Citation for published version (APA): Petrov, D. S. (2003). Bose-Einstein condensation in low-dimensional trapped gases

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1 UvA-DARE (Digital Academic Repository) Bose-Einstein condensation in low-dimensional trapped gases Petrov, D.S. Link to publication Citation for published version (APA): Petrov, D. S. (2003). Bose-Einstein condensation in low-dimensional trapped gases General rights It is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s) and/or copyright holder(s), other than for strictly personal, individual use, unless the work is under an open content license (like Creative Commons). Disclaimer/Complaints regulations If you believe that digital publication of certain material infringes any of your rights or (privacy) interests, please let the Library know, stating your reasons. In case of a legitimate complaint, the Library will make the material inaccessible and/or remove it from the website. Please Ask the Library: or a letter to: Library of the University of Amsterdam, Secretariat, Singel 425, 1012 WP Amsterdam, The Netherlands. You will be contacted as soon as possible. UvA-DARE is a service provided by the library of the University of Amsterdam ( Download date: 03 Jan 2018

2 Dmitryy S. Petrov Bose-Einsteinn Condensation in Low-Dimensionall Trapped Gases

3 Bose-Einsteinn Condensation in n Low-Dimensionall Trapped Gases ACADEMISCHH PROEFSCHRIFT terr verkrijging van de graad van doctor aann de Universiteit van Amsterdam, opp gezag van de Rector Magnificus prof.. mr. P.F. van der Heijden tenn overstaan van een door het college voor promoties ingestelde commissie,, in het openbaar te verdedigen in de Aula der Universiteit opp woensdag 9 april 2003 te 14:00 uur door r Dmitryy Sergeevich Petrov geborenn te Moskou, Rusland

4 Promotiecommissie: : Promotor:: Co-promotor:: Prof. dr. J. T. M. Walraven Prof. dr. G. V. Shlyapnikov Overigee Leden: Prof.. dr. H. J. Bakker Prof.. dr. H. B. van Linden van den Heuvell Prof.. dr. A. M. M. Pruisken Prof.. dr. C. Salomon Prof.. dr. S. Stringari Prof.. dr. B. J. Verhaar Faculteit: : Faculteitt der Natuurwetenschappen, Wiskunde en Informatica Thee work described in this thesis was part of the research program of the Stichtingg voor Fundamenteel Onderzoek der Materie (FOM), whichh is financially supported by the Nederlandsee Organisatie voor Wetenschappelijk Onderzoek (NWO) andd was carried out at the FOMM Institute AMOLF Kruislaann 407, SJ, Amsterdam, Thee Netherlands

5 Contents s 11 Introduction Background This Thesis 2 22 Overview BEC of an ideal gas in ID and 2D harmonic traps Interacting Bose gas Phase fluctuations and quasicondensates in a uniform gas Realization of BEC's in lower dimensions Quantum degenerate quasi-2d trapped gases Bose-Einstein condensation in quasi-2d trapped gases Superfluid transition in quasi-2d Fermi gases Interatomic collisions in a tightly confined Bose gas Introduction D scattering problem Scattering in axially confined geometries. General approach Quasi-2D regime Confinement-dominated 3D regime Thermalization rates Inelastic 2-body processes Concluding remarks Regimes of quantum degeneracy in trapped ID gases Phase-fluctuating 3D Bose-Einstein condensates Phase-fluctuating 3D condensates in elongated traps Observation of phase fluctuations in Bose-Einstein condensates 67 Bibliographyy 73 Summaryy 79 Samenvattingg 81 Acknowledgmentss 83 iii i

