2 + Ĥas + V̂ , (5.46) where Ĥas is the asymptotic Hamiltonian describing the separated particles in the presence of an external field. We expand the total wave function of the collision complex in a basis of product wave functions as ψ(ρ, ϕ) = ρ− 1 2 ∑ β ∑ ml Fβml(ρ) eimlϕ√ 2pi φβ, (5.47) where φβ represent the eigenfunctions of Ĥas. The diagonalization of Ĥas yields the asymptotic energies β of the interacting particles and the corresponding wave func- tions φβ. In 2D collisions, the quantum number of the orbital angular momentum l is not defined. The collision channels are therefore specified by the quantum numbers β and ml, giving the internal states of the colliding particles and the partial wave, respectively. The substitution of this expansion into the Schrödinger equation with the Hamiltonian of Eq. 5.46 results in a system of the coupled differential equations[ ∂2 ∂ρ2 + k2β + 1 4ρ2 − m 2 l ρ2 ] Fβml(ρ) = 2µ ∑ β′ Fβ′ml(ρ)〈φβ|V̂ |φβ′〉, (5.48) where kβ = √ 2µ(E − β) is the wave number of the incoming collision channel. When the separation between the colliding particles is sufficiently large, the interaction potential between them can be ignored, i.e. V = 0, which yields the set of uncoupled differential equations[ ∂2 ∂ρ2 + k2β + 1 4ρ2 − m 2 l ρ2 ] Fβml(ρ) = 0. (5.49) Each equation has a boundary condition for a particular collision channel (β,ml), namely Fβml(ρ→ 0) −→ 0, (5.50) Fβml(ρ→∞) −→ (kβρ) 1 2 [aβmljml(kβρ) + bβmlnml(kβρ)] . (5.51) The wave function for the channel (β,ml) at sufficiently large ρ therefore has the 100 5.2. Close coupling theory of collisions in two dimensions form ψβml(ρ, ϕ) −→ ν − 1 2 β (kβρ) 1 2 [aβmljml(kβρ) + bβmlnml(kβρ)] eimlϕ√ 2pi φβ, (5.52) where ν − 1 2 β is a normalization coefficient obtained by normalizing to unity the in- coming flux of the atoms in the state β. Using the asymptotic forms of jml(kβρ) and nml(kβρ), we obtain the wave func- tion for the channel (β,ml) in the asymptotic region as ψβml(ρ, ϕ) ρ→∞−→ (kβρ) 1 2 ν − 1 2 β √ 2 pikβρ × [ aβml cos(kβρ− ml 2 − pi 4 ) + bβml sin(kβρ− ml 2 − pi 4 ) ] eimlϕ√ 2pi φβ. (5.53) It can be re-written in terms of exponential functions as ψβml(ρ, ϕ) ρ→∞−→ ν− 1 2 β [ Aβmle−i(kβρ− ml 2 pi−pi 4 ) − Bβmlei(kβρ− ml 2 pi−pi 4 ) ] eimlϕ√ 2pi φβ, (5.54) where Aβml = √ 2 pi (−bβml + iaβml)/2i, (5.55) Bβml = √ 2 pi (bβml + iaβml)/2i. The S-matrix for multi-channel scattering in 2D is defined as Bβml = ∑ β′ ∑ m′l Sβml←β′m′lAβ′m′l , (5.56) where Sβml←β′m′l can be considered as the amplitude of the probability for the atoms or molecules to undergo a transition from an incident collision channel (β′,m′l) to an outgoing channel (β,ml). The sum is over all the possible incoming channels. The total asymptotic wave function for a particular channel (β,ml) is thus given by ψβml(ρ, ϕ) ρ→∞−→ ρ− 12 ν− 1 2 β ∑ β′ ∑ m′l Aβ′m′l × [ e−i(kβρ− ml 2 pi−pi 4 )δββ′δmlm′l − Sβml←β′m′le i(kβρ−ml2 pi−pi4 ) ] eim′lϕ√ 2pi φβ′ . (5.57) 101 5.2. Close coupling theory of collisions in two dimensions According to the Jacobi-Anger expansion [244], the plane wave can be written as (cf. Eq. 5.17) eikβx = ∑ ml imleimlϕ 1 2 √ 2 pikβρ [ e−i(kβρ− ml 2 pi−pi 4 ) − ei(kβρ−ml2 pi−pi4 ) ] . (5.58) The incoming wave function therefore has the form ν − 1 2 β e ikβxφβ =ν − 1 2 β ∑ ml imleimlϕ 1 2 √ 2 pikβρ × [ e−i(kβρ− ml 2 pi−pi 4 ) − ei(kβρ−ml2 pi−pi4 ) ] φβ. (5.59) In the collision systems studied in this Thesis, particles are initially prepared in a particular internal spin state β, so the amplitude of the incoming flux in channels other than β is zero. Comparing the coefficients in front of the term e−i(kβρ− ml 2 pi−pi 4 ) in Eqs. 5.57 and 5.59, we obtain Aβml = iml √ 1 kβ (in channel α) = 0 (in other channels). (5.60) Bβml is then given by Bβml = ∑ m′l Sβml←βm′li m′l √ 1 kβ′ . (5.61) In 2D, the scattered wave function is also obtained using Eq. 2.107. Here, the outgoing part of the total wave function is the sum over all energetically accessible collision channels, that is ψoutgoing = ∑ β′ (ψβ′)outgoing = ∑ β′ ∑ m′l (ψβ′m′l)outgoing = − ∑ β′ ∑ m′l ν − 1 2 β′ Bβ′m′le i(kβ′ρ− m′l 2 pi−pi 4 ) e im′lϕ√ 2pi φβ′ . (5.62) 102 5.2. Close coupling theory of collisions in two dimensions Combining Eqs. 5.61 and 5.62, we obtain ψoutgoing ψoutgoing = − ∑ β′ ∑ m′l ν − 1 2 β′ eim ′ lϕ√ 2pi φβ′e i(kβ′ρ− m′l 2 pi−pi 4 ) [∑ ml Sβ′m′l←βmli ml √ 1 kβ ] . (5.63) The outgoing part of the incident wave function is given by ψincoutgoing = − ∑ ml ν − 1 2 β i ml √ 1 2pikβρ ei(kβρ− ml 2 pi−pi 4 )eimlϕφβ = − ∑ β′ ∑ m′l ∑ ml ν − 1 2 β′ i ml √ 1 2pikβρ ei(kβ′ρ− m′l 2 pi−pi 4 )eim ′ lϕφβ′δmlm′lδββ ′ . (5.64) The scattered wave function is then obtained as ψsc(ρ, ϕ) = ψoutgoing − ψincoutgoing = ∑ β′ ∑ m′l ∑ ml ν − 1 2 β′ i mleim ′ lϕφβ′ √ 1 2pikβρ ×ei(kβ′ρ− m′l 2 pi−pi 4 )(δmlm′lδββ′ − Sβ′m′l←βml). (5.65) Following the single-channel analysis of Adhikari [242], we express the scattered wave function in terms of the scattering amplitude ψsc(ρ, ϕ) = ∑ β′ ν − 1 2 β′ √ i kβ′ fββ′ eikβ′ρ√ ρ φβ′ , (5.66) which gives the expression for the scattering amplitude fβ′←β = ∑ ml ∑ m′l √ 1 2pi i(ml− 1 2 )e(− m′l 2 pi−pi 4 )eim ′ lϕ(δmlm′lδββ′ − Sβ′m′l←βml) = ∑ ml ∑ m′l √ 1 2pi i(ml− 1 2 )e(− m′l 2 pi−pi 4 )eim ′ lϕTβ′m′l←βml , (5.67) where Tβ′m′l←βml = δmlm′lδββ′ − Sβ′m′l←βml . The differential cross section for the β → β′ transition is then obtained using 103 5.2. Close coupling theory of collisions in two dimensions Eq. 5.7, that is dσβ′←βdϕ = 1 kβ |fβ′←β|2dϕ = 1 kβ ∑ ml ∑ m′l 1 2pi |δmlm′lδββ′ − Sβ′m′l←βml | 2dϕ. (5.68) Integrating this equation over ϕ, we find the integral cross section σβ′←β(k) = 1 kβ ∑ ml ∑ m′l |δmlm′lδββ′ − Sβ′m′l←βml | 2, (5.69) where the S-matrix is constructed from the numerical solutions of the coupled dif- ferential Eqs. 5.48. In this work, we use the log-derivative method [203] to solve Eqs. 5.48, as described in Section 5.2.3. All calculations in this Thesis are for collisions of distinct particles. However, for collisions between identical particles in the same internal state, one needs to consider the symmetrization of the total wave function with respect to particle interchange. For example, the wave function for bosons must be symmetric and the wave function for fermions must be anti-symmetric under the interchange of two particles. For scattering in a centrally symmetric potential, the spatial symmetries of the wave function are completely determined by the angular momentum quantum numbers, i.e., l and ml in 3D and ml in 2D. The spherical harmonics Ylml may be written as the product of two functions, Θlml(θ) and Φml(φ), depending on the polar angle θ and the azimuthal angle φ, respectively Ylml(θ, φ) = Θlml(θ)Φml(φ). (5.70) Here, Φml(φ) = (2pi) − 1 2 eimlφ (5.71) and Θlml(θ) = (−1)ml 2ll! [ 2l + 1 2 (l −ml)! (l −ml)! ] 1 2 (sin θ)ml × [ d d(cos θ) ]l+ml (cos2 θ − 1)l (5.72) When two particles are interchanged, θ → pi − θ and φ → pi + θ, which gives the factor of (−1)l and (−1)ml for the total wave function in 3D and 2D, respectively. 104 5.2. Close coupling theory of collisions in two dimensions 5.2.5 Magnetic dipole-dipole interaction in 2D The theory presented here can be generally applied to both atomic and molecular multichannel collisions in 2D. We illustrate the theory by a calculation of elastic and inelastic cross sections for Li–Cs collisions in 2D in the presence of a magnetic field. In order to verify threshold collision laws derived by Sadeghpour and cowork- ers [193], we calculate the energy dependence of scattering cross sections for elastic and inelastic collisions in the limit of low energies in 2D. In addition, we investigate the effects of the long-range 1/r3 interaction on the threshold laws for transitions accompanied with changes of orbital angular momentum in 2D. Ĥas includes the interactions of the atoms with an external magnetic field and the hyperfine interac- tions, denoted by V̂B and V̂hf , respectively. The detailed form of the operators V̂ , V̂B, V̂hf , and the corresponding matrix elements can be found in Chapter 3. The magnetic dipole-dipole interaction between the alkali metal atoms was neglected in Chapters 3 and 4. In the present Chapter, however, we would like to investigate specifically the transitions induced by the magnetic dipole-dipole interaction. In 3D, the magnetic dipole interaction between Li and Cs can be written as V̂ 3Ddip (r) = − √ 4pi 5 √ 6 α2fs r3 ∑ q (−1)qY2−q(r̂)× [SLi ⊗ SCs](2)q , (5.73) where αfs is the fine structure constant [245], Y2−q(r̂) is the spherical harmonic, and [SLi ⊗ SCs](2)q is the tensorial product of spin angular momenta of Li and Cs. Since the magnetic dipole-dipole interaction is diagonal in nuclear spins, the matrix elements of the V̂ 3Ddip operator are given by [199] 〈SLiMSLi |〈SCsMSCs |〈lml|V̂ 3Ddip |l′m′l〉|SCsM ′SCs〉|SLiM ′SLi〉 = − √ 6 α2fs r3 [ (2l + 1)(2l′ + 1) ] 1 2 (−1)−ml ( l 2 l′ 0 0 0 ) × 2∑ q=−2 ( l 2 l′ −ml −q m′l ) × 〈SLiMSLi |〈SCsMSCs | [SLi ⊗ SCs](2)q |SCsM ′SCs〉|SLiM ′SLi〉 (5.74) where the components of [SLi ⊗ SCs](2)q have the conventional form [221] [SLi ⊗ SCs](2)q=0 = 1√ 6 [ 2SLizSCsz − 1 2 (SLi+SCs− + SLi−SCs+) ] , (5.75) 105 5.2. Close coupling theory of collisions in two dimensions [SLi ⊗ SCs](2)q=±1 = ∓ 1 2 [ SLizSCs± + SLi±SCsz ] , (5.76) [SLi ⊗ SCs](2)q=±2 = 1 2 SLi±SCs± . (5.77) In 2D, the operator describing the magnetic dipole-dipole interaction has the same form as in 3D (cf. Eq. 5.73). However, the spherical harmonics are functions of the azimuthal angle ϕ only Y 2D2−q(ρ̂) ≡ Y2−q(θ, ϕ) = Y2−q(θ = pi 2 , ϕ). (5.78) This leads to a simpler form for V̂ 2Ddip (ρ), namely V̂ 2Ddip (ρ) =− √ 4pi 5 √ 6 α2fs ρ3 { Y22( pi 2 , ϕ) [SLi ⊗ SCs](2)−2 + Y20( pi 2 , ϕ) [SLi ⊗ SCs](2)0 + Y2−2( pi 2 , ϕ) [SLi ⊗ SCs](2)2 } , (5.79) where the expressions for the components of the [SLi ⊗ SCs](2)q tensor are given by Eqs. 5.75, 5.76, and 5.77 . We evaluate the matrix elements of the operator describing the dipole-dipole interaction as 〈SLiMSLi |〈SCsMSCs |〈ml|V̂ 2Ddip |m′l〉|SLiM ′SLi〉|SCsM ′SCs〉 = − √ 4pi 5 √ 6 α2fs ρ3 {√ 15 32pi 〈ml|e2iϕ|m′l〉 × 〈SLiMSLi |〈SCsMSCs | 1 2 ŜLi−ŜCs− |SLiM ′SLi〉|SCsM ′SCs〉 − √ 5 16pi 〈ml|m′l〉〈SLiMSLi |〈SCsMSCs | 1√ 6 [ 2ŜLizŜCsz − 1 2 ( ŜLi+ŜCs− + ŜLi−ŜCs+ )] |SLiM ′SLi〉|SCsM ′SCs〉 + √ 15 32pi 〈ml|e−2iϕ|m′l〉〈SLiMSLi |〈SCsMSCs | 1 2 ŜLi+ŜCs+ |SLiM ′SLi〉|SCsM ′SCs〉 } . (5.80) The expressions for the matrix elements of the operators Ŝz, Ŝ+, and Ŝ− can be found in Ref. [221] 106 5.3. Numerical results 5.3 Numerical results Collision properties of ultracold atoms and molecules in 2D and 3D are very dif- ferent [180, 246]. For example, the energy dependence of cross sections for elastic and inelastic collisions in the limit of low energies is predicted to depend on the dimensionality of the system [193, 247]. Table 5.1 summarizes the threshold laws in 2D and 3D obtained by analytical derivations [8, 77, 180, 193, 247], where k is the collision wave number, and l (ml) and l′ (m′l) are the orbital angular momenta (pro- jections) before and after the collision in 3D (2D). In order to verify the validity of the threshold laws for collisions in 2D and elucidate the difference of the scattering dynamics near threshold in 2D and 3D, we carried out rigorous quantum calcula- tions in a 2D frame using the multichannel collision theory presented in Section 5.2.4. We numerically solved the coupled differential Eqs. 5.48 for a binary mixture of ultracold Li and Cs atoms in the spin states |2− 2〉7Li ⊗ |3, 2〉133Cs at a magnetic field of 100 G. The computed cross sections as functions of the collision energy for s- and p-wave collisions are presented in Figs. 5.1 and 5.2, respectively. Each figure shows the cross sections for elastic scattering (the first panel) and inelastic Zeeman relaxation (the second panel) in 2D (red circles) and 3D (blue diamonds) and the corresponding analytical fits (blue and red lines). Figure 5.1 also presents in the third panel the ratio of cross sections for inelastic collisions in 2D and 3D. The numerical results verify the analytical expressions for threshold collision laws in 2D. However, the calculated cross sections for s-wave elastic collisions in 2D (red circles in the first panel in Fig. 5.1) are found to deviate from the linear fit provided by the expression σ ∝ 1/(k ln2 k). The expression for the analytical fit used in Fig. 5.1 (red line in the first panel) is σ = pi2 k [ (ln k + ln d∗)2 + pi 2 4 ] . (5.81) Here, d∗ = 5.05 and the derivation of this expression and the way to obtain d∗ are given in Appendix B, which are based on the method described in Refs. [8] and [180]. The accurate representation of the energy dependence of s-wave elastic cross sections thus requires the addition of constant terms in the denominator of the expression 1/(k ln2 k), even at extremely low collision energies. Note that in the denominator of Eq. 5.81, k is an argument of a logarithm, which means that ln k will have a significant magnitude even at extremely small k. Therefore one should be cautious not to omit the significant constant terms in the derivation involving the logarithmic term. For the analysis of future experimental data on s-wave collisions 107 5.3. Numerical results Table 5.1: The energy dependence of the elastic and inelastic cross sections in 2D and 3D. k is the collision wave number, and l (ml) and l′ (m′l) are the orbital angular momenta (projections) before and after the collision in 3D (2D). Elastic Collisions 3D 2D s-wave σ = constant σ ∝ 1 k ln2 k s-wave to non-s-wave σ ∝ k2l′ σ ∝ k2|ml|−1 1 ln2 k non-s-wave to non-s-wave σ ∝ k2l+2l′ σ ∝ k2|ml|+2|m′l|−1 Inelastic Collisions 3D 2D s-wave relaxation σ ∝ 1k σ ∝ 1k ln2 k non-s-wave relaxation σ ∝ k2l−1 σ ∝ k2|ml|−1 in 2D, one should use Eq. 5.81. Wigner showed that the energy dependence of the scattering cross section near threshold in 3D is determined by the values of the orbital angular momentum of the collision complex before and after collision [77, 193]. In particular, he demonstrated that the elastic scattering cross section varies near threshold as k2l+2l ′ , whereas the cross sections for inelastic or reactive scattering vary near threshold as k2l−1 (cf. Table 5.1). Consequently, inelastic collisions in 3D are enhanced by a factor of 1/k(2l ′+1) as the collision energy approaches zero. This trend, however, changes in ultracold collisions in 2D. The cross sections for s-wave elastic and inelastic collisions in 2D have the same threshold energy dependence 1/(k ln2 k) [8, 180, 193, 247]. The numerical results shown in the third panel of Fig. 5.1 demonstrate that the cross sections for inelastic collisions in 2D are about 4 ∼ 5 orders of magnitudes smaller than those in 3D. In addition, we present in Fig. 5.3 the ratio of the inelastic and elastic cross sections in 2D (red circles) and 3D (blue diamonds) for s- (upper panel) and p-wave (lower panel) collisions. For s-wave collisions, the ratio of the cross sections for inelastic and elastic collisions in 3D is dramatically enhanced as the collision energy approaches zero, whereas the ratio of the cross sections for inelastic and elastic collisions in 2D is always less than 1. It indicates that s- wave inelastic collisions may be suppressed with respect to elastic collisions in 2D. However, inelastic collisions for p-wave scattering in 2D are greatly enhanced with respect to elastic collisions, which is the same as in 3D. At ultracold temperatures, the threshold behavior of atoms or molecules is de- termined by the inter-particle interactions, particularly, the long-range interaction 108 5.3. Numerical results 104 105 106 100 103 106 109                                                                                    Cr os s S ec tio n (a. u.) 10-14 10-12 10-10 10-8 Collision energy (cm-1) 10-5 10-4 σ 2D  /σ 3D Figure 5.1: The threshold energy dependence of cross sections for elastic (upper panel) and inelastic (middle panel) s-wave collisions of Li and Cs atoms in 3D (diamonds) and 2D (circles). Symbols – numerical calculations; lines – analytical fits based on the analysis of the threshold laws (cf. Tab.5.1 and Eq. 5.81). The lower panel shows the ratio of cross sections for inelastic collisions in 2D and 3D. The initial states are |2,−2〉7Li ⊗ |3, 2〉133Cs. The calculations were carried out in a magnetic field of 100 G. 109 5.3. Numerical results 10-20 10-15 10-10 10-5 10-14 10-12 10-10 10-8 Collision Energy (cm-1) 10-4 10-2 100                                                                                    Cr os s S ec tio n (a. u.) Figure 5.2: The threshold energy dependence of cross sections for elastic (upper panel) and inelastic (lower panel) p-wave collisions of Li and Cs atoms in 3D (di- amonds) and 2D (circles). Symbols – numerical calculations; lines – analytical fits based on the analysis of the threshold laws (cf. Tab.5.1). The initial states are |2,−2〉7Li ⊗ |3, 2〉133Cs. The calculations were carried out in a magnetic field of 100 G. 110 5.3. Numerical results 10-2 100 102 104 10-14 10-12 10-10 10-8 Collision Energy (cm-1) 100 104 108 1012 1016σ in el /σ el S-wave collision P-wave collision Figure 5.3: The ratio of inelastic and elastic cross sections in 2D (red circles) and 3D (blue diamonds) for s- (upper panel) and p-wave (lower panel) collisions. The initial states are |2,−2〉7Li ⊗ |3, 2〉133Cs. The calculations were carried out in a magnetic field of 100 G. 111 5.3. Numerical results forces. The energy dependence of collision cross sections may be altered by the long-range potentials, such as the magnetic dipole-dipole interaction which varies as r−3. Mies and Raoult [248] have shown that Wigner’s threshold laws for tran- sitions changing the orbital angular momentum of the colliding particles in 3D are modified by the dipole-dipole 1/r3 interaction. In our Letter [247], Sergey Alyaby- shev repeated their analysis for collisions in 2D and generalized it to the long-range interaction potential with a form 1/ρα. He found that the cross sections for elas- tic scattering changing the value of m (such as the s-wave to d-wave transitions) or elastic scattering in states with non-zero partial waves mediated by the long- range interaction potential 1/ρα are σsml;sm′l ∝ k2α−5s and independent of m and m′, providing ml +m′l − α + 2 > 0. We verified by numerical calculations that the ml = 0→ m′l = 2 and ml = 2→ m′l = 2 cross sections in 2D scattering induced by the dipole-dipole interaction 1/ρ3 vanish as σsml;sm′l ∝ ks. The calculations are dif- ficult to converge and the modification of the threshold law can only be observed if the maximum propagation distance of the differential equations 5.48 is large enough, say, 100,000 bohr. However, we found (to our surprise) that using the asymptotic form of the Bessel and Neumann function can converge the calculations within a reasonably small propagation range. Figures 5.4 and 5.5 show the modification of the threshold law of the cross sections for s-to-d transitions induced by the magnetic dipole-dipole 1/r3 (1/ρ3 for 2D) interactions in collisions of Li and Cs atoms in 3D and 2D, respectively. They show a gradual convergence of the calculation to the line (circles) computed using the asymptotic form of the Bessel and Neumann func- tions. In 3D, the cross sections for s-to-d transitions mediated by the dipole-dipole interaction become independent of the collision energy. Table 5.1 shows that the energy dependences of cross sections for elastic colli- sions for s-wave and s-wave to non-s-wave transitions in 2D are σs0,s0 ∝ 1/(ks ln2 ks) and σs0,sm′l ∝ k 2|ml|−1 s / ln2 ks, respectively. This indicates that s-wave collisions of ultracold molecules in a 2D gas accompanied by angular momentum change are suppressed by k2|ml|s , the same factor as in 3D. Typical temperatures of ultracold gases are about 10−7 K, so as shown in Figs. 5.4 and 5.5, collisions involving tran- sitions from s-wave to non-zero partial waves may be several orders of magnitude less probable than collisions conserving the projection of orbital angular momentum (cf. Fig. 5.1). These results were obtained for