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Renormalization approach to bound state energy computation for two ultracold atoms in an optical lattice Borzov, Dmitry 2011

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Renormalization approach to bound state energy computation for two ultracold atoms in an optical lattice Calculated intraband corrections to the bound state energies in 1D, 2D, 3D optical lattice by Dmitry Borzov  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in The Faculty of Graduate Studies (Physics)  THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) December, 2011 c Dmitry Borzov 2011  Abstract In experiments with ultra-cold gases, two alkali atoms, that interact with repulsive or attractive potentials and are confined to an optical lattice, can form bound states. In order to compute the energy of such states formed by atoms in the lowest Bloch band, one needs to take into account the intraband corrections arising from contributions by higher Bloch bands. As it is hard to implement, known calculations tend to neglect them altogether thus setting up a limit for the precision of such computations. To address the problem we apply an approach that uses renormalizationgroup equations for an effective potential we introduce. It allows for the expression of the bound state energy in terms of the free-space interaction scattering length and parameters of confining potentials. Expressions for bound state energies in 1D, 2D and 3D optical lattices are reported. We show that the method we use can be easily tailored to various cases of atoms confined by external fields of other geometries. A known result for atoms confined to a quasi-2D system is reproduced as an example. Universality of the approach makes it a useful tool for such class of problems.  ii  Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  ii  Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . .  iii  List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  v  Abstract  Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . vii Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Basic concepts . . . . . . . . . . . . . . . . . . . . . . 1.1 Ultracold atomic gases . . . . . . . . . . . . . . . 1.2 Green’s functions formalism . . . . . . . . . . . . 1.3 The low-energy limit . . . . . . . . . . . . . . . . 1.4 Bound state energy and the poles of T (k) function 1.5 Effective potentials . . . . . . . . . . . . . . . . . 1.6 Effective potential and lower dimensions . . . . .  . . . . . . .  4 4 5 8 9 10 13  2 Two atomic particles in an optical lattice . . . . . . . . . . 2.1 Interaction matrix elements . . . . . . . . . . . . . . . . . . . 2.2 Lowest-band approximation . . . . . . . . . . . . . . . . . . .  15 18 19  . . . . .  . . . . . .  22 22 25 27 28 30  . . . .  . . . .  33 38 39 41  3 An 3.1 3.2 3.3 3.4 3.5  . . . . . . .  . . . . . . .  . . . . . . .  . . . . . . .  approach based on renormalization group equations Quasi-2D system . . . . . . . . . . . . . . . . . . . . . . . Renormalization group equation in optical lattices . . . . 3D optical lattice . . . . . . . . . . . . . . . . . . . . . . 2D optical lattice . . . . . . . . . . . . . . . . . . . . . . 1D optical lattice . . . . . . . . . . . . . . . . . . . . . .  4 Computation of the energy of bound 4.1 3D optical lattice . . . . . . . . . . 4.2 2D optical lattice . . . . . . . . . . 4.3 1D optical lattice . . . . . . . . . .  states . . . . . . . . . . . . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . . . . .  1  iii  Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  45  Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  47  5 Conclusion  Appendices A Glossary of used notational conventions B Numerical calculations . . . . . . B.1 Bloch’s states . . . . . . . . . . B.1.1 Explicit expression . . . B.2 Integrating over density of states B.2.1 1D case . . . . . . . . . . B.2.2 2D case . . . . . . . . . . B.2.3 3D case . . . . . . . . . . B.3 1D optical lattice . . . . . . . . B.3.1 Details on X1D (u) . . . . R α(E) B.3.2 Computation for g1D B.3.3 Details on X2D (u) . . . .  . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . functions . . . . . .  C Low-energy limit for 2D scattering . . . . C.0.4 General solution . . . . . . . . . . C.0.5 Unitary relation . . . . . . . . . . C.0.6 T-matrix and Lippmann-Schwinger C.0.7 Asymptotic behaviour . . . . . . . C.0.8 Low energy limit . . . . . . . . . . C.1 Effective potentials . . . . . . . . . . . . C.1.1 Well potential . . . . . . . . . . .  . . . . . . . . . . .  . . . . . . . . . . .  . . . . . . . . . . .  . . . . . . . . . . .  . . . . . . . . . . .  . . . . . . . . . . .  . . . . . . . . . . .  . . . . . . . . . . .  49  . . . . . . . . . . .  . . . . . . . . . . .  50 50 51 52 52 54 54 54 54 55 55  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . equation for 2D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  . . . . . . . .  56 56 57 58 58 59 59 59  iv  List of Figures 1.1  2.1  2.2  3.1  3.2  3.3  4.1  Diagrammatic representation of the Lippmann-Schwinger equation for the two-particle scattering. T is for the scattering T-matrix and U is the potential operator. Solid lines are for the atom’s free space propagation. . . . . . . . . . . . . . . . Dispersion relation ε(q) for the three lowest Bloch bands for 1D optical lattice for the optical lattice u = 0.2 (left), and the energy ranges for these bands as a function of potential strength (right). . . . . . . . . . . . . . . . . . . . . . . . . . Example plots for the bound state energy as a function of scattering length a within the lowest-band approximation for 3D optical lattice with u= 0.2(left), 0.7(right). Blue region corresponds to the lowest band energy range, the beige one to the second band. . . . . . . . . . . . . . . . . . . . . . . . . Calculated numerical constant X3D (u) as a function of dimensionless laser potential strength u for renormalized interaction R equation. strength g3D . . . . . . . . . . . . . . . . . . . . . X2D (u) function, dimensionless constant as a function of dimensionless potential strength u in the equation for renormalized coupling strength g2D . . . . . . . . . . . . . . . . . . A calculated numerical constant X1D as function of dimensionless external potential strength u for 1D optical lattice. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The diagrammatic equation corresponding to the LippmannSchwinger equation for two-particle scattering in the 2 lowest Bloch bands of an optical lattice. . . . . . . . . . . . . . . . .  7  16  21  27  29  31  34  v  List of Figures 4.2  4.3  4.4  4.5  Bound state energy as a function of scattering length in 3D optical lattice with u=0.2. Dashed lines are for results in the single-band approximation (black for the lowest band, blue for the second). The solid lines are higher-band corrections. Lowest and second Bloch band energy ranges are denoted correspondingly with blue and beige here and in the subsequent figures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The bound state energy spectrum as a function of scattering length in 3D optical lattice with u=0.7. Dashed lines are for results in the single-band approximation (black for the lowest band, blue for the second). The solid lines are higher-band corrections. Solid blue line reflects the divergence a → ∞ energy value. . . . . . . . . . . . . . . . . . . . . . . . . . . . Computed dimensionless bound state energy for 2D optical lattice with potential strength of u = 0.2. λ - the laser field wavelength, a - scattering length, and l⊥ the characteristic length of transversal potential. . . . . . . . . . . . . . . . . . Computed dimensionless bound state energy εB as a function  35  36  41  −1  of expression of system parameters lλ⊥ la⊥ − 1.036 for 1D optical lattice with potential strength of u = 0.2 λ - the laser field wavelength, a - scattering length, and l⊥ the characteristic length of transversal potential. . . . . . . . . . . . . . .  42  B.1 Numerical values for c0m as functions of m for a number different values of F for the lowest band. . . . . . . . . . . . . . 53 B.2 Numerical values for c0m as functions of m for the second band. 53  vi  Acknowledgements I would like to express my gratitude to research supervisor Dr. Fei Zhou for his scientific guidance, sound advice and encouragement throughout my thesis-writing period. I am grateful to Mohammad Mashayekhi, a Ph.D. candidate, for helping me to clarify obscure details in some papers I struggled with. Gavin Anstee’s help with the language is greatly appreciated.  vii  Introduction Systems studied in condensed matter are notorious for the complexity of their behaviour. Many factors contribute to the dynamics of such systems and analysis without major simplifying assumptions is almost always unrealistic. One way to approach the study of such systems is to consider simple ”toy” models instead that may not always be related to the real systems directly but have an advantage of being susceptible to analytical research. A researcher could then speculate - or sometimes only hope - that the features characteristic to these toy models extend qualitatively (or even quantitatively) to the real systems that are of her ultimate interest. It is illustrative that many of the seminal historic achievements of the condensed matter theory concern the link between these toy models and real systems unapproachable with direct analysis. Thus a proposed link between at the time purely theoretical concept of Bose-Einstein phase transition in non-interacting Bose gas and experimentally found lambda-point transition in He4 shed first light on the nature of superfluidity that was crucial to the subsequent deeper understanding of the phenomenon [7]. Another example is how the general features of non-structural isolator-metal cross-sections in a wide variety of materials were understood with an example of the localization quantum phase transition in a simplistic Mott’s model [16]. The toy models demonstrate key features of the system of interest and yet can be tackled analytically. However often analytically found effects characteristic to such systems have no immediate experimental confirmation. Researchers who study such toy models may only dream of having experimental findings for these systems. Thus, Bogolyubov in attempt to explain key features of strongly-interacting superfluid He4 considered a system of bosons in the weak interaction limit but no laboratory system with such properties was available to the experimental study at the time [19]. The Hubbard model was introduced to describe transitions between conducting and insulating states and as a simplest model to manifest the existence of the Mott-insulators without any direct prototype physical system in mind [6]. During the 1990s astonishing progress in the techniques for cooling and 1  Introduction manipulation of dilute gases of alkali atoms opened up new possibilities. These systems of ultra-cold gases have unique features: ultra-low temperatures and low densities define manifestation of profoundly quantum behaviour, Feshbach resonance effect and low characteristic energy of scattering in such systems allows for the possibility to manually tune the interaction strength between atoms with the highest precision. External electric, magnetic or laser fields of almost arbitrary geometries can be used to confine the ultra-cold gases. Another important property is weak couplings to dissipative environments - in particular contrast to more classic condensed matter systems. Cold atomic gases with their flexibility of tuning represent a perfect laboratory system for simulation of many toy models and thus may provide profound insights into the most important questions of many-body and condensed matter physics. Amongst the most remarkable achievements in the field