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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 applying 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-fieldinduced 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 ultracold gaseous mixtures. The mechanisms of electric-field-induced resonances described in this Thesis allow for two-dimensional control of inter-particle interactions, leading to total control over ultracold gases. To guide future experiments, we generate 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 produces an optical lattice of ultracold particles. We develop a multichannel scattering theory for collisions of atoms and molecules in two dimensions and explore the effects 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 systems 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  ii  Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  iii  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  vi  List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  vii  Abstract  List of Tables  Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviii . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  xx  1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  1  Dedication  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  2.3  2.4  2.2.1  Typical scattering experiment  . . . . . . . . . . . . . . . . .  20  2.2.2  Time-independent Schrödinger equation . . . . . . . . . . . .  23  2.2.3  Differential cross section  . . . . . . . . . . . . . . . . . . . .  25  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  Multi-channel scattering theory 2.4.1  Multi-channel theory  . . . . . . . . . . . . . . . . . . . .  33  . . . . . . . . . . . . . . . . . . . . . .  34 iii  Table of Contents 2.4.2  Numerical integration of multi-channel equations  . . . . . .  39  3 Accurate interatomic potentials from interplay of ultracold experiment and theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1  Ultracold mixtures of  3.2  Fitting procedure  6 Li  and  87 Rb  41  . . . . . . . . . . . . . . . . . .  42  . . . . . . . . . . . . . . . . . . . . . . . . . . . .  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  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  72  4.4  Li–Rb system 4.4.1  Li–Rb collisions in combined electric and magnetic fields  4.4.2  Mechanism of electric-field-induced shifts of magnetic Feshbach resonances  . .  . . . . . . . . . . . . . . . . . . . . . . . . .  76 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  Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  87  5 Ultracold inelastic collisions in two dimensions . . . . . . . . . . .  91  4.5  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 6.1  Motivation  6.2  Ultracold quasi-2D gas  . . . . . . . . . . . . 118  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 . . . . . . . . . . . . . . . . . . . . . . . . . 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  6.6  Molecular scattering in quasi-2D . . . . . . . . . . . . . . . . . . . . 128  6.7  Numerical results  6.8  Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136  . . . . . . . . . . 127  . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130  7 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 References  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143  Appendices A  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166  B  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168  C  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170  D List of publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172  v  List of Tables 3.1 3.2  Definition of quantum numbers used in this Thesis. . . . . . . . . . . Positions and widths of  6 Li–87 Rb  46  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 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 scattering lengths of the oscillation were identified and the distance between them is indicated in parentheses. . . . . . . . . . . . . . . . . . . . . 4.1  57  The positions (B0 ) and widths (∆B) of s-wave magnetic Feshbach resonances for Li–Cs at magnetic fields below 500 G. The notation  4.2 4.3  |Fa MFa i for the atomic states is the same as in Chapter 3. . . . . . .  65  at magnetic fields below 1 kG.  69  The positions (B0 ) of p-wave magnetic Feshbach resonances for Li–Cs  . . . . . . . . . . . . . . . . . . . . .  The positions (B0 ) and widths (∆B) of s-wave resonances induced by an external electric field of 100 kV/cm for 6 Li–87 Rb 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. . . . . . . . . . . . . . . . . . . . . . . . . .  5.1  77  The energy dependence of the elastic and inelastic cross sections in 2D and 3D. k is the collision wave number, and l (ml ) and l0 (m0l ) 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 collisions 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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  1.2  3  The scheme of a laser cooling experiment, in particular, an absorptionemission 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. . . . . . . . . . .  1.3  The energy levels of a  133 Cs  6  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 highfield-seeking state, i.e., the potential energy of Cs decreases as the field strength increases. . . . . . . . . . . . . . . . . . . . . . . . . . 1.4  7  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.  1.5  . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  8  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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  1.7  12  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. . . . . . . . . . .  2.1  13  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  20  The coordinate system describing a scattering experiment. The incident 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 angle 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 = r2 dΩ. . . . . . . . . . . . . . . . . . . . . . . . .  2.3  21  Space-fixed spherical polar coordinates for two-body collisions. Collisions 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. . . . . . . .  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 | 21 , 21 i6 Li ⊗ |1, 1i87 Rb 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 ElS of the  0 0 optimal singlet and triplet potentials El=0 = −0.106 cm−1 , El=1 =  1 1 −0.0870 cm−1 , and El=0 = −0.137 cm−1 , El=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). . . . . . . . . . . . . . . . . . . . . . 3.2  The procedure of fitting the interactions potentials for ultracold 87 Rb  3.3  49  6 Li–  collisions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  51  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 1 Σ interaction potential.  3.4  . . . . . . . . .  52  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 3 Σ interaction potential.  3.5  . . . . . . . . .  53  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 | 21 , 12 i6 Li ⊗|1, 1i87 Rb 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) 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. . . . . . . 3.6  54  Magnetic field dependence of the s-wave (upper panel) and p-wave (lower panel) elastic scattering cross sections for atoms in the | 12 , 12 i6 Li ⊗ |1, 1i87 Rb 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 occur 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 1 Σ (solid lines) and 3 Σ (dashed lines) states. The interaction potentials were taken from Ref.[219] and the dipole moment functions approximate the data of Ref. [206]. . . . . . . . .  4.2  61  The interaction potentials and dipole moment functions (inset) of the LiRb molecule in the 1 Σ (solid lines) and 3 Σ (dashed lines) states. The interaction potentials were taken from Ref.[170] and the dipole  4.3  moment functions approximate the data of Ref. [206]. . . . . . . . . The coordinate system in our calculations. ζ~ and d~ represent the  62  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 ~ and ϕγ and ϕθ vector and the z-axis; χ is the angle between ζ~ and d, are the angles between the x-axis and the projections of the vectors êζ and êd on the (x, y) plane, respectively. . . . . . . . . . . . . . . . 4.4  63  Cross sections for elastic s-wave (solid curves) and p-wave (broken curves) scattering of Li and Cs atoms in the states |1, 1i7 Li ⊗|3, 3i133 Cs  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 . . . 4.5  66  Cross sections for elastic s-wave (solid curves) and p-wave (broken curves) scattering of Li and Cs atoms in the states |1, 0i7 Li ⊗|3, 3i133 Cs  computed at different electric fields: ζ = 0 kV/cm (upper panel) and ζ = 100 kV/cm (middle panel). The lower panel presents the cross 4.6  section for the s → p transition. The collision energy is 10−7 cm−1 . .  67  Cross sections for elastic s-wave (solid curves) and p-wave (broken  curves) scattering of Li and Cs atoms in the states |1, −1i7 Li ⊗|3, 3i133 Cs 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 . . . 4.7  68  Cross sections for elastic s-wave collisions of Li and Cs atoms in the |1, 0i7 Li ⊗ |3, 3i133 Cs 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 .  4.8  70  Electric-field dependence of the s-wave scattering cross section for collisions of Li and Cs atom in the |1, 0i7 Li ⊗ |3, 3i133 Cs 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, 0i7 Li ⊗ |3, 3i133 Cs 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, −1i7 Li ⊗ |3, 3i133 Cs 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 between Li and Rb in the atomic spin state | 12 , 12 i6 Li ⊗ |1, 1i87 Rb . These  results were obtained for a collision energy of 10−7 cm−1 and two different electric fields. The solid and dash-dotted curves show the sand 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 pwave 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  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 i6 Li ⊗ |1, 1i87 Rb 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, 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 10−7 cm−1 .  75  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 10−7 cm−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 ζ 2 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 i6 Li ⊗ |1, −1i87 Rb 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 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 10−7 cm−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 i6 Li ⊗  |1, 1i87 Rb . 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 10−7 cm−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 i6 Li ⊗ |1, 1i87 Rb 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 | 21 , 12 i6 Li ⊗ |1, 1i87 Rb 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 10−7 cm−1 . . . . . . . . . . . . . . . . . . . . . . . . . . . .  83  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 | 21 , 21 i6 Li ⊗ |1, 1i87 Rb 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 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 10−7 cm−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 | 21 , 21 i6 Li ⊗ |1, 1i87 Rb . 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 10−7 cm−1 . .  87  4.21 Magnetic field dependence of elastic cross sections for atoms in the atomic spin state | 21 , 12 i6 Li ⊗ |1, 1i87 Rb 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 10−7 cm−1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1  88  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, −2i7 Li ⊗ |3, 2i133 Cs . The calculations were carried out in a  magnetic field of 100 G. . . . . . . . . . . . . . . . . . . . . . . . . . 109  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 (diamonds) 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, −2i7 Li ⊗ |3, 2i133 Cs . 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) collisions. The initial states are |2, −2i7 Li ⊗ |3, 2i133 Cs . 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 sections 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, 2i7 Li ⊗ |4, 4i133 Cs . Rend specifies the propagation 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 interaction 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 maximally stretched state |2, 2i7 Li ⊗ |4, 4i133 Cs . Rend specifies the propagation 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 angular momentum. Adapted with permission from R. V. Krems, Int. Rev. Phys. Chem. 24, 99 (2005).  5.7  . . . . . . . . . . . . . . . . . . . 115  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. . . . . . . . . . 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 relaxation in s-wave collisions of 6 Li with  87 Rb:  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.  6.4  The initial states are | 12 , − 12 i6 Li ⊗ |1, 0i87 Rb . . . . . . . . . . . . . . . 131  Cross sections for s-wave inelastic collisions of 6 Li and  87 Rb  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 6.5  states are | 21 , − 12 i6 Li ⊗ |1, 0i87 Rb . . . . . . . . . . . . . . . . . . . . . 132  The ratios of inelastic and elastic cross sections for s-wave collisions of 6 Li and  87 Rb  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 i6 Li ⊗ |1, 0i87 Rb . The collision energy is 10−8 cm−1 .  6.6  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133  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 10−8 cm−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 confinement strength for 7 Li + 6 Li2 (v = 0, N = 1) collisions. The collision energy is 10−8 cm−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 questions. 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 elucidative and demonstrative. I truly thank my colleagues in our group: Timur Tscherbul, Erik Abrahamsson, 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 chemistry 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 thermal 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. Massive particles in an ideal gas can be considered as quantum-mechanical wavepackets p with an extension on the order of a thermal de Broglie wavelength Λ = h2 /2πmkB T , 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 velocities 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 momenta (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 centrifugal 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 collisions 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 coherently 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 computation 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 thermal 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 probabilities 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 independent 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 potential 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 collision 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].  C  B  A  Figure 1.2: The scheme of a laser cooling experiment, in particular, an absorptionemission 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  133 Cs  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 gradients 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 experimental 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 magnetic 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. 2×10  Energy (a.u.)  1×10 5×10  -6  A  -6 -7  0 -5×10 -1×10 -2×10  -7  -6  B  -6  0  1000  2000  3000  4000  5000  Magnetic field (G) Figure 1.3: The energy levels of a 133 Cs 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, researchers use evaporative cooling, of which the scheme is presented in Fig. 1.4. Normally, 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 remaining 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  More delicate methods: evaporative cooling  to quantum degeneracy using the sympathetic cooling technique [18, 19, 127]. Trapping potential  Elastic collisions  Text Length  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 quantum 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 = (2π~2 /mkB T )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 kB T or less. The number of these states per unit volume is approximately (mkB T /~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 ultracold 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 resonance 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 using 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 ultracold 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 Feshbach 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 dissociation 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 performing 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 groundbreaking 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]. Magnetic Feshbach resonances also offer an extremely sensitive probe of interatomic interaction potentials for collisions at ultracold temperatures [168–170]. For example, in Chapter 3 of this Thesis, we generate accurate interaction potentials for ultracold collisions in Li – Rb gaseous mixtures by fitting the experimentally measured Feshbach resonances [170]. As mentioned in the previous section, ultracold molecules can be created by tuning external magnetic fields near a magnetic Feshbach 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 numbers 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 variation 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 disso10  1.3. Feshbach resonances  Closed channel  A  B  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 scattering 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 collision threshold, forcing the dimers to form molecules by rearranging their electronic and nuclear spins. In principle, as the magnetic field varies through a Feshbach resonance, 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  Scattering length (Bohr)  1.4. Ultracold gases in restricted geometries  10 10  4  3  0  abg  ∆Β  3  -10  4  -10  B0  4  -10  1060  1062  1064  1066  1068  1070  Magnetic field (G) 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 lowdimensional quantum gases by confining the motion of ultracold particles in one or two dimensions. In low-dimensional systems, the confinement modifies the interaction 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 quantum 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 pancakeshaped 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 timeindependent 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 potentials 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 6 Li–87 Rb molecule in the singlet (1 Σ) and triplet (3 Σ) 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 resonances 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 heteronuclear 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 resonance. 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 6 Li –  87 Rb  and 7 Li –  133 Cs  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 Feshbach resonances. The electric field may also shift the positions of resonances and induce the anisotropy of ultracold scattering by rotating and spinning up the collision 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 inelastic 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 degrees 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 2 N X X ~2 ~2 Ĥ = − ∆α − ∆i 2mα 2me α=1  i=1  X X Zα Zβ e2 X X Zα e2 X X e2 + − + , R R R iα ij αβ α α β>α  i  j  (2.1)  i>j  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 N X X X Zα Zβ e2 X X Zα e2 X X e2 ~2 Ĥel = − ∆i + − + . 2me Rαβ Riα Rij α α i=1  β>α  i  j  (2.5)  i>j  One can solve the electronic Schrödinger equation Ĥel un (Ri |Rα ) = En un (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 h i 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=  X  φn (Rα )un (Ri )  (2.8)  n  where  hun |un0 i = δnn0 .  (2.9)  The Schrödinger equation for the molecular system becomes (T̂N + Ĥel )  X n  φn (Rα )un (Ri ) = E  X  φn (Rα )un (Ri ).  (2.10)  n  Multiplying Eq. 2.10 from the left by un0 (Ri ), integrating over the electronic coordinates, and using Eq. 2.9, we obtain the following set of coupled differential equations: X n0  hun (Ri )|T̂N |un0 (Ri )iφn0 (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  and trap loss. Due to the extremely low density of ultracold quantum gases, the  " of the L " · d" represents binary E collisions are dominant determining theof collision dynamics in the gases. By d where theindot product the dipole moment vector  studying ultracold collisions between two particles, one can extract parameters such " dS denotes the dipole moment functi the vector of the external electric field E, as the s-wave scattering length to explore many-body phenomena. This chapter  introduces a 2.2. time-independent scattering theory for two-body elastic and inelastic the different spin states and E is and the electric field magnitude. If the electric fi Scattering amplitude cross section collisions in a space-fixed frame and provide an approach to implement the theory " has the form at a certain angle γ with respect to the quantization axis, d" · E in numerical calculations.  Scattering detector with χ the angle between θ and γ. It can be written in terms of the first-de 2.1 Scattering amplitude and cross sections polynomial  FigThe typical set-up of a scattering experiment dΩ  =  4π 3  Scattered beam (β) & ∗ Y1−1 (γ, φγ )Y1−1 (θ, φθ ) + Y10∗ (γ, φγ )Y10 (θ, φθ )  z  cos(χ) = P1 (cos  + Y11∗ (γ, φγ )Y11 (θ, φ  Target where Incident φγ and φ(α) beam θ are the projections of angles γ and θ on the (x, y) plane and Yx Transmitted beam  harmonics.  The dipole moment functions are represented by the expressions & ' dS (R) = D exp −α(R − Re )2  Figure 2.1: The typical configuration of a conventional scattering experiment. A uniform incident beam α of particles with a certain collision energy −2 and current with the parameters Re containing = 7.2 bohr, α =centers. 0.06 bohr D = 4.57 Debye density Jinc is incoming on a target collision Particles and can then be scattered into different directions and the number of outgoing in a solid state, and R = 5.0 bohr, α = 0.045 bohr−2 and particles D = 1.02 Debye for the triple angle dΩ is detected eby a scattering detector.  