the maximally stretched spin states at zero magnetic field. The spin depolarization in these states occurs only if the change of the spin states of the collision complex is accompanied with the change of the projection of orbital angular momentum. This follows from the conservation of the total angular momentum projection on the quantization axis. 112 5.3. Numerical results 10-14 10-12 10-10 10-8 10-6 10-4 Collision Energy (cm -1) 10-18 10-12 10-6 100 Cr os s S ec tio ns  (a .u. ) R end = 1,000 a.u. R end = 5,000 a.u. R end = 10,000 a.u. R end = 50,000 a.u. R end = 100,000 a.u. Asymptotic form Figure 5.4: The modification of the threshold energy dependence of the cross sec- tions for s-to-d transitions induced by the magnetic dipole-dipole 1/r3 interaction in collisions of Li and Cs atoms in 3D. The graph shows a gradual convergence of the calculations to the line (circles) computed using the asymptotic form of the Bessel and Neumann functions. The s-to-d transitions are calculated at zero magnetic field for the maximally stretched state |2, 2〉7Li⊗|4, 4〉133Cs. Rend specifies the propagation distance of the coupled differential equation (cf. Eq. 5.48). Consider, for example, s-wave collisions of spin-1/2 2Σ molecules in the rota- tionally ground state. Before the collision, the projection of the total spin of the combined system on the z axis is 1 and after the collision it is -1, so the projection of orbital angular momentum must change from 0 to 2. Therefore, as shown in Figs. 5.4 and 5.5, such a process at zero magnetic field is strongly suppressed due to the centrifugal repulsion at large inter-particle distance in states with non-zero orbital angular momentum. It should be clarified that the suppression of the spin relaxation process only occurs in maximum spin states. In other atomic states, the changes of the spin quantum number of two particles can cancel each other out, i.e., spin relaxation can occur without changing the projection of orbital angular momentum. This spin depolarization process can be manipulated by tuning the strength 113 5.3. Numerical results 10-14 10-12 10-10 10-8 10-6 10-4 Collision Energy (cm-1) 10-20 10-15 10-10 10-5 Cr os s S ec tio n (a. u.) R end = 1,000 a.u. R end = 5,000 a.u. R end = 10,000 a.u. R end = 50,000 a.u. R end = 100,000 a.u. Asymptotic form Figure 5.5: The modification of the threshold energy dependence of cross sections for s-to-d transitions induced by the magnetic dipole-dipole 1/ρ3 interaction in col- lisions of Li and Cs atoms in 2D. The graph shows a gradual convergence of the calculations to the line (circles) computed using the asymptotic form of the Bessel and Neumann functions. The s-to-d transitions are calculated at zero magnetic field for the maximally stretched state |2, 2〉7Li⊗|4, 4〉133Cs. Rend specifies the propagation distance of the coupled differential equation (cf. Eq. 5.48). of the external magnetic fields. As shown in the upper panel of Fig. 5.6, if the magnetic field is weak, the energy splitting is less than the centrifugal energy barrier in the outgoing collision channel. In this case collisional spin relaxation of ultracold molecules initially in a maximum spin state will be strongly suppressed. Volpi and Bohn were the first to observe this suppression [100]. They found that the limiting zero-field value determines the absolute magnitude of the cross section in weak fields [100]. In the presence of large magnetic fields, however, the energy splitting between the outgoing and the incoming channels will be large so the kinetic energy of the scattering particles will be significant enough to overcome the centrifugal barrier as shown in the lower panel of Fig. 5.6. Consequently, the spin depolarization process may be enhanced and the energy dependence of the spin relaxation cross sections becomes σs0,s′l′ ∝ 1/ks for 3D and σs0,s′m′l ∝ 1/ks ln 2 ks for 2D, if the magnetic field 114 5.3. Numerical results Low field High field Figure 5.6: Collisional spin relaxation of ultracold atoms and molecules initially in a maximum spin state in 3D in the presence of a low (upper panel) and a high (lower panel) magnetic field. Solid curve – s-wave collision channel; dashed curve – collision channels with nonzero orbital angular momentum. Adapted with permission from R. V. Krems, Int. Rev. Phys. Chem. 24, 99 (2005). 115 5.4. Conclusions Figure 5.7: Collisional spin relaxation of ultracold atoms and molecules initially in a maximum spin state in 2D in the presence of a magnetic field. Left panel – the magnetic field is perpendicular to the plane of confinement; right panel – the magnetic field axis is directed at a nonzero angle with respect to the confinement plane normal. is oriented along the axis perpendicular to the plane of confinement, as shown in the left panel in Fig. 5.7. However, the symmetry of the problem changes dramatically if the magnetic field axis is rotated with respect to the confinement plane normal, as shown in the right panel in Fig. 5.7. The interaction potential matrix that drives spin-depolarization transitions remains diagonal in the total angular momentum projection. But the electron spin is no longer projected on the quantization axis. The Zeeman states become superpositions of different projection states in the coordinate system defined by the confinement axis. Therefore transitions from the maximally stretched Zeeman state no longer have to change the orbital angular momentum. We conclude that Zeeman transitions in collisions of molecules or atoms in states with maximal spin projections on the magnetic field axis must be suppressed if the magnetic field axis is perpendicular to the plane of confinement and enhanced if the magnetic field axis is directed at a nonzero angle with respect to the confinement plane normal. 5.4 Conclusions In this Chapter, we have presented a multi-channel collision theory to describe ul- tracold inelastic collisions in 2D geometry. Our numerical calculations verify the predicted threshold energy dependence of cross sections for elastic and inelastic col- 116 5.4. Conclusions lisions in the limit of low energies in 2D. We found that the accurate representation of the s-wave cross sections requires the addition of constant terms in the denomina- tor even at extremely low collision energies. For the analysis of future experimental data, one should use Eq. 5.81. Our results show that ultracold elastic collisions accompanied with changes of angular momentum m in 2D will be suppressed by the same factor as in 3D. This has important consequences for angular momentum transfer in 2D collisions of ultracold atoms or molecules. For example, this indicates that Zeeman or Stark transitions in collisions of atoms or molecules in maximum spin states will be suppressed at low external fields as in 3D collisions. In 3D, the cross section for s-wave elastic collisions is independent of the col- lision energy and the s-wave cross section for inelastic energy transfer is inversely proportional to the collision velocity. Our work shows that the cross sections for inelastic s-wave collisions have the same energy dependence as elastic s-wave col- lisions in 2D. This indicates that ultracold chemical reactions and s-wave inelastic collisions may be suppressed in ultracold collisions in 2D. For inelastic collisions in states of nonzero partial waves, however, this suppression is absent. We also found that the ratio of cross sections for inelastic and elastic collisions in 2D is always less than one. Elastic collisions ensure that atoms and molecules remain in thermal equilibrium during a cooling process whereas inelastic collisions normally lead to chemical reactions and trap loss. Our results suggest that ultracold atoms and molecules may be more stable in 2D than in 3D. This fundamental result, however, does not provide a quantitative measure of cross sections in ultracold gases under laser confinement. Atoms and molecules trapped in a 1D optical lattice are allowed to oscillate harmonically in the direction orthogonal to the plane of 2D motion, leading to the formation of a quasi-2D gas. Laser confinement therefore does not produce a purely 2D geometry. In the next Chapter, we will develop a theory to describe quantitatively the collision dynamics in quasi-2D geometry and explore the effects of the geometry change by varying the confining laser parameters on collision properties of ultracold atoms and molecules. 117 Chapter 6 Inelastic collisions in a quasi-2D trapped gas4 In this Chapter, we present a formalism for rigorous calculations of cross sections for inelastic and reactive collisions of ultracold atoms and molecules confined by laser fields in quasi-2D geometry. Our results show that the elastic-to-inelastic ratios of collision cross sections are enhanced in the presence of laser confinement and that the threshold energy dependence of the collision cross sections can be tuned by vary- ing the confinement strength and external magnetic fields. We elucidate the general features of inelastic collisions and chemical reactions in ultracold atomic and molec- ular gases in quasi-2D. Our results suggest that applying laser confinement in one dimension may stabilize ultracold systems with large scattering lengths, which may open up interesting possibilities for the research of ultracold controlled chemistry. 