were the experimental observations of Bose-Einstein condensation in 1995 [14] and of a quantum phase transition from a Mott insulator to a superfluid [10]. These initially purely theoretical concepts have found laboratory implementations. One may say that this is a physicists’ dream coming true. Amongst all classic toy models the Hubbard model occupies a unique place. Remarkably simple, it is a perfect toy system to study issues of strongly correlated many-body systems (see e.g. discussion in [1]). Ultracold gases confined to optical lattices may serve as a laboratory system for the quantum simulation of a Hubbard model [6]. Determination of the correspondence between tunable experimental parameters of atoms confined to optical lattice and model Hubbard parameters is an important first step for quantitative analysis of the laboratory systems. In order to compute the effective parameters of the Hubbard model one needs to start with a problem of two atomic particles in optical lattice. The Hubbard model with repulsive and attractive effective interaction were found to have two-body bound states by Yang in a theoretical study published in 1989 [20]. The work of Winkler et al. [12] reported evidence for experimentally observed repulsive bound states for atoms in an optical lattice. Matching the results of these two studies is a convenient way for computation of parameters of Hubbard model in terms of physically observable variables: the free space scattering length a and strength of lattice potential. The quantitative analysis used to explain the experimentally observed bound state energy in [12] had its limitations. Consideration was limited to scattering only within the lowest Bloch band and the contributions from 2  Introduction higher bands were neglected. Several approaches were proposed to tackle the issue in a number of papers: the contribution of higher bands were treated semi-classically in the limit of deep lattices [9], and numerically with some modification of renormalizational procedure for effective potential in [2]. Here we generalize the renormalization method proposed in [5]. It has some appealing comparative advantages. The method is flexible: it can be applied not only to atoms in optical lattices but also to particles confined by fields of almost arbitrary geometries. We consider the simple case of twodimensional confinement and reproduce the well-known results to illustrate it. We employ the method to calculate the bound state energies for two atomic particles in optical lattices of different dimensions: 1D, 2D, 3D. We study the expressions we obtained and highlight the similarities and differences for the cases of different dimensions. The method we study provides a useful and flexible tool for analysis of the problem of two particles in a confining field. Here we argue that it is well tailored for computation of the correspondence between model parameters and physical quantities in laboratory systems.  3  Chapter 1  Basic concepts We start with providing a concise overview of the basics of the problem of quantum scattering of two atomic particles in systems of ultra-cold dilute gases of alkali atoms. The reader will be acquainted with the concepts and terminology that are relevant to the subsequent narration. While efforts were made to keep this part self-sufficient, I suggest a reader to see [4] for the more earnest general introduction to the physics of ultra-cold gases and [1] for the discussion on the many-body physics approach towards such systems.  1.1  Ultracold atomic gases  Advancements of laser-cooling and evaporative cooling techniques over the last several decades allowed for the achievement of the incredibly low temperatures of 1mK-1µK range for dilute gases of alkali atoms. At these ultra-low temperatures and the low gas densities characteristic to such experiments, matter manifests quantum properties (see [19], p. 3). There are plenty of unique features of such laboratory systems that make them attractive subjects of research. One may name unprecedented control over system parameters, weak coupling to dissipative environments and the dominant low-energy regime of scattering (s-scattering). These properties are rare in the world of more traditional many-body quantum systems in condensed matter physics: electrons in solids or the family of quantum liquids (mixtures of He3 and He4 ). As a result, the study of ultracold gases is a field of active research and it provides unique opportunities to scientists. There have been successful experiments with the most of the alkali elements and many unique results were obtained in this field of study. External electric, magnetic and laser fields can be applied towards ultracold gases (and are used in cooling and purification techniques). For systems we are discussing here, the action of these confining fields on the atoms can be approximated with point-like elastic external potentials. We will describe the behaviour of atoms in these fields with the model particles without any  4  1.2. Green’s functions formalism inner structure in effective external fields of given specific geometry as it is sufficient for our purposes here. The ability to study dynamics of systems of many atoms in the external fields of almost arbitrary geometries provides the opportunity to see a rich variety of unique effects. Studies of many-body dynamics of atoms in confining fields start with analysis of a problem of two interacting atoms. Real interaction potential for alkali atoms is complex and is hard or impossible to retrieve in most cases. However, for the low-energy scattering that is characteristic to the temperatures and gas densities of these systems, isotropic (s-type) scattering becomes dominant and scattering can be described by a single variable Fermi scattering length a. Indeed, back of the envelope estimation suggests the characteristic length r for non-isotropic scattering (that is, modes with angular momentum quantum number l ≥ 1) to be given by kT =  2l  · (l + 1) mr2  Here kT stands for the characteristic thermal energy of the system and m is the atomic mass. It yields r to be of order of 1 nm for 1 mK and 0.5 µm for for 1 µK for the mass of 87 Rb. Characteristic lengths for the real potential are usually of several orders smaller in range [4]. Thus we have a dominant s-type (isotropic) scattering for such systems1 . We are going to introduce some notions of Green’s function formalism in our discussion of the problem of scattering of two atomic particles.  1.2  Green’s functions formalism  For the temperatures and densities that are characteristic to the systems of ultra-cold alkali atoms, collisions of the gas atoms must be treated as a problem of quantum scattering (see [19], p. 3). We are going to consider two-particle scattering problem in different settings (free space or confined by external potentials of different forms) generally, using the language of Green’s functions formalism. 1 One needs to note here that this estimation can’t be directly applied to gases with fermions. For such cases one should rather compare the term of angular momentum with characteristic Fermi energy instead of the characteristic system temperature.  5  1.2. Green’s functions formalism Generally, one can write the following Schr¨odinger equation for the problem 2  −i∂t −  2m1  ∇21 −  2  2m2  ˆ ψ(r1 , r2 ) ∇22 + V (r1 ) + V (r2 ) ψ(r1 , r2 ) = −U  (1.1) Here the indexes denote the quantities corresponding to the first and the second atoms. We have a two-particle wavefunction ψ(r1 , r2 ) We denote as V (r1,2 ) the confining potential that acts on both of atoms. It is taken to have a point-like action in the real space representation and to be of elastic form for the reasons we discussed before. The masses of the two atoms m1 and m2 can be different for the cases of atoms of different sorts. For the case of identical particles the final wave function has to be symmetrized according to the particles statistics. Here and further in our narration we treat the atoms as not identical. It allows us to keep the approach general. At the end, the solution for the wave function for the identical particles may always be restored with the procedure of symmetrization [13]. The strength of the external potential may also be different for different sorts of atoms. Generalization to such a case is also obvious. ˆ. The interaction potential for the two atoms is denoted with U Let us introduce the single-particle Green’s functions first. The defining equation will be −i∂t +  2 k2  + Vˆ (r) Gn (t|k, kI ) =  (1.2)  = −δ(t)δ(k − kI ), n = 1, 2  (1.3)  2mn  For the free space case (V (r1,2 ) = 0) energy representation of the oneparticle’s Green’s function is simply ∞  dteiEt G(t, kF |kI ) =  G(E, kF |kI ) = −∞  δ(kF − kI ) E−  2 k2 I 2m  + i0  However, here we are really after the case of atoms confined by non-trivial external fields. We need a more general expression for one-particle’s Green’s function that can be extended to such cases. Let us denote the wavefunction for the problem of a single particle confined by external field in a free space momentum representation by hq,n (k), and the energy of such state to be n (q) . We denote with q, n quantum 6  1.2. Green’s functions formalism  +  = T  U  T  U  Figure 1.1: Diagrammatic representation of the Lippmann-Schwinger equation for the two-particle scattering. T is for the scattering T-matrix and U is the potential operator. Solid lines are for the atom’s free space propagation.  numbers that define the wavefunctions we take 2 . Then the single-particle Green’s function can be represented in the so-called Lehmann representation with † dq hq,n (kI )hq,n (kF ) G(E|kI , kF ) = (2π)3 E − n (q) + i0 n∈N  Alternatively, one can generalize definitions for two-particle and one-particle Green’s functions using the basis of the set of wavefunctions hq,n (k), that we will also denote with |qI , n as ”initial” and ”final” states for the Green’s functions. In our notation we would have k|qI , n = hq,n (k) G(E|nI , qI |nF , qF ) =  δnI ,nF δ(qI − qF ) E − n (qI ) + i0  The two representations are equivalent. However the latter (orthogonal functions) representation allows for representing equations in a more compact form due to the delta functions. To shorten the expressions we are going to outline, let us also introduce notation G(E|n, q) =  1 E − n (qI ) + i0  Let us know consider the two-particle scattering T-matrix T . The LippmannSchwinger equation for T can be diagrammatically represented as 1.1 and has the following general form 2  For the case of optical lattices, we denote with n the Bloch band number, and with q the quasimomentum value. And the hq,n (k) would be the Bloch wavefunction for these quantum numbers. For a quasi-2D regime we consider in the next chapter, n is an index of quantum oscillator solution for transversal mode and q is a two-dimensional momentum.  