2.2 2.2.1  5 Scattering amplitude and cross section  16  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 spacefixed 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 = r2 dΩ  (2.12) 20  Chapter 2 #SM |V̂ (R)|S M ! = V (R) δ Material δ . mass φ! Background and note that Background Material Background Material Background Material  " has the form d" · E " = cos(χ) note that atinanumerical certain angle γ with respect totemperatures. the axis,elastic d" · E and chemistry at ultralow dS = r2and sinunderstanding θdθdφ = r2physics dΩcalculations. !quantization ! For example, ! to !chemical reactions and trap process, whereas inelastic collisions normally lead " " states |I !|I M !|S M ! as collisions play roles keeping atoms and molecules in thermal equilibrium during Li MILi !|S Li M SLiin Rb I Rb S Scattering detector |SM !d (R)#SM | V̂ (R) = −E( E · d) S S S (8) SS ! MS MS! S E Rb S S in terms dφ with χ the angle between θ andRb!γ. ! It can be written of the first-degree Legendre d! a cooling process, whereas inelastic collisions normally lead to chemical reactions loss. Due to the extremely low density of ultracold quantum gases, the binary S M #SMS |V̂ (R)|S MS ! = VSS(R) δSS ! δMS MS! . (7) Let 2.1 Scattering amplitude and cross sections and polynomial trap loss. Due to the extremely low |I density of ultracold quantum gases, the Mstudying collisions are usually dominant. By two-body ultracold collisions, one can SRb ! = Li MILi !|S Li MSLi !|IRb IRb !|SRb M The in denote a 3j-symbol. Thein ultracold operator V̂ (R) is  Eq. . product with the where scattered density Jdot A parentheses " · d"current sc binaryFigThe collisions are dominant in determining theof collision dynamics in the(6) gases. By The E represents the thedΩ dipole moment vector d" ofstudy the LiRb dimer and of collision dynamics quantum gases i typical set-up of a scattering experiment !operator ! scattering A B Thestudying parentheses Eq.extract (6) denote aS3j-symbol. Thes-wave V̂ (R) islength diagonal into the parameters such asextract the used model many-body ! !ultracold in S S collisions between two particles, one can parameters such ! ! Li Rb #SM | V̂ (R)|S M ! = V (R) δ δ . The study of collision dynamics in ultracold quantum gases is very important in unMS derstanding physics and at ultralow temperature S functions SSof(6) MSchemistry M " dSstates cos(χ) =S !P1 (cos(χ)) and  Scattering Scross 2.2. section |ILi Mland !|I Mtemperatures. !|SM (−1) (2S +properties 1)1/2 the vector of scattering the external electric E, denotes the moment LiRb inS (9)introduces nuclear spin and quantum numbers. Idipole Rbm Samplitude Li lIRb offield ultracold atomic and molecular ensembles. This Chapter derstanding physics and(") chemistry at ultralow For example, elastic Scattering beam as thespin s-wave length many-body phenomena. This chapter nuclear states and l Figure andtomexplore quantum numbers. 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Due to For the extremely low ultracold quantum the atinanumerical certain angle γ at with respect to the quantization axis, d" describes · density E hasofand the form d" ·Due E = The parentheses in Eq.and (6) a and 3j-symbol. The operator V̂low(R) isof diag understanding physics and chemistry at ultralow temperatures. For example, elastic understanding and chemistry ultralow temperatures. example, calculations. ! ! 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Due to the extremely low density of ultracold gases, " " " inelastic collisions in a quantum space-fixed and an the approach to LiRb implement theasection theory 2.1 The scattering amplitude and cross FigThe typical set-up of a scattering experiment dΩ where E · d represents the dot product of the dipole moment vector d of the dimer and The be written inthus the form e system whichThe describes aJtrap scattering experiment. The parentheses in Eq. (6) V̂denote a can 3j-symbol. The V̂ (R) is of diagonal the beam . Aloss. proportionality coefficient dσoperator can begases defined aswhereininVSunE (R) inc andoperator Due to the extremely low density ultracold quantum gases, (R) the adiabatic interaction potential ofto t inelastic collisions in adenotes space-fixed frame and an the approach in numerical calculations. The study of collision dynamics in ultracold quantum is very important " " cos(χ) P1 (cos(χ)) (9) where EE, represents dot product of theofdipole moment vector d" of the binary collisions the are vector dominant in external determining thefield collision dynamics inthe the gases. &the ' By=functions "· ddand of the electric denotes dipole moment LiRb nderstanding physics and chemistry at ultralow temperatures. For elastic 2 S.example, To elastic evaluate thein matrix elements of the interaction potenti derstanding physics and chemistry at ultralow temperatures. For example, Scattering beam (") along z-axis, thenuclear distance between the detector the S in numerical calculations. spin states and l and m quantum numbers. −2 lare The dscattering D exp −α(R − Re )the collision dynamics (10) S (R) = in 2.1.1 experiment & ∗ two ' binary dominant in the gases. By 4π collisions ∗in extract ∗such studying ultracold collisions one can parameters 2φdetermining collisions roles keeping atoms and molecules in! thermal ! states |ILi MI !|SLi M(2.14) "(θ, S !|IRb MI !|SRb MS ! as 2.1 Scattering amplitude and cross = between Y φγparticles, )Y (θ, φplay )gases + Y (γ, )Y (θ, φJθinc )electric + Yin (γ, φγsections )Yequilibrium φSθ )isduring dN = r J dΩ = dσ, the vector of the external field E, d denotes the dipole moment fun 1−1 θelectric γsc 10 11 1−1 (γ, and 10 11 the different spin states E is the field magnitude. If the electric field oriented The study of collision dynamics in ultracold quantum is very important en r and z-axis is θ, and the projection angle of " r on the The operator V̂3E (R) canatoms be written inprocess, themolecules form a cooling whereas inelastic collisions normally lead to chemical reactions "−2particles, " two ollisions instudying keeping and in thermal equilibrium during z typically |SM !d (R)#SM V̂ (R) = −E( E · d) collisions between one can extract parameters such as theplay s-waveroles scattering length toultracold explore many-body phenomena. This chapter E S S S |uniform A scattering experiment is carried out as shown in Fig. 2.1. 2.1 Scattering amplitude and cross sect |ILi M !|IR A scattering experiment is typically carried out in the way shown in Fig. 2.1. A and trap loss. Due to the extremely low density of ultracold quantum gases, the I !|SLi MS "elastic "· E with the parameters Rthe =solid 7.2the bohr, α = dΩ 0.06 bohrand and D = 4.57 Debye forcos(χ) the singletA "Ehas " = physics chemistry at ultralow For example, etemperatures.   aceunderstanding element dS subtending the scattering angle at and a certain angle γ with respect to quantization axis, d · E the form d different spin states is the electric field magnitude. If the electri Figure 2.2: ! ! Target uniform incident beam α of particles with a certain collision energy and current binary collisions are dominant in determining the collision dynamics in the gases. introduces a time-independent scattering theory for two-body elastic and inelastic ! ! where φ and φ are the projections of angles γ and θ on the (x, y) plane and Y are spherical S S S incident beam α of particles with a certain collision energy and current density J S M Incident beamknown (!) γ the θ xx Rb inc as s-wave length explore phenomena. This which is asscattering theE scattering cross section. It has the(8) dimension S M " 0.045 "differential −2to  Lichapter process, whereas inelastic collisions normally lead to chemical reactions (−1) (2S + 1)1/2 carried |SM !d |many-body V̂between (R) = −E( ·density d) Transmitted beam collisions play roles in keeping atoms and molecules in equilibrium during Li M Sand S and R = 5.0 bohr, = bohr D 1.02 Debye for the triplet state. scattering experiment is These typically out in the |I way Jinc is incoming on Sbetween a(R)#SM target containing collision centers. Particles can then By α studying ultracold collisions two particles, one can extract parameΩ.cooling eE with χstate, theframe angle θ and γ. Itthermal can be written in = terms of theA first-degree Legendre −MS of the m MS" Mpotential " where VS (R) the adiabatic interaction S Sdenotes M collisions in a space-fixed and provide an approach to implement the theory is incoming on a target containing scattering centers. Particles can then be scattered at a certain angle γ with respect to the quantization axis, d · E has the for harmonics. Sthe Minto be scattered different directions and the number of outgoing particles in a solid S of length×length and may be considered as an effective area of the incident beam ters such as s-wave scattering length to explore many-body phenomena. This uniform incident beam α of particles with a certain collisi a time-independent scattering theoryamplitude for two-body elastic and cooling process, whereasintroduces collisions normally lead to chemical 2.1. of Thereactions scattering and crossthesection S. To evaluate matrix elements of theinelastic interaction potential V̂ z low nda trap loss. calculations. Due toinelastic the extremely ultracold quantum the and note that gases, polynomial angle dΩ density is detected by a scattering detector. wheretheory V the adiabatic interaction potential ofcollision the molecu chapter introduces a time-independent scattering for denotes two-body elastic and S (R) in numerical into different directions and the number of outgoing particles in aa target solid angle dΩ " " " density Jinc is incoming on containing cent 5 The dipole moment functions are represented by the expressions where E · d represents the dot product of the dipole moment vector d of the LiRb where particles are scattered into the solid angle dΩ. Integrating dσ over all the states |I M !|S M !|I M !|S M 2 2 Li I written Li S Rb in I Rb S ! as with χ the angle and γ. It can be terms of the first and trap loss. Due to E extremely the low density of of ultracold quantum gases, " "the dS = frame rdipole sin =between r dΩframe inelastic collisions in aθdθdφ space-fixed and θ an the approach to16 implement the theory in dot aper space-fixed and provide an approach to implement the theory where · d"collisions represents product the moment vector d of the LiRb dimer and Chapter 2 S. To evaluate the matrix elements of the interaction (R), w ! potential ! numberV̂of be scattered into different directions and the outg unit timeLetcan bethe detected by a scattering detector.inThe current density ofMthe #SMS |V̂ (R)|S S ! = VS (R) δSS δM in numerical calculations. ave traveling along the z-axis inary collisions are dominant in determining collision dynamics the gases. By angles, one obtains the integral scattering cross section binary collisions are dominant in determining the collision in the gases. By Scattered beamdynamics z |I M !|S M !|I M & ' Li I Li S Rb I " cos(χ) = P (cos(χ)) (9) angle dΩ is detected by a scattering detector. |ILifunctions MILi !|S !|I !|SRb MSRb !  as 1 Li Mof "JRscmoment Rb MIin the the vectorvector of external electric fielddbeam E,with delectric denotes the dipole LiRb  polynomial the scattered current density .Assuming A denotes 2sc inthe numerical calculations. Li Rb S= scattered β denoted as .Jstates thatSthe the incident beam is coming of the external field E, dsuch dipole function Dis −α(R S S (R) e) The! in Eq.moment (6) denote The oper ! = r(10) 2 dΩ 2.1 ultracold Scattering amplitude and2.1cross sections SLi aS3j-symbol. S Rb & ∗two particles, ' parentheses Zexp Z −and Background Material studying collisions between one can extract parameters dS = r2 sin θdθdφ 4π M 1/2  |ILi MI !  ∗ ∗ (−1) (2S + 1) Scattering amplitude cross sections tudying ultracold collisions between two particles, one can extract parameters such dσ the different spin states andφalong E is the electric field magnitude. If the electric field is oriented = Y1−1 (γ, )Y (θ, φ ) + Y (γ, φ )Y (θ, φ ) + Y (γ, φ )Y (θ, φ ) nuclear spin states and l and m quantum numbers. a space-fixed z-axis, the distance between the detector and the target is r, lMand M 1−1 θ γ 10 θ γ 11 θ S 2.1. −MRb 10 11 Scattering amplitude sections S Mof S Licross |ILi MIinteraction !|S MSSpotential !|I !|Smo M R IRb Let ikz length3 to exploreγ many-body where VS (R) denotes adiabatic the Li Li σ= dσ the = This ( chapter )dΩ. (2.15) r and as the s-wave phenomena. the ψspace-fixed Aethe , different (2.5) −2 spin states E is electric field magnitude. If electric field The operator V̂E (R) canthe beultracold written inthe gases formis very inc =scattering A particles scattering experiment isB typically carried out in= the way shown in Fig. 2.1. The study ofAcollision dynamics in quantum dΩ " " with the parameters R = 7.2 bohr, α = 0.06 bohr and D 4.57 Debye for the singlet " " and the angle between r and the z-axis is θ, the configuration of the scattering In a frame, collisions between two A and can be treated as a e cos(χ) = P 4π S.·and To evaluate the2matrix elements of the interaction potentialimport V̂(c (R with the scattered current density J . A ! ! at a certain angle γ with respect to the quantization axis, d · E has the form d E = cos(χ) note that 1 sc S S S 2 Figure 2.2: Li Rb s the s-wave scattering length to uniform explore many-body phenomena. This understanding chemistry at ultralow temperatures. For example, = rMsin θdθdφ = 1/2 r physics dΩ Sand incident beam α of particles with a certain collisiondS energy current   andchapter ! introduces a time-independent scattering theory for two-body elastic andthe inelastic |I! (−1) (2S + !|S 1) Li MILi !|IRb M where φγand and φθ=are projections of angles andD θ yon (x, y) plane and YChapter are "as in " ·!d) states MI spherical M !|Ikeeping !|S MSmolecules −2γ xx|Icollisions play roles and equilibrium Lipolar Licoordinate S in I atoms Rb |SM = system, −E( E be projected on awhere space-fixed spherical asthermal S !dS (R) &and problem of awith particle interacting with aα fixed scattering center acollision dθ Particles Rbetween 5.0the bohr, = It 0.045 bohr = 1.02 Debye for the triplet state. 4π MRb Mpotential density Jcan incoming on a target containing can then2 ! ! S of the molecule "These " in e "V̂LiME(R) Vcentral denotes the adiabatic interaction inc is S SRb −M Scenters. M χstate, theCangle θexperiment and γ. can be written inthrough terms of the first-degree Legendre S (R)  unctions m m are represented by the expressions (2.1) m +m & ' Chapter d (R) 2= D exp −α(R − R )  Background Material  (10)  Material =Background 7.2 bohr, α = 0.06 bohr and D = 4.57 Debye for the singlet  r, α = 0.045 bohr  and D = 1.02 Debye for the triplet state. These Li  Li  Rb  Rb  Li  5  Li  S  Li  S  Li  Li  Rb  Rb  Rb  !  Li  S  Li  R  S  Li  Li  S  2.1  Scattering amplitude and cross sections  Rb  ∗to the ∗acorresponding ∗ SVSform #SM |V̂ M != (R) δM d .· cooling process, inelastic collisions lead δtoSSchemical Shas M rea at a Itframe certain angle γ approach with quantization axis, dwhereas · inelastic E the M Snormally can and be interpreted as an respect effective area the incident toθ(R)|S collisions inaa space-fixed provide an todifferent implement the theory = Yare (γ,of φtwo-body )Y (θ, φnumber +particles Y10 (γ, φ (θ, φ )low+by Y (γ, φquantum ( Let γthe 1−1 θ )beam γ )Y 10 γ )Y11 be scattered into directions and number of outgoing a solid 1−1for 11 ntroduces scattering theory elastic and andinparticles trap loss. Due beam to (denoted the extremely density of!|S ultracold gas harmonics. show in Fig.a2.2. We interested in the of dN ) Scattered |I M M !|I M potential (r), where C hasofa the reduced mass φ n factor. TheVtime-independent interaction wave with scattering Li I Li S Rb I S. To evaluate the matrix elements of the interaction potential V̂ (R), we w 3 by a scattering detector.and note thatrwith The z angle dΩ is detected  moment . product the where scattered density Jdot A " · d"current sc binary E collisions areEq. dominant determining theof collision dynamics in the gasv polynomial represents theindenote the dipole parentheses in (6) a 3j-symbol. The operator in numerical calculations. scattering in all directions. 52 between In athe space-fixed frame, collisions two particles ARbstudying and B can be treated asextract a paramete ! !subtends Material SLiparticles, SRb ultracold collisions between two one S can unit time passing a surface element dS Background which vector M M 1/2  states |ILi M16 !|I M !|S !the as "solid The χ dipole moment functions by the expressions  |ILi dSare = r2represented sinθ θdθdφ =through r dΩγ. I !|S Li MS nuclear I (−1) Rb Sl+and with angleper between and It can be written in terms of the first-degre MIdipol !|I (2S 1)theory the vector of the external electric field E, dS denotes the spin states and m numbers. ollisions in ψascspace-fixed frame an approach to implement the l quantum as2.1. the s-wave scattering length to explore many-body phenomena. This c tered wave . If the scattering isand isotropic, i.e. the m mBprovide Scattering amplitude and cross sections Figure 2.1: A Chapter 2 −M M M ! ! where V (R) denotes the adiabatic interaction potential of the molecu S S S S M S LetdSφis where and φ are the projections of angles γ and θ on the (x, y) plane and #SM | V̂ (R)|S M ! = V (R) δ δ . introduces a time-independent scattering theory for two-body elastic and µ = (2.1) angle dΩ. then given by S S SS x the different spin states and E is the electric field magnitude. M M γ θ problem of a m particle C interacting with a' fixed center through y inbeSultracold The operator V̂E (R) can writtenquantum inathecentral formis very important ini Scattered Axis of incident beam The study of collision dynamics gases = P1scattering (cos(χ)) (9) Chapter 2 mbeam & collisions in a space-fixed frame provide an approach thew S. To evaluate the matrix elements ofand the interaction V̂Rb (R), with the scatteredzcurrent density Jsccos(χ) . A A + can B be MIrespect !|S MtoI implement !|Saxis, M Chapter 2Rbpotential 16|I 2and 2physics Li Li M Sthe!|I note that 2 at a certain angle γ with to quantization d"S· s is equally probable, the scattered wave described understanding and chemistry at ultralow temperatures. example, elasti dS = r sin θdθdφ = r dΩ ! ! For calculations. polynomial in numerical n numerical calculations. dS (R) = D exp −α(R − Re ) (10) 2.1 Scattering amplitude and cross sections & ∗ ' 4π ! " " states |I M !|S M !|I M !|S M ! as potential V (r), where C φ collisions play roles in keeping atoms and molecules in thermal equilibrium durin Li I Li S Rb I Rb S harmonics. Scattering detector |SM !d (R)#SM V̂ (R) = −E( E · d) ∗ has a reduced mass ∗2The! Ea 3j-symbol. S S inV̂term parentheses in Eq. (6) denote The operator (R dφ ! 2 with χ the angle between θ and γ. It can be written d! S S S = Y1−1 (γ, φγ )Y1−1 (θ, φθ ) + Y10 (γ, φγ )Y10 (θ, φθdS ) +=Y11 φθdθdϕ φrθ )dΩ Li inelastic Rb γ )Y11 (θ, ! S(2.1) Background Material a cooling process, normally lead to δchemical reaction r (γ, sin =M  whereas #SMS |V̂collisions (R)|S ! M = Li VM (R) δMMM . S SSRb S ! cross |I M (−1) (2S + 1)1/2 3 Let I !|I I !| wavefunction r 2.1 Scattering amplitude and sections Background Materi polynomial trap Due toquantum the extremelynumbers. low |I density of ultracold quantum gases, th nuclear2.1. spin states and and l loss. and mlM LiSMI !|SLi MS !|IRb MI !|SR −M M Scattering amplitude and cross sections −2 S S Background Material S M In a space-fixed frame, collisions betweenRtwo particles B moment can be treated as a TheAdipole functions are represented by the molecule . potential with the scattered density Jdot A expressions "current " where V (R) denotes the adiabatic interaction of the in sc binary collisions are dominant in determining the collision dynamics in the gases. B with the parameters αand = 0.06 bohr and D = 4.57 Debye for the singlet where E · d represents the product of the dipole moment vecto S e = 7.2 bohr, FigThe typical set-up a scattering experiment dΩ mA m Boperator The parentheses in Eq. ofbetween (6) denote aS3j-symbol. Theparameters operatorsucV ! ! SThe The V̂are (R) can be written in the form ultracold collisions two one S candynamics extract in Liparticles, Rb study of ultracold quan Estudying Multracold 1/2 whereCφγinteracting and φθ are the projections of angles γcenter and θ on the (x, y) plane and Y spherical The study of collision dynamics in quantum gases is collision very important in!|IRb "=  µ = (2.1) xx x |I M M (−1) (2S + 1) the vector of the external electric field E, d denotes the dipole mo cos(χ) P (cos(χ) Li I S S. To evaluate the matrix elements of the interaction potential V̂ (R), we writ problem of a particle with a fixed scattering through a central derstanding physics and1 chemistry This at ultralow te and noteAxis that Scattering beam (") as thespin s-wave scattering many-body phenomena. chapte nuclear states and l Figure andtomexplore quantum numbers. of incident beam ikr and Re = 5.0 bohr, α = 0.045 bohr−2 and yD = dS lM 2 sin 2.1: 1.02 forB triplet state. & ∗length −M MSplayFor S understanding and atThese ultralow temperatures. example, elastic m m S chemistry M = rDebye θdθdφ =the r2physics dΩ ∗inS keeping estate, collisions atoms andinelasti A+ ! &4π introduces a time-independent scattering theory two-body elastic and the different spin states and E)Y is the field magnitude. = Y (γ, φγ! (θ, electric φθ')for +roles Y10 (γ,very φ (θ, φθmolecu )If+t 1−1 γ )Y10 1−1 The study of collision dynamics in ultracold quantum gases is important harmonics. 2 " " The operator V̂ (R) can be written in the form states |I M !|S M !|I M !