6.1 Motivation Atomic ensembles cooled to ultracold temperatures can be confined by optical forces of counter-propagating laser beams to form a periodic lattice structure. Optical lat- tices can be used to produce low-dimensional quantum gases by confining the motion of ultracold particles in one or two dimensions [35]. Ultracold atoms can be combined to form ultracold molecules. Molecules confined in low dimensions may undergo in- elastic and chemically reactive collisions, which suggests the possibility of studying chemistry in restricted geometries. Several previous studies showed that collision dy- namics of ultracold molecules restricted to move in two dimensions is different from scattering processes in an unconfined 3D ultracold gas [180, 193, 249, 250]. The effect of the confining laser force on inelastic and reactive collisions of molecules in an optical lattice, however, remains unknown. Ultracold atoms and molecules in re- stricted geometries may be used for quantum simulations of fundamental many-body systems [38, 49, 51, 182, 237, 251] and the development of novel schemes for quan- tum computation [39, 40, 42, 43, 171, 172, 252]. An analysis of inelastic scattering 4A part of this Chapter was presented in Ref. [5] of Appendix D. 118 6.2. Ultracold quasi-2D gas in a quasi-2D trapped gas is necessary to understand the feasibility of the quantum simulation proposals [39, 40, 42, 43, 171, 172, 252]. Ultracold atoms and molecules trapped in a quasi-2D geometry can also be used as controllable model systems of excitons and exciton polaritons in microcavity semiconductors [253–255]. Studies of inelastic interactions in quasi-2D ultracold gases may thus find applications for new research in chemical physics, quantum condensed-matter physics, quantum optics of semiconductors and quantum information science. Collision dynamics of atoms and molecules at ultracold temperatures is deter- mined by Wigner’s threshold laws [77, 193], which give the energy dependence of the scattering cross sections in the limit of low energies. As demonstrated in the previous Chapter, the threshold laws change with the dimensionality of the system [193]. However, the interaction of molecules confined by a harmonic laser force in one dimension cannot be described as a collision process in two dimensions [180, 249]. Particles confined in quasi-2D geometry move freely in two dimensions and oscillate harmonically in the third dimension. At the same time, the interaction forces be- tween the colliding molecules are much stronger than the laser confinement. The reactive complex of molecules is therefore unconstrained and the reaction process occurs in 3D. An inelastic collision or chemical reaction releases a lot of energy and accelerates the collision products, which are therefore free to move in 3D as well. The effect of the confining laser force is thus only to restrict the motion of molecules before the collision. Petrov and Shlyapnikov developed a theory of elastic collisions between atoms in a quasi-2D gas based on the renormalization of the scattering wave function [180]. In this Chapter, we extend their work to develop the formalism for quantum calculations of probabilities for inelastic and chemically reactive collisions of molecules confined in quasi-2D geometry. Our work leads to an important con- clusion that the ratio of cross sections for elastic and inelastic collisions is enhanced in the presence of laser confinement. 6.2 Ultracold quasi-2D gas In a quasi-2D system, the motion of particles is confined in one direction (e.g., along the z axis) by a harmonic potential to zero-point oscillations. Particles are only allowed to move freely in a pancake shaped trap (e.g. the (x, y) plane with a certain oscillation length along z-axis). The strength of the confinement can be described by the axial extension of the wave functions of atoms and molecules l0 = √ ~/µω0, where ω0 is the frequency of the harmonic potential and µ is the reduced mass for the collision complex. As shown in Fig. 6.1, particles are confined to the ground 119 6.2. Ultracold quasi-2D gas state of a harmonic potential and the oscillation length of the confining potential is usually much larger than the characteristic radius re of inter-particle interaction potentials [180]. For example, the strongest confinement in experiments corresponds to about l0 ≈ 200 Å, while for alkali metal atoms, re ranges from 20 Å for Li2 to 100 Å for Cs2. Therefore, as demonstrated in Fig. 6.2, the collision coordinate can be divided into three regions: (i) short interparticle separations r < re, where the interaction between collision partners is not affected by the confining potential and the collision occurs in 3D; (ii) the region of r between re and the characteristic de Broglie wavelength of the particles Λ̃ε, where the wave function is proportional to the 3D s-wave scattering wave function [180]; (Λ̃ε ∼ ~/ √ µ(ε+ ~ω0/2) with ε denoting the collision energy of the particles.) (iii) the asymptotic region r > Λ̃ε, where the wave function is the product of a circular wave function and the wave function for the ground state harmonic motion in the confining potential. The theory of Petrov and Shlyapnikov [180] relates the quasi-2D scattering wave function in the region re  r  Λ̃ε to the 3D wave function by a proportionality coefficient. In this Chapter, we assume that the temperature of the confined gas is much smaller than ~ω0 and molecules prepared in the initial state are trapped in a quasi-2D geometry. !ω (27) IV. CONCLUSIONS Acknowledgments The work was supported by the Natural Sciences and Engineering Research Council (NSERC) of Canada. 16 !ω (27) l0 (28) IV. CONCLUSIONS Acknowledgments The work was supported by the Natural Sciences and Engineering Research Council (NSERC) of Canada. 16 The Collision Energies ! !ω Th Collisio Energies! !ω (27) l0 " re (28) "α′ − "α " !ω (29) VH(z) = µω 2z2/2 (30) IV. CONCLUSIONS Acknowledgments The work was supported by the Natural S i nces and Engineering Research Council (NSERC) of anada. 16 6.6. Appendix Collision Energy ! !ω 125 Figure 6.1: The schematic diagram of a quasi-2D system. Particles are confined in the ground state of a harmonic potential with the oscillation length of the confining potential much larger than the characteristic radius re of inter-particle interaction potentials. For inelastic collisions of ultracold atoms and molecules in quasi-2D geometry, the relative momentum of the collision complex for a particular channel is calculated as k2α = 2µ(ε − α), where α are the channel energies corresponding to different 120 6.3. Elastic collisions in quasi-2D geometry 5 10 15 20 -0.03 -0.02 -0.01 0 (iii) Asymptotic region (ii) The wave function is proportional to the regular 3D wave function (i) 3D collision core η re ! r ! Λ̃ε r re Λ̃ε η re ! r ! Λ̃ε ψα(r) = i √ piηϕ0(0) kαr [ e−ikαr − Sααeikαr ] φαY00(r̂) kα α φα Sαα S α Y00(r̂) η S η = 1 (1−Sαα)ω($/2!ω0)√ 2ikl0 + √ pi(1 + Sαα) η η′ √ 4pi η = η′√ 4pi Y00(r̂) α r > re r > re s α ψα = ν − 12 α r −1 [Aαe−ikαr −Bαeikαr]φαY00(r̂) z (x, y) l0 = √ !/µω0 ω0 µ l0 ≈ 500 Re Re R << l0 k2α = 2µ(E − "α) E "α k2α << !ω0 "α′ − "α >> !ω0 R > Re R > Re R > Re α α′ Re l0 s Re < R << l0 Figure 6.2: The schematic diagram of an elastic collision in quasi-2D geometry: (i) at short interparticle separations r < re, the collision occurs in 3D; (ii) in the region of r between re and the characteristic de Broglie wavelength of the particles Λ̃ε, the wave function is proportional to the 3D s-wave scattering wave function [180]; (iii) in the asymptotic region, the wave function is the product of a circular wave function and the wave function for the ground state harmonic motion in the confining potential. internal states of the particles. The index α is used to describe the internal energy as well as the chemical identity of the colliding particles, i.e. it specifies the collision channels. The channel is considered to be confined when k2α  ~ω0. In general, an inelastic collision or chemical reaction releases a lot of energy which is much larger than the confinement potential, i.e. α′ − α  ~ω0. As a result, any transition α→ α′ results in the acceleration of the collision products and they are free to move in 3D. At the inter-particle distance r > re, the interaction between the colliding particles can be omitted from the Schrödinger equation. Therefore, there are no couplings between different collision channels in the coupled-channel equations at r > re. The differential equations can in this region be solved independently and the confined and unconfined channels can be treated separately. 6.3 Elastic collisions in quasi-2D geometry The detailed theory of elastic collisions in the presence of a confinement was de- scribed in Ref. [180], where the authors obtained the scattering amplitude in terms of 3D scattering length. Here, we briefly repeat their derivation using the S matrix 121 6.3. Elastic collisions in quasi-2D geometry formulation. The relative motion of particles confined in quasi-2D geometry is described by the Schrödinger equation[ − 1 2µ 4+V + VH(z)− 12ω0 ] ψ(~r) = εψ(~r), (6.1) where ε is the collision energy and VH(z) = µω20z 2/2 is the confinement potential. This equation can be re-written in the form[ − 1 µ 4+2V + µν 2 0z 4 − 1 2 ν0 ] ψ(~r) = Eψ(~r), (6.2) where ν0 = 2ω0 and E = 2ε. So we can use the method described in Ref. [180] to solve Eq. 6.2. The solution of Eq. 6.2 with V = 0 can be expressed using Green’s function GE(~r, 0), that is ψα(~r) = [ϕ0(z)J0(kρ) +A0GE(~r, 0)]φαY00(r̂), (6.3) where ρ is the projection of inter-particle distance r on the (x, y) plane, r̂ indicates the orientation of r, ϕ0(0) is the eigenfunction of the ground state of the harmonic confinement potential, φα is the eigenfunction of the Hamiltonian of separated par- ticles, and J0 is the Bessel function. The scattering amplitude f00 for elastic collisions in quasi-2D geometry is defined through the asymptotic form of the quasi-2D wave function, which is the product of the eigenfunctions of the ground state of the harmonic potential and the scattered circular wave function in the (x, y) plane, namely ψα(~r) = [ ϕ0(z)ei ~k·~ρ − f00(E)ϕ0(z) √ i 8pikρ eikρ ] φαY00(r̂). (6.4) Matching Eqs. 6.3 and 6.4 at ρ→∞, we obtain the scattering amplitude f00 = −A0ϕ0(0)Θ(E − ~ν0), (6.5) where Θ(E − ~ν0) is the step function. At small inter-particle separation, the interaction forces between the colliding molecules are much stronger than the laser confinement. The collision complex of molecules is therefore unconstrained and the collision occurs in 3D. Accordingly, 122 6.3. Elastic collisions in quasi-2D