7  1.3. The low-energy limit  T (E, EA , ω|qI , qF ) = U (E, EA , ω|qI , qF ) + +i  dω1 2π  T (E, EA , ω|qI , q)U (E, EA , ω|q, qF )G1 (E + ω1 |q)G2 (−ω1 | − q) q  Here the two particles start off with the opposite momenta qI , −qI and the final states are qF , −qF . E is the total energy of the system, EA , ω are relative off-shell energy shifts for the initial and final system state. We are going to focus on the case of instantaneous and elastic (that is, not affecting the inner structure of our particles) interaction potentials. That means that we have no energy parameter dependence for U U (E, EA , ω|qI , qF ) = U (qI , qF ) and no explicit dependency on ω, EA parameters in the equation and they can be dropped. We have then  +i  1.3  dω1 2π  T (E|qI , qF ) = U (qI , qF ) +  (1.4)  T (E|qI , q)U (q, qF )G1 (E + ω1 |q)G2 (−ω1 | − q)  (1.5)  q  The low-energy limit  Let us consider the case of scattering in free space, without any constraining fields (V (r1,2 ) = 0). This problem is extensively covered in a standard textbook material, one may see e.g. [13]). The straightforward approach to this case would be to make a transformation of coordinates to reformulate a problem in the frame of the center of mass. The two-particle scattering problem is then equivalent to the problem of scattering of a single particle in an external stationary potential. The Lippmann-Schwinger equation 1.5 in this form is equivalent to an integral equation for the scattering amplitude for a problem of single particle scattering on a potential. Let us remind ourselves the properties of such a problem for a low energy scattering limit. The single particle scattering amplitude f (k) is defined by the template wave function form we use to solve the problem of singleparticle scattering ψ(k, r) = eik·r +  1 f (k) ei|k|·|r| |r|  (1.6) 8  1.4. Bound state energy and the poles of T (k) function 2 here r = r1 −r can be interpreted as the vector of relative position between 2 the two atoms, k(and k = |k|) is the vector of the initial momentum of the relative motion. f k, k·r is a scattering amplitude that we are to |k||r| solve for. We discussed in section 1.1 that the dominant scattering is s-type (isotropic) and we neglect all the higher modes scattering and thus drop the dependence on direction (the second parameter) and take f k, k·r =  |k||r|  f (k). It can be shown [13] that the asymptotic behaviour for f (k) for k → 0 limit is −1 f (k) = 1 1 ik + a − 2 re k 2 + o(k 2 ) where the two parameters of the length dimension are introduced. a is a Fermi scattering length and re is called the effective range of the interacting potential. These two characteristic parameters are determined by the real interaction potential. One sees that for small k satisfying kre < 1 the 21 re k 2 term might be dropped and the scattering amplitude is described by the scattering length parameter only. These length parameters were measured and reported for the most of alkali atoms in different states (using a variety of methods, see [4]) and are known with high precision. The energy spectrum of different internal states for atoms shifts under action of external magnetic field allowing for the phenomenon of Feshbach resonance [3]. Thanks to it, the scattering length can be tuned with highest precision by the manipulation of externally applied magnetic field. A comparison of the definition of scattering amplitude, eq. 1.6, to the expansion for two-particle Green’s function gives the relationship between the scattering amplitude and T -matrix for the case of free space [13] f (k) =  1.4  µ 2π  2  kI |Tˆ|kF =  1 −ik −  1 a  + o(k)  (1.7)  Bound state energy and the poles of T (k) function  In our discussion we √ treated T -matrix as a function of scattering k = 2µE. Here the parameter k was corresponded to the physical scattering amplitude. analytical continuation of the function T (k) to the  of the relative energy real and positive and It turns out that the ”non-physical” range 9  1.5. Effective potentials for k is linked with the solutions for bound states that the two interacting particles can form [19]. One can say that the two-particle Green’s function K is equivalent to the T vertex function (and hence to the T -matrix) up to the trivial free propagation term. Specifically , we√can say that the poles of K(E) function correspond to the poles for T (k = 2µE) function. Let us write the general form for the precise two-particle Green’s function K in Leehmann’s representation ˆ α|K(E)|β = m  + j1 ,j2 ∈N  dk1 (2π)D  α|m m|β + E − Em + i0  dk2 α|k1 , j1 ; k2 , j2 k1 , j1 ; k2 , j2 |β (2π)D E − b (k1 ) − b (k2 ) + i0  ˆ is written in general form for the arbitrary bra and ket states α| Here K , |β . The first term in the right-hand side of the equation comes out of all the bound states that the two particles can form with energies Em . The second term reflects the continuous part of the spectrum corresponding to the free relative movement of the two particles. Explicit computation of the limit ˆ lim (E − En ) α|K(E)|β =0  E→En  shows that the poles of K(E) as a function of energy parameter are linked with energies of bound states for the two particles. As there are no poles in the free propagation term for two particles, poles of the exact Green’s function K(E) must be reproduced by the poles of the function T (E). This link lays foundation to the approach we are going to use to determine the bound state energies for two particles in various setups. We will try to solve the Lippmann-Schwinger equation for the effective potential and then perform the analytical continuation. The condition for the poles of the 1 function T (E = 0 shall give us the energy spectrum of bound states of two b) interacting particles.  1.5  Effective potentials  In our discussion so far we did not specify the explicit expression for the ˆ , noting only that we consider it to be elastic and interaction potential U 10  1.5. Effective potentials instantaneous. In general, the actual potential with which alkali atoms interact has complex nature. It may be hard to approach the general problem formulated with the explicit form of the interaction potential. However, one may use the fact that the solution of the scattering problem is completely defined by the expression for T -matrix. We have seen before that the T matrix has the asymptotic form eq. 1.7 in the low-energy scattering limit. We may introduce an effective potential that yields the same vertex function T and yet is easy for us to tackle. As T uniquely defines the two-body Green’s function K for the scattering particles, we would solve the problem of scattering for the limit of low energy. We can use an arbitrary potential as long as it reproduces the scattering amplitude for the real potential. One of the choices for the effective scattering potential that is often used in the literature is the so-called Fermi-Huang’s potential [11], which is effectively equivalent to the hard-sphere potential. The potential is defined in the form of the function in the real coordinates and is usually used for considerations in space coordinates. We are going to use here a modified version of this potential that is defined in the momentum space by expression ˆeff |k3 , k4 = gδ(k1 + k2 )δ(k3 + k4 )θ(Λ − k1 )θ(Λ − k3 ) k1 , k2 |U  (1.8)  Here we introduced g, the scalar for interaction potential strength, and Λ, a ”cut-off” momentum for our potential. The Lippmann-Schwinger equation for such a potential has the form T (kI , kF ) = gθ(Λ − kI ) − gθ(Λ − kI )  1 Ω  T (k, kF ) k,k<Λ  2 k2 I  2µ  −  2 k2  2µ  −1  + i0  (1.9) Here µ is the two interacting particles’ reduced mass, Ω is the total volume of the system. The expression on the right-hand side has no explicit dependency on the direction of kI . As the solution must depend only on the relative angle between kF and kI , we conclude that it means that there is no dependency on direction of either of the two parameters for T (kI , kF ). We should obtain the rotationally invariant solution, i.e. , T (kI , kF ) = T (k) only is a function of the absolute value of k. The equation then can be presented as 1 1 = + T (kI ) g  d3 k (2π)3  2 k2 I  2µ  −  2 k2  2µ  −1  + i0  (1.10) 11  1.5. Effective potentials Or, taking the integral, 1 k+Λ Λ − k log 2 k−Λ  1 µ 1 = + 2 2 T (k) g π  And for the limit of sufficiently high cut-off frequency Λ T-matrix of the form 2π µ  T (k) =  2  1 2π 2 µg  + π2 Λ − ik  (1.11) k, we get the  (1.12)  We see that indeed the asymptotic form for T -matrix reproduces one for low energy limit in the form of T (k) dependency. Comparison to the eq. 1.7 suggests that the T-matrix is equivalent to the physical one for lowenergy scattering if the parameters of the effective potential are gauged by the scattering length a value 1 2π 2 2 = + Λ a µg(Λ) π  (1.13)  Apparently, there is freedom of choice for g and Λ. We may treat them to be ”coupled”, having a value of g for each taken Λ and thus, defining a function g(Λ). Then the g(Λ) is a solution for the differential equation, obtained by taking a derivative of both sides of our expression with Λ d dΛ  1 g  =−  1 µ π2 2  On the other hand, we may obtain the renormalization group equation for our given potential 1.8, using the cumulant expansion for the g(Λ) constant [18] 1 δg(Λ) 1 δ 1 = 2 g δΛ Ω δΛ 2ε(k) k<Λ  1 is the contribution from the kinetic term of the Hamiltonian, and Here ε(k) the ε(k) corresponds to the energy spectrum that is for free space equal to 2 k2 2µ .For the free space, the sum part can be represented as  d dΛ  1 g  =−  1 1 4πΛ2 3 (2π) (Λ)  We see, that we have the same equation as the one we obtained with our ”gauge” condition. 12  1.6. Effective potential and lower dimensions For the free space scattering within the low-energy scattering limit the interaction potential between the atoms can approximated by the short range 2 pseudo potential g(Λ → 0) = 2πaµ with as the s-wave scattering length and m the mass of the atoms [3]. Solving the free space renormalization equation gives an expression of the relation between the low energy limit value expressed in terms of the scattering length T0 = 2πa µ and high energy limit value U (Λ → +∞) µ 2π  1.6  2a s  =  1 1 + g(Λ → +∞) Ω  1 ∀k  2ε(k)  (1.14)  Effective potential and lower dimensions  To what extend are the results we obtained for the free space special to the case of 3-dimensional space? We are interested in the 1D and 2D (lower dimensional) optical lattices as well and are going to need to consider the low-energy limit for scattering in 2D and 1D space, so let us find out. It turns out that the general analysis for comparative contributions from different angular momenta can be done in 2D in a way quite similar to the 3D case. However, while one can say that the low-energy limit in the 3D is a part of general background in quantum mechanics, 2D case is somewhat less known. We provide an accurate consideration for the different modes in the appendix to this thesis and we encourage the reader to use it as a reference for results outlined here. It turns out, that for sufficiently small k the dominant contribution is for the isotropic mode. We define the isotropic part of scattering amplitude in 2D by the following wave function template ψ(r) = eikr − f2D (k) √  eikr −i8πkρ  The scattering amplitude can be linked to the T -matrix in 2D by explicit computation 2  T (k) =  2µ  f2D (k)  The analysis shows that the dependency on the momentum k for scattering amplitude in the low-energy limit is determined by the expression f2D (k) =  2π for 2D space. log(dk) + i π2  (1.15) 13  1.6. Effective potential and lower dimensions Here we introduced the notation for d - a length parameter that is defined by the real potential. It is sometimes referred to as the bound state size. One may introduce the effective potential defined in a way similar to the 3D case ˆeff |kF , −kF = g2D θ(Λ − kI )θ(Λ − kF ) kI , −kI |U (1.16) Here we introduced the coupling constant g2D and cut-off momentum Λ similarly to the 3D case. The Lippmann-Schwinger equation is then d2 k (2π)2  1 1 = − T (k1 ) g  1 2 k2 1  −  2µ  2 k2 2µ  (1.17) + i0  And, taking the integral, we get the following expression for T (k) T (k) =  1 1 g  +  which for sufficiently small k (k T (k) =  π 2 µ  µ 2π 2  log 1 −  Λ2 k2  (1.18)  Λ) it becomes 1 π 2 gµ  + log  Λ k  + i π2  (1.19)  reproducing the low-energy form of dependency on k from eq. 1.15. Gauging the coupling constant g(Λ) to d gives d=  1 π 2 exp − Λ g(Λ)µ  (1.20)  In contrast to 3D and 2D case, for 1D scattering, the high cut-off frequency limit Λ → +∞ does not give divergent T1 value and we can just use delta function effective potential for this case. We thus don’t need the 1D free space renormalization group equations for the effective potential.  