|S M ! as a thermal cooling process, whereas inelastic collisions no 3 Chapter 2 E collisions play roles in keeping atoms and molecules in equilibrium during Li I Li S Rb I Rb S potential V (r), where C has a reduced mass φ |SM !d (R)#SM | V̂ (R) = −E( E · d) collisions in a space-fixed frame − and provide an approach to the ψsc,iso = A , (2.6) dθ Chapter z E S 17 S implement S theor dS (R) exp −α(R Rtemperatures. )the 4π & ∗ 2and= 2 Due "elas !with ! respect note that etrap the extremely atinD a#SM certain angle γ at to quantization axis, low d" · density E h and For ∗Chapter |V̂calculations. (R)|S Multralow != VSand (R) δloss. δareMtodominant .example, numerical cooling process,physics whereas inelastic collisions normally lead to reactions Schemistry SSchemical M S M STarget r ! ! binary collisions in θdetermining th whereφ φγ θand φ+ the projections of angles γ and onφ theθ(x = functions Y1−1 (γ, φγ )Y1−15(θ, φθexpressions )Let+ Y10∗aandunderstanding (γ, φ )Y (θ, ) (γ, φ )Y (θ, ) Incident beam (!) θ areY Scattering detector " " The dipole moment are represented by the γ 10 γ 11 |SM (R)#SM V̂density (R) molecules = −E( Estudying ·It d)can Transmitted beam collisions play roles in keeping atoms and in thermal equilibrium duri 11 with χ the angle between θ and γ. be written terms of E S !din Sbetween Sp trap loss. Due to the extremely low of ultracold quantum gases, the By ultracold collisions two !Li M ! S !|IRb MI Scattered beam mA mB |I M !|S !|S M Rb S Li I #SM | V̂ (R)|S M ! = V (R) δ δ . 3 S S SS M M harmonics. M S  r ters such as Sthe s-wave scattering length to expl 17  with the scattered current density J . A " " 2.1 Scattering amplitude and cross sections " Background Material a cooling process, whereas inelastic collisions normally lead to chemical reactio sc binary collisions are dominant in determining the collision dynamics in the gases. By µ = (2.1) where E · d represents the denote dot product of thechapter dipole moment vector d is of x polynomial TheA parentheses in Eq. (6) a 3j-symbol. The operator V̂ (R) introduces a time-independent scatterin Axis of incident beamtwo −2 Background Material n a space-fixed frame, collisions particles and BThe can be treated as arepresented ! ! mA +between mB & elastic The dipole functions are bysuch the expre Sbohr Sdot Scan with the parameters R' Background = 7.2 bohr, α = and D = 4.57 Deb studying ultracold collisions between two particles, one parameters Limoment Rb and trap loss. Due to E the extremely of extract ultracold gases, unction has the same normalization factor A for "arepresents "Material collisions in a quantum space-fixed frame and an t 16 FigThe typical set-up of alow scattering experiment dΩ dipole Mdescribes where ·0.06 d1/2 the product of the moment vector "density   Figure 2.2:−α(R The coordinate system which scattering experiment. The parentheses in Eq. field (6) denote ainelastic 3j-symbol. The operator V̂!|S (R |I M !|I M (−1) (2S + 1) the vector of the external electric E, d denotes the dipole moment dS (R) = D exp − Re )2 enuclear (10) Li I Rb I S spins-wave states and and mto quantum numbers. in phenomena. numerical calculations. The study of collision dynamics in ultracold quantum ga l in asbinary the scattering length explore many-body This chapter cos(χ collisions are lvector dominant determining the collision dynamics in the gases. Scattering and cross sections &the ' Figure 2.1: " dand −M M Mm theamplitude of theand external electric field E, denotes dipole mom where VS (R)2.1. the adiabatic interaction potential of the molecule in the S S and Sl quantum derstanding and chemistry at ultralow Scattering beam (") Sdenotes M beam is coming along z-axis, the distance between the detector the S= nuclear spin states numbers. −2 dSphysics (R) D exp −α(R −xx Rtempera )2 a whereCφanisotropic. theincident projections of angles γ θ on the (x, y) plane and Y eelec & is∗ lD introduces aand time-independent scattering theory for two-body elastic and inelastic 4π the different spin states and E the electric field magnitude. If the γ and φθ areThe roblemwave of a isparticle interacting with a fixed scattering center through a central ∗in extract ∗su attered normally anisotropy of studying ultracold collisions between two particles, one can parameters state, and R = 5.0 bohr, α = 0.045 bohr and = 1.02 Debye for the y collisions play roles keeping atoms and molecules in The operator V̂ (R) can be written in the form e 2.1 Scattering amplitude and cr E =ultracold Y (γ, and φγ )Y (θ, φisθelectric )gases + Y (γ, φthe (θ, φθ ) + Ytri (γ 1−1 γ )Y 10 The study of collision dynamics in gases very in 1−1quantum 10important 11 the different spin states E is the field magnitude. If the Chapter 2 The study of collision dynamics in ultracold quantum is very important in target is r, the angle S. between r and z-axis is θ, and the projection angle of " r on The operator V̂3Ethe (R) can be inprocess, theimplement form a cooling whereas inelastic collisions normally collisions in a space-fixed frameofand provide anwritten approach to theory To evaluate the2s-wave matrix elements potential V̂ (R), we write th zthe as the scattering length tointeraction explore many-body phenomena. chap 2 " bohr "lowThis note −2 at athat certain angle γand with respect totemperatures. theRquantization ·E has with the parameters Re = 7.2 bohr, α = 0.06 bohr−2dSand Dsinunderstanding = 4.57 Debye forchemistry the singlet A scattering experiment isdtypically out and atChapter ultralow For example, elastic loss. Due to the density of ul = r2and θdθdφ = r physics dΩphysics "carried with the parameters =trap 7.2 bohr, α axis, = extremely 0.06 and D " the understanding chemistry at ultralow temperatures. For example, elastic eand  2.1  Li  Li  Rb  Rb  S  !  S  Li  Li  Li  S  Rb  Li  Li  Li  Rb  Li  Li  !  S  Li  Rb  ! S  Rb  Rb  S  !  S  Li  Li  S  Scattering amplitude and cross sections 16  Li  Li  Rb  Rb  S  Li  ! S  S  Li  Rb  S  Li  ! SRb  Rb  S  Li  Li  S  Rb  Rb  Rb  !  S  Li  Li  ! S  S  ! Rb S  S  ! S  Rb  S  Li  Li  S  Rb  Rb  !solid y) is ϕ. The surface element dScalculations. subtending thescattering scattering angle dΩ at a certain angle γ with ! respect to the quantization axis, d · E has in numerical Figure 2.2: ! ! γ and may be described by a C modulation factor(x,fmass (k,plane θ, φ ϕ). In ain time-independent theory two-body elastic inelas where φ=and φ molecules projections ofequilibrium angles θ on |and the (x, y) p "as "are states |Icollisions M introduces !|S !|I M !|S M otential V (r), where has a reduced play M rolesroles keeping atoms and in thermal during harmonics. "for "α· = |SM !d |SM (R)#SM V̂between (R) −E( ·!the d) !d (R)#SM | of V̂E (R) = −E( E d) collisions in keeping and molecules thermal dφ= r 2 dΩ. and R = 5.0 0.045 D 1.02 Deb is dS−2= r2 sin θdθdϕ with χ play the angle θ and γ. Itbohr, be written in and terms ofduring the fit with χstate, the atoms angle between θcan and γ.in It canbohr be equilibrium written in = terms  state, and Re = 5.0 bohr, α = 0.045 bohr  Li  SLi  Target  incident beam αinofdetermining particles with cer binaryuniform collisions are dominant the acolli −2 Sinc S beam S Transmitted  beam (!) γ Rb θSRb Scattering detector Rb E IRb Incident  ILi  Li  2.1  S cross S S z sections Scattering amplitude and Background Material  S on Sbetween density ultracold J is incoming a target containing co By studying collisions two Sparticle eE 5 and inwhereas a space-fixed provide an approach todifferent implement the theo ! normally ! S ters and D = 1.02 Debye forcollisions the triplet state. These a cooling process, inelastic|frame collisions lead chemical reactions harmonics. S M!into beto scattered and the num as δ the scattering length to explore m #SM M ! =normally VM (R) δtoM S V̂ (R)|S SSsuch SSs-wave M ! . directions a cooling process, whereas inelastic collisions lead chemical reactions  Let  S  polynomial angle dΩ is detected by a scattering detector.theo chapter introduces a time-independent scattering in loss. numerical and polynomial trap Due tocalculations. the extremely low moment density ultracold 5bythe The dipole functions represented the != |I M of !|S M arequantum !|I M gases, !|S Mexpressio  Material The dipole moment functions by the expressions mABackground mB are represented (2.1) x µ= mA + mB & ' Chapter d (R) 2= D exp −α(R − R )  2 sin θdθdφ SRb Li ILiproduct Li of of Sultracold IRb and trap loss. Due to E extremely the lowdot density gases, "the dS = rRb = r2 dΩRb Li inelastic collisions in a quantum space-fixed frame and an the approa where ·in d"Jrepresents the dipole moment vector d" of  . product the where scattered density A " · d"current sc binary collisions are dominant determining theof collision dynamics in the gases. By " E represents the thedΩ dipole Letmoment in numerical calculations. vector d of the FigThe typical set-up of(6) a dot scattering experiment be considered as awith plane wave traveling along the z-axis 5 binary collisions are dominant in determining the collision dynamics in the gases. Bydi Scattered beam The parentheses in Eq. denote a 3j-symbol. The operator V̂ (R) & '.is " ! !ultracold collisions the vector of the external electric field E, d denotes the dipole moment S S S with the scattered current density J2sccos(χ) Ais ve studying between two particles, one can extract parameters such S Li Rb The study of collision dynamics in ultracold quantum gases dS (R) =sections D exp −α(R cos(χ) − Re ) = P M S "particles,  2.1 Scattering amplitude and cross & field |ILi M !|Ichapter Mtemperatures. !|SM ! (−1) (2S +length 1)1/2 the vector of the external electric dS denotes the dipole func studying ultracold collisions between twoE, one can extract parameters such 4π derstanding Rbmoment IRb Liamplitude ∗ IThis ∗ S Fs 17 2.1 Scattering and cross and chemistry at ultralow Scattering beam as thespin s-wave scattering phenomena. the spin states and Enumbers. is the electric field magnitude. nuclear states and l Figure andto mexplore quantum = Y ∗ many-body (γ, φγphysics )Y−M (θ, φ(") (γ, φ Y (γ, Axis of incident beam 1−1 θ ) + Y10 γ )Y10 (θ, φθ )If+the lM 2.1: 11 elec & different M ikz 4π S keeping SLi1−1 Splay Sas M 3 r Rb S s-wave the scattering length to explore many-body phenomena. This chapter ∗ ∗ ∗ collisions roles in atoms and molecules in thermal ψinc ,(γ, (2.5) −2 16=inatAe introduces time-independent scattering theory two-body elastic and inelastic A scattering experiment isφ typically carried out the the different spin states and is electric field magnitude. If+the electric = Y φangle )Y (θ, )for Yto (γ, φ (θ, )can (γ, )Y with the parameters = 7.2 bohr, αA10 =and 0.06 bohr and Dφinthe "treated γE 1−1 θR γ )Y θaxis, γ= In aaspace-fixed frame, collisions between particles B be a e+two 10 a1−1 certain γ the with the quantization d"Y ·11 E has The study of collision dynamics ultracold quantum gases is very important in Figure 2.2: 2φrespect The operator V̂3E (R) can be written in the form a cooling process, whereas inelastic collisions normally uniform incident α of particles with a lead certaintoc introduces a time-independent scattering theory for elastic and inelastic collisions in a space-fixed frameφand provide an approach totwo-body implement the theory z beam where and φθ=are the projections of angles and θthrough on the (x, y) pla −2γ γand problem of a particle C interacting with a fixed scattering center a cent Chapter 2 y " state, R 5.0 bohr, α = 0.045 bohr and D = 1.02 Debye " density J is incoming on a target containing collision and note that S e e inc and trap loss. Due to the extremely low density of ultracold with χultralow therespect angle and between and γ. ItFor canexample, be in terms of the fifq at acollisions certain γ at with toprovide theθ quantization axis, d·E has form understanding physics and chemistry temperatures. elastic inangle a space-fixed frame an approach towritten implement thethe theory numerical calculations. be scattered into different directions and the number of ! ! Target harmonics. collisions are in θdetermining the y) collision dyna potential Vbeam (r), C has reduced mass φdominant where A is a normalizationinfactor. interaction ofabinary the wave with a scattering where φThe φ(!) are the projections of angles γ and on the (x, plane Incident γ and θ where z Scattering detector " " polynomial dΩ is detected by a scattering detector. calculations. |SM (R)#SM | the first-d V̂Eand (R) molecules = −E( E ·It d)can Transmitted beam collisions play roles inin keeping atoms in thermal equilibrium during with χnumerical the angle between θdipole and γ. beangle written terms of 5 two S !din Sbetween Sparticles, By ultracold collisions one ca ! studying ! The moment functions dSare = r2represented sin θdθdφ = r2 dΩby the expressions 1 ! . the ! δM i.e. center gives rise to a scattered wave ψsc . #SM If the scattering is isotropic, Msuch V (R) δSS mSA mSBLet S |V̂ (R)|S S M S! = SM ters the s-wave scattering length to explore many-body S 2.1harmonics. Scattering amplitude and sections =as lead (2P a cooling process, polynomial whereas inelastic collisions normally to chemical reactions x µ cross Scattered beam Axis of incident beam cos(χ) = m + m & ' with current density Jtheory chapter introduces a time-independent scattering for tw A B the scattered sc . A −2functions scattering into all directions is2.1 equally probable, the scattered wave can d (R)be = described D −α(R −" R e )2 Scattering amplitude and cross sections The dipole moment are represented byexpthe expressions & experiment 4π product and trap loss. Due to E extremely density ofφof ultracold gases, "the collisions in aSquantum space-fixed and an the approach to Li ∗ inelastic ∗ e FigThe alow scattering dΩ dipole ·typical d" represents the dot the d is of11∗ (γ, the = Y1−1 (γ, )+ Ymoment (γ, φγframe )Y10vector (θ, φ(R) φγim )Y The where parentheses inset-up Eq. of(6) denote a 3j-symbol. V̂ diagon γ )Y1−1 (θ, φθThe θ) + Y 10operator 3 The study by a spherically symmetric wavefunction r in numerical calculations. of collision dynamics in ultracold quantum gases is very imp cos(χ) =DPas= binary collisions are dominant in determining the collision dynamics in the gases. By 1 (co −2 & ' In a space-fixed frame, collisions between two particles A and B can be treated a " with the parameters R = 7.2 bohr, α = 0.06 bohr and 4. the spin vector of theand external electric field E, de beam denotes the dipole moment functi 2 derstanding and chemistry at ultralow temperatures. For exa Scattering (") S= nuclear states l andφm numbers. −2 l quantum dSphysics (R) D exp −α(R R where and φθ are the projections of angles γcenter and−θthrough one )the (x,a y) plane a 4πa&particle problem between C γinteracting acan central ∗ ∗in −2 molecules yD =∗such studying ultracold collisions two one extract parameters collisions play roles keeping atoms in thermal ikr state, and R(θ, =φwith 5.0 bohr, α(γ, =scattering 0.045 bohr 1.02 Scattering amplitude and cross = of Y φγparticles, )Y )gases + Yfixed φγ )Y (θ,and φθand )+ Yin (γ, φDebye )Yequilib (θ e 2.1 1−1 θelectric 10 γsection 11for e 1−1 (γ, 10is 11 the different spin states and E is the field magnitude. If the electric fi The study of collision dynamics in ultracold quantum very important The operator V̂3VE(r), (R) can be inprocess, the form aa cooling whereas inelastic collisions normally lead to chemi harmonics. potential C has reduced mass φ ψsc,iso = A ,written (2.6) z as the s-wave scattering length towhere explore many-body phenomena. This chapter −2 A scattering is typically carried out in= the4.57 way sho and trap loss. Dueexperiment to the extremely low 5 density of ultracold quantu r " with the parameters R = 7.2 bohr, α = 0.06 bohr and D " understanding physics and chemistry at ultralow temperatures. For example, elastic e at a certain γ with respect to the quantization axis, dby· E the formD The dipole moment functions are represented thehas expressions 16angle Figure 2.2: ! ! Target theory m incident αinofθdetermining particles certain collision e binaryuniform collisions are the ay) collision dynamics A mdominant B beam introduces a time-independent for elastic inelastic where Incident φγ and φscattering oftwo-body angles γ−2and on and thewith (x, plane and Y beam (!) θ are the projections µ = (2.1) " " x 2 2 |SM !d (R)#SM | V̂ (R) = −E( E · d) Transmitted beam play roles in keeping atoms and molecules inIt thermal equilibrium during Axis of incident beam Sand Sbetween Sparticles, and the Rbetween =same 5.0 bohr, = 0.045 bohr D 1.02 Debye for the density Jinc is incoming on a target containing collision centers. By α studying ultracold collisions two can extr m m ' onefirst-de eE A+ B with χstate, theframe angle θnormalization and γ. can be written in&= terms of the wherecollisions thecollisions scattered has factor A for elastic 2 in wavefunction a space-fixed and provide an approach to implement the theory d (R) = D exp −α(R − R ) emany-body harmonics. Minto be scattered directions and the number of outgoing S Sdifferent ters such as Sthe s-wave length to explore phen a cooling process, whereas inelastic collisions normally lead to scattering chemical reactions zanisotropic. collisions. scattered wave is normallychapter The oftheory for two-bod polynomial angle dΩ is detected by aanisotropy scattering detector. introduces a time-independent scattering inHowever, numericalthe calculations. 5 The dipole moment functions are represented by the expressions −2 27.2 and trap loss. Due to E extremely the low density of of ultracold quantum gases, "the withdot theproduct parameters R bohr, 0.06 bohr D to=16 4.57 Dd dS rdipole sin =αr2 = dΩframe inelastic collisions in aθdθdφ space-fixed and an the approach implemen e== where · d" represents the moment vector d"and of the LiRb  19  Background Material  Material withFigure the parameters R system =Background 7.2describing bohr, αa scattering = 0.06 bohr and = 4.57 Debye for 2.2: The coordinate experiment. The D incident  beam is directed along z-axis, the distance between the detector and the target is state, and R = 5.0 bohr, α = 0.045 bohr and D = 1.02 Debye for the triplet s r, the anglee 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 subtending the scattering solid angle dΩ is dS = r sin θdθdϕ = r dΩ. 5  the scattered wavefunction may be described by a inmodulation factor f (k, θ, ϕ). In Letcalculations. numerical  16  −2 state, and Recollision = 5.0"bohr, α = 0.045& bohr and D =cos(χ) the binary collisions the are vector dominant in determining the in the gases. By Debye Scattered beamdynamics '1.02 = P1for (cos( 21 of the external electric field E, dS=the denotes dipole with scatteredthe current densitymoment d (R) D exp −α(R − Re )J2sc . A functions o S 2.1 ultracold Scattering and2.1cross studying collisions amplitude between oneScattering can sections extract parameters such 4π & ∗two particles, ∗ ∗ amplitude cross 5 and Y the different = spin states andφγE)Yis the field magnitude. i Y1−1 (γ, (θ, electric φθ ) + Y10 (γ, φ (γ,sections φγ )Yfield 1−1 γ )Y10 (θ, φθ )If+the 11 (θ, φ 11 electric r phenomena. This as the s-wave scattering length3 to explore many-body chapter −2 A scattering experiment is typically carried out in way shown "be· and " · in with the parameters Re two = to 7.2particles bohr, αA=and 0.06Baxis, bohr D the =theas4.57 Debye "treated " In a space-fixed collisions between can a at a frame, certain angle γ with respect the quantization d E has form d E Figure 2.2: uniform incident beamelastic α of particles with a certain collision energy introduces a time-independent scattering theory for two-body andthe inelastic where and φθ=are projections of angles andD θthrough on y) plane andtrip Yx −2γ γand problem of awith particle Cφangle interacting with aα fixed scattering center acollision central y= state, Rbetween 5.0the bohr, = It 0.045 bohr and 1.02(x, Debye for the density Jinc is incoming on a target containing centers. Parti e χ the θ and γ. can be written in terms of the first-degree collisions in a space-fixed frame and provide an approach todifferent implement thethetheory 19 be scattered into directions and number of outgoing partic harmonics. potential V (r), where C has a reduced mass φ z angle polynomial dΩ is detected by a scattering detector. in numerical calculations. 5  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 = mA mB /(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  " ddynamics ary collisions are the collision gases. By functions thedominant vector of in thedetermining external electric field E, thethe dipole moment S denotes in  dying ultracoldthe collisions between two and particles, oneelectric can extract parametersIf such different spin states E is the field magnitude. the electric field  the s-wave scattering length to explore many-body phenomena. axis, Thisd" ·chapter " has the form d" · E at a certain angle γ2.2. with respectamplitude to the quantization E 2.3. Multi-channel scattering theory Scattering and cross section roduces a time-independent scattering theory for two-body elastic and inelastic with χ the angle numerical between θ and By γ. examining It can be written in terms of the first-degre calculation. those matrices, one can extract the probability z approach lisions in a space-fixed frame and provide an to implement the theory of collisional energy transfer and thereby obtain the cross sections for both elastic polynomial numerical calculations. and inelastic collisions. C  mA mB mA + mB  cos(χ) = P(2.74) 1 (cos(χ)) ' 4π & ∗ 1 Scattering amplitude and cross sections = Y1−1 (γ, φγ )Y1−1 (θ, φθ ) + Y10∗ (γ, φγ )Y10 (θ, φθ ) + Y11∗ (γ, φγ )Y11 (θ, φθ ) 3 r a space-fixed frame, collisions between two particles A and B can be treated as a where φγ and φθ are the projections of angles γ and θ on the y (x, y) plane and Yxx ar blem of a particle C interacting with a fixed scattering center through a central harmonics. tential V (r), where C has a reduced mass φ The dipole moment functions are represented by the expressions mA mB µ= & ' (2.1) mA + mB 2 x d (R) = D exp −α(R − R ) µ=  S  e  2.3: Space-fixed coordinates for two-body Colliwith Figure the parameters Rspherical bohr, α = 0.06 bohr−2 collisions. and D = 4.57 Debye for e = 7.2polar  sions between particles A and B can be treated as a problem of a virtual particle −2 with R a ereduced µ interacting with a fixed scattering state,Cand = 5.0mass bohr, α = 0.045 bohr and D =center 1.02 through Debye the for 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 5 θ and ϕ. r and the orientation of the vector ~r is specified by angles  2.2.2  the triplet st  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  1 1 T̂ = − ∇2 = − 2µ 2µ    ∂2 ∂2 ∂2 + + ∂x2 ∂y 2 ∂z 2    .  