6 Thiss Thesis is based on the following papers: Chap.. 3 D.S. Petrov, M. Holzmann, and G.V. Shlyapnikov, Bose-Einstein condensationdensation in quasi-2d trapped gases, Phys. Rev. Lett. 84, 2551 (2000) D.S.. Petrov, M.A. Baranov, and G.V. Shlyapnikov, Superfluid transition inin quasi-2d Fermi gases, accepted for publication in Phys. Rev. A Chap.. 4 Chap.. 5 Chap.. 6 D.S. Petrov and G.V. Shlyapnikov, Interatomic collisions in a tightly confinedconfined Bose gas, Phys. Rev. A 64, (2001) I.. Bouchoule, M. Morinaga, C. Salomon, and D.S. Petrov, Cesium gas stronglystrongly confined in one dimension: Sideband cooling and collisional properties,properties, Phys. Rev. A 65, (2002) D.S. Petrov, G.V. Shlyapnikov, and J.T.M. Walraven, Regimes of quantumtum degeneracy in trapped ID gases, Phys. Rev. Lett. 85, 3745 (2000) D.S. Petrov, G.V. Shlyapnikov, and J.T.M. Walraven, Phase-fluctuating 3D3D Bose-Einstein condensates in elongated traps, Phys. Rev. Lett. 87, (2001) S.. Dettmer, D. Hellweg, P. Ryytty, J.J. Arlt, W. Ertmer, K.Sengstock, D.S.. Petrov, G.V. Shlyapnikov, H. Kreutzmann, L. Santos, and M.. Lewenstein, Observation of phase fluctuations in elongated Bose- EinsteinEinstein condensates, Phys. Rev. Lett 87, (2001) Otherr papers to which the author contributed: M.A. Baranov and D.S. Petrov, Critical temperature and Ginzburg-Landau equation forfor a trapped Fermi gas, Phys. Rev. A 58, R801 (1998) M.A. Baranov and D.S. Petrov, Low-energy collective excitations in a superfluid trappedtrapped Fermi gas, Phys. Rev. A 62, (R) (2000) P. Rosenbusch, D.S. Petrov, S. Sinha, F. Chevy, V. Bretin, Y. Castin, G.V. Shlyapnikov,, and J. Dalibard, Critical rotation of a harmonically trapped Bose gas, Phys. Rev.. Lett. 88, (2002) I. Shvarchuck, Ch. Buggle, D.S. Petrov, K. Dieckmann, M. Zielonkowski, M. Kemmann,, T.G. Tiecke, W. von Klitzing, G.V. Shlyapnikov, and J.T.M. Walraven, Bose-EinsteinBose-Einstein condensation into non-equilibrium states studied by condensate cusing,cusing, Phys. Rev. Lett. 89, (2002) D.S. Petrov, Three-body problem in Fermi gases with short-range interparticle interaction,teraction, Phys. Rev. A 67, (R) (2003)

7 Chapterr 1 Introduction 1.11 Background Naturee provides us with three spatial coordinates to describe various phenomena which happenn in our daily life. We often do not use all the coordinates as it is sufficient to buildd a description or a model in 2-dimensional or even in 1-dimensional terms using eitherr symmetry or necessity considerations. Low-dimensional systems were always used ass textbook examples aimed to make the description of phenomena technically easier andd the phenomena themselves physically transparent. This does not mean, however, thatt all low-dimensional problems are only poor dependent relatives of the rich family off colorful and challenging 3D physics. Dilute low-dimensional gases present an example off many-body systems that stand by themselves and for which the dimensionality is essential.. Bose-Einsteinn condensation (BEC) is a quantum statistics phenomenon which occurs inn a 3D system of bosons (particles with an integer spin) when the characteristic thermal dee Broglie wavelength of particles exceeds the mean interparticle separation. Under this conditionn it is favorable for particles to occupy a single ground state, and below a critical temperaturee the population of this state becomes macroscopic. This phenomenon has beenn predicted as a result of the work of Bose [1] and Einstein [2] in the mid twenties. Sincee that time, a number of phenomena have been considered as manifestations of BEC: superfluidityy in liquid helium, high-t c superconductivity in some materials, condensation off hypothetical Higgs particles, BEC of pions and so on. Bose-Einstein condensation in dilutee gases has been observed in 1995 in pioneering experiments with clouds of magneticallyy trapped alkali atoms at JILA [3], MIT [4] and RICE [5]. Sincee then the field of ultracold quantum gases has developed from the point of proof off principle to a mature field, and hundreds of BEC experiments with different atoms, atomm numbers, temperatures, interatomic interactions and trapping geometries have been performed.. Many experiments focus attention on creating quasi2d and quasild trapped gasess by tightly confining the particle motion to zero point oscillations in one or two directions.. Then, kinematically the gas is 2D or ID, and the difference from purely 2D orr ID gases is only related to the value of the interparticle interaction which now depends onn the tight confinement. The presence of a shallow confinement in the other direction(s) allowss one to speak of a trapped 1D(2D) gas. Recently, several groups have realized quasi2dd and quasild regimes for trapped condensates [6; 7; 8; 9]. One-dimensional systemss of bosons become especially interesting in view of ongoing experiments on atom laserss and atom chip interferometry. These experiments deal with very elongated atomic cloudss where the problem of phase coherence is of fundamental importance. Uniformm purely ID and 2D many-body systems are relatively well understood and described.. However, the presence of the trapping potential adds new features to the wellknownn problems and attracts our attention to the low-dimensional systems again. The trappingg geometry introduces a finite size of the system and significantly modifies the 1 1