14  Chapter 2  Two atomic particles in an optical lattice Optical lattice is a conventional term for patterns of EM standing waves that are induced by the AC Stark effect of interacting laser beams. Experimental techniques allow for the confinement of the ultra-cold gases to optical lattices. A good overview of main results of studies of such systems may be found in [15]. What is the effect of optical lattice field on the atoms of ultracold gases? An isolated neutral alkali atom alone have no net charge and no permanent electric dipole moment. In the presence of electric field, however, the atoms develop an electric dipole moment which can then interact with the electric field. When the field is off-resonant with the atomic transitions, the dominant effect is that the atomic energy levels undergo an energy shift proportional to the square of electric field magnitude. We treat this effect semi-classically, by introducing an effective confining potential of the corresponding form. Let us consider the scattering problem for two atomic particles in an optical lattice in the context of language introduced in the previous chapter. Before solving for the T -matrix, we need to obtain single-particle Green’s functions for such systems. That is, we want to get G given by eq. 1.2, but with V (r) being the potential of the atom in an optical lattice. The field intensity of standing waves would give us the potential of the form Vi sin2  V (x) = i  2πxi λ  in the coordinate representation, with λ being the wavelength of the laser emission. The summation is performed over all Cartesian coordinates, and the Vi values give the intensity of the lattice potential along different directions. By varying the Vi intensity one may obtain the quasi 2D or 1D optical lattice configuration. 15  Εq  Chapter 2. Two atomic particles in an optical lattice  2.0  2.0  2.0  1.5  1.5  1.5  1.0  1.0  1.0  0.5  0.5  0.5  0.0 0.4  0.2  0.0  0.2  0.4  0.0 0.2  0.4  qΛ  0.6  0.8  0.0 1.0  u  Figure 2.1: Dispersion relation ε(q) for the three lowest Bloch bands for 1D optical lattice for the optical lattice u = 0.2 (left), and the energy ranges for these bands as a function of potential strength (right). Vi reflect physical quantities such as the polarizability of atoms and the intensity of laser light. It is also convenient to introduce dimensionless parameters for potential strength 2  Vi =  µ  4π λ  2  u  The parameter u comes out naturally if we reformulate the equation for Green’s function (eq. 1.2) in the dimensionless form. Factoring out the quantity of the energy dimension, that is proportional to the recoil energy, allows us to put the problem in a more universal form. The solutions for the quantum problem of single atom in this periodic  16  Chapter 2. Two atomic particles in an optical lattice potential are Bloch functions that we will denote as cn (q + Q)ei(q+Q)r  hn,q (r) =  (2.1)  Q  Here q is a quasimomentum, n ∈ N is the index for Bloch band and Q are the reciprocal lattice vectors. hn,q (r) gives the Bloch’s function with quasimomentum q and the band number n presented as expansion in terms of Wannier wave function in the momentum space, cn (q + Q) (alternatively, we can think of it simply as the Fourier expansion for periodic part of the Bloch’s function). Quasi 1D configuration corresponds to deep confinement along the two other directions (Vx,y Vz ). It is reasonable to approximate the confinement along the 2 deep limit lattice directions with characteristic length of harmonic potential l⊥ . Equivalently, for quasi 2D optical lattice potential along only one direction is so large it must be treated as harmonic potential. The dispersion relations εn (q) for Bloch’s functions for 1D lattice are presented in Fig. 2.1 (left). The energy values are presented in the dimensionless form, normalized for the recoil energy in the same manner as the lattice potential strength. Atom’s energy must be shifted by the total contribution form the two harmonic directions as well. The dispersion relation εn (q) changes as we change the lattice potential strength u. Higher values of u correspond to ”deeper” lattices, lower values - to more ”shallow” ones. On the Fig. 2.1 (right) the energy ranges for three first Bloch bands are depicted as a function of u. We see that for deeper lattices the energy gap between the two lowest bands becomes bigger. It is going to be useful for us to introduce notations that characterize the energy range for Bloch bands. Let us denote as gn (E) the density of single atom states for atoms in the lattice in the nth band. Then we may introduce the upper and lower bounds for each Bloch band ∆U n = Upper limit gn (E) ∆U L = Lower limit gn (E) and the ”center of mass” energy for the band ∆L =  Egn (E)dE  Technical aspects related to numerical computation are put into the appendix. 17  2.1. Interaction matrix elements Generally, one can see that in the limit of ”deep” lattices the energy range for lower bands become narrow, and the gap between the bands widens. It corresponds to the limit where the coupling between the nearby lattice cells weakens and we have degenerate superpositions of independent solutions of atoms confined to specific cells.  2.1  Interaction matrix elements  Now that we had a look at single-particle solutions in an optical lattice, we can proceed to the formulation of Lippmann-Schwinger equation for two interacting particles. We have seen in the chapter on Green’s functions that it is most convenient to use the basis of solutions for specific single particle Hamiltonian, in this case of Bloch states. We will need to consider the interaction matrix elements for the effective potential we introduced with eq. 1.8 for Bloch states. Specifically, we take ˆeff. | {qF , nF } , {−qF , nF } = Mn ,n (qI , qF )g {qI , nI } , {−qI , nI } |U I F Here we have the matrix element for the state of two atoms in Bloch states with quasimomenta qI and −qI in the nI th band and the state of quasimomenta qF and −qF in the nF th band. We introduced the notation of Mm,n (k1 , k2 ) to separate the contribution due to the overlap of the wavefunctions with the effective potential strength parameter. Expressing the Bloch states in Wannier functions (with eq. 2.1), we get   MnI ,nF (qI , qF ) =  cnF (qI + G)cnF (Q − qI − G) ·   Q  G      ·  cnI (k2 + G)cnI (Q − k2 − G) G  Here the summation Q, G is over vectors of the reciprocal lattice. Using this expression, the coefficients MnI ,nF (q1 , q2 ) can be numerically calculated for any Bloch states. Numerical calculations of the matrix elements for M1,1 (qI , qF ), M1,2 (qI , qF ) and M2,2 (qI , qF ) as functions of dimensionless lattice potential strength u are discussed in the appendix. There spread for values of the matrix elements with different values of quasimomentum qI , qF increases for the limit of ”shallow” lattices (smaller 18  2.2. Lowest-band approximation u values). Also it is much smaller than the band number-defined value and it is justified to drop this dependence and approximate values with ones averaged over the whole band M1,1 (qI , qF ) ≈ M1,1 for the subsequent approximation. One can also calculate matrix elements for the states of atoms starting off in different bands, or with quasimomenta not obeying q1 + q2 = 0 for bra or ket states. However, explicit numerical calculations show that such matrix elements are negligibly small when compared to the ones we considered. There are symmetrical reasons for it. Using this result, we are going to drop all such matrix elements regarding them as zero. Taking the approximations and notations we had into account, we get the Lippmann-Schwinger equation for two atoms in optical lattice of the form  +g  MnI ,b b∈N  nI |Tˆ(E)|nF = MnI ,nF g + 1 1 b|Tˆ(E)|nF Ω E − 2 b (q) + i0  (2.2) (2.3)  q  Here we presented the T -matrix as an analytical function of energy parameter E. Since we neglect the dependency on quasimomentum values q1 for matrix elements of effective potential, we may drop the quasimomentum indexes for the T -matrix too. It is similar to the case of the isotropy of T -matrix in free space we discussed before. We see the summation over all Bloch band indices b in the right-hand side of the equation.  2.2  Lowest-band approximation  Lippmann-Schwinger equation gets the form of infinite system of equations for matrix elements of T -matrix with discrete indices of the Bloch bands of bra and ket states. In order to solve for, e.g., the element 1|T (E)|1 as a function of energy parameter E, we need to take into account contributions from all the higher n|T (E)|n elements. Physically one may think of this link as of the contributions from the higher bands to scattering within the lowest-bands. The simplest way to approach such a system is to drop the contributions from higher bands altogether by manually chopping off the corresponding terms out of the defining expression. The justification for such approximation is following. We are interested in values of |E| being of order of 19  2.2. Lowest-band approximation lower bands for the bound state energies of such a range. The denominator for higher energy terms is thus dumped down due to the large energies 1 1 E−εn (q) ∼ εn (q) . One can neglect their contribution. We can cut-off only bands of very high energy or all of them except one. The more bands we neglect the simpler is the system of equations. The simplest case to consider is to leave only one band. In this case we have the following equation for the single T-matrix element that is left 1|Tˆ(E)|1 = M1,1 g + M1,1 g1,1 1|Tˆ(E)|1  dk (2π)D E −  1 1 (k)  + i0  The pole position for the T-matrix as function of energy and, as a consequence, an energy of the bound state Eb is defined by 1 M1,1 g  dq 1 D (2π) Eb − 1 (q) + i0  =  (2.4)  An example of bound state energy spectrum for 3D optical lattice as a function of scattering length is presented on Fig. 2.2. Let us now estimate the precision and applicability of the method. In order to do that we estimate the contribution of the neglected terms. The pole of the T -matrix corresponds to the pole of the two-particle ˆ That is, we are looking for the pole in approximated Green’s function K. K≈ (E) with the condition lim (E −  E→  n)  α|K≈ (E)|β = 0  n  The approximation we used is equivalent to the dropping of the contributions that correspond to the higher bands for the two-particle Green’s function. ˆ We drop the higher-bands contribution term δ K(E) ˆ ˆ ≈ (E) + δ K(E) ˆ K(E) =K The condition now looks like lim (E −  E→  n)  ˆ α|δ G(E)|β  n  Order ˆ lim (E− n ) α|δ G(E)|β ≈ lim (E− n )  E→  n  E→  n  α|2 2|β E−  2 (k)  + i0  C  ≈ n  − min  (k) ∀k 2  Here C is some constant of ord(1). 20  2.2. Lowest-band approximation  1.6  1.5 3.5  ΕB  ΕB  1.4  1.3  3.0 1.2  1.1 2.5 1.0 0.000 0.002 0.004 0.006 0.008 0.010 0.012  0.000 0.005 0.010 0.015 0.020 0.025 0.030  a Λ  a Λ  Figure 2.2: Example plots for the bound state energy as a function of scattering length a within the lowest-band approximation for 3D optical lattice with u= 0.2(left), 0.7(right). Blue region corresponds to the lowest band energy range, the beige one to the second band. We see that the first order approximation is inapplicable for states with energy within the range close to the lower limit of the second band. We also see that while the approximation is reasonable for optical lattice in ”deep” limit (u → ∞), it can’t be applied for lattices in the ”shallow” limit (u → 0).  21  Chapter 3  An approach based on renormalization group equations In the previous chapter we made an estimation for the bound state energy of atoms in the optical lattice in the lowest-band approximation. We have seen that its precision is low in the limit of shallow lattices and for bound state energy ranges close to the energy ranges of higher Bloch bands. In order to go beyond this crude approximation we need to take into account the contributions from those higher Bloch bands. Here we employ the renormalization method used in [5] to introduce a unified approach of relating low energy behaviour of particles in arbitrary confining fields to the physical quantities of the system such as free space scattering length and parameters of confining fields. We reproduce a known result for effective interaction in quasi-2D systems to illustrate the method. We then proceed to the case of optical lattices.  