16  (2.20)  29  23  2.2. Scattering amplitude and cross section Here and throughout this Thesis, we will use the atomic units with ~ = 1. Transforming Eq. 2.20 into the spherical polar coordinates, we obtain 1 1 ∂ 2∂ r 2µ r2 ∂r ∂r   1 ∂ 1 ∂ 1 ∂2 , − (sin θ ) + 2µr2 sin θ ∂θ ∂θ sin2 θ ∂ϕ2  T̂ = −  (2.21) (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 ˆl2 (θ, ϕ) describing the angular motion of the collision complex is defined as ˆl2 (θ, ϕ) = −     1 ∂ ∂ 1 ∂2 , (sin θ ) + sin θ ∂θ ∂θ sin2 θ ∂ϕ2  (2.23)  The time-independent Schrödinger equation then becomes "  # ˆl2 (θ, ϕ) 1 1 ∂ 2∂ − r + + V̂ ψ(r, θ, ϕ) = Eψ(r, θ, ϕ). 2µ r2 ∂r ∂r 2µr2  (2.24)  Multiplying Eq. 2.24 by −2µ we obtain "  # ˆl2 (θ, ϕ) 1 ∂ 2∂ 2 r − + k − 2µV̂ ψ(r, θ, ϕ) = 0, r2 ∂r ∂r r2  (2.25)  where k 2 = 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  "  # ˆl2 (θ, ϕ) 1 ∂ 2∂ r − + k 2 ψ(r, θ, ϕ) = 0. r2 ∂r ∂r r2  (2.26)  By solving Eq. 2.26, one can obtain the total wave function ψ(r, θ, ϕ) in the asymptotic 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, r→∞    ψ(r) −→ ψinc (r) + ψsc (r) = A e  ikz   eikr + f (k, θ, ϕ) . 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 1 [ψ(r)∗ (∇ψ(r)) − (∇ψ(r))∗ ψ(r)] 2iµ 1 = Im [ψ(r)∗ (∇ψ(r))] . µ  J(r) =  (2.28)  In spherical polar coordinates, the gradient operator is given by ∇=  ∂ ∂ 1 ∂ 1 r̂ + θ̂ + ϕ̂. ∂r r ∂θ r sin θ ∂ϕ  (2.29)  ∂ r̂, ∂r  (2.30)  Asymptotically it becomes ∇=  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 = |A|2 ν, µ  (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 |f (k, θ, ϕ)|2 . µr2  (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 = |f (k, θ, ϕ)|2 . dΩ  (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 ˆl2 (θ, ϕ) are spherical harmonics Ylml (θ, ϕ) ˆl2 Ylm (θ, ϕ) = l(l + 1)Ylm (θ, ϕ), l l  (2.34)  where l is the quantum number for the orbital angular momentum and ml is the projection of ˆl 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) = =  XX l  ml  l  ml  XX  ψlml (r, θ, ϕ) 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) =  ∞ X  Fl (k, r)Pl (cos θ).  (2.37)  l=0  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 d 2d d2 2 d r = + r2 dr dr dr2 r dr  (2.38)  26  2.3. Single-channel scattering theory lead to the radial equation    2 d d2 l(l + 1) 2 + Fl (k, r) = 0. +k − dr2 r dr r2  (2.39)  With the substitution ρ = kr, Eq. 2.39 is transformed into the spherical Bessel differential equation    2 d l(l + 1) d2 + Fl (ρ) = 0. + 1 − dρ2 ρ dρ ρ2  (2.40)  The solutions of Eq. 2.40 are Fl (k, r) = Bl jl (kr) + Cl nl (kr),  (2.41)  where jl (kr) and nl (kr) are the spherical Bessel and Neumann functions, respectively. In the asymptotic region, this radial wave function becomes r→∞  Fl (k, r) −→    lπ lπ 1 Bl sin(kr − ) − Cl cos(kr − ) . kr 2 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)  Cl = −Al sin δl .  (2.44)  and  The radial wave function (Eq. 2.42) thus has a form   1 lπ lπ Fl (k, r) −→ Al sin(kr − ) cos δl + cos(kr − ) sin δl kr 2 2 1 lπ = Al sin(kr − + δl ), kr 2 r→∞  (2.45) (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 r→∞  ψ(r) −→ eikz + f (k, θ)  eikr , 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 expanded in Legendre polynomials as [8] ikz  e  =  ∞ X  (2l + 1)il jl (kr)Pl (cos θ),  (2.49)  l=0  f (k, θ) =  ∞ X  fl (k)Pl (cos θ).  (2.50)  l=0  Asymptotically, Eq. 2.49 becomes, eikz =  ∞ X sin(kr − (2l + 1)il kr  lπ 2)  Pl (cos θ).  (2.51)  l=0  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 1 lπ eikr fl (k) sin(kr − ) + kr 2 r r→∞ l 1 −→ (2l + 1)i kr     lπ kfl (k) lπ kfl (k) × sin(kr − ) 1 + i + cos(kr − ) . 2 2l + 1 2 2l + 1  r→∞  Fl (k, r) −→ (2l + 1)il  (2.52)  (2.53)  Comparing this expression with Eq. 2.45, we get   kfl (k) (2l + 1)il 1 + i = Al cos δl , 2l + 1  (2.54)  28  2.3. Single-channel scattering theory and (2l + 1)il  kfl (k) = Al sin δl . 2l + 1  (2.55)  The constant Al and the expansion coefficient fl are thus given by Al = (2l + 1)il eiδl (k)  (2.56)  and fl (k) = =  2l + 1 iδl (k) e sin δl (k) k   2l + 1 2iδl (k) e −1 . 2ik  (2.57) (2.58)  We then obtain the expression for the scattering amplitude ∞ 1X (2l + 1)eiδl sin δl Pl (cos θ). f (k, θ) = k  (2.59)  l=0  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 r→∞  Fl (k, r) −→ (2l + 1)il eiδl  1 lπ sin(kr − + δl ), kr 2  (2.60)  which gives rise to the asymptotic form of the scattering wave function ∞ X sin(kr − lπ 2 + δl ) ψ(r, θ) −→ (2l + 1)il eiδl Pl (cos θ). kr r→∞  (2.61)  l=0  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 =|f (θ)|2 dΩ ∞ X ∞ X = fl∗ (k)Pl (cos θ)fl0 (k)Pl0 (cos θ) =  l=0 l0 =0 ∞ X ∞ X  1 k2  (2l + 1)(2l0 + 1)ei(δl −δl0 )  l=0 l0 =0  sin δl sin δl0 Pl (cos θ)Pl0 (cos θ).  (2.62)  Taking into account the orthogonality property of the Legendre polynomials [202] Z  0  π  dθ sin θPl (cos θ)Pl0 (cos θ) =  2 δll0 , 2l + 1  (2.63)  we obtain the integral cross section by integrating the above expression over all angles to give σ(k) =  Z  dΩ|f (θ)|2 Z π Z 2π dϕ dθ sin θ|f (θ)|2 = 0  0  ∞ 4π X = 2 (2l + 1) sin2 δl (k). k  (2.64)  l=0  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 the K and S matrix elements and the S and T matrix elements are Sl (k) = [1 + iKl (k)] [1 − iKl (k)]−1  (2.68)  Sl (k) = 1 + 2iTl (k).  (2.69)  and  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) = =  ∞ 1X (2l + 1)Tl (k)Pl (cos θ) k l=0 ∞ X  1 2ik  l=0  (2l + 1) [Sl (k) − 1] Pl (cos θ)  (2.70) (2.71)  and the integral cross section may be expressed as σ(k) =  ∞ 4π X (2l + 1)|Tl (k)|2 , k2  (2.72)  l=0  or in terms of the S-matrix element as σ(k) =  ∞ π X (2l + 1)|Sl (k) − 1|2 . k2  (2.73)  l=0  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  Eq. 2.25 becomes    1 d2 d2 2 d r = + , 2 2 r dr dr r dr  (2.74)   d2 + W (r) φl (r) = 0 dr2  (2.75)  31  2.3. Single-channel scattering theory where W (r) = k 2 − 2µVeff (r, l), Veff (r, l) = V (r) +  l(l + 1) , 2µr2  (2.76)  (2.77)  and φl (r) = krFl (r).  (2.78)  The logarithmic derivative yl is defined as yl =  φ0l . φl  (2.79)  Equation 2.75 can then be re-written in terms of yl as yl0 (r) + W (r) + yl2 (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 /Bl nl (kr)]  r→∞  −→ krBl [jl (kr) − Kl nl (kr)] h i r→∞ −→ Bl ĵl (kr) − Kl n̂l (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 h i φ0l (r) = Bl ĵl0 (kr) − Kl n̂0l (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 = φ0l φ−1 leads to: l h i φ0l (r) = yl φl = Bl ĵl (kr) − Kl n̂l (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 = (yl n̂l − n̂0l )−1 (yl ĵl − ĵl0 ).  (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 differential 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 representation [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. Therefore 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 Smatrix 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 Ĥas φα = α φα ,  (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 = −  ˆl2 (θ, ϕ) 1 1 ∂ 2∂ r + + V̂ + Ĥas . 2µ r2 ∂r ∂r 2µr2  (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 XXX Fα0 l0 m0l (r)φα0 Yl0 m0l (r̂). r 0 0 0 α  l  (2.89)  ml  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 ∗ (r̂), integrating over θ and ϕ, and using the orthonormality of φ and Y φ∗α Ylm α lml (r̂), l  we obtain a set of coupled-channel differential equations (the derivation is presented in Appendix A),    X ∂2 l(l + 1) 2 − + k F (r) = 2µ hφα |V̂ |φα0 iFα0 lml (r), αlm α l ∂r2 r2 0  (2.91)  α  34  2.4. Multi-channel scattering theory where kα2 = 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 l(l + 1) 2 − + k α Fαlml (r) = 0, ∂r2 r2  (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) → kα r [aαlml jl (kα r) + bαlml nl (kα r)] .  (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) → Aα kα [aαlml jl (kα r) + bαlml nl (kα r)] φα Ylml (r̂), −1/2  where Aα = να  (2.94)  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 (kα r) and nl (kα r) r→∞  aαlml jl (kα r) + bαlml nl (kα r) −→   1 lπ lπ aαlml sin(kα r − ) + bαlml cos(kα r − ) , kα r 2 2  (2.95)  we obtain the asymptotic form of the channel wave function r→∞  ψαlml −→ Aα    1 lπ lπ aαlml sin(kα r − ) + bαlml cos(kα r − ) φα Ylml (r̂), r 2 2  (2.96)  which can be re-written in terms of exponential functions as r→∞  ψαlml −→ Aα lπ  i lπ lπ 1h Aαlml e−i(kα r− 2 ) − Bαlml ei(kα r− 2 ) φα Ylml (r̂), r  (2.97)  lπ  where e−i(kα r− 2 ) and ei(kα r− 2 ) describe the incoming and outgoing waves, respectively, and  A αlml = −(aαlml + ibαlml )/2i B = (−a + ib )/2i. αlml  αlml  (2.98)  αlml  35  2.4. Multi-channel scattering theory The relationship between A and B defines the scattering matrix Bαlml =  XXX α0  l0  m0l  (2.99)  Sαlml ←α0 l0 m0l Aα0 l0 m0l ,  where Sαlml ←α0 l0 m0l can be considered as the probability amplitude of atoms or molecules to go from one incoming channel α0 l0 m0l 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 1 r→∞ ψαlml −→ Aα φα Ylml (r̂) r  lπ  × Aαlml e−i(kα r− 2 ) −  XXX α0  l0  m0l  lπ    Sαlml ←α0 l0 m0l Aα0 l0 m0l ei(kα r− 2 )  .  (2.100)  Assuming that particles in the incoming channel α are scattered into all directions, the incident plane wave can be expanded as [8] ~  eik·~r = 4π  XX  ∗ (k̂)Ylml (r̂) il jl (kα r)Ylm l  ml  l  i lπ i2π X X l h −i(kα r− lπ ) ∗ 2 (k̂)Ylml (r̂). = i e − ei(kα r− 2 ) Ylm l kα r m l  (2.101)  l  The incident wave function thus has a form ~  Aα φα eik·~r = Aα φα  i lπ i2π X X l h −i(kα r− lπ ) ∗ 2 i e − ei(kα r− 2 ) Ylm (k̂)Ylml (r̂). (2.102) l kα r m l  l  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 lπ  different from α is zero. Comparing the coefficients in front of the term e−i(kα r− 2 ) in Eqs. 2.100 and 2.102, we obtain Aαlml Aαlml =  i2π l ∗ i Ylml (k̂) kα  =0  (in  all  (in  channel α)  other  channels).  (2.103)  36  2.4. Multi-channel scattering theory Bαlml is then given by (cf. Eq. 2.99) Bαlml =  XX l0  m0l  Sαlml ←αl0 m0l  i2π l0 ∗ i Yl0 m0 (k̂). l kα  (2.104)  Note that there is no sum over α0 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 → ψ inc + ψ sc ,  (2.105)  so the scattered part of the wave function is given by ψ sc = ψ − ψ inc  inc inc = ψincoming + ψoutgoing − (ψincoming + ψoutgoing ).  (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 inc ψ sc = ψoutgoing − ψoutgoing .  (2.107)  The outgoing part of the total wave function is the sum over all energetically accessible collision channels ψoutgoing =  X (ψα0 )outgoing α0  =  XXX α0  =−  l0  m0l  (ψα0 l0 m0l )outgoing  XXX α0  l0  m0l  lπ 1 Aα0 Bα0 l0 m0l ei(kα0 r− 2 ) φα0 Yl0 m0l (r̂). r  (2.108)  Here, the normalization factor Aα0 is equal to Aα only for elastic collisions. Combining Eqs. 2.104 and 2.108 we obtain ψoutgoing XXX  l0 π 1 Aα0 φα0 Yl0 m0l (r̂)ei(kα0 r− 2 ) r α0 l0 m0l " # XX i2π l ∗ × Sα0 l0 m0l ←αlml i Ylml (k̂) . kα m  ψoutgoing = −  l  (2.109)  l  37  2.4. Multi-channel scattering theory The outgoing part of the incident wave function is given by i2π X X l i(kα r− lπ ) ∗ 2 Y ie lml (k̂)Ylml (r̂) kα r l ml XXXXX i2π l i(kα0 r− l0 π ) 2 =− Aα0 φα0 ie kα r 0 0 0 m  inc ψoutgoing → − Aα φ α  α  l  ml  l  l  ∗ (k̂)Yl0 m0l (r̂)δαα0 δll0 δml m0l . Ylm l  (2.110)  The scattering wave function is then obtained as ψ sc =ψ − ψ inc XXXXX i2π l i(kα0 r− l0 π ) ∗ 2 Y Aα0 φα0 = ie lml (k̂)Yl0 m0l (r̂) kα r α0 l0 m0l l ml h i × δαα0 δll0 δml m0l − Sα0 l0 m0l ←αlml .  (2.111)  The scattering wave function can also be written in terms of the scattering amplitude ψ sc =  X α0  Aα0 fα0 ←α  eikα0 r φ α0 . r  (2.112)  Comparing Eqs. 2.111 and 2.112, we extract the expression for the scattering amplitude fα0 ←α =  X X X X i2π l0  m0l  l  ml  ml  l  l  kα  il0 π 2  i × δαα0 δll0 δml m0l − Sα0 l0 m0l ←αlml X X X X i2π 0 ∗ = il−l Ylm (k̂)Yl0 m0l (r̂)Tα0 l0 m0l ←αlml , l k α 0 0 m l  h  ∗ (k̂)Yl0 m0l (r̂)e− il Ylm l  (2.113)  where Tα0 l0 m0l ←αlml = δαα0 δll0 δml m0l − Sα0 l0 m0l ←αlml .  The differential cross section for α → α0 transition is given by (cf. Eq. 2.33) dσα0 ←α = |fα0 ←α |2 . dΩ  (2.114)  Integrating the above equation over all orientations and dividing the result by 4π to account for the random orientation of the incoming flux, we obtain the total cross  38  2.4. Multi-channel scattering theory section [199] σα0 ←α =  2.4.2  π XXXX |δαα0 δll0 δml m0l − Sα0 l0 m0l ←αlml |2 . kα2 0 0 m l  ml  l  (2.115)  l  Numerical integration of multi-channel equations  The coupled-channel equations of Eq. 2.91 can be written in the form of a matrixvector equation    X ∂2 2 + k − U (r) F (r) = Unm (r)Fm (r), nn n n ∂r2  (2.116)  m6=n  where n corresponds to the incoming channel α whereas m 6= n corresponds to the outgoing channel α0 6= α, and  kn2 = 2µ(E − n ),  (2.117)  Unm (r) = 2µhφn |V̂ |φm i.  (2.118)  and  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 matrixmatrix 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 y0 (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 = kn2 δmn − 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) = 1030 I, 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 − N0 )−1 (yJ − J0 ),  (2.121)  where −1  [J]nm = δnm kn 2 ĵl (kn r)  (2.122)  − 21  [N]nm = δnm kn n̂l (kn r).  (2.123)  −1/2  As we include the normalization coefficient An = νn  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−1/2  plying the matrix elements in J and N by the factor kn  −1/2  ∝ νn  . 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 calculations 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 positions and widths of the resonances observed in experiments. In this Chapter, we describe a procedure of fitting the experimental data and generate accurate interaction potentials for the 6 Li–87 Rb molecule in 1 Σ and 3 Σ 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. 1  A part of this Chapter was presented in Ref. [1] of Appendix D.  41  3.1. Ultracold mixtures of 6 Li and  3.1  Ultracold mixtures of 6 Li and  87  87 Rb  Rb  The Li–Rb system is very important for the study of both ultracold atomic and molecular gases. 6 Li and 87 Rb 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 ultracold polar molecules and for the experimental study of electric-field-induced Feshbach 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  6 Li  and  87 Rb  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 1 Σ and 3 Σ symmetries. The molecular states are characterized by the corresponding scattering lengths. The interspecies triplet scattering lengths for the 6 Li–87 Rb mixtures have been evaluated through measure6,87 7,87 +19 ments of cross-thermalization: |atriplet | = 20+9 −6 bohr [204] and |atriplet | = 59−19 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 ultracold 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 6 Li–87 Rb scattering properties in any combination of spin states. We find that the sign of the 6 Li–87 Rb 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 6 Li–87 Rb spin combinations where 6 Li  and  3.2  87 Rb  are in the lower hyperfine manifold.  Fitting procedure  Our starting point for this work is to model the triplet a3 Σ and singlet X1 Σ interaction 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) with G(r) =  8 X  5 X C2i i=3  r2i  gl rl ,  ,  (3.1)  (3.2)  l=0  and T (r) =  1 [1 + tanh(1 + T r)] . 2  (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 determined 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 resonances 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 crossings 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 interaction of the collision complex with external magnetic fields, and V̂hf represents the hyperfine interactions. Ĥrel can be written as Ĥrel = −  ˆl2 (θ, ϕ) 1 ∂2 r + + V̂ , 2µr ∂r2 2µr2  (3.5)  where µ is the reduced mass of the colliding atoms, r is the interatomic distance, ˆl 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 =  XX S  MS  |SMS iVS (r)hSMS |,  (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  S 0 MS0  bound states ψlSMS in the basis |RlSMS i|SMS i, the matrix elements hψlSMS |Ĥrel |ψl  are given by  S 0 MS0  hψlSMS |Ĥrel |ψl  i  S 0 MS0  = hSMS |hRlSMS |Ĥrel |Rl  i  i|S 0 MS0 i  ˆl2 (θ, ϕ) 1 ∂2 S 0 MS0 r + + V̂ |R i|S 0 MS0 i l 2µr ∂r2 2µr2 ˆl2 (θ, ϕ) 1 ∂2 r + = hSMS |hRlSMS | − 2µr ∂r2 2µr2 XX S0M 0 + |S 00 MS00 iVS (r)hS 00 MS00 ||Rl S i|S 0 MS0 i = hSMS |hRlSMS | −  S 00 MS00  = hRlSMS | −  ˆl2 (θ, ϕ) 1 ∂2 S0M 0 r + + VS (r)|Rl S iδSS 0 δMS MS0 2 2 2µr ∂r 2µr  = ElS δSS 0 δMS MS0 ,  (3.7)  where RlSMS 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 hRl0 |Rl1 i = 1. Here,  Rl0 and Rl1 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 . Sa ,MSa ,Sb ,MSb  We expand the wave function ψl  Sa ,MSa ,Sb ,MSb  in the basis Rl  |Sa MSa i|Sb MSb i.  45  3.3. Asymptotic bound state model Table 3.1: Definition of quantum numbers used in this Thesis. l ml F MF I MI S MS Fa MFa Fb MFb Ia MI a Ib MI b Sa MSa Sb MSb  orbital angular momentum of a collision complex projection of l on the space-fixed quantization axis total spin angular momentum of a two-atom system projection of F on the space-fixed quantization axis total nuclear spin angular momentum of a two-atom system projection of I on the space-fixed quantization axis total electronic spin angular momentum of a two-atom system projection of S on the space-fixed quantization axis total spin angular momentum of atom a projection of Fa on the space-fixed quantization axis total spin angular momentum of atom b projection of Fb on the space-fixed quantization axis nuclear spin angular momentum of atom a projection of Ia on the space-fixed quantization axis nuclear spin angular momentum of atom b projection of Ib on the space-fixed quantization axis electronic spin angular momentum of atom a projection of Sa on the space-fixed quantization axis electronic spin angular momentum of atom b projection of Sb on the space-fixed quantization axis  The matrix elements of Ĥrel in the uncoupled representation thus have the form Sa ,MSa ,Sb ,MSb  hψl  Sa ,MS0 a ,Sb ,MS0  |Ĥrel |ψl  b  Sa ,MS ,Sb ,MS  i  Sa ,MS0 ,Sb ,MS0  a a b b |Ĥrel |Rl = hSa MSa |hSb MSb |hRl |Sa MS0 a i|Sb MS0 b i X X hSMS |Sa MSa Sb MSb ihS 0 MS0 |Sa MS0 a Sb MS0 b i =  SMS S 0 MS0  S0M 0  hSMS |hRlSMS |Ĥrel |Rl S i|S 0 MS0 i X = hSMS |Sa MSa Sb MSb ihSMS |Sa MS0 a Sb MS0 b iElS ,  (3.8)  SMS  where hSMS |Sa MSa Sb MSb i and hSMS |Sa MS0 a Sb MS0 b i are the Clebsch-Gordan coefficients 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 = 2µ0 B(ŜZa + ŜZb ) − B     µa ˆ µb ˆ IZ + IZ , Ia a Ib b  (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 = γa Iˆa · Ŝa + γb Iˆb · 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) = Iˆa(b) + Ŝa(b) .  (3.11)  i 1h 2 2 2 F̂a(b) − Iˆa(b) − Ŝa(b) , Iˆa(b) · Ŝa(b) = 2  (3.12)  Squaring this expression yields  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)    1 Ia(b) + 2    .  (3.13)  In this calculation, we use γa = 152.14 MHz for the 6 Li atom and γb = 3417.34 MHz for the  87 Rb  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 Iˆa2 , Iˆb2 , IˆZa and IˆZ as follows: b  Sa ,MSa ,Sb ,MSb  |Ψl i = |Rl  i ⊗ |αi,  (3.14)  where |αi = |Ia MIa i|Sa MSa i|Ib MIb i|Sb MSb i.  (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 |Ia MIa i|Sa MSa i|Ib MIb i|Sb MSb i. The matrix elements of the hyperfine interaction operators can be readily evaluated using the relations  and  1 Iˆa · Ŝa = IˆZa ŜZa + (Iˆa+ Ŝa− + Iˆa− Ŝa+ ) 2  (3.16)  1 Iˆb · Ŝb = IˆZb ŜZb + (Iˆb+ Ŝb− + Iˆb− Ŝb+ ), 2  (3.17)  ˆ Ŝ)a(b)+ and I( ˆ Ŝ)a(b)− are the raising and lowing operators, respectively. where I( The matrix elements of the raising and lowering operators can be evaluated as [202] hjmj |j± |j 0 m0j i =  q j(j + 1) − m0j (m0j ± 1)δjj 0 δmj m0j ±1 .  (3.18)  Since the long-range part of the potential is known, the energies ElS of the higher l > 0 states are uniquely determined by the l = 0 singlet and triplet energies. Therefore, 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 calculations are presented in Section 2.3 of Chapter 2. We note that the matrix of the Hamiltonian in the basis |αlml i 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 Smatrix 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 6 Li and  87 Rb  in the presence of magnetic fields.  