8 2 2 ChapterChapter 1. Introduction structuree of energy levels. The effective interparticle interaction plays an important role inn low dimensions and in trapped gases it strongly depends on the tight confinement. AA number of questions arise naturally: How do these features modify the well-known conclusionss about the BEC phase transition, superfluidity and phase coherence properties off the low-dimensional quantum gases? What are the regimes of quantum degeneracy in thesee novel low-dimensional systems? Tryingg to find answers to these questions we rely on a long prehistory of the subject. Inn uniform low-dimensional Bose systems the true BEC and long-range order are absent att finite temperatures [10; 11]. At sufficiently low temperatures the density fluctuations aree suppressed as in the 3D case. However, long-wave fluctuations of the phase of the bosonn field provide a power law decay of the density matrix in 2D and an exponential decayy in ID (see [12] for review). The characteristic radius of phase fluctuations exceeds thee healing length, and locally the system is similar to a true condensate. This state of thee system is called quasicondensate or condensate with fluctuating phase (see [13]). Thee presence of the quasicondensate is coupled to the phenomenon of superfluidity. Inn uniform 2D systems one has the Kosterlitz-Thouless superfluid phase transition associatedd with the formation of bound pairs of vortices below a critical temperature. The Kostelitz-Thoulesss transition has been observed in monolayers of liquid helium [14; 15]. Recently,, the observation of KTT in the 2D gas of spin-polarized atomic hydrogen on liquid-heliumm surface has been reported [16]. BEC of excitons - electron-hole bound pairs inn semiconductors - has been discussed for a long time (see [17; 18; 19; 20] for review). Theree are now promising prospects for reaching quantum degeneracy in quasi2d trapped excitonn gases [21; 22; 23; 24]. Bose-Einstein condensation of composite bosons formed outt of tightly bound pairs of electrons (bipolaron mechanism) is one of the explanations off the high critical temperature in superconducting copper-oxide layers and other HTSC materialss [25]. Non-BCS mechanisms of superconductivity have been reported in quasild wiress [26]. Fromm a theoretical point of view, the situation in dilute gases is unique compared withh liquids or HTSC materials. In a gas the characteristic radius of interaction between particles,, R e, is much smaller than the mean interparticle separation, giving rise to a small parameterr (gaseous parameter). In the ultra-cold limit, collisions are dominated by the s-wavee scattering and the scattering amplitude is characterized by a single parameter of thee dimension of length. In low dimensions this quantity is an analog of the 3D scattering lengthh and replaces the full knowledge of the interaction potential. The existence of the smalll gaseous parameter allows an ab initio theoretical description of gases and makes thee physical picture more transparent than in phenomenological theories of liquids This Thesis Inn this Thesis we develop a theory for describing regimes of quantum degeneracy and BECC in ultra-cold low-dimensional trapped gases. The main emphasis is put on the phase coherencee and on the role of interparticle interaction in trapped degenerate gases. The resultss of the Thesis are linked to the ongoing and future experimental studies. Inn Chapter 2 we introduce fundamental concepts related to BEC in low dimensions andd give a brief overview of literature on low-dimensional quantum gases.