3.1  Quasi-2D system  A new technique must reproduce previous results. We consider a case of two particles in the strong transversal field that confines particles to a plane and thus forms a quasi-2D system. This problem was already studied by Fedichev et al. in [8]. Here we check if we can reproduce the established results with a new technique. The physical system that corresponds to such a case is an experimental setup of the ultra-cold gas confined by a laser standing wave pattern along one direction with sufficiently strong field strength that effectively locks atoms in one directions and limits their movement to the quasi 2D regime. We are going to describe the transversal movement potential with a po2 2 tential of harmonic form V (r) = mω2 z , z = r3 here is a Cartesian coordinate of a particle and ω reflects the frequency of harmonic field. 22  3.1. Quasi-2D system The Schr¨ odinger equation for the two particles in such a confining field is 2  −  2m  ∇1 +  2 mω 2 z12 mω 2 z22 − ∇2 + + gδ(r1 − r2 ) ψ(r1 , r2 ) = (3.1) 2 2m 2 = i∂t ψ(r1 , r2 ) (3.2)  We used the free space effective potential 1.8 in the limit of Λ → ∞. We 1 have seen that in 3D this limit corresponds to divergent T (k) , so g here has no finite value. Let us now change the reference frame by introducing new variables r = r1 − r2 and reduced mass notation µ = m 2 for identical particles. The Schr¨ odinger equation for the relative motion now gets the form 2  −  2µ  µω 2 z 2 + gδ(r) ψ(r) = i∂t ψ(r) 2  ∇+  (3.3)  we see that the Hamiltonian could be split for the z-variable part in the cylindrical coordinates’ frame. The equation for the transversal coordinate is then just a harmonic oscillator and that allows us to use the expansion ψ(r) =  φn (z)ψn (ρ, φ) n  where φn (z) is an eigenfunction for the harmonic oscillator with quantum number n. Now we can use the Fourier transform for the motion within the plane d2 k ψm (r) = eikr ψm (k) (2π)2 to get −i ∂t + here  2 k2  2µ 2 Λ2  +  1 +n 2  ω ψn (k) = gφn (0)  φj (0) j  2 Λ2 µ  2Λ  1 = ( + N ) ω, = − 2µ 2 2µ 2µ  1 +n 2  d2 k1 ψj (k1 )(3.4) (2π)2  ω = (N − n) ω(3.5)  We see that for the energy of relative motion E < 32 ω particles are confined to the lowest state of harmonic oscillator. System for such energy range is quasi-2D. And for the two dimensional system, one can introduce 2D 2 2 effective potential with cut-off momentum Λ given by the condition 2µΛ < 3 2 ω and strength parameter gR (renormalized g). 23  3.1. Quasi-2D system Renormalization group equation for the effective potential strength is gR = gφ20 (0) − ggR  φ2n (0)  d2 k (2π)2  |k|<Λn  1<n<N  1 2 k2  2µ  + ( 12 + n) ω  (3.6)  that we express as 1 φ20 (0) = + gR g  φ2n (0)  |k|<Λn  1<n<N  d2 k (2π)2  1 2 k2  2µ  (3.7)  + ( 12 + n) ω  we link it to the free space renormalization potential and get µ φ20 (0) = + gR 2π 2 a  φ2n (0)  d2 k (2π)2  |k|<Λn  1<n<N  1 2 k2  − |k|<Λ 2 Λ2  for 2 Λ2 µ  with  2µ  2Λ  =  2m  2µ  1 +n 2  −  −  (3.8)  d3 k 1 (2π)3 2 k2  (3.9)  + ( 12 + n) ω  2µ  2µ  1 = ( + N ) ω (3.10) 2  ω = (N − n) ω (3.11)  with taking integrals we get φ2n (0)  =  n! 2n  n 2  µω ! π  1 2  =  n! 2n  n 2  !  √  1 for even n, and 0 for odd (3.12) 2πl  to get φ0 (0)2 µ B µ = + 2√ 2 2π a gR 2ππ Introducing notation of transversal length l = equation as 2  µgR  (3.13) mω ,  we represent this  1 l 1 =√ − B a π 2π  (3.14)  B here is a dimensionless constant given by 1 √ B=√ 1 + 2N − π  n! even n≤N  2n+1 1 − log 2  n 2  !2  log  1 2 +N 1 2 +1  1 2 +N 1 2 +n  −  (3.15)  = 0.98891  (3.16) 24  3.2. Renormalization group equation in optical lattices The 2D effective Hamiltonian gives the scattering amplitude (see chapter 1) of the form 2 Λ2 1 ω= , so Λ = 2µ l0 so we get d = l0 exp(B) exp −  π l0 · 2 a  , exp(B) = 2.688  (3.17)  Comparing the result reported here to the one obtained in [8] and taking into account different definitions of l we see that results are very close, π √1 exp(B) = 1.901 here and 0.915 ∼ 1.853 in the paper. 2 Yet the approach used here was different from the one used in the original paper. Let us generalize and review what was done. We used an effective potential and obtained the renormalization group equation for interacting particles in the external confining harmonic potential. Then we linked the potential in the high cut-off momentum limit Λ → ∞ to the renormalization group equation for an effective potential in the free space. This allowed us to parametrize the effective potential with physical quantities of the system: characteristic length of the transversal field, and the free space scattering legnth.  3.2  Renormalization group equation in optical lattices  Our example sets out the approach we are going to use for the analysis of the case of optical lattice. Now in a similar fashion we are going to employ the renormalization-group approach to find the effective lowest-band scattering T -matrix and to study the energy spectrum of the bound states. We consider the renormalization of the interaction pseudo potential to the lowest-energy bands scattering. Then we consider the T -matrix within the lowest energyscattering. The poles for the T -matrix scattering allows for the study of the bound state energy dependency for different effective dimensions: 1,2,3 dimensional optical lattice with strong transverse potentials. We write down the Hamiltonian for the two atoms in an optical lattice (eq. 1.1)in the Bloch functions representation. We are using our results for the interaction matrix elements that we discussed in the previous chapter and the notations we introduced.  25  3.2. Renormalization group equation in optical lattices The Schr¨ odinger equation then has the form +∞  [i∂t − εn (q)] ψn (q) = g  dq1D ψm (q1 ) (2π)D  Mm,n −∞  m<N  (3.18)  Here N is a cut-off number that can be linked to the cut-off momentum Λ for the free space with 2 Λ2  N : min EN +1 (q) >  2m  ∀q  .  Let us denote as g(n) the interaction strength with N = n fixed cut-off band number. It is equivalent to the notation of g(Λ) we used before for the momentum cut-off. Then we get the renormalization group equation for g(n) of the form g(n) = g(n + 1) − g(n + 1)g(n)  1 Ω  q  1 2εn (q)  (3.19) (3.20)  In order to derive this equation, we neglected the matrix interaction terms M1,m ≈ 1. As we demonstrated earlier, it is justified for m ≥ 3. We can represent the equation in the form 1 1 1 = − g(n + 1) g(n) Ω  q,m  1 2εm (q)  (3.21)  1 2εm (q)  (3.22)  Taking the n → ∞ limit, we get 1 1 1 = + g(n) g(∞) Ω  q,m>n  This equation gives the relationship for non-renormalized effective potential without cut-off band g(n → ∞) and effective potential defined only for lower Bloch bands g(n), n small being enough. Our aim now will be to calculate the constant of effective potential strength for the two lowest Bloch bands, g(n = 2) for optical lattices in different dimensions. The justification for our choice of cut-off band number n = 2 in the presented consideration is following. We neglected the M1,n terms in our derivation of renormalization equation. It is justified for sufficiently high Bloch bands but is not that reasonable for lower bands. 26  3.3. 3D optical lattice  20  X3 D  15 10 5 0 0  1  2  3  4  u Figure 3.1: Calculated numerical constant X3D (u) as a function of dimenR sionless laser potential strength u for renormalized interaction strength g3D equation.  Another argument in favor of this approach is that our interest is in twoparticle bound states of energy range between the first and second Bloch bands. As a consequence, approximation to the lowest-band g(n = 1) looses precision for bound state energies close to the range of the second band, in the way we discussed in the section on lowest-band approximation. Let us consequently discuss each specific dimensional case of optical lattice, starting with 3D optical lattice.  3.3  3D optical lattice  The effective potential strength parameter g(n → ∞) in the context of 3D optical lattice can be related to the parametrization of effective potential g(Λ → ∞) in free space. While the specific value g(Λ → ∞) diverges in 3D, the renormalization group equation for free space 1.14 allows us to link it to the finite physical  27  3.4. 2D optical lattice quantity of the free space scattering length a 1 1 1 µ − = = g(n → ∞) g(Λ → +∞) 2πas Ω  ∀k  1 2ε(k)  This way we can express effective interaction constant renormalized to arbitrary band   1 1 1  µ 1 +  = − 2 g(1) 2π as Ω 2εi (k) 2ε(k) k  k,i>1    µ 1 1 = +  g(2) 2π 2 as Ω  k,i>2  1 − 2εi (k)    k  1  2ε(k)  (3.23)  The approach discussed allowed us to relate the effective lowest-band scattering interaction strength and free space s-wave scattering length as the behaviour of which is extensively studied for different atoms [3]. We introduce a dimensionless constant X3D (u) that only depends on the potential strength and can be calculated numerically X3D (u) =  2π 2 λ µ  k∈R3  d3 k 1 − (2π)3 2 k2 2µ  m>2  d3 q 1 3 (2π) 2εm (q)  (3.24)  X3D (u) as a function of dimensionless external potential strength as a parameter. It can be calculated numerically. We use the notation for the equation for the renormalized potential 3.23 to get R g3D =  2π 2 a a 1 − X3D (u) µ λ  −1  (3.25)  The function X3D (u) is numerically calculated and plotted in Fig. 3.3. For R = 2π 2 a the limit λ → ∞ the renormalized potential strength value g3D µ reproduces the free space effective potential we have seen before. The result is what one would expect.  3.4  2D optical lattice  In 2D optical lattice, atoms of the gas are confined to a plane by external field like in a quasi 2D system we discussed. Additionally, there is an external potential that forms the 2D optical lattice. 28  3.4. 2D optical lattice  7 6  X2 D  5 4 3 2 1 0 0.0  0.5  1.0  1.5  2.0  2.5  3.0  u Figure 3.2: X2D (u) function, dimensionless constant as a function of dimensionless potential strength u in the equation for renormalized coupling strength g2D  Such a system can be obtained by tuning the potentials in a 3D optical lattice. Potential of the laser standing wave along one direction is made so strong, that the tunnelling between the two lattice rows along this direction is negligibly small. We get a 2D optical lattice. Let us consider the renormalization procedure for the 2D optical lattice. We evaluate the eq. 3.22 for our case. We need to relate the coupling constant in the limit of high cut-off energy with the quasi 2D system regime. For the quasi 2D effective potential we have the integral of the form 1 k  2 k2 2µ  ∼  d2 k ∼ k2  dk 2 k2  For the 3D case we had a divergence in the limit of high energies. Here the integral diverges logarithmically, but for both high energy and low energy limits. Thus, we need to modify the approach used for the 3D case a little bit. We need to use finite cut-off momentum Λ as a reference in order to express g2D (∞) in terms of the physical parameters. Let us take Λ to be Λ = 4π λ . 29  3.5. 