48  3.4. Results  FESHBACH RESONANCES IN ULTRACOLD  85  RB-…  0.1  -1  Energy (cm )  0 -0.1  A  B  C  -0.2 -0.3 -0.4 0  500  1000  Magnetic Field (G)  2000  1500  FIG. 3. #Color online$ Molecular bound state energies versus  Figure 3.1: The s- and p-wave molecular bound state energies as functions of magmagnetic field computed with the asymptotic bound state model. netic fields for all the states with 1 1 MF = 3/2 computed within the asymptotic bound 6Li !threshold The threshold for the %1 , 1& 87Rbforcollision shown state model. The solid line %shows the | 12 , 12 ichannel collision 6 Li ⊗ |1, 1iis 87 Rb 2 , 2 & the channel (see text)line while the dashed (dotted)#dotted$ lines indicate s-wave the (p-wave) states. by the solid while the dashed linesthe indicate s-wave These molecular state energies were computed given the least bound states ElS of #p-wave$ states. These molecular state energies were computed 0 0 = −0.0870 = −0.106 cm−1 , El=1 the optimal singlet and triplet potentials El=0 l of the optimal singlet and given the least bound states E −1 −1 1 −1 1 S = −0.116 cm . The predicted triplet resonance cm , E cm , and El=0 = −0.137 l=1 l=0 l=1 l=0 −1 −1 potentials of E = −0.106 cm , E = −0.0870 cm , and E positions are close to0 the actual positions determined by the full coupled-channel 0 1 = l=1 −1 −1 −0.137 cm , E The(A,predicted calculation and are indicated by thecm solid. dots B, and C).resonance loca1 = −0.116  tions are close to the actual locations determined by the full coupled-channel 3.4 Results calculation and are indicated by the solid dots. Near 890 G #A$ the threshold crosses a p-wave molecular state and Deh al. have found in their experiments two Feshbach for atoms theet corresponding p-wave elastic scattering crossresonances section shown in in 1 1 theFig. | 2 , 25i6 Li |1, 1i87 Rb state with positions 882.02 G andto1066.92 G, and 0 ) ofthen is⊗observed to rapidly diverge(Band return the backwidths (∆B) of 1.27 G and 10.62 respectively Here, |Fa , crosses MFa i|Fb ,both MFb i is ground level. Likewise, nearG,1070 G #B$ [205]. the threshold theausual notation atomicmolecular hyperfine states. ands-wave Fb are not good p-wave andfor anthe s-wave state Note and that bothFathe and quantum numbers the presencecross of a magnetic low and moderate p-wave elasticin scattering sectionsfield. areHowever, affected.at Finally, near magnetic fields they can be used to specify atomic states correlating with particular 1300 G #C$ a second s-wave-induced Feshbach resonance occurs. hyperfine states at zero magnetic field. By fitting these two resonances, we want to 6 Li–87 Rb collisions. At generate accurate ultracold 882 G. For interaction each of potentials the fivedescribing candidate regions, the corre-  first, we thought fitting these two potential curves an easy problem sponding potentials were generated andwould the be predicted elasticand could be solved in two days. The original procedure of we planned to follow shown in scattering cross sections as a function magnetic fieldiswere Fig. 3.2. However, after obtaining one pair of the bound energies Esinglet and Etriplet 0  I  iplet  -1  ( cm )  -0.1  -0.2  II  V  49  computed us dition, the co computed. E out based o resonance at width than th experimental tively !16". I are #as in #Esinglet , Etripl wide #!5 G not observed the ordering mental meas lower resona #!10 G$ s-w experiment, resonance at two observe length for r −0.377 cm−1 with the ex ments of the 6 Li- 87Rb mix scattering len In order constructed = C12 / R12 − C same numbe energies #Esi regions cons the Feshbach essentially th sitive to the provides an the four pure that the exp inconsistent sider the pos nates from a Region V  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 i6 Li ⊗ |1, 1i87 Rb 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 coupledchannel 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 A: The asymptotic bound state model  The optimal least bound energies for singlet and triplet potentials B: Discrete variable representation calculations  Adjust the short-range repulsive wall to generate the singlet and triplet potential curves C: Full coupled-channel calculations  Calculate the experimentally measured scattering lengths  Reproduce both the positions and width of the resonances  Make consistent  Figure 3.2: The procedure of fitting the interactions potentials for ultracold 6 Li– collisions.  87 Rb  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 a6,87 triplet = 105 bohr. This value is in disagreement with the experimentally  determined value from the measurements of the cross thermalization in magnetically trapped 6 Li–87 Rb mixtures which indicate that the interspecies triplet scattering +9 length is |a6,87 triplet | = 20−6 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 potentials, we verified that the Feshbach resonance locations and scattering lengths  51  3.4. Results  4  3  Scattering Length (x10 Bohr)  6  2 0 -2 -4  -6  1.641  1.644  1.647  1.65  1.653  1.656  1.659  bsinglet 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 1 Σ interaction potential. are essentially the same as for the fitted potentials and are insensitive to the shortrange 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 ) 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. 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  4  3  Scattering Length (x 10 Bohr)  6  2 0 -2 -4  -6 2.06  2.065  2.07  2.075  2.08  btriplet 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 3 Σ 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 i6 Li ⊗ |1, 1i87 Rb state show divergences at magnetic fields of 1065 and 882 G, in excellent  agreement with the experimentally determined Feshbach resonance positions. In addition, the triplet scattering length from the fine-tuned triplet potential was found to be |a6,87 triplet | = −19.8 bohr, also in excellent agreement with the experimentally determined value. For the reduced mass corresponding to the 7 Li–87 Rb complex,  the optimal fine-tuned triplet potential predicts a7,87 triplet = 448 bohr, in disagreement 7,87 with the experimental measurement of |atriplet | = 59+19 −19 bohr [208]. This is because  the slope of the short range repulsive wall of the interaction potentials is not welldefined. 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 interaction potentials thus generated are not unique. The reduced mass of the 7 Li–87 Rb 53  882 G. For each of the five candidate regions, the corresponding potentials were generated and the predicted elastic scattering cross sections as3.4.a function of magnetic field were Results 0  I -1  Etriplet ( cm )  -0.1  -0.2  -0.3  -0.4 -0.4  V  II  III IV -0.3  -0.2  -1  -0.1  0  Esinglet ( cm )  FIG. 4. 3.5:#Color points in the #Esinglet , Etriplet$ Figure Locus of online$ points in theLocus (Esinglet , Eof s-wave triplet ) parameter space where an resonance occurswhere at one of an the two experimentally determined locationsat 882.02 G of the parameter space s-wave resonance occurs one 1 1 [gray (green)] or 1066.92 G [dark (red)] for atoms in the | 2 , 2 i Li ⊗ |1, 1i Rb state. two experimentally determined locations #green$" or The dotted lines indicate the approximate values for 882.02 Esinglet and!gray Etriplet beyond which a new bound state enters the potential at zero energy. 1 1There are four regions 1066.92 G indicated !dark on #red$" atoms inresonance the % 2occures , 2 & 6Lisimultaneously ! %1 , 1& 87Rb (I-IV) the plotfor where an s-wave at state. 882.02 G and at 1066.92 G. Region V indicates a range of values for which an s-wave The dotted lines indicate the approximate values for Esinglet and resonance occurs at 1066.92 G while a p-wave resonance (not presented in this plot) at 882.00 G. Etriplet occurs beyond which a new bound state enters the potential at zero energy.complex Thereis about are four regions #I–IV$ indicated on the plot where an 16% different from that of the 6 Li–87 Rb dimer, so the interaction s-wavepotentials resonance simultaneously at 882.02 and at for 6 Li–87 Rboccurs may not be reliable to describe the collisions between atoms 7 Li and 87 Rb. Close inspection of the potential also reveals that there is a bound 1066.92 G. Region V indicates a range of values for which an state very close to the dissociation threshold for the 7 Li–87 Rb system. Therefore, a s-wavesmall resonance occurs at 1066.92 while a p-wave resonance #not uncertainty in the exact location of this very weakly bound state translates 7 Li–these 87 Rb represented inlarge this plot$ inoccurs at 882 For length eachforof five into a very uncertainty the predicted triplet G. scattering mixtures. candidate regions, the character of the predicted elastic cross secthe refined potentials we calculated the s- and p-wave scattering cross tions assections aUsing function of magnetic field was studied, and the results of as a function of magnetic field for all spin combinations where 6 Li and 87 Rb this analysis in the text. are in the are lower discussed hyperfine manifold, and the location and widths of all resonances 6  87  t e s p t t i s n  # r r F c t t t t n p r 1 s = l t E  c  below 2 kG are summarized in Table 3.2. In experiments with 6 Li–87 Rb mixtures, no Feshbach resonances were observed below 1.2 kG for the | 12 , 12 i6 Li ⊗ |1, 0i87 Rb , 54  022710-  10  4  10  2 0  10 -2 10 10  -4  10  -6  -8  10 0  (Esinglet , Etriplet ) (E as singlet the fitted potential each , Etriplet ) as theinfitted these potentials, we Feshbach verified thr these potentials, we verified that the lengths essentially the samep lengths are essentially theare same as for the fitted details of thecheck poten short-range detailsshort-range of the potentials. This of ourofcharacterization the fo of our characterization the four purelyofs-wave is that the experimentally is that the experimentally observed Feshbachobser res  10  6  roughly the same roughly number the of bound states, and t same number of bo  2  σpσp(units of a2B)) (Bohr  2 2  (Bohr s(units σσ of aB) ) s  In order to verifyInthe robustness these fi order to verifyofthe robu Lennard-Jones potentials V (r) =potentials C12 /r12 − C6 Lennard-Jones V (r)  the cross thermalization in thermalization magnetically trapped the cross in mag that the interspecies scattering triplet length scat is | thattriplet the interspecies  in disagreement with the experimentally determin in disagreement with the experi  there is an additional resonance s-wave at approx theres-wave is an additional re the two observed resonances. In addition, the trip the two observed resonances. In −1 i and IV corresponding to E and IV corresponding to cm Etriplet triplet = −0.377  LI et al.  3.4. Results Results 3.4.  1000  500  Magnetic Field (G)  2000  1500  TA for m chann nance for co full w relate nance oscill and c with scatte betwe  A FIG. 5. %Color online& Magnetic field dependence of the s-wave Figure 3.6: panel& Magnetic field dependence of the s-wave (upper panel) p-wave (lower Figure 3.6: Magnetic field dependence of the s-wave (upper panel)and andcross p-wavesec%upper and p-wave %lower panel& elastic scattering 1 11 1 elastic scattering sections for atoms in the i Li⊗⊗|1, |1,1! 1i87 Rb panel) (lower elasticpanel) scattering cross sections for atoms in the | 2 , |22!, 62Li 1 cross 1 Rb state. 6Li ! !1 , 1" 87calculations tions for atoms in the ! , " state. These results !f , m state. These results are from the coupled-channel for a collision energy Rb 2 2 These results are from the coupled-channel calculations for a collision energy are of 144 of 144 nK and using the optimal singlet and triplet potentials. Only the m = 0 l thethecoupled-channel for Only a collision of nk from and using optimal singlet andcalculations triplet potentials. the ml = energy 0 contribution 1 1 contribution of the p-wave elastic scattering cross section is shown. Two s-wave ! of the elastic scattering cross section is shown. Two s-wave resonances occur 144p-wave nk and using the optimal singlet and triplet potentials. Only the 2 , 2" resonances occur at 1065 and 1278 G, while two p-wave resonances occur at 882 and at 1065 resonances at 882cross and 1066 G. is G.1278 G, while 0and contribution totwo thep-wave p-wave elastic occur scattering section ml =1066 shown. Two s-wave resonances occur at 1065 and 1278 G, while | 12 , 21 i Li ⊗|1, −1i Rb , and | 23 , 32 i Li ⊗|1, 1i Rb states. The results presented in Table bound state arising from uncertainties the exact shape the potential translates two3.2 p-wave resonances occur atin of882 and 1066 of G. are in agreement with the last two these observations but not the first. It 6  6  87  6  87  87  7 87 into a very large uncertainty in the predicted triplet 1 1 scattering length for Li– Rb  is possible that, because the resonances present in | 2 , 2 i6 Li ⊗ |1, 0i87 Rb combination  mixtures. are very similar in position and width to those of1the1 | 12 , 12 i6 Li ⊗ |1, 1i87 Rb state, 6Li ! !1 , 1" 87Rb state scattering crosspotentials sections for the the ! 2 ,s-2 "and Using we erroneously calculated concluded p-wave theythe mayrefined have been observed and to arise from scattering an impure cross  1  1  1  1  !2 , 2"  87 Rb show atwasmagnetic fields ofDeh, 1065 and 882 state recently by combinations B. the lead author of G, the sections as divergences apreparation. function ofThis magnetic field confirmed for all spin where 6Li andin  experiment papers [205]. with the experimentally determined Fesareexcellent in the loweragreement hyperfine manifold, and the location and widths of all resonances 7 Li–scattering 87 Rb mixtures, hbach positions. In addition, the with triplet below 2 kGresonance are summarized in Table 3.2. In experiments  1 the fine-tuned was| 21 ,found to be no length Feshbachfrom resonances were observedtriplet below potential 1.2 kG for the 2 !6 Li ⊗ |1, 0!87 Rb , 6,87 = −19.8a , also withpresented the experi| 12 , a12 !triplet −1!87 Rb ,Band | 32 , 32in !6 Liexcellent ⊗|1, 1!87 Rb agreement states. The results in Table 6 Li ⊗|1, 3.2mentally are in agreement with the last two of these observations not the first. It determined value. For the reducedbutmass corre55 7 87 is possible that, because the resonances present in | 1 , 1 !6 ⊗ |1, 0!87 combination  !2 , 2"  1 1 !2 ,−2 fine-tuned tripsponding to a Li- Rb complex, the optimal Rb 2 2 Li 7,87 arelet very similar inpredicts position and width to those of the | 12 , 12 ! Li ⊗ |1,with 1! Rbthe state, = 448a potential atriplet B, in disagreement 7,87 +19 they may have been measurement observed and erroneously concluded toaarise#41$. from an impure ! = 59−19 Close experimental of !atriplet B 6  87  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 6 Li–87 Rb triplet scattering length is a6,87 triplet = −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 wellknown 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 6 Li–87 Rb 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 6 Li–87 Rb 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 |f, mf i6 |f, mf i87 | 12 , 21 i |1, 1i  | 12 , 21 i  |1, 0i  | 12 , 21 i |1, −1i | 21 , − 21 i |1, 1i  | 12 , − 21 i  |1, 0i  | 21 , − 21 i  |1, −1i  | 23 , 23 i | 23 , 23 i | 32 , 23 i  |1, 1i |1, 0i |1, −1i  | 23 , − 23 i  |1, 1i  | 23 , − 23 i  |1, 0i  | 23 , − 23 i  |1, −1i  Theory B0 (G) 882 1065 1066 1278 889 1064 1096 1308.5 1361.7 1348 773 923 926 1108.6 1119.5 923 1105 1150 1362 1408 1611 None None 953 1236.6 809 960 971 1156 973 1149 1609  4B (G) p-wave 11.5 p-wave 0.07 p-wave 17 p-wave (3) p-wave (4) p-wave < 0.001 p-wave 11 p-wave p-wave 16.3 p-wave (3) (4) 0.06  Experiment [205] B0 (G) 4Bexpt (G) 882.02 1.27 1066.92 10.62  None below 1.2 kG  None below 1.2 kG  None below 1.2 kG 48.5 p-wave p-wave < 0.001 p-wave 11.7 p-wave 16.7 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 ultracold 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 research 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 interac2  A 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 electricfield-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. Electricfield-induced Feshbach resonances may therefore be more versatile for quantum computation schemes than magnetic resonances. Magnetic field control of interatomic interactions is limited to para-magnetic species. The possibility of inducing scattering 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 molecular 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 ultracold 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 modifies 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 = −  ˆl2 (θ, ϕ) 1 ∂2 r + + V̂ + V̂ζ + V̂B + V̂hf , 2µr ∂r2 2µr2  (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 =  XX S  MS  |SMS iVS (r)hSMS |,  (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 |Ia MIa i|Sa MSa i|Ib MIb i|Sb MSb i as |Ia MIa i|Sa MSa i|Ib MIb i|Sb MSb i = ×  XX (−1)MS (2S + 1)1/2  Sa  Sb  S  MSa  MSb  −MS  S  !  MS  |Ia MIa i|Ib MIb i|SMS i  (4.3)  and note that hSMS |V̂ |S 0 MS0 i = VS (r) δSS 0 δMS MS0 ,  (4.4) 60  4.2. Atomic collisions in combined electric and magnetic fields  Dipole moment (Debye)  6  -1  Energy (cm )  5000  5 4 3 2 1 0 9  6  12  15  r (Bohr)  0  3  C6 = 2.9338x10 a.u. 5  C8 = 3.1253x10 a.u. 8  -5000  C10 = 1.7515x10 a.u.  8  12  r (Bohr)  16  20  Figure 4.1: The interaction potentials and dipole moment functions (inset) of the LiCs molecule in the 1 Σ (solid lines) and 3 Σ (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 )  XX S  MS  |SMS idS (r)hSMS |,  (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  Dipole moment (Debye)  5  -1  Energy (cm )  6000  3000  4 3 2 1 0 5  10 r (Bohr)  20  15  0  3  C6 = 2.5457x10 a.u.  -3000  5  C8 = 2.2825x10 a.u. 7  C10 = 2.5647x10 a.u.  5  10  r (Bohr)  15  20  Figure 4.2: The interaction potentials and dipole moment functions (inset) of the LiRb molecule in the 1 Σ (solid lines) and 3 Σ (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 1 Σ 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]  4π ∗ [Y (γ, ϕγ )Y1−1 (θ, ϕθ ) 3 1−1 ∗ ∗ (γ, ϕγ )Y10 (θ, ϕθ ) + Y11 (γ, ϕγ )Y11 (θ, ϕθ )], +Y10  êζ · êd = cos(χ) = P1 (cos(χ)) =  (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  ecule ground lines) (dashed lines) states. 3 excited oundin1 the(solid lines) (solid andin excited (dashed lines) states. # ϕand coordinate system detail. θ ∗ action potentials were taken and the dipole moment functions + (γ, ϕθγfrom )Y (θ,(x, ϕfrom ) Ref.[4] als taken Ref.[4] and the dipole moment 11 θy) esYwere γ11 and on the plane and Yxx functions m in detail. tween the electric field and the ate the data of Ref. [3]. on the (x, y) plane and Y ϕ xx the collision dynamics in a system of θ f Ref. [3]. e coordinate system in detail. ! and!d! represent the vector ordinate system in our calculations. E system in detail. ngle between the electric field and the −2 ! amics in a system of 4.2. Atomic collisions in combined electric and magnetic fields bohr and D = 1.02 Debye forand the triplet state. TheseE analytical 4.2. Theory dinate system in our calculations. and dexpresrepresent the vector on1.02 the (x, y)dynamics plane YThese xx the collision in a system of Doximate = Debye for the triplet state. analytical expres! ! tric field and the dipole moment vector, respectively; γ specifies the numerical data for the dipole moment functions by The coordinate system inour our calculations. andd!drepresent representthe the vector !Ecomputed n dynamics in a system of ic field and the dipole moment vector, respectively; γ specifies he coordinate system in calculations. E and vector electric field and the umerical data for the dipole moment functions computed by enal system incoordinate detail. angle between the field andto the d Dulieu [3]. Figs. 4.1 andelectric 4.2 show thein interaction potentials VSζ and d !(r) ! represent the electric field with respect thevector, quantization axis; θ isvector the 59 electric field and the dipole moment respectively; γ specifies ure 4.3: The system our calculations. and the al:Figs. electric field and the dipole moment vector, respectively; γ specifies enmoment electric field with respect to the quantization axis; θ is the 1 3 4.1 and 4.2 show the interaction potentials V (r) and !LiCs SE the electric field and functions dS system (r) in thein Σour andthe Σ states for the and LiRb The calculations. and represent thebetween vector ! !isd!represent !the he external electric field and the dipole moment vector, respectively; γθ specifies ion ofcoordinate the electric field with respect to59E quantization axis; is the!E on dynamics in a system of dipole moment vector and the z-axis; χ the angle The coordinate system in our calculations. and d the vector 1 3 n of the electric field with respect to the quantization axis; θ is the tions d (r) in the Σ and Σ states for the LiCs and LiRb respectively. pole vector and the z-axis; χ the isχrespectively; the angle E Smoment ernal electric and thevector dipole moment vector, γ between specifies !the orientation offield the electric field with respect to quantization axis; θ is!the ! ! en the dipole moment and the z-axis; is the angle between E nal electric field and the dipole moment vector, respectively; γ specifies 3: The coordinate system in our calculations. ζ and d represent ϕ are the projections of angles γ and θ on the (x, y) plane. electric and θ the neen the dipole moment vector and z-axis; χon is the the angle between E ζ! vector electric field is oriented atfield a field certain anglethe γrespect withthe respect to thequantization quantization ation of the electric with to the axis; θ is the 59 are the projections of angles γ and θ (x, y) plane. le between the dipole moment vector and the z-axis; χ is the angle between θ ϕ and ϕθ electric are the projections ofangle angles and θγ.vector, on theaxis; (x, y) ofthethe field with to theγ quantization θ plane. is the! γ specifies γhas ternal electric and dipole moment respectively; êion form êEthe · êmoment =projections cos(χ) withthe χrespect the between θ and It the canangle d! dfield and ϕ are of angles γ and θ on (x, y) plane. oriented at a certain angle γ with respect to the quantization ween the dipole vector and the z-axis; χ is the between θ 59 d, the and dipole ϕγ andmoment ϕθ are the projections ofz-axis; anglesχγisand θangle on thebetween (x, y) plane. !E en vector and the the E n in terms of the first-degree Legendre polynomial 59 the electric fieldbetween with respect thethequantization m êEγ ·and êd =of cos(χ) the angle θ and Itto can dtation ϕ ϕ the χprojections of angles γ γ. and θ on (x, y) plane. axis; θ is the θ are with ϕthout are the projections of angles γ and θ on the (x, plane. γ and ϕloss θ ofloss generality, we we cancan assume that êêχEy)isis in the the x−− z he first-degree Legendre polynomial tween the dipole moment vector and the z-axis; the angle between ζ! eld. Without of generality, assume that is in x z E hout loss of generality, we can assume that ê is in the x − z 59 ê · ê = cos(χ) = P (cos(χ)) (4.7) E 1 assume E we d d. Without loss of generality, canweassume that that êE isêEinisthe x −xz− z gnetic field. Without loss of generality, can in the # nd ϕ and ϕ are the projections of angles γ and θ on the (x, plane. The matrix elements of V̂ (r) are therefore evaluated using the θϕ loss E)we ∗γ =γ0. ∗ elements ∗ ϕfield. V̂EY can (r) assume ê evaluated is in the using x − y) z the (γ, ϕWithout )YThe (θ,matrix ) + Y of (γ,generality, ϕ )Y (θ, ϕof + (γ, ϕare )Y therefore (θ, ϕthat )  E evaluated γ ϕ 1−1 γ =10 γ 11are θThe 10 θ = P θ 1−1 11 The matrix elements V̂of are using the êgenerality, cos(χ) (4.7) ne, 0.loss matrix elements of V̂ (r)therefore therefore the E (r) 1 (cos(χ)) = i.e., 0. The V̂E can (r) are evaluated E · êdof γ =matrix Etherefore ld. Without ofelements we assume that êevaluated is in the using x −using z the E  # ϕγ = 0. The matrix elements∗ of V̂E (r) are therefore evaluated using the ∗ (θ, ϕ ) + Y (γ, ϕ )Y (θ, ϕ ) + Y (γ, ϕ )Y (θ, ϕ ) ϕressions = 0. The matrix elements of V̂ (r) are therefore evaluated using the 1and γ 10 γ 11 θ θ θ ϕ are the projections of angles γ and θ on the (x, y) plane and Yxx 10 11 γ E θ scalfield. Without loss the of coordinate generality, can assume that êE is in the x − z harmonics. Fig. 4.3 depicts systemwe in detail. ! ! ! γfield e projections on the (x, y)dynamics plane and m |ê · !rotating ê#d0. m xx ffect of!Em a"lm the collision in aY system of lϕ |êof ·angles ê# d |lmatrix m!l #andonθ elements lelectric !l|l !The ê |l E ., = of V̂ (r) are therefore evaluated d l ! ! γ E " # " ### using the |ê · ê |l m # " # " E ! ! l.al d √"lm lmon √ " # " |lFig. m # 4.3 depicts the coordinate system in detail. atoms depends the relative angle between the electric field and the |ê · ê |l # ! ! " # " # E l d !!! l 1 l!