geometry there will be a region between re and Λ̃E , where the wave function of the confined channel is proportional to the usual 3D s-wave scattering wave function. At r > re, different collision channels are uncoupled. We express the s-wave component of the wave function for the confined channel α at re  r  Λ̃E as a regular single-channel wave function in 3D multiplied by a proportionality coefficient ηϕ0(0). In order to maintain the consistency between the derivation of the single-channel and the multi- channel scattering theory in quasi-2D, we choose the form of the single-channel 3D wave function as (the detailed derivation is given in Appendix C) ψα(~r) = i √ pi kαr [ e−ikαr − Sα←αeikαr ] φαY00(r̂), (6.6) where Sα←α is the usual 3D S matrix element for elastic collisions in channel α and Y00(r̂) is the spherical harmonic depending on the orientation angles of the vector r. Therefore, the confined quasi-2D wave function in the region re  r  Λ̃E can be written as ψα(~r) = i √ piηϕ0(0) 1 kαr [ e−ikαr − Sα←αeikαr ] φαY00(r̂). (6.7) In order to find A0, we need to match Eq. 6.3 with Eq. 6.7 at r → 0. Equation 6.7 can be written in terms of trigonometric functions as ψα(~r) =i √ piηϕ0(0)φαY00(r̂) × [ cos kαr kαr − i sin kαr kαr − Sα←α cos kαr kαr − iSα←α sin kαr kαr ] . (6.8) For kαr → 0, ψα(~r)→ i √ piηϕ0(0) [ 1 kαr − i− Sα←α 1 kαr − iSα←α ] φαY00(r̂) = [ 1 r i √ piηϕ0(0) kα (1− Sα←α) + √ piηϕ0(0)(1 + Sα←α) ] φαY00(r̂). (6.9) For small arguments, J0 is about 1 and the Green’s function has the form GE(~r, 0) ≈ 14pir + 1 4pi √ pil0 ω( E 2~ν0 ), (6.10) where ω( E 2~ν0 ) = ln(B~ν0/piE) + ipi (6.11) 123 6.4. Inelastic collisions in quasi-2D geometry with B ≈ 0.915. So Eq. 6.3 for r → 0 has the form ψα(~r)→ [ϕ0(0) +A0GE(~r, 0)]φα ≈ [ ϕ0(0) + 1 r A0 4pi + A0ω(E/2~ν0) 4pi √ pil0 ] φα. (6.12) Comparing Eqs. 6.9 and 6.12, we get A0 = i √ pi4piηϕ0(0) kα (1− Sα←α)Y00(r̂), (6.13) and η = √ 4pi (1−Sα←α)ω(E/2~ν0) ikαl0 + √ pi(1 + Sα←α) . (6.14) In the limit kα → 0, the matrix element Sα←α is related to the scattering length a of ultracold particles by a = − (Sα←α − 1) /2ikα. (6.15) Equation 6.14 can be re-written in terms of ω0, ε, and l0 as follows η = √ 4pi (1−Sα←α)ω(ε/2~ω0) ikαl0 + √ pi(1 + Sα←α) , (6.16) and ϕ0(0) is given by ϕ0(0) = ( 1pil20 ) 1 4 . 6.4 Inelastic collisions in quasi-2D geometry In this section, we consider collision processes that induce transitions from state (α; l = 0) (denoted hereafter by α00 ) to another state α′l′m′l, where l ′ is the rotational angular momentum (partial wave) of the collision complex in state α′ and m′l is its projection on the quantization axis. We assume that particles are initially confined in quasi-2D geometry (l = 0,ml = 0) and any transition α → α′ results in loss of confinement. In the region re  r  Λ̃ε, the wave function (Eq. 6.7) for channel α can be generally written as ψα00 = ν − 1 2 α r−1√ 4pi [ Aα00e −ikαr −Bα00eikαr ] φα, (6.17) where Aα00 and Bα00 are the amplitudes of the incoming and outgoing scattering waves, and να is a normalization constant [195]. At re  r  Λ̃ε, the amplitude 124 6.4. Inelastic collisions in quasi-2D geometry of the incoming scattering wave is modified by the confinement and proportional to the amplitude of incoming wave in 3D Aα00 = χAα00, (6.18) where Aα00 is the amplitude of the incoming scattering wave for s-wave collisions in 3D. Comparing the coefficient in front of the term e−ikαr in Eq. 6.7 with that in Eq. 6.17 and using the conventional form [199] Aα00 = i √ pi/kα, (6.19) we obtain the coefficient χ χ = i √ piηϕ0(0) kαAα00 = ηϕ0(0). (6.20) Since the asymptotic motion of the collision products after a reactive or inelastic process is unconstrained, a combination of the exponential functions and 3D spheri- cal harmonics should be used to describe the wave function in the outgoing collision channels ψα′l′m′l = −ν − 1 2 α′ r −1Bα′l′m′le i(kα′r−l′pi/2)φα′Yl′m′l(r̂). (6.21) The 3D wave function after a collision (ψout) is related to the 3D wave function before the collision (ψin) by the S-matrix operator ψout = Ŝψin. Therefore, the amplitudes of the outgoing scattering waves Bα′l′m′l are related to the amplitude of the incoming wave Aα by the S-matrix elements Bα′l′m′l = Sα′l′m′l←α00Aα00. (6.22) where the elements Sα′l′m′l←α00 of the scattering S-matrix describe the probability of inelastic or chemically reactive collisions in 3D. Since the colliding particles are initially prepared only in state α, the scattered 125 6.4. Inelastic collisions in quasi-2D geometry part of the wave function for all outgoing channels α′ 6= α is given by ψscinel = ψ α′ 6=α outgoing = − ∑ α′ 6=α ∑ l′ ∑ m′l ν − 1 2 α′ r −1Sα′l′m′l←α00Aα00e i(kα′r−l′pi/2)φα′Yl′m′l(r̂) = − ∑ α′ 6=α ∑ l′ ∑ m′l ν − 1 2 α′ r −1Sα′l′m′l←α00χAα00e i(kα′r−l′pi/2)φα′Yl′m′l(r̂) = − ∑ α′ 6=α ∑ l′ ∑ m′l ν − 1 2 α′ r −1Sα′l′m′l←α00χ i √ pi kα ei(kα′r−l ′pi/2)φα′Yl′m′l(r̂). (6.23) After an inelastic collision or chemical reaction, the particles initially confined in quasi-2D geometry lose confinement and move in 3D. Therefore Eq. 6.4 cannot be applied as a boundary condition for inelastic collisions in quasi-2D. The boundary condition now should be in terms of 3D spherical wave functions with the scatter- ing amplitude modified by the confining potential. The scattering amplitudes for inelastic collisions are defined by ψscinel = ∑ α′ 6=α ν − 1 2 α′ fα′←α eikα′r r φα′ . (6.24) Comparing Eq. 6.23 with Eq. 6.24, we obtain fα′←α = − ∑ l′ ∑ m′l Sα′l′m′l←α00χ i √ pi il′kα Yl′m′l(r̂). (6.25) The differential cross section can thus be obtained using dσα′←α dΩ = |fα′←α|2. (6.26) Integrating Eq. 6.26 over all orientations, we get the integral cross sections σα′←α = ∑ l′ ∑ m′l pi k2α χ2|Sα′l′m′l←α00| 2, (6.27) where η is given by Eq. 6.14 and ϕ0(0) = ( 1pil20 ) 1 4 . 126 6.5. Threshold laws for inelastic collisions in quasi-2D 6.5 Threshold laws for inelastic collisions in quasi-2D Combining Eq. 6.14 and Eq. 6.27, we can write the integral inelastic cross section in quasi-2D as σα′←α = ∑ l′ ∑ m′l piϕ0(0)2 |Sα′l′m′l←α00|2 k2α η2 = ∑ l′ ∑ m′l piϕ0(0)2 |Sα′l′m′l←α00|2 k2α × 4pi[ (1−Sαα)[ln(B~ω0/piε)+ipi]√ 2ikαl0 + √ pi(1 + Sαα) ]2 . (6.28) According to Wigner [77], the square of the elastic T matrix element is proportional to the square of the wave number kα as kα → 0, that is |Tαα|2 = |1− Sαα|2 ∼ k2α, (6.29) so |1− Sαα| ∼ kα (6.30) and |1 + Sαα| ∼ constant. (6.31) Therefore η2 ∼ 4pi[ kα ln kα kα + constant ]2 . (6.32) The constant in the denominator can be omitted at very small kα, so the dependence of η2 on kα is η2 ∼ 1/ ln2 kα. (6.33) When kα → 0, the inelastic cross section in 3D is proportional to 1/kα σ3Dα′l′m′l←α00 ∼ 1/kα, (6.34) and |Sα′l′m′l←α00| 2 ∼ kα. (6.35) From Eqs. 6.28, 6.33, and 6.35 we obtain the threshold law for s-wave inelastic collisions in quasi-2D |σα′←α| ∼ 1/kα ln2 kα. (6.36) 127 6.6. Molecular scattering in quasi-2D This result is consistent with a qualitative analysis in Ref. [250]. 6.6 Molecular scattering in quasi-2D In order to explore the effect of laser confinement on collisions of molecules, we con- sider elastic and rotationally inelastic H2–H2 scattering. We assume that molecules are prepared in a particular vibrational and rotational state (va = 0, Na, mNa ; vb = 0, Nb, mNb), where v and N specify the vibrational and rotational quan- tum numbers, respectively, and a and b label the molecules. After scattering, the molecules will still remain in the ground vibrational state, so we are only interested in the inelastic cross sections with respect to different rotational states σN ′aN ′b←NaNb . The S-matrix elements in Eq. 6.27 are calculated in the uncoupled representa- tion while the S-matrix elements for transitions between ro-vibrational states are best calculated in the coupled total angular momentum representation [256]. The matrix elements in these two representations are related by the Clebsch-Gordan transformation [221] |NamNa〉|NbmNb〉 = ∑ N ∑ mN 〈NmN |NamNaNbmNb〉|NmN 〉, (6.37) where 〈NmN |NamNaNbmNb〉 are the Clebsch-Gordan coefficients. The orthonor- mality of angular momentum functions leads to the orthogonality relation of the Clebsch-Gordan coefficients:∑ mNa ∑ mNb 〈NmN |NamNaNbmNb〉〈NamNaNbmNb |N ′m′N 〉 = δNN ′δmNm′N . (6.38) We reformulate the theory described in Section 6.4 in the total angular momentum representation. The relation between σN ′aN ′b←NaNb and σN ′am′NaN ′bm′Nb←NamNaNbmNb is as follows: σN ′aN ′b←NaNb = 1 (2Na + 1)(2Nb + 1)∑ mNa ∑ mNb ∑ m′Na ∑ m′Nb σN ′am′NaN ′ bm ′ Nb ←NamNaNbmNb = 1 (2Na + 1)(2Nb + 1) ∑ l′ ∑ m′l ∑ mNa ∑ mNb ∑ m′Na ∑ m′Nb pi k2 χ2|SN ′am′NaN ′bm′Nb l′m′l←NamNaNbmNb00| 2, (6.39) 128 6.6. Molecular scattering in quasi-2D where 1(2Na+1)(2Nb+1) accounts for the degeneracy factor of the initial internal states of the molecules. Using the Clebsch-Gordan transformation, we rewrite Eq. 6.39 as σN ′aN ′b←NaNb = 1 (2Na + 1)(2Nb + 1) ∑ l′ ∑ m′l ∑ mNa ∑ mNb ∑ m′Na ∑ m′Nb pi k2 χ2 ∑ J ∑ mJ ∑ J ′ ∑ m′J ∑ N ∑ mN ∑ N ′ ∑ m′N ∑ N ′′ ∑ m′′N ∑ N ′′′ ∑ m′′′N 〈NmN |NamNaNbmNb〉〈N ′′m′′N |NamNaNbmNb〉 〈N ′m′N |N ′am′NaN ′bm′Nb〉〈N ′′′m′′′N |N ′am′NaN ′bm′Nb〉 〈JmJ |NmN00〉〈J ′m′J |NmN00〉〈JmJ |N ′m′N l′m′l〉 〈J ′m′J |N ′m′N l′m′l〉 × SJ∗N ′aN ′bN ′l′←NaNbN0S J ′ N ′′′a N ′′′b N ′′′l′′′←N ′′a N ′′bN ′′0 = 1 (2Na + 1)(2Nb + 1) pi k2 χ2∑ l′ ∑ J ∑ mJ ∑ J ′ ∑ m′J ∑ N ∑ N ′ ∑ N ′′ ∑ m′′N ∑ N ′′′ ∑ m′′′N δNN ′′δmNm′′N δN ′N ′′′δm′Nm ′′′ N (δJJ ′δmJm′J ) 2 × SJ∗N ′aN ′bN ′l′←NaNbN0S J ′ N ′′′a N ′′′b N ′′′l′′′←N ′′a N ′′bN ′′0 = 1 (2Na + 1)(2Nb + 1) × ∑ l′ ∑ J ∑ mJ ∑ N ∑ N ′ pi k2 χ2|SJN ′aN ′bN ′l′←NaNbN0| 2 = ∑ l′ ∑ J ∑ N ∑ N ′ 2J + 1 (2Na + 1)(2Nb + 1) × pi k2 χ2|SJN ′aN ′bN ′l′←NaNbN0| 2, (6.40) where J(J ′) = N + 0(N ′ + l′). The inelastic cross sections for H2-H2 collisions in the coupled total angular momentum representation are thus given by σN ′aN ′b←NaNb = ∑ l′ ∑ J ∑ N ∑ N ′ 2J + 1 (2Na + 1)(2Nb + 1) × pi k2 χ2|SJN ′aN ′bN ′l′←NaNbN0| 2. (6.41) 129 6.7. Numerical results 6.7 Numerical results Equation 6.27 shows that the cross sections for inelastic or reactive collisions in a quasi-2D