1D optical lattice For the renormalization of the potential in quasi 2D regime, we have 1 g2D (Λ =  4π λ )  =  1 + g2D (Λ → ∞)  1 k∈R2 ,k> 4π λ  On the other hand, we may express g2D (Λ = 1 g2D (Λ =  4π λ )  µ µ log(dΛ) = 2 2π π  =−  2π λ )  2 k2  2µ  as  π l⊥ l⊥ −B − log 4π 2 a λ  And the renormalized effective optical lattice potential in 2D is then 1 R g2D  =  1 g2D (Λ =  4π λ )  −  µ (X2D (u) − B) 2π  (3.26)  here X2D (u) is a ”universal” numerically calculated function given by the expression X2D = B +  π 2 µ  k∈R2 ,k> 2π λ  d2 k 1 − (2π)2 2 k2 2µ  m>2  d2 q 1 2 (2π) 2εm (q)  (3.27)  We put the constant B for the convenience in subsequent expressions. It is calculated numerically and presented on the Fig. 3.2. Universality here means that the function does not depend on the mass of the atomic particles, the transversal length l0 , or laser wavelength λ and only depends on the adjusted potential strength. Thus one may say that the function affects geometrical properties of the system. We have expressed the renormalized coupling constant for a 2D optical lattice system in terms of the physical variables: free space scattering length a, the characteristic length of the transverse confining potential l⊥ , and the wavelength of the laser that forms the optical lattice. The equation is R g2D =  3.5  2π  µ  π l⊥ l⊥ − log 4π − X2D (u) 2 a λ  −1  (3.28)  1D optical lattice  1D optical lattice, just like the name suggests, is a system with a gas confined to movement only along one single direction with an additional optical lattice potential field applied along it. 30  3.5. 1D optical lattice 1.0  0.8  X1 D  0.6  0.4  0.2  0.0 0  1  2  3  4  5  6  7  u Figure 3.3: A calculated numerical constant X1D as function of dimensionless external potential strength u for 1D optical lattice.  A conventional way to obtain such a system is to take a 3D optical lattice and tune the potential strength of the laser field along two directions out of the three to be big enough for the potential barrier to become particularly impenetrable by tunnelling (see e.g. [1] and a relevant figure on p.896). We start with the analysis of the quasi-1D system. For potential strength strong enough, we may approximate the confining fields with a harmonic field. The analysis for such a system was reported in the paper [17]. We introduce the effective potential of the delta-function form and link it to free space scattering length a and characteristic length of the harmonic confining 2 = potentials along the two directions l⊥ µω (potentials are assumed to be equivalent for the two transversal directions) V (x) = g1D (a)δ(x), g1D (a) =  2 ω⊥ a 1 − 1.036 la⊥  (3.29)  The potential is not diverging with Λ → 0 and we can use instead the form  31  3.5. 1D optical lattice  1 R g1D  =  1 1 + g1D (a) Ω  k,m>2  1 2εm (k)  (3.30)  Substituting the expression for renormalization gives us the following expression for the renormalized coupling constant R g1D (a) =  1−  2 ω⊥ a + 2π1 2 aλ X1D (u) l2  1.036 la⊥  (3.31)  ⊥  Here X1D (u) is a dimensionless constant that only depends on the potential strength and can be calculated numerically X1D = 2π 2  2  µλ  m>2  dq 1 2π 2εm (q)  (3.32)  X1D (u) as a function of dimensionless external potential strength is plotted in Fig. 3.3 Let us once again consider the deep lattice limit, V → ∞. We see that X1D → 0 for 1D optical lattice and the renormalized potential strength R → g . Once converges to the quasi-1D expression eq. 3.29, that is g1D 1D again, physically, one can interpret this as ”decoupling” of higher bands for deeper lattices.  32  Chapter 4  Computation of the energy of bound states Calculations employing the renormalization group equations for the effective potential allowed us to reduce the problem of scattering of two atoms in an optical lattice to the problem of scattering within the two lowest Bloch bands of the system. Here we solve this reduced problem for the bound state energies of the two atomic particles. We solve the problem for scattering within the two lowest Bloch bands. We use the expressions obtained in the previous chapter for the renormalized potential. This is our method to take into account the contributions from the higher Bloch bands, which was our ultimate goal. We take the Lippmann-Schwinger system of equations for the optical lattice (eq. 2.3). We cut the contributions from bands higher than the second one and rearrange terms. One gets the following system of equations 1|T |1 = M1,1 g R + 1|T |2 = M1,2 g R +  1 Ω 1 Ω  q  q  1|T |1 M1,1 g R 1 + E − ε1 (q) + i0 Ω 1|T |1 M1,2 g R 1 + E − ε1 (q) + i0 Ω  q  q  1|T |2 M1,2 g R E − ε2 (q) + i0  (4.1)  1|T |2 M2,2 g R E − ε2 (q) + i0  (4.2)  This system of equations can also be represented in the diagrammatic form (Fig. 4.1). We did not introduce the diagrammatic rules in any form so one must note that the figure is provided only as a scheme to illustrate the conceptual structure of the system of equations. The hope is that it may facilitate understanding of the system of equations we reported here. It can be seen that one can factor out T -matrix and g out of the summa-  33  Chapter 4. Computation of the energy of bound states  T11  +  =  +  =  T12 ⋆ M12 U  M11 U  M11 U T12  +  T11  +  T11  M12 U  T12  M22 U  M12 U  Figure 4.1: The diagrammatic equation corresponding to the LippmannSchwinger equation for two-particle scattering in the 2 lowest Bloch bands of an optical lattice.  tion terms for the system of equations. We can then introduce the notation α(E) = β(E) = −  1 Ω 1 Ω  q  q  1 E − ε1 (q) + i0  (4.3)  1 E − ε2 (q) + i0  (4.4) (4.5)  Generally speaking, these functions are specific to each dimensional case. We see that only these two terms contain explicit dependency on E energy parameter. Expressing the consequent expressions in terms of α(E), β(E) allows us to keep them general to systems of different dimensions. Now we can solve for 1|T |1 in a general form 1|T |1 (E) =  M1,1 g R 1 − M2,2 g R β(E) + M1,2 g R [1 − M2,2  g R β(E)] [1  + M1,1  g R α(E)]  + (M1,2  2  β  g R )2 α(E)β(E)  Here we consider 1|T |1 (E) as a function of energy parameter. In accordance with our conceptual approach, one can make an analytical continuation to the values of energies beyond the ”physical” range. Then the poles of the T -matrix should give us the energy for the bound state. We see that the poles for our specific T -matrix element 1/ 1|T |1 (Eb ) = 0 are given by the equation gR  2  2 α(Eb )β(Eb ) M1,2 − M1,1 M2,2 +  +g R M1,1 α(Eb ) − M2,2 β(Eb ) + 1 = 0  (4.6) (4.7) 34  Chapter 4. Computation of the energy of bound states 1.6  ΕB  1.4  1.2  1. 0.05  0.03  0.01  0  0.01  a Λ Figure 4.2: Bound state energy as a function of scattering length in 3D optical lattice with u=0.2. Dashed lines are for results in the single-band approximation (black for the lowest band, blue for the second). The solid lines are higher-band corrections. Lowest and second Bloch band energy ranges are denoted correspondingly with blue and beige here and in the subsequent figures.  Or one can represent it in the partly factorized form (1 − M2,2 β(Eb )g R )(1 + g R M1,1 α(Eb )) + α(Eb )β(Eb ) g R  2  2 M1,2 =0  or we can use the form 1 + M2,2 β(Eb )g R  1 − M1,1 α(Eb )g R  2 + M1,2 =0  (4.8)  Note that here the β(Eb )g R , α(Eb )g R terms are dimensionless. Here the values Eb that satisfy this equation, give us the energy of the two-particle bound states. Note, that for the case M1,2 → 0 the equations become equivalent to the ones we have seen in the discussion of the lowestband approximation, for gR → g. We obtain it when we neglect the quantum amplitude for hopping from the lowest band- to the higher (second) band. This is the result one may expect. 35  Chapter 4. Computation of the energy of bound states  ΕB  3.5  3.0  2.5  0.1  0.0  0.1  0.2  0.3  a Λ Figure 4.3: The bound state energy spectrum as a function of scattering length in 3D optical lattice with u=0.7. Dashed lines are for results in the single-band approximation (black for the lowest band, blue for the second). The solid lines are higher-band corrections. Solid blue line reflects the divergence a → ∞ energy value.  The last two equations allow for computation of the bound energy from known system parameters. One cannot represent the solution in form of the explicit function Eb . The reason for that is that we have the implicit functions α(E) in such form. However, one can solve the equation 4.7 for the value of renormalized potential  R  g =  M2,2 β(E) − M1,1 α(E) ±  M1,1 α(E) + M2,2 β(E)  2 2αβ M1,1 M2,2 − M1,2  2  2 − 4αβM1,2 (4.9)  In this form for a given bound state energy we get the corresponding value of the renormalized potential. One can use this expression for plotting the Eb as a function of g R , or using the expressions from the previous chapter, as a function of the free space scattering length a. Let’s refer to such a function as a energy spectrum. 36  Chapter 4. Computation of the energy of bound states Let us give a closer look at the functions α(E) and β(E). It turns out that functions α(E) and β(E) have a simple asymptotic behaviour for two limits of energy value that are of interest to us. One corresponds to the energy parameter value being ”distant” enough from the energy range of the corresponding band. Indeed, let us consider the following expansion α(E) = +  1 Ω  q  g(E )dE 1 1 + = E − ε(q) Ω E − E g(E )dE  2 g(E  E )dE 1 +O Ω (E − E g(E )dE )3  1 Ω (E −  1 E g(E )dE )4  (4.10) (4.11)  Equivalent expansion for second band can be performed for β(E) function. At what energy values is it applicable? We estimate the range of energy values for which the second term is much smaller than the first one. One obtains the following condition E − Ei  1 Ω  (E 2 − Ei )G(E )dE  (4.12)  This approximation has the same form for different dimensions. However the volume factor 1/Ω and g(E)dE combinations are dimension-specific. They are defined by the volume of one Brillouin zone in the reciprocal space in each dimension 2 D 1/Ω g(E)dE = λ Here D = 1, 2, 3 is the dimension index. Another asymptotic limit that needs to be considered is for the energy parameter value range E being close enough to the corresponding Bloch L band energy range E → ∆U 1 for α(E) function and E → ∆2 for β(E). This limit is dimension-specific and we are going to consider it for each case separately. One may note that the α(E), β(E) functions are monotonous in the region intermediate eith regard to these to E value limits. Thus the two asymptotics can give us a qualitative understanding of the behaviour of these functions in the whole energy parameter range that is of interest to us. Let us now consider each dimensional case separately. We start with 3D optical lattice.  37  4.1. 3D optical lattice  4.1  3D optical lattice  The two-particle bound state energy Eb in 3D optical lattice can be computed numerically and is plotted at Fig. 4.2 and 4.3 for different fixed u R using eq. 4.9. values) as a function of renormalized potential strength g3D To obtain dependency of bound state energy on system parameters, we use eq. 3.25. Thus the scattering length a is given by λ 2π 2 λ = + X3D (u) R a µg3D B We have seen that for energy parameter values ∆U 1 < E < ∆2 not too close to the boundary energy regions we can use the expansion given by eq. 4.11 for α(E) function, and the equivalent one for β(E) function. On the other hand, the renormalized effective potential strength parameter for 3D optical lattice is given by the eq. 3.32. Applying these two approximations, we get the following equations  1 µλ2 = (E − ∆1 ) R 2π 2 α(E)g3D  λ − X3D (u) a  1 µλ2 (∆2 − E) = R 2π 2 β(E)g3D  λ − X3D (u) a  for this energy range. We substitute these expression to the eq. 4.8 that should give us the bound state energy. Let us introduce yet one more notation for characteristic energy parameter C(a) =  a π 2 2 µλ λ − aX3D (u)  Then we have the following explicit expression for the energies of the bound state 1 Eb± = (∆1 + ∆2 ) − (4.13) 2    −C(a) M1,1 + M2,2 ±  2 + 4M1,2  (∆2 − ∆1 ) − M2,2 + M1,1 2C(a)  2   (4.14)  Now to the limit of E → ∆U 1 energy parameter range. Here we can’t apply the expansion given by eq. 4.11 for α(E) because this is the very range for 38  4.2. 