# l# 1 1~ !l#! l! # 2l |êE l· ê! 1"" l! l l lζ~ 1and 2|l! mm !! "l system " m #l l ! m d √ ! ns l ! ! Figure 4.3: The coordinate in our calculations. thevector vector 1 l ! l sinelectric γ(−1) 1)(2l + 1)] ! 1and d!drepresent ting field on[(2l the + collision in a1 system of l l# sin [(2l + dynamics 1)(2l 1)] ! γ(−1) !# " ! represent " + lsystem Figure 4.3: The coordinate calculations. E m !inllour !l ! the m !external l 1 l 1 l ! ! 2= ! !! l l (−1) [(2l + 1)(2l + 1)] of the electric field and the dipole moment vector, respectively; γ specifies 2 l 1 l l 1 l 22sin m ! γ(−1) 1)(2l + 1)] ! −m 0l mvector, −m 01field 0l!0and 00 the ! electric l [(2l l respectively; l1 m−1 of + the+ external field the dipole moment m sin γ(−1) [(2l 1)(2l + 1)] l 0and l−1 !! ! γl specifies !orientation ! laxis; nds on relative angle between the electric l the m the of the electric field with respect to the quantization θ is the ! 59 ! 1) [(2l + 1)(2l + 1)] 0 0 m −1 −m 0 0 0 m −1 −m " # " # l " # " # 2sin γ(−1) l l [(2lthe+orientation 1)(2l +of1)] 0 0 0 m −1 −m the electric field with respect to the quantization axis; θ is the~ l l between l z-axis; χ! is !the ! l ! ! ! angle between the dipole moment vector and the angle ! ! 0l 0moment 010l l# 0# m " "l andm # !ζ 1" l1−1−1 1χl−m l−m 0# ! angle " " " between the dipole vector the is#the angle between E l l lz-axis; "lm |ê ê |l # ! ·! mm l l# E l d ! ~ m l l ! ! ! and d, and ϕ and ϕ are the angles between the x-axis and the projections of the +!cos γ(−1) [(2l +" 1)(2l γ " + 1)] l ![(2l + ! !# !on # ! and " oflllangles os √ γ(−1) 1)(2l + "" # 1the l !# l θl# and d, ϕγ 1)] and ϕθlare projections γ# and lθthe 1(x,ly) 11 l!" ! l 1 1 ! m!l m ! the!l (x, y) plane. ! ! 0 0 0 m 0 −m vectors ê and ê on plane, respectively. ! ! ! ζ d m l l ! ! 59 γ(−1) [(2l + 1)(2l + 1)] ! ! ! 0 0 0 m 0 −m cos γ(−1) [(2l + 1)(2l + 1)] l 1 l l 1 l ! ! l )! !2 [(2l +[(2l 1)(2l 1 "l# 1# # l ! +! 1)] l 100! "00l 0" l lm 10l0"−m l ! ! l!l m!1)(2l √mlγ(−1) + + 1)] m −m # l l l l √sl γ(−1) sin [(2l + 1)(2l + 1)] ! ! l [(2l2+ 1)(2l + 1)]depends 00 " 0relative m 0−m −m magnetic field. Without generality, we assume in the x − z! l between llthe l!l1fieldthat 1and !! E lis !1 0loss l0 0ofangle m 0can # l lê!# ! l" # " #magnetic m !m √ atoms on the electric the−1 field. !" 2 1 l l 2− ! l 0 0 0 −m sin γ(−1) [(2l + 1)(2l + 1)] (4.8) l m 0 0 0 m 0 −m ! ! " # " # ! ! ! " # " # l plane, i.e., ϕ = 0. The matrix elements of V̂ (r) are therefore evaluated using thel l ! ![(2l ! ! γ E l 1 l l 1 l 2sin γ(−1) + 1)(2l +generality, 1)] l" (4.8) 1 l l 1 l 2 !m Without loss of we can assume that ê is in the (x, z) plane, i.e., ϕ = 0 0 0 m 1 −m # " # ! γ ζ ! ! ! ! l m ! l l (4.8) 0. ! ++ ! " # " # sin γ(−1) [(2l + 1)(2l 1)] (4.8) l ! ![(2l 22sin γ(−1) l 1 l l 1 l 0 0 0 m 1 −m l 1 l l 1 l expressions + 1)(2l 1)] ! l ml ! The matrix m elements the expressions 2sin γ(−1) ! +! + 0ltherefore 111l −m ! 001 evaluated [(2l1)(2l + 1)(2l 1)] lof V̂0ζ01are0 l! mlml l using 1l!l! ! l (4.8) l! (4.8) l ! + ! (−1) [(2l 1)] −m m l ! ! ! l ! mlγ(−1) l [(2l + mlm1 −m cos 0!0 00 00 1 l−m 1) [(2l + 1)(2l! +1)(2l 1)] + !1)] (4.8)  0 −m 0 0 00" 0 0 m #1m" −m  "lml |êE ·0 êd0|l ml # ! l #" hlml√ |êζ · êd |l ml i !l " l l! !# ! ! ! l √2   l 1 l l 1 l ! l l0 l = [(2l + 1)(2l! + 1)] l 1 l0 l 1 2 sin γ(−1)mm0 p ! ! l   = 2 sin γ(−1) [(2l + 1)(2l0 + 1)] && 0 !0 0 ml −1 −m 0 l ! 2 !! !! !! !! ! ! 0 0 0 m −1 −m l ! # "! δ  ml |S MS #dS !! "S "SM !!MS | |S MS "# =l d1S lδ!!SS |& . ! # l (4.9) &  S& M & SM l 1 S 0l ! ! !! !! !! !! ! ! m 0 ! !! Mm!!0 p !+ 1)(2l ! !#+= 1δSS l ! δ ! l. !1. l !(4.9) +S!!!!cos γ(−1) [(2l !! !!M !! !! #d & M M "S | M (4.9) !δ && & S|S SM | |S M "S M!!SSl|l |S|S M #S0 +=1)]1)] dSdSlδ0SS S |S l −m S cos + γ(−1) [(2l + 1)(2l M MSSM S SS 0 0 m 0 l SM S l !! !! !! ! ! 0 !! !! !! !! ! ! l 0 δ"SS 0! δ !0δM ! M m ! .0 −ml(4.9) "S MSS| ||S|SMM (4.9) #l" # M|S |  SS!!!! MM|S MSS#dS!!!! "S #S #==dSdSδSS . !!|S !! M √ M M M S S S !! S SS √2 l 1 lS l 1 0 l! ! !! ! + 1)] l 1 l0 !! !! M l! l 1 l !!S!!S!! !! !! γ(−1)mm0l p − !! 2 sin [(2l! + 1)(2l (4.8) SS M ! !! − 2 sin [(2lS+ 1)(2l0 + S 1)] SS (4.8) γ(−1) 0 ! 0 M0S M ! ml 1 −m S 0 S S l 2 0 0 0 m S l 1 −ml !! !! !! !! ! ! 60 !! M !! S SS ! MS MS! S !! S S S and S 60 60 MS!!   60 60 && !! !! !! !! !! !! ! !! ! ! ! !! !! !  |S MS # S "SM |S S MS #dS !! "S MS |S = dSSS δSS ! δM . ! (4.9) S M S S |S SS!M SM  √  ! 2 sin γ(−1) [(2l + 1)(2l + 1)] 2   1 l  l  0 0 0  m  l   && & |S M #d "S M | |S M # = d δ δ |SM #d "S M | |S M #  =d δ δ  SM |   && S !! MS!!  |S M #d  "S M | |S M # = d δ  S !! MS!!  .  1  1 −m  (4.8)  (4.9) (4.9)  .  δ  l  #  S  .  (4.9)  60 63 60 60 60  4.3. Li–Cs system and   hSMS |   XX S 00 MS00    |S 00 MS00 idS 00 hS 00 MS00 | |S 0 MS0 i = dS δSS 0 δMS MS0 .  (4.9)  If the electric field is directed along the z-axis, i.e., γ = 0, Eq. 4.8 reduces to hlml | cos θ|l0 m0l i = (−1)  ml  l  1  l0  −ml 0 ml  !  l  1 l0  0 0 0  !   1 (2l + 1)(2l0 + 1) 2 δml m0l .  (4.10)  The numerical approach for solving the coupled differential equations and constructing 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 laboratory of Weidemüller in Freiburg [217, 218] for the formation of ultracold polar LiCs molecules through photoassociation [222]. An alternative method of producing ultracold molecules is based on linking ultracold atoms with magnetic-field-induced Feshbach resonances. As mentioned in Chapter 1, Feshbach resonances may also enhance 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 important 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 i 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 resonances for Li–Cs at magnetic fields below 500 G. The notation |Fa MFa i for the atomic states is the same as in Chapter 3. Atomic States |1, 1i7 Li ⊗ |3, −2i133 Cs |1, 1i7 Li ⊗ |3, −3i133 Cs |1, 0i7 Li ⊗ |3, −3i133 Cs |2, −2i7 Li ⊗ |4, 4i133 Cs |2, −2i7 Li ⊗ |3, 3i133 Cs |2, −2i7 Li ⊗ |3, 2i133 Cs |2, −2i7 Li ⊗ |3, 1i133 Cs  |2, −2i7 Li ⊗ |3, 0i133 Cs |2, −2i7 Li ⊗ |3, −1i133 Cs  B0 (G) 2.03 1.49 21.50 387.81 4.92 20.05 0.86 2.27 7.06 1.02 2.64 7.20 0.47 1.25 3.15 6.65 0.59 1.65 4.47 0.63 2.49  4B (G) > 2.00 > 2.00 > 2.00 > 2.00 > 2.00 0.70 0.03 0.16 1.68 0.06 0.34 2.00 0.02 0.12 0.42 1.50 0.04 0.22 901.06 0.10 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  Cross section (a.u.)  of 100 kV/cm. The collision energy is 10−5 cm−1 (full curve) , 10−6 curve) and 10−7 cm−1 (dotted-dashed curve). The magnetic field i 4.3 Li–Cs system G. 4.3. Li–Cs system Ultracold mixtures of Li and Cs gases have recently been created in the laboratory of 4.3. Li–Cs system We report for the firstLiCs timemolecules the observation Weidemüller [185, 186] for the formation of ultracold polar through that the coupling indu photoassociation [188]. An alternative method producing ultracold into molecules is resonances for states fields splits of Feshbach resonances multiple based on linking ultracold atoms together magnetic-field-induced Feshbach res- a complementary way gular with momenta. This new phenomenon offers 8 onances. Feshbach resonances enhance the probability for photoassociation tune an anisotropic interaction and to study its effect on the many-b 10may also [189] and provide detailed information for the analysis of interatomic interaction poheteronuclear atomic gases. tentials [190]. Experimental measurements of the positions and widths of magnetic 4 10 Feshbach resonances are therefore very important dynamical in studies of ultra-electric and magne 4.4.1 Li–Rbforcollisions combined cold gases. To guide future experiments in the search of Feshbach resonances, we 0 10 ζ==00kV/cm kV/cm E present in Table 4.1 the positions and widths of purely magnetic s-wave resonances ζ = 100 kV/cm calculated with the potentials of Staanum et al. [2]. -4 As discussed in Section 4.2, in the absence of electric fields, differe 10  4.3.1  states |lml "resonances of the Li–Rb collision complex are uncoupled and s-w Electric-field-induced Feshbach  Cross section (a.u.)  -8 is dominant in ultracold collisions. The interaction with an electr 10 In the absence of electric fields, different partial wave states |lml " of collision com700 800 900 1000 1100orbital 1200angular momenta with 600 couplings between states of different plex are uncoupled and s-wave scattering entirely determines the collision cross a result, a resonant enhancement of the s-wave cross section appea sections at ultralow kinetic energies. The interaction of the atoms with electric fields near intrinsic p-wave resonances – as demonstrated in the prev 8 10 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 of104the 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) and0 at E ζ = 100 (lower panel). The examination 100kV/cm kV/cm 10  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  22  -4  resonances at the location of10magnetic p- (s-) wave Feshbach resonances, i.e., s- and -8  10  600  700  800  900  1000  1100  61 1200  Magnetic field (G) Figure s-wave (solid (solid curves) curves)and andp-wave p-wave(broken (broken Figure4.4: 4.4: Cross Crosssections sections for for elastic elastic s-wave curves) scattering curves) scatteringofofLiLiand andCsCsatoms atomsininthe thestates states|1, |1,1!1i7 Li ⊗|3, |3,3!3i133 computed at 7 Li⊗ 133 Cs computed Cs at different electric fields: ζ = 0 kV/cm (upper panel) and ζ = 100 kV/cm (lower zero electric field (upper panel) and an electric field strength of 100 kV/cm (lower −7 −1 −7 −1 panel).The Thecollision collisionenergy energy isis 10 10 cm . panel).  66  63  −ml 0 ml  0 0 0  We enhance report for the first time the observation that the coupling induce onances. Feshbach resonances may also the probability for photoassociation The approach of propagating the coupled differential equations and constructing [189] and provide detailed information for the analysis of interatomic interaction pofields splits Feshbach resonances into multiple resonances for states of  the scattering S-matrix is the same as described in Chapter 2. Here, we also apply  tentials [190]. Experimental measurements of the positions and widths of magnetic gular momenta. This new phenomenon offers a complementary way to  an additional transformation that diagonalizes the matrix of V̂ζ + V̂B + V̂hf before 4.3. for Li–Cs system studies of ultraFeshbach resonances are therefore tune very important dynamical an anisotropic interaction and to study its effect on the many-bo constructing the S-matrix [173].  cold gases. To guide future experiments in the search of Feshbach heteronuclear atomic gases. resonances, we present in Table 4.1 the positions and widths of purely magnetic s-wave resonances 8  Cross section (a.u.)  10 4.3 Li–Cs calculated withsystem the potentials of Staanum et al. [2]. 4.4.1 Li–Rb collisions in combined electric and magneti 4 10gases have Ultracold mixtures of Li and Cs E = recently 0 kV/cmbeen created in the laboratory of  4.3.1 Electric-field-induced resonances ζFeshbach = kV/cm Weidemüller [185, 186] for the formation of 0 ultracold polar LiCs molecules through 0 10  ζ= 100 kV/cm photoassociation An alternative method of producing ultracold is comIn the absence [188]. of electric fields, different partial wave states |lml "molecules of collision -4 based ultracold atoms with res-absence 10 together As magnetic-field-induced discussed in Section Feshbach 4.2, collision in the plex on arelinking uncoupled and s-wave scattering entirely determines the cross of electric fields, different onances. Feshbach resonances may also enhance the probability for photoassociation states " of theofLi–Rb collision sections at ultralow kinetic -8energies. The|lm interaction the atoms with complex electric are uncoupled and s-wa l  10 700 for the800 [189] and provide detailed information analysis of 900 interatomic1000 interaction 1100 poCross section (a.u.)  1200 fields (Eq. 4.5), however, induces couplings between different angular momentum is dominant in ultracold collisions. The interaction with an electric tentials [190]. Experimental measurements of the positions and widths of magnetic states and may thus affect the length. Figs. 4.4, 4.5 and 4.6 display the angular momenta with ∆ 8 scattering couplings between states of different orbital Feshbach resonances are therefore 10 very important for dynamical studies of ultramagnetic field dependence of the s-wave (solid curve) and p-wave (dashed curve) a result, a resonant enhancement of the s-wave cross section appears cold gases. To guide future experiments in the search of Feshbach resonances, we 4  scattering cross sections calculated for various Zeeman states of Li and Cs at zero  10 fields near intrinsic p-wave resonances present in Table 4.1 the positions and widths of purely magnetic s-wave resonances – as demonstrated in the previou electric field (upper panel) and at ζ = 100 kV/cm (lower panel). The examination  E = 100 kV/cm calculated with the potentials10 of0 Staanum et al. [2].  of Figs. 4.4, 4.5 and 4.6 leads to two important observations:  -4 The couplings between and p-wave scattering induce new s- (p-) wave 10 s-Feshbach 4.3.1(i) Electric-field-induced resonances resonances at the location of magnetic p- (s-) wave Feshbach resonances, i.e., s- and -8 In the absence of electric fields, different partial wave states |lml " of collision com10 700 800 900 1000 1100  plex are uncoupled and s-wave scattering entirely determines the collision cross  61  Cross section (a.u.)  sections at ultralow kinetic energies. The interaction of the atoms with electric fields (Eq. 4.5), however, induces 3 couplings between different angular momentum  10  states and may thus affect the scattering length. Figs. 4.4, 4.5 and 4.6 display the  1200  22  magnetic field dependence of the s-wave (solid curve) and p-wave (dashed curve) 0 scattering cross sections calculated for various Zeeman states of Li and Cs at zero 10  electric field (upper panel) and at ζ = 100 kV/cm (lower panel). The examination  E = 100 kV/cm  of Figs. 4.4, 4.5 and 4.6 leads to two important observations: -3  10 s- and p-wave scattering induce new s- (p-) wave (i) The couplings between 700  800  900  1000  1100  resonances at the location of magnetic p- (s-) wave Feshbach resonances, Magnetic field (G)i.e., s- and  1200  Figure 4.5: Cross sections for elastic s-wave (solid curves)61and p-wave (broken curves) scattering of Li and Cs atoms in the states |1, 0i7 Li ⊗ |3, 3i133 Cs 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  G.  Cross section (a.u.)  Ultracold mixtures of Li and Cs gases have recently been created in the laboratory of We report for the first time the observation that the coupling indu Weidemüller [185, 186] for the formation of ultracold polar LiCs molecules through fields splits Feshbach resonances into multiple resonances for states 4.3. of Li–Cs system ultracold molecules is photoassociation [188]. An alternative method producing gular momenta. This new phenomenon offers a complementary way based on linking ultracold atoms together with magnetic-field-induced Feshbach res12 tune an anisotropic interaction and to study its effect on the many10 onances. Feshbach resonances may also enhance the probability for photoassociation heteronuclear atomic gases. 8 [189] and provide detailed information for the analysis of interatomic interaction po10 tentials [190]. Experimental measurements of the positions and widths of magnetic 4  4.4.1  Li–Rb collisions in combined electric and magne  10 Feshbach resonances are therefore very important for dynamical studies of ultrakV/cm ζE==0 0kV/cm cold gases. To guide future experiments in the search of Feshbach resonances, we 0 10 = 100ofkV/cm present in Table 4.1 the positions and ζwidths purely magnetic s-wave resonances  As discussed -4 of Staanum calculated with the potentials et al. [2].in Section 4.2, in the absence of electric fields, differe 10 states |lml " of the Li–Rb collision complex are uncoupled and s-w  4.3.1  -8 isFeshbach dominant in ultracold collisions. The interaction with an electr Electric-field-induced resonances 10  couplings between states of different orbital angular momenta with 800 900 1100 1300 In the absence of electric fields, different partial1000 wave states |lml " 1200 of collision coma result, a resonant enhancement of the s-wave cross section appea plex are uncoupled and s-wave scattering entirely determines the collision cross fields near intrinsic p-wave resonances – as demonstrated in the prev 12 sections at ultralow kinetic10 energies. The interaction of the atoms with electric  Cross section (a.u.)  fields (Eq. 4.5), however, induces couplings between different angular momentum 8 10 scattering length. Figs. 4.4, 4.5 and 4.6 display the states and may thus affect the  magnetic field dependence of4 the s-wave (solid curve) and p-wave (dashed curve) 10  scattering cross sections calculated for various Zeeman states of Li and Cs at zero  electric field (upper panel) and ζ= = 100 100 kV/cm 0 atE kV/cm(lower panel). The examination 10  of Figs. 4.4, 4.5 and 4.6 leads to two important observations:  22  (i) The couplings between -4 s- and p-wave scattering induce new s- (p-) wave 10  resonances at the location of magnetic p- (s-) wave Feshbach resonances, i.e., s- and -8  10 800  900  1000  1100  Magnetic field (G)  1200  1300  61  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, −1i7 Li ⊗ |3, 3i133 Cs 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 |1, −1i7 Li ⊗ |3, 3i133 Cs |1, 0i7 Li ⊗ |3, 3i133 Cs |1, 1i7 Li ⊗ |3, 2i133 Cs |1, 1i7 Li ⊗ |3, 3i133 Cs  Bres (G) 953.54 862.74, 907.55, 965.61 998.79 785.57, 862.47  P -wave magnetic resonances are essential for the electric-field-induced resonances 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 Feshbach 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 swave 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  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 . 4.3. Li–Cs system  e  8  Cross section (a.u.)  10  a  ab c  d  c  b  d  e  4  10  0  10  1100  1050  1200  1150  Magnetic field (G) Figure 4.7: Cross sections for elastic s-wave collisions of Li and Cs atoms in the |1, 0i7 Li ⊗ |3, 3i133 Cs 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 0 4π 2 X X X X X X X X l10 −l1 +l2 −l20 Yl1 ml1 (r̂i ) i = 2 dr̂i dr̂ kβ l m l m 0 0 0 0 1  ×  l1  2  Yl∗2 ml (r̂i )Yl∗0 m0 1 l 2  1  l2  l1 ml  1  l2 ml  2  ∗ (r̂)Yl20 m0l (r̂)Tβl 0 0 0 1 ml1 →β l1 ml 2  1  Tβl2 ml  2  →β 0 l20 m0l  2  (4.11)  where β and β 0 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 π = 2 {|Tl=0→l0 =0 |2 + 3 cos2 θ|Tl=0→l0 =1 |2 + 2 3 cos θ dr̂ 4kα  × [Re(Tl=0→l0 =0 )Im(Tl=0→l0 =1 ) − Im(Tl=0→l0 =0 )Re(Tl=0→l0 =1 )]},  (4.12)  where θ is the angle between the initial and final collision fluxes. The first term is 70  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.  4.3. Li–Cs system  9  Cross section (a.u.)  10  6  10  3  10  0  10  -3  10  0  20  40  60  80  100  120  Electric field (kV/cm) Figure 4.8: Electric-field dependence of the s-wave scattering cross section for collisions of Li and Cs atom in the |1, 0i7 Li ⊗ |3, 3i133 Cs 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 pstate. The rate constant for this transition therefore vanishes in the limit of zero  71  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 . 4.4. Li–Rb system  9  Cross section (a.u.)  10  6  10  3  10  0  10  -3  10  0  20  40  80  60  100  120  140  Electric field (kV/cm) Figure 4.9: Variation of the cross sections for s-wave collisions of Li and Cs atoms in the states |1, 0i7 Li ⊗ |3, 3i133 Cs 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 πµ  1/2  (kB T )3/2 2!,  (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 fea72  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  Differential scattering cross section (a.u.)  field is fixed at 1162 G.  4.4. Li–Rb system  2  10  0  10  0  50  100  150  Scattering angle (degrees)  Figure 4.10: Differential scattering cross sections for ultracold collisions of Li and Cs atoms in the |1, −1i7 Li ⊗ |3, 3i133 Cs 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 generated 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 mixtures. 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 angular momenta.  73  4.4. Li–Rb system  Cross section (a.u.)  10  8  4  10  C  B  A  0  10  -4  10  E  D  -8  10  900  1000  1100  Magnetic field (G)  1200  1300  Figure 4.11: Magnetic field dependence of the elastic cross section for collisions between Li and Rb in the atomic spin state | 12 , 12 i6 Li ⊗ |1, 1i87 Rb . These results were obtained for a collision energy of 10−7 cm−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  8  Cross section (a.u.)  10  4  10  0  10  -4  10  -8  10  874  876  878  880  Magnetic field (G)  882  Figure 4.12: Magnetic field dependence of s- and p-wave elastic cross sections for atoms in the atomic spin state | 12 , 12 i6 Li ⊗ |1, 1i87 Rb 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 10−7 cm−1 .  75  4.4. Li–Rb system  8  Width (G)  6  4  2  0 0  50  100  150  Electric field (kV/cm)  200  250  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 10−7 cm−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 ζ 2 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 i 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 i6 Li ⊗ |1, 1i87 Rb 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 6 Li–87 Rb 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 interaction 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 6 Li–87 Rb 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 |f, mf i6 |f, mf i87 | 12 , 21 i |1, 1i | 21 , 21 i  |1, 0i  | 12 , − 21 i |1, 1i | 21 , − 21 i |1, 0i | 32 , 23 i  |1, −1i  B0 (G) 536.65 877.5 654.52 555.88 885.8 578.58 707.70 770.50 596.01 926.8 1242.5  (d)  (d) (d)  (d)  4B (G) 0.01 2.3 < 0.01 < 0.01 2.6 0.01 < 0.01 < 0.01 < 0.01 2.6 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 electricfield-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 , − 21 i6 Li ⊗ |1, −1i87 Rb . 