gas depend on the 3D scattering length of the colliding particles in state α as well as the confinement strength. In order to illustrate the effect of these parameters on inelastic scattering, we present in Fig. 6.3 the results of rigorous calculations for collisions of 87Rb atoms in the mf = 0 state with 6Li atoms in the mf = −12 state, leading to Zeeman relaxation in a magnetic field. The calculations are based on accurate interaction potentials for the 6Li–87Rb molecule generated as described in Chapter 3. The scattering length a of the 6Li–87Rb system is tuned by varying an external magnetic field near the Feshbach resonance at 1104.9 G. According to Wigner’s threshold laws (see Table 5.1), the cross sections for inelastic transitions in the limit of low collision energy vary as ∼ 1/k in 3D [77] and as ∼ 1/(k ln2 k) in 2D [250]. Figure 6.3 shows that the energy dependence of the cross sections for inelastic scattering in a quasi-2D gas resembles the 3D threshold law if |a|/l0  1. It becomes similar to the energy dependence in 2D when |a|/l0 > 1. This suggests that the threshold energy dependence of inelastic cross sections in quasi-2D systems can be tuned by varying the ratio |a|/l0. Figure 6.3 also demonstrates that the laser confinement reduces the magnitude of the inelastic cross sections and that the suppression is more significant for the larger value of |a|/l0. In order to examine the dependence of the suppression on the scattering length, we present in Fig. 6.4 the cross sections for inelastic Zeeman relaxation in 3D and quasi-2D collisions of 6Li and 87Rb atoms as functions of the magnetic field varying through the resonance. The position of the Feshbach resonance is shifted by the confinement in agreement with previous calculations of elastic cross sections [146, 180, 257]. The inset shows the ratio of the cross sections for inelastic collisions in quasi-2D with l0 = 104 and 3D. The effect of the confinement is enhanced near the resonances due to the large absolute value of the scattering length. The inelastic cross section is suppressed in quasi-2D geometry because in the limit of zero collision energy it must smoothly approach the threshold energy de- pendence for scattering in a purely 2D geometry [250]. In order to quantify the suppression of inelastic scattering, it is necessary to consider the ratio of cross sec- tions for elastic and inelastic collisions. The elastic-to-inelastic ratio is of paramount importance for experiments with ultracold atoms and molecules. Elastic collisions determine the macroscopic dynamics of quantum gases. Inelastic and chemically re- active collisions destroy ultracold atoms and molecules. The ratio of cross sections 130 6.7. Numerical results 10-13 10-12 10-11 10-10 10-9 10-8 10-7  Collision Energy (K) 10-4 10-3 10-2 10-1 100 101 102 σ  (a. u.) Figure 6.3: The threshold energy dependence of cross sections for inelastic relax- ation in s-wave collisions of 6Li with 87Rb: filled circles−purely 2D geometry; filled squares−3D scattering cross section reduced by a factor of 4 × 104; open circles−quasi-2D with |a|/l0 > 1 cross section reduced by a factor of 30; open squares−quasi-2D with |a|/l0  1. The initial states are |12 ,−12〉6Li ⊗ |1, 0〉87Rb. for inelastic and elastic scattering in a quasi-2D gas can be written as σquasi−2Dinel σquasi−2Del = γ σ3Dinel σ3Del , (6.42) where σ3D denote the cross sections in an unconfined 3D gas. The scattering amplitude for elastic collisions in quasi-2D geometry can be writ- ten in terms of the 3D scattering length a and η [180] as f00 = 4piϕ20(0)aη, (6.43) which yields the cross section for elastic collisions in quasi-2D as [180] σquasi−2Del = |f00|2 4kα = 16pi2ϕ40(0)a 2η2 4kα . (6.44) 131 6.7. Numerical results 1095 1098 1101 1104 1107 B (G) 10-15 10-10 10-5 100 105 !  ( a . u . ) 1095 1098 1101 1104 1107B (G) 10-8 10-6 10-4 ! q u a s i - 2 D / ! 3 D Figure 6.4: Cross sections for s-wave inelastic collisions of 6Li and 87Rb atoms in 3D (solid curve) and quasi-2D scattering with a weak confinement (l0 = 104 bohr – dotted curve) and a strong confinement (l0 = 103 bohr – dot-dashed curve) as functions of the magnetic field. The inset shows the ratio of the cross sections for inelastic collisions in quasi-2D with l0 = 104 and 3D. The collision energy is 10−8 cm−1. The initial states are |12 ,−12〉6Li ⊗ |1, 0〉87Rb. Using Eq. 6.27, we obtain the ratio of cross sections for elastic and inelastic collisions in quasi-2D as σquasi−2Dinel σquasi−2Del = pi k2α η2ϕ20 ∑ l′ ∑ m′l |Sα′l′m′l←α00|2 16pi2ϕ40(0)a 2η2 4kα = ∑ l′ ∑ m′l |Sα′l′m′l←α00|2 4pikαϕ20(0)a2 . (6.45) The elastic cross sections in 3D can be written in terms of the scattering length as σ3Del = 4pia 2, (6.46) 132 6.7. Numerical results while the cross section for inelastic collisions is given by σ3Dinel = pi k2α ∑ l′ ∑ m′l |Sα′l′m′l←α00| 2. (6.47) Therefore, the relation for inelastic-to-elastic ratio in 3D is σ3Dinel σ3Del = ∑ l′ ∑ m′l |Sα′l′m′l←α00|2 4a2k2α . (6.48) Using Eq. 6.45 and Eq. 6.48 and the expression ϕ0(0) = ( 1pil20 ) 1 4 , we obtain γ = kαl0√ pi = √ 2ε pi~ω0 . (6.49) 0 1 2 3 l0 (in units of 10 4 Bohr) 10-4 10-3 10-2 10-1 100 ! i n / ! e l Figure 6.5: The ratios of inelastic and elastic cross sections for s-wave collisions of 6Li and 87Rb atoms as functions of l0 for |a| = 13.58 bohr (B = 200G) (circles) and |a| = 1704.43 bohr (B = 1104.9G) (triangles). The initial states are |12 ,−12〉6Li⊗|1, 0〉87Rb. The collision energy is 10−8cm−1. Because the collision energy is necessarily much smaller than the confinement potential in a quasi-2D gas, i.e., ε/~ω0  1, the ratio of elastic and inelastic cross sections must always be enhanced under laser confinement and must increase as the 133 6.7. Numerical results 0 2 4 6 8 10 l0 (in units of 10 4 Bohr) 10-6 10-4 10-2 100 ! q u a s i - 2 D / ! 3 D  ( B o h r- 1 ) Figure 6.6: The ratio of cross sections for elastic (circles) and inelastic (diamonds) collisions in quasi-2D and 3D as functions of l0 for the H2–H2 system. The collision energy is 10−8cm−1. The initial states are v1 = 0, N1 = 2; v2 = 0, N2 = 2 laser confinement increases. The degree of the enhancement is given quantitatively by the equation above. This general result is illustrated by a numerical calculation in Figs. 6.5 and 6.6. Figure 6.5 shows the ratios of inelastic and elastic cross sections in quasi-2D for s-wave collisions of 6Li and 87Rb atoms as functions of l0 for small and large scattering lengths. The inelastic-to-elastic ratio is always less than 1 and decreases as the strength of the confinement increases (the oscillation length l0 decreases). The suppression of inelastic collisions is more significant for the larger scattering length, which is consistent with the observation in Fig. 6.4. In order to explore the effect of laser confinement on collisions of molecules, we consider elastic and rotationally inelastic H2–H2 scattering. Figure 6.6 presents the ratio of the cross sections for elastic collisions and rotationally inelastic scattering of H2 molecules in a quasi-2D gas and in 3D as functions of l0. As predicted above, the ratios are always smaller than one, which means that both elastic and inelastic col- lisions of molecules are suppressed when the geometry changes from 3D to quasi-2D while the inelastic collisions are suppressed much more significantly than the elas- tic scattering. Another interesting observation: the magnitude of the cross section for both elastic and inelastic collisions increases as the strength of the confinement 134 6.7. Numerical results 0 2 4 6 8 10 12 14 l0 (in units of 10 3 Bohr) 10-4 10-3 10-2 10-1 ! q u a s i - 2 D / ! 3 D  ( B o h r- 1 ) Figure 6.7: The ratios of cross sections for elastic scattering (circles) and chemical reaction (triangles) in quasi-2D and 3D as functions of the confinement strength for 7Li + 6Li2(v = 0,N = 1) collisions. The collision energy is 10−8cm−1. increases. This happens because the extension of the wave function l0 decreases with the increase of the confinement potential, which leads to a higher probability to detect scattering atoms and molecules. Yet the suppression of inelastic collisions when the geometry changes from 3D to quasi-2D is due to the modification of the scattering wave function. The formalism presented in this Chapter can be applied to describe chemical reactions in an ultracold molecular gas under laser confinement. The index α′ in Eq. 6.27 must then include outgoing channels in different chemical reaction ar- rangements. To explore the effects of laser confinement on chemical interactions of ultracold molecules, we consider an illustrative example of the reaction 7Li + 6Li2(v = 0, N = 1) →7Li6Li + 6Li. The cross sections for elastic scattering and chemical rearrangement transitions in 7Li + 6Li2 collisions have been calculated by Cvitaš et al. [81]. Using their results and Eq. 6.27 of this Chapter, we evaluate the cross sections for the chemical reaction in a confined gas (Fig. 6.7). We note that the results presented in this Chapter apply to a gas of particles trapped in the ground state of the laser confinement potential. The limit of 3D scattering therefore cannot be obtained from our results simply by increasing l0. As described by Petrov 135 6.8. Conclusions and Shlyapnikov [180], the limit of 3D scattering is obtained by heating the system so that the particles populate a manifold of states in the confinement potential. As demonstrated by Cvitaš et al. [81], rigorous quantum calculations of cross sections for ultracold reactive scattering in 3D are computationally very demanding, though not impossible. Breaking the symmetry of space by applying an external field increases the complexity of the scattering problem to a great extent and converged reactive scattering calculations in the presence of external fields are at present pro- hibitively difficult [258]. The theory presented here makes the analysis of reactive scattering of molecules in a quasi-2D gas feasible. 