2D optical lattice which the condition 4.12 isn’t satisfied. However, the equivalent expansion for β(E) function is still applicable for such a range. For α(E) function in such a limit we use another approximation. We now that the dispersion relation has to satisfy ∇ε(qU ) = 0 for ε(qu ) = E1U 2 the band boundary. We can use expansion of ε(qu ) = E1U − i 2Πi qi sort. If we denote with Π characteristic ”effective mass” of such expansion, we get the following approximation for g(E) density of states function 3  1 1 Π2 U g1 (E) = √ 2 3 ∆U 1 − E, for E → ∆1 − 0 limit Ω 2π Substituting this expression into the definition of α(E) gives us  α(E) = √  2Π3/2 π2 3  = √ →  2Π3/2 π2 3  √  ∆U 1 ∆U 1 −∆  √ √  ∆−−  ∆−  π 2  E − ∆U 1 Arctan E − ∆U 1  E dE = (E − E )  ∆ E − ∆U 1  (4.15)  →  (4.16)  with E → ∆U 1 + 0 limit.  (4.17)  Using this expression and neglecting the dependency of β(E) on energy parameter, we get the following asymptotic expression for bound state energy E=  ∆U 1  4 + 2 π  √  2  1  M2,2 − βgR π2 3 ∆− √ 3/2 2 2Π g R (M1,2 − M1,1 M2,2 ) +  M1,1 β  (4.18)  4∆ For the g R → ∞ limit, we get the minimum (threshold) value E = ∆U 1 + π2 . The expression for renormalized potential strength 3.1 gives us that this limit is reached for the scattering length value reaching limit a = λ/X3D (u). That is , to sum up, we have the following limit  E = ∆U 1 +  4.2  λ 4∆ , a= 2 π X3D (u)  (4.19)  2D optical lattice  The two-particle bound state energy Eb in 2D optical lattice can be computed numerically and plotted at Fig. 4.4, a) as a function of renormalized R using eq. 4.9. potential strength g2D 39  4.2. 2D optical lattice To obtain dependency of bound state energy on system parameters, we use eq. 3.31. Thus the scattering length a is given by 2π π l⊥ l⊥ = + log 4π 2 a µg2D (Eb ) λ  + X2D (u)  Let us start with the approximation for energy ranges on a distance from the Bloch energy ranges. Using eq. and eq. , we have 1 (E − ∆1 ) = R 2π α(E)g2D 2  π l⊥ l⊥ − log 2π 2 a λ  − X2D (u)  1 (∆2 − E) = R 2π β(E)g2D 2  π l⊥ l⊥ − log 2π 2 a λ  − X2D (u)  µλ  µλ  Substituting these expression to eq. gives us an expression for the bound state energy equivalent to eq. , except that we need to define the characteristic energy as 2  C2D (a, λ, l⊥ ) = 2π  π l⊥ l⊥ − log 2π 2 a λ  µλ2  −1  − X2D (u)  The expansion still holds. However for E → ∆U 1 the expansion would be Π  g(E) =  2  , for E → ∆U 1 − 0 limit  and α(E) has a diverging contribution α(E) =  Π 2π  ∆U 1 ∆U 1 −∆  dE Π E − ∆U 1 → − 2 log (E − E ) ∆  for E → ∆U 1 + 0 we get   E = ∆U 1 + ∆ exp  2  Πg R     M2,2 gR β − 1 2 −M M gR β M1,2 1,1 2,2 + M1,1     For g R → 0 the exponential factor becomes negative and we have E = ∆U 1 + ∆ exp −  2  Πg R M1,1 40  4.3. 1D optical lattice  1.2  1.1  ΕB  1.0  0.9  0.8  0.7  0.6 0.2  0.1  0.0  0.1  0.2  1  2 a  log  Λ  Figure 4.4: Computed dimensionless bound state energy for 2D optical lattice with potential strength of u = 0.2. λ - the laser field wavelength, a scattering length, and l⊥ the characteristic length of transversal potential.  The limit g R → 0 corresponds to the limit a → 0, and thus we get the following asymptotic expression for bound state energy in 2D optical lattice √ 2π 2 a R g → µ l⊥ E = ∆U 1 + ∆ exp − √  4.3  1 2πM1,1  l⊥ a  µ Π  for a → 0 limit.  1D optical lattice  The two-particle bound state energy Eb in 1D optical lattice is calculated numerically and plotted at Fig. 4.5, a) as a function of renormalized potential R using eq. 4.9. strength g1D  41  4.3. 1D optical lattice 0.9  0.8  ΕB  0.7  0.6  0.5  0.4  0.3 10  5  0  5  Λ a  10 1  15  20  1.036  Figure 4.5: Computed dimensionless bound state energy εB as a function −1  of expression of system parameters lλ⊥ la⊥ − 1.036 for 1D optical lattice with potential strength of u = 0.2 λ - the laser field wavelength, a - scattering length, and l⊥ the characteristic length of transversal potential. To obtain dependency of bound state energy on system parameters, we use eq. 3.31. Thus the scattering length a is given by 2 ω⊥ l⊥ λ a = 1.036 + − l⊥ g1D (Eb ) l⊥ X1D (u)  −1  ⊥ l⊥ Dimensional analysis shows that the term 2 gω1D is proportional to lλ⊥ (see R has the discussion of the technical details in appendix). We see that the g2D a certain range. Depending on the value of lλ⊥ relation of laboratory setup a→∞ for 1 → 0 given by parameters we can have the limit energy value EB a the equation 2 ω⊥ l⊥ λ = 1.036 − (4.20) − a→∞ g1D (EB ) l⊥ X1D (u)  or a finite range of a values for which we have bound states. 42  4.3. 1D optical lattice This two cases are illustrated with Eb (a) computations plotted for different lλ⊥ values at Fig. 4.5 b.), c.). Another interesting property evident from eq. 4.20 is that if laboratory system parameters are tuned so that λ = 1.036X1D (u) l⊥ then the renormalized interaction strength is simply proportional to the scattering length g R ∼ a. The Fig. 4.5 a) then gives us a dependency for bound state energy in such a case. Let us now proceed to the discussion of applicability of the approximation given by eq. 4.11 for α(E) and of the equivalent one for β(E) function. Equivalently to the other cases we already discussed, we get the follwoing expressions for 1 = R α(E)g1D  E − ∆1 ω⊥  λ a  1+  aλ a 2 X1D (u) − 1.036 λ l⊥ ⊥  1 = R β(E)g1D  ∆2 − E ω⊥  λ a  1+  aλ a 2 X1D (u) − 1.036 λ l⊥ ⊥  If we introduce notation for characteristic energy in the form C1D =  2 ω⊥ a λ 1 − 1.036 λa + aλ X1D (u) l2 ⊥ ⊥  the eq. 4.14 would give us the bound state energy. As we discussed before, this approximation is not applicable for region 2 q2 L of E → ∆U 1 , ∆2 For = 2Π we have 2ΠU 1Ω  g(E) = 2π  E−  ∆U 1  2ΠL 2Ω  g(E) = 2π  ∆L 2 −E  , for E → ∆U 1 − 0 limit  , for E → ∆L 2 + 0 limit  43  4.3. 1D optical lattice The α(E) then diverges like ∆U 1  α(E) ∼  ∆U 1 −∆  √ g(E )dE 2Π ∆ → = √ arctan (E − E ) E − ∆U π ∆ 1 √ Π → , for E → ∆U 1 + 0 limit E − ∆U 1  For the region E → ∆U 1 + 0 we have  E=  ∆U 1  +  Π  g2 2 R  2  2 M1,2  M2,2 −  1  − M1,1  for E → ∆U 1 + 0 limit  gR β  One can see that the E → ∆U 1 + 0 bound energy range corresponds to R → 0, and for this limit we get the asymptotic expression g1D 2 E = ∆U 1 + M1,1  Π  g2 2 R  R for g1D →0  R → 0 corresponds to scattering length limit a → 0. We get the Limit g1D asymptotic expression R 2 2 2 E = ∆U 1 + 4M1,1 Πa ω⊥ for g1D → 0  44  Chapter 5  Conclusion Physical implementation of some theoretical models may help answer fundamental problems of different branches of physics. Ultra-cold dilute gases of alkali atoms confined by externally applied fields are excellent candidates of laboratory systems to simulate many of these important models. In order to relate a laboratory system to a model, we need to express the model parameters in terms of physical quantities of the physical setup: properties of gas atoms and of the external potential. A renormalization group equation approach to such computations was discussed in this thesis. The basic scheme behind the method is as follows. One can introduce effective potential in such a way, so that the scattering amplitude in a two-body problem is recovered for low-energy scattering. We consider then the Shr¨ odinger equation for atoms with and without action of external potential. The renormalization group equations for the effective potential were computed for each case. In the limit of contact interaction (Λ → ∞) they can be matched thus giving us the relation we are after. We started with the quasi 2D systems to demonstrate the method on an already studied example. The free space scattering length was matched to the low-energy scattering parameter for 2D regime. We successfully reproduced the well-known result reported in [8] with our approach (eq. 3.17). We then proceeded to the problem of two atoms in optical lattices. In general, we get the unapproachable system of equations for T -matrix. One may see neglecting the intra-band scattering terms as a reasonable approximation. The approach was used in [12]. We call this approach the lowestband approximation and consider its advantages and limits. We then move on to apply our approach to the problem. The renormalization group equations were obtained for the effective potential in 1D, 2D, 3D for optical lattices. The method let us to express parameters of the effective interaction potential renormalized to the two lowest bands cut in terms of physical quantities of the system (eq. 3.31 for 1D, eq. 3.28 for 2D, eq. 3.25 for 3D). We report calculations for the energy of bound states in terms of parameters of confining potentials and scattering length in different dimension as the main results (plotted for fixed u at figures 4.2 for 3D, fig. 4.4 45  Chapter 5. Conclusion for 2D, and eq. 4.5 for 1D). We considered asymptotic behaviour of bound state energies in different limits. Dimensional effects were investigated with analysis of similarities and differences. We mentioned in the course of our narration excellent works with computation of model parameters performed for specific geometries of confining potential with a variety of approaches. Thus Olshanii et al. provided analysis for gases confined to quasi 1D regime [17], Petrov et al. for the case of quasi 2D regime [8] and gases in 3D optical lattices were addressed in [2], [9] and other works. The method considered here is an alternative to these methods. We employed a more generic renormalization group equations based approach. It may be applied to cold atoms confined by fields of almost arbitrary geometries with a uniform procedure. By contrast, the other known solutions need substantial tailoring for each of the cases. It makes the method a powerful and universal tool for calculation of the model parameters.  46  Bibliography [1] Immanuel Bloch, Jean Dalibard, and Wilhelm Zwerger. Many-body physics with ultracold gases. Rev. Mod. Phys., 80(3):885–964, Jul 2008. [2] Hans Peter B¨ uchler. Microscopic derivation of hubbard parameters for cold atomic gases. Phys. Rev. Lett., 104(9):090402, Mar 2010. [3] Cheng Chin, Rudolf Grimm, Paul Julienne, and Eite Tiesinga. Feshbach resonances in ultracold gases. Rev. Mod. Phys., 82(2):1225–1286, Apr 2010. [4] H.Smith C.J. Pethick. Bose-Einstein Condensation in Dilute Gases. Cambridge University Press, 2002. [5] Xiaoling Cui, Yupeng Wang, and Fei Zhou. Resonance scattering in optical lattices and molecules: Interband versus intraband effects. Phys. Rev. Lett., 104(15):153201, Apr 2010. [6] P. Zoller D. Jaksch. The cold atom hubbard toolbox. Annals of Physics, 315(1):52 – 79, 2005. [7] London F. The λ-phenomenon of liquid helium and the bose-einstein degeneracy. Nature, 141:643–644, 1938. [8] Fedichev. Extended molecules and geometrical scattering resonances in optical lattices. Phys. Rev. Lett., 81(5):938–941, Aug 2004. [9] P. O. Fedichev, M. J. Bijlsma, and P. Zoller. Extended molecules and geometric scattering resonances in optical lattices. Phys. Rev. Lett., 92(8):080401, Feb 2004. [10] Esslinger T. H¨ ansch T. W. Bloch I. Greiner M., Mandel O. Quantum phase transition from a superfluid to a mott insulator in a gas of ultracold atoms. Nature, 415:39–44, 2001. [11] Kerson Huang and C. N. Yang. Quantum-mechanical many-body problem with hard-sphere interaction. Phys. Rev., 105(3):767–775, Feb 1957. 47  [12] F.Lang K.Winkler, G. Thalhammer. Repulsively bound atom pairs in an optical lattice. [13] Lifshitz E.M. Landau L.D. Quantum mechanics: non-relativistic theory. Elsevier Science, 1991. [14] M. R. Matthews C. E. Wieman M. H. Anderson, J. R. Ensher and E. A. Cornell. Observation of bose-einstein condensation in dilute atomic vapor. Science, 269(5221):198–201, 1995. [15] Oberthaler M. Morsch O. Dynamics of bose-einstein condensates in optical lattices. Rev. Mod. Phys., 78:179–215, 2006. [16] Mott N.F. Metal-insulator transitions. Taylor&Francis, 1990. [17] M. Olshanii. Atomic scattering in the presence of an external confinement and a gas of inpenetrable bosons. Phys. Rev. Lett., 81(5):938–941, Aug 1998. [18] Shankar R. Renormalization-group approach to interacting fermions. Rev. Mod. Phys., 66(1):129–192, 1994. [19] D. Van Neck W. H. Dickhoff. Many-body Theory Exposed!: Propagator Description of Quantum Mechanics in Many-Body Systems, 2 Ed. World Scientific Publishing Co. Pte., 2008. [20] Chen Ning Yang. n- pairing and off-diagonal long-range order in a hubbard model. Phys. Rev. Lett., 63(19):2144–2147, Nov 1989.  