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 9  Cross section (a.u.)  10  6  10  3  10  0  10  1602  1604  1606  1608  1610  Magnetic field (G)  1612  1614  Figure 4.14: Magnetic field dependence of the s-wave elastic cross section for atoms in the atomic spin state | 12 , − 21 i6 Li ⊗ |1, −1i87 Rb 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 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 10−7 cm−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 scattering length by tuning into and out of an intrinsic magnetic field resonance. Figure 4.16 presents two such tuned resonances arising from the variation of the electric field for atoms in the atomic spin state | 21 , 12 i6 Li ⊗ |1, 1i87 Rb . 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 res79  4.4. Li–Rb system  l=0  l=1  Closed channel  l=2  Open channel  d-wave  ml = −1  ml = 0  ml = 1  ml = 2  ml = −2  ml = 0  p-wave  Energy  ml = 1  ml = −1  s-wave  ml = 0  Figure 4.15: A schematic illustrating the mechanism of the shifts and splitting of pand 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  9  Cross Section (a.u.)  10  6  10  3  10  0  10  -3  10 0  20  40  60  80  Electric field (kV/cm)  100  120  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 i6 Li ⊗ |1, 1i87 Rb . 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 10−7 cm−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 resonance 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  0  Cross section (a.u.)  10  10  -3  -6  10  -9  10  ml = 0  |ml| = 1  -12  10 874  876  878  880  882  Magnetic field (G)  884  886  888  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 , 21 i6 Li ⊗ |1, 1i87 Rb 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  Cross Section (a.u.)  10  10  10 10  -12  -14  -16  -10  ml = 0 ml = -1  -12  ml = 1 ml = 2  ml = -2  10 10  -14  -16  10  -18  ml = 0  |ml| = 2  |ml| = 1  537  538  539  Magnetic field (G) 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 i6 Li ⊗ |1, 1i87 Rb 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 10−7 cm−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 components) 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  40 K  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 three ml 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 same ml 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 , 21 i6 Li ⊗ |1, 1i87 Rb at ζ = 100 kV/cm and with three orientations of the electric 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 particularly 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  -4  10  -6  10  Cross Section (a.u.)  -8  10  -5  10  -6  10  0  30  60  90  120  ! (Degree)  150  180  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 | 21 , 12 i6 Li ⊗ |1, 1i87 Rb 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 10−7 cm−1 .  86  4.5. Conclusions 0  Cross Section (a. u.)  10  -3  10  -6  10  -9  10  874  876  878  880  882  Magnetic field (G)  884  886  888  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 i6 Li ⊗|1, 1i87 Rb . 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 10−7 cm−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 ensemble would not affect their visibility either. Nevertheless, since the positions of the resonances do depend on the electric field strength, any inhomogeneities in the magnitude 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 ultracold collisions in Li–Cs and Li–Rb mixtures in the presence of superimposed electric 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 0  Cross section (a.u.)  10  -2  10  -4  10  -6  10  876  878  880  882  884  Magnetic field (G)  886  888  Figure 4.21: Magnetic field dependence of elastic cross sections for atoms in the atomic spin state | 12 , 12 i6 Li ⊗ |1, 1i87 Rb 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 10−7 cm−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-fieldinduced Feshbach resonances. We have shown that external electric fields can also shift the positions of s-wave resonances significantly. These shifts lead to the variation 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 mixtures 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 magnetic 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 involving 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 electric fields as shown in this Chapter. Electric-field-induced Feshbach resonances may thus allow for two-dimensional control of interatomic interactions with both magnetic 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 dominated 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 electric 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 condensates released from the trap [229]. Measurements of the differential scattering cross sections may probe the anisotropy of interatomic interactions [230] and provide detailed 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 couplings 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 Feshbach 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” BoseEinstein 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 3  The 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 quasi2D 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 cylindrical polar coordinates referred to the quantization axis directed along the normal to the confinement plane. The time-independent Schrödinger equation in 2D is "  # ˆl2 (ϕ) 1 d d − ρ + z 2 + V̂ (ρ) ψ(ρ, ϕ) = Eψ(ρ, ϕ), 2µρ dρ dρ 2µρ  (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, ˆlz 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]: ρ→∞  "  ψ(ρ) −→ A eikx +  r  # i eikρ f (k, ϕ) √ . k 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]   eikρ ρ→∞ ψ(ρ) −→ A eikx + f (k, ϕ) √ ρ  (5.3)  1  by the factor of (i/k) 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 k jinc = |A|2 , µ  jsc = |A|2  (5.4)  1 |f (k, ϕ)|2 . µρ  (5.5)  The differential cross section in 2D is defined as dσ(k, ϕ)dϕ =  jsc ρdϕ 1 = |f (k, ϕ)|2 dϕ, jinc k  (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 |f (k, ϕ)|2 . k  (5.7)  The scattering amplitude f (k, ϕ) can be expanded in terms of the eigenfunctions of the operator ˆlz (ϕ), ∞ X  f (k, ϕ) =  fml (k)eiml ϕ ,  (5.8)  ml =−∞  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 1 σ(k) = k 1 = k = =  1 k  Z  2π  0  ∞ X  ∞ X  ml =−∞ m0 =−∞ ∞ X  ml =−∞ m0 =−∞  2π k  ∞ X  ml =−∞  ∗ fml (k)fm 0 (k) l  l  ∞ X  0  ∗ −iml φ dϕ fm 0 (k)e  m0l =−∞  ml =−∞  ∞ X  ∞ X  fml (k)eiml ϕ  Z  2π  l  0  ei(ml −ml )φ dϕ  0  ∗ fml (k)fm 0 (k)2πδm m0 l l l  l  |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 ˆlz ψ(ρ, ϕ) =  ∞ X  Fml (k, ρ)eiml ϕ .  (5.10)  ml =−∞  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 radial 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) = Bml Jml (kρ) + Cml Nml (kρ).  (5.11)  In the asymptotic region, Fml (k, ρ) becomes: r   2 cos kρ − πkρ r  2 +Cml sin kρ − πkρ  ρ→∞  Fml (k, ρ) −→ Bml  ml π π  − 2 4 ml π π  . − 2 4  (5.12)  We define δml as the 2D phase shift of the ml th 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 ρ→∞  Fml (k, ρ) −→ Aml  r    2 ml π π cos kρ − − + δml . πkρ 2 4  (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 ikx  e  =  ∞ X  iml jml (kρ)eiml ϕ .  (5.17)  ml =−∞  95  5.2. Close coupling theory of collisions in two dimensions At ρ → ∞ it becomes e  ∞ X  ikx ρ→∞  −→  ml iml ϕ  i e  ml =−∞  r   ml π π  2 cos kρ − − . πkρ 2 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 ρ→∞  ψ(ρ, ϕ) −→  ∞ X  "  i  ml =−∞  ml  r   2 ml π π  cos kρ − − + πkρ 2 4  r  # i eikρ iml ϕ (. 5.19) fm (k) √ e k l ρ  From there we obtain the radial part of the wave function as r  2 ml π π  i eikρ + Fml (k, ρ) = i cos kρ − − fml (k) √ πkρ 2 4 k ρ r r     2 π 1 −ml ml π π ml ikρ 2 =i cos kρ − − + i fml (k)e πkρ 2 4 2 r r   ml π π  2 π cos kρ − − + = iml πkρ 2 4 2        1 π 1 π × cos − ml + i sin − ml fml (k) (cos kρ + i sin kρ) 2 2 2 2 r   2 ml π π  ml cos kρ − − =i πkρ 2 4 r   h  π ml π π i ml π π  + fm (k) − sin kρ − − + i cos kρ − − 2 l 2 4 2 4 r r      2 ml π π π cos kρ − − 1+i fm (k) = iml πkρ 2 4 2 l r   π ml π π  − fm (k) sin kρ − − . (5.20) 2 l 2 4 ml  r  From Eq. 5.16, we have ρ→∞  Fml (k, ρ) −→ Aml  r  2 πkρ     ml π π  cos kρ − − cos δml − 2 4   ml π π  sin kρ − − sin δml . 2 4  (5.21)  96  5.2. Close coupling theory of collisions in two dimensions Comparison of Eqs. 5.21 and 5.20 leads to ml  i    r  π 1+i fm (k) = Aml cos δml , 2 l r π ml i fm (k) = Aml sin δml . 2 l  (5.22) (5.23)  We thus obtain the coefficient Aml and the expansion coefficient fml (k) as (5.24)  Am = iml eiδml , r l 2 iδm fml (k) = e l sin δml . π  (5.25)  The asymptotic form of the scattering wave function is therefore given by the expression (cf. Eq. 5.16) ψ(ρ, ϕ) =  ∞ X  ml iδml  i e  ml =−∞  r    2 ml π π cos kρ − − + δml . πkρ 2 4  (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) =  ∞ X  2π 4 |fml (k)|2 = k k m =−∞ l  ∞ X  sin2 δml .  (5.27)  ml =−∞  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 = eiδml sin δml ,  (5.28)  Sml = e2iδml ,  (5.29)  Kml = tan δml = −Cml /Bml .  (5.30)  and  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  ∞ X  ml =−∞  |Tml |2 =  1 k  ∞ X  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 d ρ dρ      m2l d 2 ρ Fml (ρ) + k − Û (ρ) − 2 Fml (ρ) = 0, dρ ρ  (5.33)  where k 2 = 2µE and Û (ρ) = 2µV̂ (ρ). Noting that 1 d ρ dρ    d ρ dρ    1 1  ρ2  1  = ρ− 2  1 5 d2 + ρ− 2 , 2 dρ 4  (5.34)  we can write Eq. 5.33 as    d2 + W (ρ) φml (ρ) = 0, dρ2  (5.35)  where W (ρ) = k 2 − 2µV̂eff (ρ, ml ), 4ml 2 − 1 V̂eff (ρ, ml ) = V̂ (ρ) + , 8µρ2  (5.36) (5.37) (5.38)  and 1  φml (ρ) = (kρ) 2 Fml (ρ).  (5.39)  98  5.2. Close coupling theory of collisions in two dimensions The logarithmic derivative y of φml is defined as yml =  φ0ml , φm l  (5.40)  which leads to a new form of Eq. 5.35 in terms of yml as 0 2 ym (ρ) + W (ρ) + ym (ρ) = 0. l l  (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 1  φml (ρ) = (kρ) 2 [Bml Jml (kρ) + Cml Nml (kρ)] 1  = (kρ) 2 Bml [Jml (kρ) + Cml /Bml Nml (kρ)] 1  where  = (kρ) 2 Bml [Jml (kρ) − Kml Nml (kρ)] h i = Bml Jˆml (kρ) − Kml N̂ml (kρ) , 1 Jˆml (kρ) = (kρ) 2 Jml (kρ), 1 2  N̂ml (kρ) = (kρ) Nml (kρ).  (5.42)  (5.43) (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 0 −1 0 Kml = (yml N̂ml − N̂m ) (yml Jˆml − Jˆm ). l l  5.2.4  (5.45)  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 representation [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 collisions 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 = −  ˆl2 (ϕ) 1 ∂ ∂ ρ + z 2 + Ĥas + V̂ , 2µρ ∂ρ ∂ρ 2µρ  (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 1  ψ(ρ, ϕ) = ρ− 2  XX β  ml  eiml ϕ Fβml (ρ) √ φβ , 2π  (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 functions 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    X m2l ∂2 1 2 + + k − F (ρ) = 2µ Fβ 0 ml (ρ)hφβ |V̂ |φβ 0 i, βm β l ∂ρ2 4ρ2 ρ2 0  (5.48)  β  where kβ =  p 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    m2l ∂2 1 2 + kβ + 2 − 2 Fβml (ρ) = 0. ∂ρ2 4ρ ρ  (5.49)  Each equation has a boundary condition for a particular collision channel (β, ml ), namely Fβml (ρ → 0) −→ 0, 1  Fβml (ρ → ∞) −→ (kβ ρ) 2 [aβml jml (kβ ρ) + bβml nml (kβ ρ)] .  (5.50)  (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 1 eiml ϕ −1 ψβml (ρ, ϕ) −→ νβ 2 (kβ ρ) 2 [aβml jml (kβ ρ) + bβml nml (kβ ρ)] √ φβ , 2π  − 21  where νβ  (5.52)  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 (kβ ρ) and nml (kβ ρ), we obtain the wave function for the channel (β, ml ) in the asymptotic region as ρ→∞  ψβml (ρ, ϕ) −→ (kβ ρ)  1 2  −1 νβ 2  s  2 πkβ ρ  h ml π ml π i eiml ϕ × aβml cos(kβ ρ − − ) + bβml sin(kβ ρ − − ) √ φβ . (5.53) 2 4 2 4 2π It can be re-written in terms of exponential functions as − 12  ρ→∞  ψβml (ρ, ϕ) −→ νβ  h i eiml ϕ ml ml π π Aβml e−i(kβ ρ− 2 π− 4 ) − Bβml ei(kβ ρ− 2 π− 4 ) √ φβ , (5.54) 2π  where Aβml = Bβml  r  2 (−bβml + iaβml )/2i, π r 2 = (bβml + iaβml )/2i. π  (5.55)  The S-matrix for multi-channel scattering in 2D is defined as Bβml =  XX β0  m0l  (5.56)  Sβml ←β 0 m0l Aβ 0 m0l ,  where Sβml ←β 0 m0l can be considered as the amplitude of the probability for the atoms or molecules to undergo a transition from an incident collision channel (β 0 , m0l ) 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 ρ→∞  1  − 12  ψβml (ρ, ϕ) −→ ρ− 2 νβ h  −i(kβ ρ−  × e  XX β0  m0l  ml π− π4 ) 2  Aβ 0 m0l  δββ 0 δml m0l − Sβml ←β 0 m0l e  i(kβ ρ−  ml π− π4 ) 2  i eim0l ϕ √ φβ 0 . 2π (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) eikβ x =  X ml  1 iml eiml ϕ 2  s  i m 2 h −i(kβ ρ− ml π− π ) i(kβ ρ− 2l π− π4 ) 2 4 − e e . πkβ ρ  (5.58)  The incoming wave function therefore has the form −1 νβ 2 eikβ x φβ  X  −1 =νβ 2  ml iml ϕ 1  i e  s  2 πkβ ρ  2 ml h i ml ml π π × e−i(kβ ρ− 2 π− 4 ) − ei(kβ ρ− 2 π− 4 ) 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(kβ ρ−  ml π− π4 ) 2  in Eqs. 5.57 and 5.59, we obtain Aβml = iml  s  =0  1 kβ (in  (in other  Bβml is then given by Bβml =  X m0l  m0l βml ←βm0l  S  i  channel  α)  channels).  (5.60)  s  (5.61)  1 . kβ 0  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 =  X  (ψβ 0 )outgoing  β0  =  XX (ψβ 0 m0l )outgoing β0  =−  m0l  XX β0  m0l  −1  νβ 0 2 Bβ 0 m0l ei(kβ0 ρ−  m0l π− π4 ) 2  0  eiml ϕ √ φβ 0 . 2π  (5.62)  102  5.2. Close coupling theory of collisions in two dimensions Combining Eqs. 5.61 and 5.62, we obtain ψoutgoing ψoutgoing = −  XX β0  m0l  −1 νβ 0 2  s # " 0 X m0l π eiml ϕ 1 √ φβ 0 ei(kβ0 ρ− 2 π− 4 ) Sβ 0 m0l ←βml iml . (5.63) kβ 2π ml  The outgoing part of the incident wave function is given by X  s  ml π 1 ei(kβ ρ− 2 π− 4 ) eiml ϕ φβ 2πkβ ρ ml s X X X −1 m0 1 i(kβ 0 ρ− 2l π− π4 ) im0l ϕ m 2 l e φβ 0 δml m0l δββ 0 . (5.64) =− νβ 0 i e 2πkβ ρ 0 0 m  inc ψoutgoing =−  β  −1 νβ 2 iml  ml  l  The scattered wave function is then obtained as inc ψ sc (ρ, ϕ) = ψoutgoing − ψoutgoing  =  XXX β0  m0l ml  ×ei(kβ0 ρ−  0 −1 νβ 0 2 iml eiml ϕ φβ 0  m0l π− π4 ) 2  s  1 2πkβ ρ  (δml m0l δββ 0 − Sβ 0 m0l ←βml ).  (5.65)  Following the single-channel analysis of Adhikari [242], we express the scattered wave function in terms of the scattering amplitude ψ sc (ρ, ϕ) =  X β0  −1 νβ 0 2  s  i eikβ0 ρ fββ 0 √ φβ 0 , kβ 0 ρ  (5.66)  which gives the expression for the scattering amplitude fβ 0 ←β = =  XX ml m0 l  XX ml m0 l  r r  1 (ml − 1 ) (− m0l π− π ) im0 ϕ l (δ 2 e 2 4 e i ml m0l δββ 0 − Sβ 0 m0l ←βml ) 2π 1 (ml − 1 ) (− m0l π− π ) im0 ϕ l T 0 0 2 e 2 4 e i β ml ←βml , 2π  (5.67)  where Tβ 0 m0l ←βml = δml m0l δββ 0 − Sβ 0 m0l ←βml .  The differential cross section for the β → β 0 transition is then obtained using  103  5.2. Close coupling theory of collisions in two dimensions Eq. 5.7, that is 1 |fβ 0 ←β |2 dϕ kβ 1 XX 1 = |δml m0l δββ 0 − Sβ 0 m0l ←βml |2 dϕ. kβ m 2π 0  dσβ 0 ←β dϕ =  l  (5.68)  ml  Integrating this equation over ϕ, we find the integral cross section σβ 0 ←β (k) =  1 XX |δml m0l δββ 0 − Sβ 0 m0l ←βml |2 , kβ m 0 l  (5.69)  ml  where the S-matrix is constructed from the numerical solutions of the coupled differential 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, 1  Φml (φ) = (2π)− 2 eiml φ  (5.71)  and 1  (−1)ml 2l + 1 (l − ml )! 2 Θlml (θ) = l (sin θ)ml 2 (l − ml )! 2 l! l+ml  d × (cos2 θ − 1)l d(cos θ)  (5.72)  When two particles are interchanged, θ → π − θ and φ → π + 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 coworkers [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 interactions, 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 r  3D (r) = − V̂dip  2 X 4π √ αfs 6 3 (−1)q Y2−q (r̂) × [SLi ⊗ SCs ](2) q , 5 r 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 3D operator are given by [199] elements of the V̂dip 3D 0 0 |l ml i|SCs MS0 Cs i|SLi MS0 Li i hSLi MSLi |hSCs MSCs |hlml |V̂dip ! 0 2  √ αfs 1  l 2 l = − 6 3 (2l + 1)(2l0 + 1) 2 (−1)−ml r 0 0 0 ! 2 X l 2 l0 × −ml −q m0l q=−2  0 0 × hSLi MSLi |hSCs MSCs | [SLi ⊗ SCs ](2) q |SCs MSCs i|SLi MSLi i  (5.74)  where the components of [SLi ⊗ SCs ](2) q have the conventional form [221] [SLi ⊗  (2) SCs ]q=0    1 1 = √ 2SLiz SCsz − (SLi+ SCs− + SLi− SCs+ ) , 2 6  (5.75)  105  5.2. Close coupling theory of collisions in two dimensions (2)  [SLi ⊗ SCs ]q=±1 = ∓   1 SLiz SCs± + SLi± SCsz , 2  1 (2) [SLi ⊗ SCs ]q=±2 = SLi± SCs± . 2  (5.76)  (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 2D Y2−q (ρ̂) ≡ Y2−q (θ, ϕ) = Y2−q (θ =  π , ϕ). 2  (5.78)  2D (ρ), namely This leads to a simpler form for V̂dip  2D (ρ) V̂dip  r  2  4π √ αfs π (2) 6 3 Y22 ( , ϕ) [SLi ⊗ SCs ]−2 5 ρ 2  π π (2) (2) , + Y20 ( , ϕ) [SLi ⊗ SCs ]0 + Y2−2 ( , ϕ) [SLi ⊗ SCs ]2 2 2  =−  (5.79)  where the expressions for the components of the [SLi ⊗ SCs ](2) tensor are given q  by Eqs. 5.75, 5.76, and 5.77 . We evaluate the matrix elements of the operator describing the dipole-dipole interaction as 2D |m0l i|SLi MS0 Li i|SCs MS0 Cs i hSLi MSLi |hSCs MSCs |hml |V̂dip r r 2  4π √ αfs 15 =− 6 3 hml |e2iϕ |m0l i 5 ρ 32π 1 × hSLi MSLi |hSCs MSCs | ŜLi− ŜCs− |SLi MS0 Li i|SCs MS0 Cs i 2 r 5 − hml |m0l ihSLi MSLi |hSCs MSCs | 16π   1 1 √ 2ŜLiz ŜCsz − ŜLi+ ŜCs− + ŜLi− ŜCs+ |SLi MS0 Li i|SCs MS0 Cs i 2 6 r  15 1 −2iϕ 0 0 0 + hml |e |ml ihSLi MSLi |hSCs MSCs | ŜLi+ ŜCs+ |SLi MSLi i|SCs MSCs i . 32π 2 (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 different [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 l0 (m0l ) are the orbital angular momenta (projections) 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 calculations 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 − 2i7 Li ⊗ |3, 2i133 Cs 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=  π2 h k (ln k + ln d∗ )2 +  π2 4  i.  (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 l0 (m0l ) are the orbital angular momenta (projections) before and after the collision in 3D (2D). Elastic Collisions s-wave  3D σ = constant  s-wave to non-s-wave  σ ∝ k 2l  non-s-wave to non-s-wave  σ ∝ k 2l+2l  Inelastic Collisions s-wave relaxation  3D σ∝  non-s-wave relaxation  σ ∝ k 2l−1  0  1 k  2D σ∝  1 k ln2 k  σ ∝ k 2|ml |−1 ln12 k 0  0  σ ∝ k 2|ml |+2|ml |−1 2D σ∝  1 k ln2 k  σ ∝ k 2|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 0  that the elastic scattering cross section varies near threshold as k 2l+2l , whereas the cross sections for inelastic or reactive scattering vary near threshold as k 2l−1 (cf. Table 5.1). Consequently, inelastic collisions in 3D are enhanced by a factor of 0  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 swave 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 determined by the inter-particle interactions, particularly, the long-range interaction 108  5.3. Numerical results  Cross Section (a.u.)  10 10 10  6 5 4  10 10  6  10  10  3  0  -4  σ  2D  /σ  3D  10  9  10  -5  10  -14  10  -12  10  -10  -1  10  -8  Collision energy (cm ) 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, −2i7 Li ⊗ |3, 2i133 Cs . The calculations were carried out in a magnetic field of 100 G.  109  5.3. Numerical results  10  10  -10  Cross Section (a.u.)  10  10  -5  -15  -20  10  10  0  -2  -4  10 -14 10  10  -12  10  -10  -1  10  -8  Collision Energy (cm ) 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 (diamonds) 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, −2i7 Li ⊗ |3, 2i133 Cs . The calculations were carried out in a magnetic field of 100 G.  110  5.3. Numerical results  10  10  S-wave collision 2  0  10  σ  inel  /σ  el  10  4  10 10  -2  P-wave collision  16  12  10 10  8  4  0  10 -14 10  10  -12  10  -10  -1  10  -8  Collision Energy (cm ) 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, −2i7 Li ⊗ |3, 2i133 Cs . 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 transitions 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 Alyabyshev 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 elastic 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 longrange interaction potential 1/ρα are σsml ;sm0l ∝ ks2α−5 and independent of m and  m0 , providing ml + m0l − α + 2 > 0. We verified by numerical calculations that the  ml = 0 → m0l = 2 and ml = 2 → m0l = 2 cross sections in 2D scattering induced by the dipole-dipole interaction 1/ρ3 vanish as σsml ;sm0l ∝ ks . The calculations are difficult 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 functions. 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 collisions for s-wave and s-wave to non-s-wave transitions in 2D are σs0,s0 ∝ 1/(ks ln2 ks ) 2|ml |−1  and σs0,sm0l ∝ ks  / ln2 ks , respectively. This indicates that s-wave collisions of  ultracold molecules in a 2D gas accompanied by angular momentum change are 2|ml |  suppressed by ks gases are about  , the same factor as in 3D. Typical temperatures of ultracold  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  Cross Sections (a.u.)  10  10  10  0  -6  -12  10  Rend = 1,000 a.u. Rend = 5,000 a.u. Rend = 10,000 a.u. Rend = 50,000 a.u. Rend = 100,000 a.u. Asymptotic form  -18  10  -14  10  -12  10  -10  10  -8  -1  10  -6  10  -4  Collision Energy (cm ) Figure 5.4: The modification of the threshold energy dependence of the cross sections 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, 2i7 Li ⊗|4, 4i133 Cs . Rend specifies the propagation distance of the coupled differential equation (cf. Eq. 5.48). Consider, for example, s-wave collisions of spin-1/2 2 Σ molecules in the rotationally 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  Cross Section (a.u.)  10  10  -10  10  10  -5  -15  Rend = 1,000 a.u. Rend = 5,000 a.u. Rend = 10,000 a.u. Rend = 50,000 a.u. Rend = 100,000 a.u. Asymptotic form  -20  10  -14  10  -12  10  -10  10  -8  -1  10  -6  10  -4  Collision Energy (cm ) 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 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 maximally stretched state |2, 2i7 Li ⊗|4, 4i133 Cs . 