6.8 Conclusions We have developed a formalism for rigorous calculations of cross sections for in- elastic and reactive collisions of ultracold atoms and molecules confined in quasi-2D geometry. The approach provides expressions for inelastic and reactive scattering cross sections in terms of the S-matrix elements for collisions in 3D and the laser confinement parameters. Our theory makes the analysis of reactive collisions of molecules in confined geometries feasible. Otherwise, one would have to calculate the cross sections by solving numerically the scattering problem in the presence of laser fields, which is at present prohibitively difficult. Using the formalism, we elu- cidate the general features of inelastic scattering and chemical reactions in ultracold quasi-2D gases of atoms and molecules. We have found that the cross sections for inelastic and chemically reactive collisions are suppressed by the confinement forces. This suppression is generally more significant than the effect of the laser confine- ment on the probability of elastic scattering. The elastic-to-inelastic collision ratios are therefore enhanced in the presence of a laser confinement. Our results suggest that applying laser confinement in one dimension may stabilize ultracold systems. Moreover, we have found that the threshold energy dependence of cross sections for both elastic and inelastic collisions in quasi-2D gases depends on the scattering length of the collision partners in the confined state and the confinement strength. Therefore the threshold laws for inelastic collisions can be tuned by varying the confinement forces and an external magnetic field, which suggests new mechanisms for controlling inelastic collision dynamics of atoms and molecules. The results presented in this Chapter should be of significant immediate interest to researchers of ultracold atoms and molecules since it is nowadays quite easy to create atomic and molecular systems confined in quasi-2D. Our studies may also stimulate new experimental studies as the suppression of inelastic processes may 136 6.8. Conclusions allow for the creation of ultracold atoms in quantum states that are unstable in the usual experiments. In addition, our work suggests new research directions for the study of collisional decoherence in quantum information science, fundamental physics of threshold collisions, many-body systems, condensed-matter physics and quantum optics of semiconductors. There are unique parallels between reactive collisions of molecules in a quasi-2D geometry and inelastic scattering of excitons and exciton polaritons in microcavity semiconductors. Our work therefore indicates that inelastic scattering of excitons in condensed-matter systems should be suppressed if confined to 2D. If our predictions are confirmed experimentally, they may therefore have applications reaching beyond the field of cold atoms and molecules. 137 Chapter 7 Outlook The development of experimental methods for controlling atomic and molecular dynamics at ultralow temperatures offers interesting and powerful tools for new fundamental research in chemistry and physics. For example, magnetic Feshbach resonances [57, 61–65, 133, 144, 145] provide a mechanism to tune the magnitude and the sign of the scattering length of atoms and molecules in ultracold gases. This can be used to improve the efficiency of evaporative or sympathetic cooling and develop new experimental models of solid-state physics phenomena, such as Cooper pairing and superfluidity [164, 165, 259]. Magnetic Feshbach resonances can also be used to create ultracold molecules. The electric dipole moments of ultracold polar molecules give rise to anisotropic intermolecular interactions which may find applications in quantum computation research and lead to intriguing dynamics of ultracold many-body systems [66, 67, 173]. The creation of cold molecular beams with precisely tunable energies [7, 260–262] offers another way to control molecular dynamics and can be employed to study cold collisions. The investigation of cold molecular collisions will elucidate molecular dynamics in interstellar clouds, advance the cooling and trapping experiments for molecules, and lead to the study of cold controlled chemistry [5, 6]. The development of experiments for trapping ultracold atoms and molecules in optical lattices opened up new opportunities to study quan- tum many-body systems in a highly controllable fashion. Both the inter-particle and particle-field interactions can be tuned by varying the trap parameters of optical lat- tices [35, 51]. This can be used to explore novel quantum phase transitions [38, 51], design quantum simulators [35, 59], and develop new schemes for quantum infor- mation processing [39, 43]. Optical lattices can also be used to control the spatial dimensionality of ultracold gases by confining the motion of atoms and molecules in one or two dimensions. The confinement modifies the scattering properties of ultracold particles, leading to new states of matter and dynamics not observable in 3D gases. This Thesis presents a theoretical study of new control mechanisms of micro- scopic interactions in ultracold gases. The Thesis extends the magnetic-field control of binary atomic interactions to control mechanisms using superimposed magnetic 138 Chapter 7. Outlook and electric fields. The new mechanisms based on electric-field-induced resonances may allow for two-dimensional control of inter-particle interactions, leading to total control over microscopic interactions in ultracold gases [216]. This Thesis demon- strates that electric fields induce anisotropic scattering in ultracold gases which may be used for the developments of novel experiments to explore many-body dynamics of heteronuclear atomic mixtures. This Thesis also extends the study of the scatter- ing dynamics of elastic collisions in ultracold gases confined in quasi-2D geometry to inelastic and reactive collisions of atoms and molecules in optical lattices. The theory presented in this Thesis provides a rigorous method to explore inelastic dy- namics of ultracold particles in experimentally realizable confined systems, which should be of significant interest to researchers of collisional decoherence in quantum information science, quantum many-body systems, and quantum optics of semi- conductors. Numerical calculations based on the theory demonstrate a new control mechanism for inelastic collisions in quasi-2D geometry and show that applying laser confinement in one dimension may stabilize ultracold systems with large scattering lengths. This result should be of immediate practical importance for the develop- ment of experimental studies of complex atomic and molecular systems in confined geometries. The effects predicted in this Thesis should be easy to measure in cur- rent experiments with ultracold atoms and molecules and Madison’s group at UBC has already begun to develop an experiment to test our predictions. If our results are confirmed experimentally, this work may lead to a new research field of low- dimensional chemistry and may open up interesting opportunities for new studies of ultracold controlled chemistry [6], condensed-matter physics [38, 49, 182, 237, 251], and quantum optics of semiconductors [253–255]. In many experiments with optical lattices, atoms and molecules populate a man- ifold of states in the confinement potential at temperatures T ∼ ~ω. In this confine- ment dominated 3D regime [180], the 2D character of the scattering dynamics is not important, yet the confinement can still affect inter-particle interactions to a great extent. Petrov and Shlyapnikov have recently studied the effect of the confining potential on elastic scattering in the confinement-dominated 3D regime [180]. They found that collisions in this regime resemble regular 3D scattering when the mag- nitude of the scattering length is much smaller than the oscillation length |a|  l0, whereas a big deviation from 3D scattering was observed for large scattering length |a|  l0. However, the dynamics of inelastic collisions in the confinement dominated 3D regime remain unknown. The work presented in this Thesis can be (and will be) extended to the study of inelastic scattering of atoms and molecules in this regime. In particular, it might be interesting to examine the threshold behavior of atoms 139 Chapter 7. Outlook and molecules in the confinement dominated 3D regime and calculate the ratio of cross sections for elastic and inelastic collisions. Most of the experimental and theoretical studies in the field of cold and ul- tracold molecules to date have focussed on diatomic molecules. An emerging di- rection in this field aims to extend the cooling techniques and study of ultracold diatomic molecules to polyatomic molecules. The research of ultracold chemistry may also require ultracold complex molecules. However, due to the complicated internal structure of large molecules, the possibility of cooling polyatomic molecules to ultracold temperatures still remains an open question and will be an important research topic in the near future. Among all cooling techniques, buffer-gas cooling is the most powerful and versatile method and could potentially produce molecules in translationally and internally cold states [108–111]. Noble gas atoms (e.g., He and Ne) are normally used as refrigerants in this technique and molecules of inter- est are cooled by elastic collisions with the buffer gas atoms that lead to energy thermalization. Once the molecules are cooled to cold temperatures, one could use evaporative cooling to further cool them to ultracold temperatures or generate cold molecular beams [7, 260, 261]. However, molecules may diffuse and stick to the cell walls during the thermalization process. Also, momentum transport required for energy thermalization may not be efficient in collisions of small buffer gas atoms with large complex molecules. The lifetime of atom-molecule collision complexes may be very long and the buffer gas