48  Appendix A  Glossary of used notational conventions Some of the notations that are used frequently throughout the text are outlined here. The appendix may be used as a quick reference and help to comprehend main equations out of the notational context. To avoid confusion, as a general rule quasimomentum vectors are denoted by q letter (with specific subtitles), and free space momentum by k. l is a characteristic length of harmonic potential. l2 = mω . Ω stands for a volume of the system. The summation over all quantum states can be represented using the notation of continuous integral 1 Ω  ∞  f (k) → k  ∞  dk D f (k) (2π)D  Or, more generally 1 Ω  f (q, n) → q,n  n  dq D f (q, n) (2π)D  Here D is the number of dimensions in the space or subspace we consider. ˆ is a physical interaction potential for the two atoms, and Uef f is an U effective potential we introduce. It has the form ˆeff |k3 , k4 = gδ(k1 + k2 )δ(k3 + k4 )θ(Λ − k1 )θ(Λ − k3 ) k1 , k2 |U throughout the text, every time adjusted for the dimensional space we discuss. g here is the strength of effective interaction. For the renormalization group equations we consider g to be a function of the cut-off frequency Λ: g(Λ) or of the cut-off band number n: g(n). a is reserved for the (free space) Fermi scattering length between the two atoms. 49  Appendix B  Numerical calculations Some numerical calculations were presented. Some important technical details are outlined in this section. All the computations were done in the Wolfram Mathematica software package.  B.1  Bloch’s states  We denote 1D Bloch’s wavefunction for a particle confined to optical lattice with ψ qx ,n (x) = uqx ,n exp [iqx x] Here uqx ,n (x + λ2 ) = uqx ,n (x) is a periodic function. It can be represented in the Fourier series form. It would give us 2 exp [iqx x] λ  ψ qx ,n (x) =  with the normalization condition  cqfx ,n exp i f =−∞..∞  4π fx λ  dxψ † (x)ψ(x) = 1 gets the form c2f =  1 N  Here N is the number of cells in the lattice. The periodic potential can be represented in the Fourier terms x = λ 1 x 1 x 1 − exp i4π − exp −i4π 2 λ 2 λ V (x) = V0 sin2 2π  =  V0 2  We have the following equation for the set cf 2  2m  qx +  4π f λ  2  1 1 1 cqfx + V0 cqfx − V0 cqfx+1 − V0 cqfx−1 = ε(qx )cqfx 2 4 4  50  B.1. Bloch’s states We introduce notations for dimensionless energy (q) and optical lattice potential strength u 2 2 4π 2 V0 = u m λ 2  ε(qx ) =  4π λ  2m  2  (qx )  And the equation for Hamiltonian eigenvalues now has dimensionless form qλ +m 4π  2  + 2υ cqfx − υcqfx+1 − υcqfx−1 = (qx )cqfx  qλ The range for S = 4π is S ∈ [− 12 , 12 ]. Asymptotic behaviour of the exact solution is of interest to us. For x x by −c (m) and − υcqm−1 m2 1 (around 10) we approximate 2υcqmx − υcqm+1 we have the equation of the form  −υc (m) + m2 c(m) = εc(m) It is an equation of one dimensional harmonic oscillator and the solutions are well known. Specifically, it gives us that for m ⇒ ∞ the leading order of 2 √ , and the matrix equation span can c(m) is exponential, c(m) ∼ exp − 2m υ √ be safely ”cut” for big m values, m2 υ, when obtaining the numerical solutions. The cutting m value can be taken lower for low energy bands.  B.1.1  Explicit expression  Let us calculate the interaction matrix element between the two initial Bloch states for the short-range potential approximated by delta function U (x) = U0 δ(x). We need to calculate ˆeff | {qx,3 , s3 }{−qx,4 , s4 } M1,2,3,4 g = {qx,1 , s1 }{−qx,2 , s2 } | U By substituting the Fourier representation, we obtain the expression of the form M1,2,3,4 =  2 λ  2  q  cfx,1 1  ,s1 qx,1 ,s2 qx,2 ,s3 qx,2 ,s4 cf2 cf3 cf4  ·  f1 ,f2 ,f3 ∈N  ·  dx exp  i4πx (f3 + f4 − f1 − f2 ) λ 51  B.2. Integrating over density of states and after the integration the total sum can be factorized to (here Q = f1 + f2 = f3 + f4 effectively) M1,2,3,4 =  2 λ  Q Q M1,2 M3,4  (B.1)  ,s1 qx,2 ,s2 cQ−f  (B.2)  Q∈N q  Q M1,2 =  cfx,1 f ∈N  If m = n due to the symmetry M Q is symmetric M Q = M −Q as function of Q. An example of M Q dependency on Q for I = 4 states is depicted on Fig. ??.  B.2  Integrating over density of states  We have seen that the poles of T-matrix satisfy the following equation for one-band approximation (see 2.4)  B.2.1  1D case 1 M1,1 g  dq 1 (2π) Eb − ε1 (k) + i0  =  (B.3)  Presenting the right-side integral in dimensionless computable form, although trivial, needs utmost care as we are making the quantitative prediction and every factor is valuable. Eb = 1 M1,1 g  2 2π2 λ2  =  2 2π2 λ2 dq (2π)  b  1 b  −  1 (k)  (B.4)  + i0  For a finite lattice the quasimomentum q=  2π n , for n = −N.. + N λ N  We switch to dimensionless energy and substitute the integral with the sum over discrete values of k. 1 M1,1 g  2 2π2 mλ2  =  1 1 λN  1 n∈{−N...N }  −  2π n 1( λ N )  + i0  (B.5)  52  B.2. Integrating over density of states  Figure B.1: Numerical values for c0m as functions of m for a number different values of F for the lowest band.  Figure B.2: Numerical values for c0m as functions of m for the second band. From here we get 1 M1,1 g  2 2π2 mλ  =  1 N  n∈{−N...N }  1 n − 1 ( 2π λ N)  (B.6)  One can define a dimensionless function y( b ) =  1 N  n∈{−N...N }  y( b ) =  1 M1,1 g  1 n − 1 ( 2π λ N) 2 2π2 mλ  (B.7)  (B.8)  53  B.3. 1D optical lattice  B.2.2  2D case 1 M1,1 g  1 d2 q 2 (2π) Eb − ε1 (q) + i0  =  2 2π2 1 = 2 M1,1 gm N  B.2.3  1 n  M1,1 g  (B.10)  1 (n)  d3 q 1 3 (2π) Eb − ε1 (q) + i0  =  1 2 2π2λ = 3 M1,1 gm N  B.3.1  −  3D case 1  B.3  (B.9)  (B.11)  1 n  −  (B.12)  1 (n)  1D optical lattice Details on X1D (u)  Similarly to the previous considerations 2  X1D (u) =  µλ  m>2  dq 1 1 = 2 2π 2εm (q) 4π  m  1 S  r=−S..S  1 (r)  One can see that the function X1D (u) is always positive and monotonic and X1D (u) → 0 with u → ∞. For u = 0 limit we have the case of free space dispersion relation X1D (0) =  2 2 µλ  ∞ 4π λ  2 dq 1 2 q2 = (2π) πλ 2µ  λ 4π  =  1 2π 2  54  B.3. 1D optical lattice  B.3.2  R Computation for g1D α(E) functions  R α(E) function Let us consider the g1D R g1D α(Eb ) =  dq 1 = 2π E − (q) + i0 1 ω⊥ a = 2 4π b − ε(S)  2 ω⊥ a 1 − ..  (B.13)  =  (B.14)  2m  λ  S  1 λa 1 = 2 2 4π l⊥ (1 − ..) 1 = 2 4π  B.3.3  S  (B.15)  −1  2 l⊥ 1 l⊥ X (u) + − 1.036 1D 2 2π aλ λ  1 b − ε(S)  S  1 = − ε(S) b  (B.16)  Details on X2D (u)  It is reasonable to integrate over the rectangle shape for the free space density integral part. Let us calculate it explicitly f (Λ) = |kx,y |<Λ Λ  = 2π 0  d2 k = 2π k2  Λ 0  √  dk +4 k √  dk − π log 2 + 8 k  2Λ  Λ  2Λ  Λ  ArcSin( Λ k)  dk k  dk ArcSin k Λ  = 2π 0  ( )  π −ArcSin Λ 2 k  Λ k  Λ  = 2π 0  dφ = (B.17) dk = (B.18) k  dk − 4π log 2 + 4β(2) (B.19) k  Here β(2) ≈ 0.915.. is the Catalan’s constant. Integral for the small rectangle is √  h=  √  5a  2a  √  =2  √  √ 2 2a  ArcCos( ka )  dk k  ( )  π −ArcCos ka 2  5a  2a  dφ +  √  5a  dk a ArcCos +2 k k π − log 2  dk k  √ 2 2a √  5a  √ 2 2 √ 2  ArcSin( 2a k )  ( )  π −ArcSin 2a 2 k  dφ =  (B.20)  dk ArcSin k  2a k  = 1.3201.. −  π log 2 (B.22) 2  −  (B.21)  The overall integral is equal to f (Λ) − f (2a) − 4h = 2π log  Λ 2Λ − 5.2804.. = 2π log − 5.2804.. a λ 55  Appendix C  Low-energy limit for 2D scattering The low-energy limit for the problem of quantum scattering in 2 dimensions is considered. We will obtain the limiting form for the scattering amplitude and calculate the scattering amplitude for a well potential as an example. These results are used in the main narration of the thesis. While the similar analysis for 3D is well known and often is a part of any standard course on quantum mechanics , the 2D case is of a somewhat special interest. For the convenience of the reader, we reflect the main results here.  C.0.4  General solution  We consider a 2D elastic quantum scattering problem. We take the orthogonal coordinates x, y and define the cylindrical in a standard way x = ρ cos(φ)  (C.1)  y = ρ sin(φ)  (C.2)  Hamiltonian can be reduced to a single particle scattering in a harmonic potential 2  ∂ 2 + ∂y2 ψ(r) + U (r)ψ(r) = Eψ(r) 2m x We consider separate modes −  Rm (ρ)eimϕ  ψ= m∈N  The normalization coefficients will be dropped in the following. Here the functions Rm (ρ) are the solutions of the equation ∂ 2 R(ρ) 1 ∂R(ρ) m2 2m 2mE + − 2 R(ρ) − 2 Um (ρ)R(ρ) = − 2 R(ρ) 2 ∂ρ ρ ∂ρ ρ 56  Appendix C. Low-energy limit for 2D scattering 2mE  We define the momentum value k = span of the potential we have  2  and for the ρ range beyond the  ρ2 R (ρ) + ρR (ρ) + k 2 ρ2 − m2 R(ρ) = 0 The solution is a linear combination of the Bessel functions of the first and second kind Rm (ρ) = cos(δm )Jm (kρ) + sin(δm )Ym (kρ) Here we have δm = δ−m We need to represent it in form ψ2D (z) = eikz − f (φ) √  eikρ −i8πkρ  The expansion eikz = eik cos φ =  im J(kρ)eimϕ  and eimϕ im J(kρ) − √  ψ= m∈N  For kρ ψ=√  fm eikρ −i8πkρ  1 it goes like π 1 1 im−1 im−1 π fm 1 − e−ikρ √ ei 2 (m− 2 ) eimϕ eikρ √ e−i 2 (m− 2 ) − √ πkρ m∈N −i8 2 2  We obtain (e2iδm − 1)eimϕ =  f (φ) = 2i  m∈N  m∈N  C.0.5  2π eimϕ cot(δm )  π 2  i π2 −  Unitary relation  Just like in the 3-dimensional case we can define S-matrix for our scattering ikρ ˆ ikz = eikz − f (φ) √ e Se −i8πkρ  and use the unitary of the S-matrix Sˆ† Sˆ = ˆ1 to obtain the relationship f (ϕ) exp i  π π + f ∗ (−ϕ) exp −i 4 4  =√  1 8πkρ  2π 0  dθ f (ϕ)f ∗ (θ − ϕ) 2π  The total scattering section is given by σ0 =  |f (q)|2 4q 57  Appendix C. Low-energy limit for 2D scattering  C.0.6  T-matrix and Lippmann-Schwinger equation for 2D  We rewrite the Schr¨ odinger equation in the integral form ψ(r) = exp(ikr) +  dr G(r, r )U (r )ψ(r )  Here G(r, r ) is the Green’s function for free motion Schr¨odinger equation G0 (E, r) =  eikρ E−  p2 2m  d2 p ∼− 2 + i0 (2π)  √ m i √ eikρ 2 2πkρ  By comparing it to the scattering amplitude definition, we obtain the relationship 2m f (ϕ) = − 2 dr exp ikr cos(ϕ) r |Tˆ|k Here Tˆ is two-dimensional T-matrix for which holds the Lippmann-Schwinger operator equation ˆ + TˆG ˆU ˆ Tˆ = U  C.0.7  Asymptotic behaviour  We characterize a potential by the asymptotic behaviour of Bessel functions on the large distances eiδm im  ψ= m∈N  2 π 1 sin kr − (m − ) + δm eimφ πkρ 2 2  Here we define the range for the angle φ to be φ ∈ [0, 2π). For the range of x low we have m = 0 : Rm (ρ) =  1 (m − 1)!2m m (kρ) cos(δ ) + m 2m m! π  1 kρ  m  sin(δm )  and for the s-scattering (m = 0) R0 (ρ) = cos(δm ) +  2 log π  eγ kρ 2  sin(δm )  Here we use the notation γ is the Euler’s constant.  58  C.1. Effective potentials  C.0.8  Low energy limit  Within the low-energy limit the s-scattering behaviour at kρ  1  ρ2 R (ρ) + ρR (ρ) − m2 R(ρ) = 0 The solution 1 + C2m ρm ρm m = 0 : Rm (ρ) = C1 + C2 log(ρ)  m = 0 : Rm (ρ) = C1m  (C.3) (C.4)  The limiting behaviour is m = 0 : cot(δm ) ∼  1  (C.5)  k 2m  2 m = 0 : cot(δ0 ) = − log (dk) π  (C.6)  The s-scattering amplitude f (q) =  i π2  2π + log(dk)  For the s-scattering σ(k) =  C.1 C.1.1  π2 k  1 π 2 2  +  π 2  cot δ0 (k)  2  Effective potentials Well potential  Denoting κ2 = k 2 −  2m 2  U , the boundary condition  J (kl) cos(δ0 )J (kl) + sin(δ0 )Y (kl) = J(kl) cos(δ0 )J(kl) + sin(δ0 )Y (kl) From it (kl  1) 2 π cot(δ0 ) ∼ − − log 2 2mU l2  eγ kl 2  One can introduce the notation 2  d = exp  2mU l2  eγ l 2 59  C.1. Effective potentials d has dimension of length and is of order of the bound state size for U < 0 when there is a bound state. Then phase shift is simply given by π cot(δ0 ) ∼ − log dk 2 We obtain the scattering amplitude f (k) =  − π2  2π = cot(δ0 ) + i π2  2π 2  2mU l2  + log  eγ 2 kl  + i π2  60  

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