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,s0 l0 ∝ 1/ks for 3D and σs0,s0 m0l ∝ 1/ks ln2 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 ultracold inelastic collisions in 2D geometry. Our numerical calculations verify the predicted threshold energy dependence of cross sections for elastic and inelastic col116  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 denominator 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 collision 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 collisions 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 varying the confinement strength and external magnetic fields. 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.  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 lattices 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 inelastic and chemically reactive collisions, which suggests the possibility of studying chemistry in restricted geometries. Several previous studies showed that collision dynamics 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 restricted 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 quantum computation [39, 40, 42, 43, 171, 172, 252]. An analysis of inelastic scattering 4  A 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 determined 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 between 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 conclusion 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 p 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 p 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  The Collisionand Energies ! [180] !ω relates the quasi-2D scattering wave function in the region Shlyapnikov 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  6.6. Appendix  ~ω0 and molecules prepared in the initial state are! trapped TheCollisionEnergies !ω in a quasi-2D geometry.  Collision Energy !!ω !ω  (28)  l0 " re  !ω " −" α!  USIONS IV. CONCLUSIONS IV.  α  " !ω  (27)  l0  (29)  (30)  VH (z) = µω 2z 2 /2  Figure 6.1: The schematic diagram of a quasi-2D system. Particles are confined in CONCLUSIONS  the ground state of a harmonic potential with the oscillation length of the confining  Acknowledgments ments potential much larger than the characteristic radius re of inter-particle interaction potentials. Acknowledgments  For inelastic collisions of ultracold atoms and molecules in quasi-2D geometry, The work was supported by the Natural Sciences and Engineering the relative momentum of the collision complex for a particular channel is calculated  as supported bysupported the Natural Sciences Engineering Research The work was by the Natural Sciencesand and Engineering Research Council as kα2 = 2µ(ε − α ), where α are the channel energies corresponding to different  (NSERC) of Canada.  (NSERC) of Canada. nada.  120  l0 = µ  !  !/µω0  ω0  6.3. Elastic collisions inl0quasi-2D ≈ 500 geometry  Re Re  0  The wave function R <<is l0 re (ii) ! proportional to the  (i) 3D collision core η  (iii) Asymptotic region  regular 3D wave function  √ -0.01 " 2 i πηϕ0 (0) ! −ikα r = 2µ(E − "α ) α (r̂) e − Sαα eikα r φα Yk00 = "kα r  E  α  kα2 << !ω0  -0.02  α  α φα Sαα  S  α  -0.03  "α! − "α >> !ω0  Y00 (r̂)  R schematic > Reη diagram of anSelastic collision in quasi-2D geometry: Figure 6.2: The  η=  (i) at 5short interparticle10separations r < re15 , the collision occurs 20 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 theR3D > s-wave Re scattering R >wave Re function 1 [180]; (iii) in the asymptotic region, the wave function is the product of a circular √ (1−Sαα )ω($/2!ω 0 ) function and the wave function for the ground state harmonic motion in the wave √ + π(1 + Sαα ) 2ikl0 confining potential.  η α η#  α!  √ states of the particles. The index α is used to describe the internal energy internal  as well as4π the chemical identity of the colliding particles, i.e. it specifies the collision channels. The channel is considered to be confined when kα2  ~ω0 . In general, an η # collision or chemical reaction releases a lot of energy which is much larger inelastic  η= √ than 4π the confinement potential, α→  Y00 (r̂)  α  α0  i.e. α0 − α  ~ω0 . As a result, any transition  results in the acceleration of the collision products and they are Refree to move l0  in 3D. At the inter-particle distance r > re , the interaction between the colliding  s  particles can be omitted from the Schrödinger equation. Therefore, there are no couplings between different collision channels in equations at Rethe < coupled-channel R << l0 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 described 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 1 − 4 +V + VH (z) − ω0 ψ(~r) = εψ(~r), 2µ 2  (6.1)  where ε is the collision energy and VH (z) = µω02 z 2 /2 is the confinement potential. This equation can be re-written in the form   1 µν02 z 1 − 4 +2V + − ν0 ψ(~r) = Eψ(~r), µ 4 2  (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) + A0 GE (~r, 0)] φα Y00 (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 particles, 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)e  i~k·~ ρ  s  − f00 (E)ϕ0 (z)  # i ikρ e φα Y00 (r̂). 8πkρ  (6.4)  Matching Eqs. 6.3 and 6.4 at ρ → ∞, we obtain the scattering amplitude f00 = −A0 ϕ0 (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 multichannel 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) √ i i π h −ikα r ψα (~r) = e − Sα←α eikα r φα Y00 (r̂), kα r  (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  i √ 1 h −ikα r e − Sα←α eikα r φα Y00 (r̂). ψα (~r) = i πηϕ0 (0) kα r  (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 πηϕ0 (0)φα Y00 (r̂)   cos kα r i sin kα r Sα←α cos kα r iSα←α sin kα r × − − − . kα r kα r kα r kα r  (6.8)  For kα r → 0,    √ 1 1 ψα (~r) → i πηϕ0 (0) − i − Sα←α − iSα←α φα Y00 (r̂) kα r kα r  √  √ 1 i πηϕ0 (0) (1 − Sα←α ) + πηϕ0 (0)(1 + Sα←α ) φα Y00 (r̂). = r kα  (6.9)  For small arguments, J0 is about 1 and the Green’s function has the form GE (~r, 0) ≈ where ω(  1 1 E + √ ω( ), 4πr 4π πl0 2~ν0  E ) = ln(B~ν0 /πE) + iπ 2~ν0  (6.10)  (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) + A0 GE (~r, 0)] φα   1 A0 A0 ω(E/2~ν0 ) √ ≈ ϕ0 (0) + φα . + r 4π 4π πl0  (6.12)  Comparing Eqs. 6.9 and 6.12, we get √ i π4πηϕ0 (0) A0 = (1 − Sα←α )Y00 (r̂), kα  (6.13)  and η=  √  4π  (1−Sα←α )ω(E/2~ν0 ) ikα l0  +  √  π(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=  √ 4π (1−Sα←α )ω(ε/2~ω0 ) ikα l0  +  √  π(1 + Sα←α )  ,  (6.16)  1  and ϕ0 (0) is given by ϕ0 (0) = ( πl12 ) 4 . 0  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 α0 l0 m0l , where l0 is the rotational angular momentum (partial wave) of the collision complex in state α0 and m0l 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 → α0 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  −1 i να 2 r−1 h Aα00 e−ikα r − Bα00 eikα r φα , = √ 4π  (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−ikα r in Eq. 6.7 with that in Eq. 6.17 and using the conventional form [199] √ Aα00 = i π/kα ,  (6.19)  √ i πηϕ0 (0) χ = kα Aα00 = ηϕ0 (0).  (6.20)  we obtain the coefficient χ  Since the asymptotic motion of the collision products after a reactive or inelastic process is unconstrained, a combination of the exponential functions and 3D spherical harmonics should be used to describe the wave function in the outgoing collision channels −1  0  ψα0 l0 m0l = −να0 2 r−1 Bα0 l0 m0l ei(kα0 r−l π/2) φα0 Yl0 m0l (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 Bα0 l0 m0l are related to the amplitude of the incoming wave Aα by the S-matrix elements  Bα0 l0 m0l = Sα0 l0 m0l ←α00 Aα00 .  (6.22)  where the elements Sα0 l0 m0l ←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 α0 6= α is given by 0  α 6=α sc ψinel = ψoutgoing X X X −1 0 =− να0 2 r−1 Sα0 l0 m0l ←α00 Aα00 ei(kα0 r−l π/2) φα0 Yl0 m0l (r̂) α0 6=α l0  =− =−  m0l  X XX  α0 6=α l0  m0l  X XX  α0 6=α  l0  m0l  −1  0  να0 2 r−1 Sα0 l0 m0l ←α00 χAα00 ei(kα0 r−l π/2) φα0 Yl0 m0l (r̂) −1  να0 2 r−1 Sα0 l0 m0l ←α00 χ  √ i π i(kα0 r−l0 π/2) e φα0 Yl0 m0l (r̂). kα  (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 scattering amplitude modified by the confining potential. The scattering amplitudes for inelastic collisions are defined by sc = ψinel  X  α0 6=α  −1  να0 2 fα0 ←α  eikα0 r φ α0 . r  (6.24)  Comparing Eq. 6.23 with Eq. 6.24, we obtain fα0 ←α = −  XX l0  m0l  √ i π Sα0 l0 m0l ←α00 χ l0 Yl0 m0l (r̂). i kα  (6.25)  The differential cross section can thus be obtained using dσα0 ←α = |fα0 ←α |2 . dΩ  (6.26)  Integrating Eq. 6.26 over all orientations, we get the integral cross sections σα0 ←α =  XX π χ2 |Sα0 l0 m0l ←α00 |2 , 2 k α 0 0 l  (6.27)  ml  1  where η is given by Eq. 6.14 and ϕ0 (0) = ( πl12 ) 4 . 0  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 σα0 ←α = =  XX l0  ×h  |Sα0 l0 m0l ←α00 |2  πϕ0 (0)2  |Sα0 l0 m0l ←α00 |2  kα2  m0l  XX l0  πϕ0 (0)2  η2  kα2  m0l  4π  (1−Sαα )[ln(B~ω0 /πε)+iπ] √ 2ikα l0  +  i2 . √ π(1 + Sαα )  (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 ∼ kα2 ,  (6.29)  |1 − Sαα | ∼ kα  (6.30)  |1 + Sαα | ∼ constant.  (6.31)  so  and  Therefore η2 ∼ h  4π kα ln kα kα  + constant  i2 .  (6.32)  The constant in the denominator can be omitted at very small kα , so the dependence of η 2 on kα is η 2 ∼ 1/ ln2 kα .  (6.33)  When kα → 0, the inelastic cross section in 3D is proportional to 1/kα σα3D 0 l0 m0 ←α00 ∼ 1/kα ,  (6.34)  |Sα0 l0 m0l ←α00 |2 ∼ kα .  (6.35)  l  and  From Eqs. 6.28, 6.33, and 6.35 we obtain the threshold law for s-wave inelastic collisions in quasi-2D |σα0 ←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 consider 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 quantum 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 σNa0 Nb0 ←Na Nb . 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] |Na mNa i|Nb mNb i =  XX N mN  hN mN |Na mNa Nb mNb i|N mN i,  (6.37)  where hN mN |Na mNa Nb mNb i are the Clebsch-Gordan coefficients. The orthonormality of angular momentum functions leads to the orthogonality relation of the Clebsch-Gordan coefficients: XX  mNa mNb  hN mN |Na mNa Nb mNb ihNa mNa Nb mNb |N 0 m0N i = δN N 0 δmN m0N .  (6.38)  We reformulate the theory described in Section 6.4 in the total angular momentum representation. The relation between σNa0 Nb0 ←Na Nb and σNa0 m0N is as follows:  1 σNa0 Nb0 ←Na Nb = (2Na + 1)(2Nb + 1) XXXX σNa0 m0N mNa mNb m0 Na  =  m0N b  a  a  Nb0 m0N ←Na mNa Nb mNb b  Nb0 m0N ←Na mNa Nb mNb b  XX X X X X 1 (2Na + 1)(2Nb + 1) 0 0 m 0 0 m l  ml  Na  Nb  mNa mN  π 2 χ |SNa0 m0N Nb0 m0N l0 m0l ←Na mNa Nb mN 00 |2 , b a b k2  b  (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 σNa0 Nb0 ←Na Nb =  XX X X X X π 1 χ2 2 (2Na + 1)(2Nb + 1) 0 k 0 m 0 0 m l  ml  Na  Nb  m Na m N  b  XXXXXXXXXXXX J  mJ  J 0 m0J  N mN N 0 m0N N 00 m00 N 000 m000 N N  hN mN |Na mNa Nb mNb ihN 00 m00N |Na mNa Nb mNb i  0 0 0 0 hN 0 m0N |Na0 m0Na Nb0 m0Nb ihN 000 m000 N |Na mNa Nb mNb i  hJmJ |N mN 00ihJ 0 m0J |N mN 00ihJmJ |N 0 m0N l0 m0l i hJ 0 m0J |N 0 m0N l0 m0l i  0  J∗ S J 000 000 000 000 × SN 0 0 0 0 00 00 00 a N N l ←Na Nb N 0 Na N N l ←Na N N 0 b  b  b  1 π 2 = χ (2Na + 1)(2Nb + 1) k 2 XXXXXXXXXXX l0  J  mJ  J 0 m0J  N  N 0 N 00 m00 N 000 m000 N N  δN N 00 δmN m00N δN 0 N 000 δm0N m000 (δJJ 0 δmJ m0J )2 N 0  J∗ S J 000 000 000 000 × SN 0 0 0 0 00 00 00 a N N l ←Na Nb N 0 Na N N l ←Na N N 0 b  b  b  1 = (2Na + 1)(2Nb + 1) XXXXX π J × |2 χ2 |SN 0 0 0 0 2 a Nb N l ←Na Nb N 0 k J mJ N N 0 l0 XXXX 2J + 1 = (2N + 1)(2Nb + 1) a J N N0 l0 π J |2 , × 2 χ2 |SN 0 0 0 0 a Nb N l ←Na Nb N 0 k  (6.40)  where J(J 0 ) = N + 0(N 0 + l0 ). The inelastic cross sections for H2 -H2 collisions in the coupled total angular momentum representation are thus given by σNa0 Nb0 ←Na Nb =  XXXX l0  J  N  N0  2J + 1 (2Na + 1)(2Nb + 1)  π J × 2 χ2 |SN |2 . 0 0 0 0 a Nb N l ←Na Nb N 0 k  (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  87 Rb  atoms in the mf = 0 state with 6 Li atoms in the  mf = − 21 state, leading to Zeeman relaxation in a magnetic field. The calculations are based on accurate interaction potentials for the 6 Li–87 Rb molecule generated as  described in Chapter 3. The scattering length a of the 6 Li–87 Rb 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 6 Li and  87 Rb  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 dependence 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 sections 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 reactive collisions destroy ultracold atoms and molecules. The ratio of cross sections  130  6.7. Numerical results 10  2  10  σ (a.u.)  10 10 10  0  -1  -2  10 10  1  -3  -4  10  -13  10  -12  10  -11  10  -10  10  -9  10  -8  10  -7  Collision Energy (K) Figure 6.3: The threshold energy dependence of cross sections for inelastic relaxation in s-wave collisions of 6 Li with 87 Rb: 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 | 21 , − 21 i6 Li ⊗ |1, 0i87 Rb . for inelastic and elastic scattering in a quasi-2D gas can be written as quasi−2D σinel quasi−2D σel  =γ  3D σinel , 3D σel  (6.42)  where σ 3D denote the cross sections in an unconfined 3D gas. The scattering amplitude for elastic collisions in quasi-2D geometry can be written in terms of the 3D scattering length a and η [180] as f00 = 4πϕ20 (0)aη,  (6.43)  which yields the cross section for elastic collisions in quasi-2D as [180] |f00 |2 4kα 16π 2 ϕ40 (0)a2 η 2 = . 4kα  quasi−2D σel =  (6.44)  131  6.7. Numerical results  5  10  0  ! (a.u.)  10  -5  /!  3D  10  -4  quasi-2D  10  -10  -6  10  !  10  -8  10  1095  -15  10  1095  1098  1101  B (G)  1098  1101  B (G)  1104  1104  1107  1107  Figure 6.4: Cross sections for s-wave inelastic collisions of 6 Li and 87 Rb 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 | 21 , − 21 i6 Li ⊗ |1, 0i87 Rb . Using Eq. 6.27, we obtain the ratio of cross sections for elastic and inelastic collisions in quasi-2D as π 2 2P P 2 2 η ϕ0 l0 m0l |Sα0 l0 m0l ←α00 | kα = quasi−2D 16π 2 ϕ40 (0)a2 η 2 σel 4kα P P 2 l0 m0l |Sα0 l0 m0l ←α00 | . = 4πkα ϕ20 (0)a2  quasi−2D σinel  (6.45)  The elastic cross sections in 3D can be written in terms of the scattering length as 3D σel = 4πa2 ,  (6.46)  132  6.7. Numerical results while the cross section for inelastic collisions is given by 3D σinel =  π XX |Sα0 l0 m0l ←α00 |2 . kα2 0 0 l  (6.47)  ml  Therefore, the relation for inelastic-to-elastic ratio in 3D is 3D σinel = 3D σel  P P l0  m0l  |Sα0 l0 m0l ←α00 |2  4a2 kα2  (6.48)  . 1  Using Eq. 6.45 and Eq. 6.48 and the expression ϕ0 (0) = ( πl12 ) 4 , we obtain 0  kα l0 γ= √ = π  r  2ε . π~ω0  (6.49)  0  10  -1  -2  10  in  ! /!  el  10  -3  10  -4  10  0  1  4  2  3  l0 (in units of 10 Bohr) Figure 6.5: The ratios of inelastic and elastic cross sections for s-wave collisions of 6 Li and 87 Rb 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 , − 21 i6 Li ⊗|1, 0i87 Rb . The collision energy is 10−8 cm−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  10  -4  10  !  quasi-2D  /!  3D  -1  (Bohr )  10  -6  10  0  2  4 6 4 l0 (in units of 10 Bohr)  8  10  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 10−8 cm−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 6 Li and  87 Rb  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 collisions of molecules are suppressed when the geometry changes from 3D to quasi-2D while the inelastic collisions are suppressed much more significantly than the elastic 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 -1  -2  10  -3  10  !  quasi-2D  /!  3D  -1  (Bohr )  10  -4  10  0  2  4  6  8  3  10  12  14  l0 (in units of 10 Bohr) 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 7 Li + 6 Li (v = 0, N = 1) collisions. The collision energy is 10−8 cm−1 . 2 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 α0 in Eq. 6.27 must then include outgoing channels in different chemical reaction arrangements. To explore the effects of laser confinement on chemical interactions of ultracold molecules, we consider an illustrative example of the reaction 7 Li + 6 Li  2 (v  = 0, N = 1) →7 Li6 Li + 6 Li. The cross sections for elastic scattering and  chemical rearrangement transitions in 7 Li + 6 Li2 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 prohibitively 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 inelastic 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 elucidate 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 confinement 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 quantum 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 lattices [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 information 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 microscopic 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 demonstrates 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 scattering 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 dynamics 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 semiconductors. 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 development of experimental studies of complex atomic and molecular systems in confined geometries. The effects predicted in this Thesis should be easy to measure in current 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 lowdimensional 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 manifold 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 magnitude 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 ultracold molecules to date have focussed on diatomic molecules. An emerging direction 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 interest 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 unimolecular 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 reaction 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 excellent 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. Manipulating 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 experiments 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]. In conclusion, quantum effects and high degree of control over collision dynamics at low temperatures make ultracold atoms and molecules an extremely interesting playground for new research in physics and chemistry. The study of atomic and 141  Chapter 7. 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The Schrödinger equation with the Hamiltonian given by Eq. 2.88 and the total wave function given by Eq. 2.89 is "  # XXX ˆl2 (θ, ϕ) 1 ∂2 − + + V̂ + Ĥ Fα0 l0 m0l (r)φα0 Yl0 m0l (r̂) as 2µ ∂r2 2µr2 α0 l0 m0l XXX Fα0 l0 m0l (r)φα0 Yl0 m0l (r̂). =E α0  l0  (A.1)  m0l  ∗ (r̂), integrating over θ and ϕ, and Multiplying Eq. A.1 from the left by φ∗α Ylm l  using the relations  ˆl2 (θ, ϕ)Yl0 m0 = l0 (l0 + 1)Yl0 m0 l l  (A.2)  Has φα0 = α0 φα0 ,  (A.3)  and  we get XXX α0  + + +  l0  m0l  XXX α0  l0  m0l  α0  l0  m0l  α0  l0  m0l  XXX  XXX  =E    1 ∂2 hφα |hYlml (r̂)|Yl0 m0l (r̂)i|φα0 i − Fα0 l0 m0l (r) 2µ ∂r2 Fα0 l0 m0l (r)hφα |hYlml (r̂)|Yl0 m0l (r̂)i|φα0 i  Fα0 l0 m0l (r)hYlml (r̂)|Yl0 m0l (r̂)ihφα |V̂ |φα0 i Fα0 l0 m0l (r)hφα |hYlml (r̂)|Yl0 m0l (r̂)i|φα0 iα0  XXX α0  l0  l0 (l0 + 1) 2µr2  m0l  Fα0 l0 m0l (r)hφα |hYlml (r̂)|Yl0 m0l (r̂)i|φα0 i.  (A.4)  The orthonormality of the Yl0 m0l (r̂) and φα0 functions hYlml (r̂)|Yl0 m0l (r̂)i = δll0 δml m0l  (A.5)  166  Appendix A. and hφα |φα0 i = δαα0 ,  (A.6)  reduces Eq. A.4 to 1 ∂2 l(l + 1) Fαlml (r) + Fαlml (r) 2µ ∂r2 2µr2 X + hφα |V̂ |φα0 iFα0 lml (r) + Fαlml (r)α = EFαlml (r). −  (A.7)  α0  Multiplying Eq. A.7 by −2µ and rearranging the terms, we get    X ∂2 l(l + 1) 2 − + kα Fαlml (r) = 2µ hφα |V̂ |φα0 iFα0 lml (r), 2 2 ∂r r 0  (A.8)  α  where kα2 = 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 ρ→∞  ikx  ψ(ρ) −→ e  +  r  i eikρ f (k, ϕ) √ . k ρ  (B.1)  When ρ  ρe (where ρe is the characteristic distance of inter-particle interaction  potentials), we have [8]  eikρ √ =i ρ  r  1 (1) πkH0 (kρ), 2  (B.2)  (1)  where H0 (kρ) is the Hankel function of the first kind. When kρ  1, eikx ≈ 1,  (B.3)  (1)  and H0 (kρ) has the following approximate expression (1) H0 (kρ)  ≈−    2i π    ln    2i γkρ    ,  (B.4)  where γ = eC and C ≈ 0.577 is the Euler constant. Combining Eqs. B.1 – B.4, we have  r  r    i 2i 2i ψ(ρ) −→ 1 + f (k, ϕ) i − log k π γkρ r r 2i 2i 2i =1 + f (k, ϕ) ln − f (k, ϕ) ln ρ. π γk π ρ→∞  1 πk 2  (B.5)  This should agree with the general solution of the equation −  1 1 d dψ (ρ ) = 0, 2µ ρ dρ dρ  (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 = − ln d, c2  (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+  q  2i 2i π f (k, ϕ) ln γk  −f (k, ϕ)  The scattering amplitude is then given by f (k, ϕ) =  pπ  2i  ln d − ln  2i γk  q  =  2i π  ln γ  pπ  (B.9)  .  2i  ln kd∗ −  iπ 2  ,  (B.10)  where d∗ = (d/2)eC . We thus obtain the expression for the integral cross section 2π |f (k, ϕ)|2 k π2 = h k (ln k + ln d∗ )2 +  σ=  π2 4  i.  (B.11)  One can evaluate d∗ using the value of the calculated cross section σ as ln d∗ = −π  r  1 1 − − ln k. σk 4  (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] −1  Fαl (r → ∞) = να 2    1 l l aαl sin(kr − π) − bαl cos(kr − π) . r 2 2  (C.1)  It can also be written in terms of exponential functions as (cf. Eq. 2.97) i h −1 Fαl (r → ∞) = να 2 r−1 Aαl e−ikα r − Bαl eikα r ,  with  Aαl = −  aαl + ibαl . 2i  (C.2)  (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 → ∞) =  −1 να 2   l l Bαl sin(kα r − π) − Cαl cos(kα r − π) , kα r 2 2 1    (C.4)  where Bαl = Aαl cos δαl  (C.5)  Cαl = −Aαl sin δαl ,  (C.6)  and  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 cos δαl kα  (C.7)  170  Appendix C. and bαl = −  Aαl sin δαl . kα  (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 − i Akαl sin δαl α 2i  =−  Aαl e−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 ∗ (r̂i )eiδαl . Aαl = 4πil Ylm l  (C.10)  Substituting Eq. C.10 into Eq. C.9, we obtain the amplitude of the incoming wave function in Eq. C.2 Aαl =  ∗ (r̂ ) i2πYlm i l  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, ∗ Y00 (r̂i ) = 1/  p (4π),  (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 π/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 85 Rb–87 Rb and 6 Li–87 Rb mixtures”, Physical Review A 78, 022710 (2008). [2] Z. Li and R.V. Krems “Electric-field-induced Feshbach resonances in ultracold 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 6 Li–87 Rb 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  

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