atoms may stick to molecules, which may lead to clustering. Therefore, the study of the time scales of the thermalization process for complex molecules in a buffer-gas cooling experiment is of great importance. To address this problem, one could use the theory of unimolecular chemical reactions. In a simple two-state model [263], a unimolecular reaction is considered to occur in two steps: A+M  A∗ +M (7.1) and A∗ → P (7.2) where A and A∗ denote a regular and an activated reactant molecule, respectively, M represents a medium atom or molecule, and P is a final product of the unimolec- ular reaction. First, the reactant molecule A is promoted to an activated state A∗ via collisions with the medium particle M (cf. Eq. 7.1). The reaction then occurs when the internal energy of the reactant is sufficiently large to overcome the re- action barrier (cf. Eq. 7.2). A more complicated model of unimolecular reactions is based on the so-called master equation [264]. It extends the two-state model 140 Chapter 7. Outlook to a theory considering all internal energy states of reactant molecules and gives the time evolution of the molecular energy redistribution. In buffer-gas cooling, molecules of interest are de-activated, i.e., the internal energy of the molecules is taken away by the buffer-gas atoms. One therefore only needs to consider the first step – collisional energy transfer process – to obtain the time scale by solving the master equation. I believe the results of this research would play a significant role in guiding future experiments on cooling polyatomic molecules and plan to pursue this research direction after graduation. If polyatomic molecules can be cooled to ultracold temperatures, they will be ex- cellent candidates for the study of external field control of chemical reactions, which may provide a novel tool to explore mechanisms of chemical reactions and open up a new regime of molecular dynamics research to address fundamental problems of modern chemical physics. At cold and ultracold temperatures, perturbations due to interactions of molecules with external fields are comparable with the kinetic energy of the molecules. At the same time, chemical reactions of molecules can be enhanced by resonances, threshold phenomena, tunneling, and collective dynamics. Manipu- lating chemical reactions with external fields can therefore be easily achieved at low and ultralow temperatures. Studies of cold chemistry in the presence of external fields have already become the subject of both experimental [4, 5] and theoretical research [265]. Staanum et al. [266] and Zahzam et al. [267] have recently measured the rate coefficients for inelastic and reactive collisions of Cs2 molecules with Cs atoms confined in an optical field. Researchers have also been developing exper- iments to study chemical reactions using slow molecular beams [262]. Tscherbul and Krems [265] developed a theory for rigorous quantum scattering calculations of cross sections for chemical reactions in the presence of an external electric field. However, the study of ultracold chemistry has just begun. Current experimental work is limited to the study of collisions between atoms and diatomic molecules and rigorous quantum calculations of reaction processes in the presence of external fields are at present computationally very challenging. With the development of new experimental techniques and efficient computational algorithms, the research field of cold controlled chemistry is expected to become very dynamic and may expand very rapidly in the near future [6, 7]. The theories developed in this Thesis can be extended to address some of the most interesting problems of cold controlled chemistry [6, 7]. 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Lett., 96:023202, 2006. 165 Appendix A In this appendix the coupled-channel differential equations of Eq. 2.91 are derived from the Schrödinger equation of Eq. 2.90. The Schrödinger equation with the Hamiltonian given by Eq. 2.88 and the total wave function given by Eq. 2.89 is[ − 1 2µ ∂2 ∂r2 + l̂2(θ, ϕ) 2µr2 + V̂ + Ĥas ]∑ α′ ∑ l′ ∑ m′l Fα′l′m′l(r)φα′Yl′m′l(r̂) = E ∑ α′ ∑ l′ ∑ m′l Fα′l′m′l(r)φα′Yl′m′l(r̂). (A.1) Multiplying Eq. A.1 from the left by φ∗αY ∗lml(r̂), integrating over θ and ϕ, and using the relations l̂2(θ, ϕ)Yl′m′l = l ′(l′ + 1)Yl′m′l (A.2) and Hasφα′ = α′φα′ , (A.3) we get ∑ α′ ∑ l′ ∑ m′l 〈φα|〈Ylml(r̂)|Yl′m′l(r̂)〉|φα′〉 ( − 1 2µ ∂2 ∂r2 ) Fα′l′m′l(r) + ∑ α′ ∑ l′ ∑ m′l Fα′l′m′l(r)〈φα|〈Ylml(r̂)|Yl′m′l(r̂)〉|φα′〉 l′(l′ + 1) 2µr2 + ∑ α′ ∑ l′ ∑ m′l Fα′l′m′l(r)〈Ylml(r̂)|Yl′m′l(r̂)〉〈φα|V̂ |φα′〉 + ∑ α′ ∑ l′ ∑ m′l Fα′l′m′l(r)〈φα|〈Ylml(r̂)|Yl′m′l(r̂)〉|φα′〉α′ =E ∑ α′ ∑ l′ ∑ m′l Fα′l′m′l(r)〈φα|〈Ylml(r̂)|Yl′m′l(r̂)〉|φα′〉. (A.4) The orthonormality of the Yl′m′l(r̂) and φα′ functions 〈Ylml(r̂)|Yl′m′l(r̂)〉 = δll′δmlm′l (A.5) 166 Appendix A. and 〈φα|φα′〉 = δαα′ , (A.6) reduces Eq. A.4 to − 1 2µ ∂2 ∂r2 Fαlml(r) + l(l + 1) 2µr2 Fαlml(r) + ∑ α′ 〈φα|V̂ |φα′〉Fα′lml(r) + Fαlml(r)α = EFαlml(r). (A.7) Multiplying Eq. A.7 by −2µ and rearranging the terms, we get[ ∂2 ∂r2 − l(l + 1) r2 + k2α ] Fαlml(r) = 2µ ∑ α′ 〈φα|V̂ |φα′〉Fα′lml(r), (A.8) where k2α = 2µ(E − α). 167 Appendix B In this appendix, we show how to obtain Eq. 5.81 and the value of d∗. In 2D geometry, the asymptotic form of the total wave function is ψ(ρ) ρ→∞−→ eikx + √ i k f(k, ϕ) eikρ√ ρ . (B.1) When ρ  ρe (where ρe is the characteristic distance of inter-particle interaction potentials), we have [8] eikρ√ ρ = i √ 1 2 pikH (1) 0 (kρ), (B.2) where H(1)0 (kρ) is the Hankel function of the first kind. When kρ 1, eikx ≈ 1, (B.3) and H(1)0 (kρ) has the following approximate expression H (1) 0 (kρ) ≈ − ( 2i pi ) ln ( 2i γkρ ) , (B.4) where γ = eC and C ≈ 0.577 is the Euler constant. Combining Eqs. B.1 – B.4, we have ψ(ρ) ρ→∞−→ 1 + f(k, ϕ) √ 1 2 pik √ i k i ( −2i pi log 2i γkρ ) =1 + √ 2i pi f(k, ϕ) ln 2i γk − f(k, ϕ) √ 2i pi ln ρ. (B.5) This should agree with the general solution of the equation − 1 2µ 1 ρ d dρ (ρ dψ dρ ) = 0, (B.6) which is valid in the range 1/k  ρ  ρe. In this region, both V and E terms in the Schrödinger equation 5.1 are negligible and ψ(ρ) has a form ψ(ρ) ≈ c1 + c2 ln ρ. (B.7) 168 Appendix B. The ratio c1/c2 is real and independent of energy. Let c1 c2 = − ln d, (B.8) where d can be considered as a characteristic length given by the exact solution of Eq. 5.1 with E = 0. Equalizing Eq. B.5 and Eq. B.7 and using Eq. B.8, we obtain − ln d = 1 + √ 2i pi f(k, ϕ) ln 2i γk −f(k, ϕ) √ 2i pi ln γ . (B.9) The scattering amplitude is then given by f(k, ϕ) = √ pi 2i ln d− ln 2iγk = √ pi 2i ln kd∗ − ipi2 , (B.10) where d∗ = (d/2)eC . We thus obtain the expression for the integral cross section σ = 2pi k |f(k, ϕ)|2 = pi2 k [ (ln k + ln d∗)2 + pi 2 4 ] . (B.11) One can evaluate d∗ using the value of the calculated cross section σ as ln d∗ = −pi √ 1 σk − 1 4 − ln k. (B.12) 169 Appendix C In conventional multi-channel and single-channel collision theories [195, 196], the radial wave functions have different forms. In this appendix, we show that the amplitude of the incoming wave in these expressions are consistent and how we obtain the coefficient in Eq. 6.6. The radial 3D wave function in multi-channel collision theory has a form [195] Fαl(r →∞) = ν− 1 2 α 1 r [ aαl sin(kr − l2pi)− bαl cos(kr − l 2 pi) ] . (C.1) It can also be written in terms of exponential functions as (cf. Eq. 2.97) Fαl(r →∞) = ν− 1 2 α r −1 [ Aαle−ikαr − Bαleikαr ] , (C.2) with Aαl = −aαl + ibαl2i . (C.3) The coefficient Aαl can be obtained using both the multi-channel and the single- channel wave functions. Here, we give the derivation of Aαl from the single-channel wave function and compare it with the expression obtained in multi-channel collision theory. The radial part of the asymptotic single-channel wave function in 3D can be written in terms of trigonometric functions as [196] Fαl(r →∞) = ν− 1 2 α 1 kαr [ Bαl sin(kαr − l2pi)− Cαl cos(kαr − l 2 pi) ] , (C.4) where Bαl = Aαl cos δαl (C.5) and Cαl = −Aαl sin δαl, (C.6) and δαl is the phase shift. Equalizing Eq. C.2 and Eq. C.4 and using Eqs. C.5 and C.6, we find aαl = Aαl kα cos δαl (C.7) 170 Appendix C. and bαl = −Aαl kα sin δαl. (C.8) The substitution of Eqs. C.7 and C.8 into Eq. C.3 gives the relation between Aαl and Aαl Aαl = − Aαl kα cos δαl − iAαlkα sin δαl 2i = −Aαle −iδαl 2ikα . (C.9) In the single-channel collision theory [196], Aαl is obtained by expanding the wave function and the scattering amplitude in terms of Legendre polynomials. Since in multi-channel collision theory [195] Aαl is obtained by expanding the wave function in terms of spherical harmonics, here, we re-write Aαl using spherical harmonics to obtain Aαl = 4piilY ∗lml(r̂i)e iδαl . (C.10) Substituting Eq. C.10 into Eq. C.9, we obtain the amplitude of the incoming wave function in Eq. C.2 Aαl = i2piY ∗lml(r̂i) kα . (C.11) Equation C.11 is consistent with the expression for Aαl from the derivation based on the multi-channel theory [199]. For l = 0, Y ∗00(r̂i) = 1/ √ (4pi), (C.12) and we can reduce the multi-channel wave function (Eq. C.2) to a single-channel s-wave wave function (Eq. 6.6) with the incoming amplitude given by Aαl = i √ pi/kα. (C.13) 171 Appendix D List of publications This Thesis is based on the work presented in the following publications: [1] Z. Li, S. Singh, T. V. Tscherbul, and K. W. Madison, “Feshbach resonances in ultracold 85Rb–87Rb and 6Li–87Rb mixtures”, Physical Review A 78, 022710 (2008). [2] Z. Li and R.V. Krems “Electric-field-induced Feshbach resonances in ultra- cold alkali-metal mixtures”, Physical Review A 75, 023709 (2007). [3] Z. Li and K.W. Madison, “Effects of electric fields on heteronuclear Feshbach resonances in ultracold 6Li–87Rb mixtures”, Physical Review A 79, 042711 (2009). [4] Z. Li, S. V. Alyabyshev, and R. V. Krems, “Ultracold inelastic collisions in two dimensions”, Physical Review Letters 100, 073202 (2008). [5] Z. Li and R. V. Krems, “Inelastic collisions in an ultracold quasi-two-dimensional gas”, Physical Review A 79, 050701(R) (2009). 172