9 SectionSection L2. This Thesis 3 3 Inn Chapter 3 we investigate the interparticle interaction in the quasi2d regime and discusss the influence of the tight confinement on the properties of degenerate quasi2d Bosee and Fermi gases. The first section is dedicated to the mean-field interaction in quasi2dd Bose gases. We show that it is sensitive to the strength of the tight confinement, andd one can evenn switch the sign of the interaction by changing the confinement frequency. Wee find that well below the BEC transition temperature T c the equilibrium state of a quasi2dd Bose gas is a true condensate, whereas at intermediate temperatures T < T c onee can have a quasicondensate. Afterr the recent progress in cooling 3D Fermi gases to well below the temperature off quantum degeneracy, achieving superfluidity is a challenging goal and it may require muchh lower temperatures. In the second section we show that quasi2d atomic Fermi gasess are promising for achieving superfluid regimes: the regime of BCS pairing for weak attractionn between atoms, and the regime of strong coupling resulting in the formation off weakly bound quasi2d bosonic dimers. For the BCS limit, we calculate the transition temperaturee T c and discuss how to increase the ratio of T c to the Fermi energy. In the otherr extreme, we analyze the stability of a Bose condensate of the quasi2d dimers. Activee studies on achieving the quasi2d regime [27; 28; 29; 30; 31; 6; 7] for cold gases raisee a subtle question of how 3D collisions and interactions transform into 2D. In a nondegeneratee gas, the crossover from 3D to 2D should take place when the level spacing forr the tightly confining potential becomes comparable with the mean kinetic energy of particles.. In Chapter 4 we develop a theory of pair interatomic collisions at an arbitrary energyy in the presence of tight confinement in one direction. We identify regimes in whichh collisions can be no longer regarded as three-dimensional and the 2D nature of thee particle motion is important. We describe the collision-induced energy exchange betweenn the axial and radial degrees of freedom and analyze the recent experiments on kineticc properties, thermalization and spin relaxation in a tightly confined gas of Cs atomss [27; 28; 29; 30]. Chapterr 5 is dedicated to quasild trapped Bose gases. We discuss the regimes of quantumm degeneracy and obtain the diagram of states for this system. We identify three regimes:: the BEC regimes of a true condensate and quasicondensate, and the regime of aa trapped Tonks gas (gas of impenetrable bosons). We show that the presence of a sharp cross-overr to the BEC regime requires extremely small interaction between particles. Thee phase coherence in a degenerate 3D Bose gas depends on the geometry. If the samplee is sufficiently elongated, long-wave thermal fluctuations of the phase in the axial directionn acquire ID character and can destroy BEC. In Chapter 6 we show that in very elongatedd 3D trapped Bose gases, even at temperatures far below the BEC transition temperaturee T c, the equilibrium state is a quasicondensate. At very low temperatures the phasee fluctuations are suppressed and the quasicondensate turns into a true condensate. AA simple and efficient method of observing phase fluctuations in an elongated trapped Bose-condensedd gas has been recently demonstrated in Hannover. The method relies on thee measurement of the density distribution after releasing the gas from the trap. In thee second section of Chapter 6 we analyze this experiment and show how the phase fluctuationss in an elongated BEC transform into the density modulations in the course off free expansion.

10

11 Chapterr 2 Overview InIn this chapter we give a brief overview of literature on low-dimensional gases and introduceduce important concepts and methods used in this Thesis BEC of an ideal gas in ID and 2D harmonic traps Thee analysis of trapped gases has been extended mostly to the noninteracting case. Bagnatoo and Kleppner [32] have found that in a harmonically trapped ideal 2D Bose gas occupationn of the ground state becomes macroscopic (ordinary BEC transition) below aa critical temperature which depends on the number of particles and trap frequencies. Ketterlee and van Druten have shown that the BEC-like behavior is also present in an ideall trapped ID gas [33]. Wee start with thermodynamic description of an ideal ID or 2D gas of N bosons trappedd in a harmonic external potential. The gas sample is assumed to be in thermal equilibriumm at temperature T. We will calculate thermodynamic averages for the grand canonicall ensemble, where the system is characterized by a fixed chemical potential \i andd fluctuating number of particles N. In the thermodynamic limit (N - oo) this is equivalentt to the description in the canonical ensemble (fixed N and fluctuating /i). Generally,, in an arbitrary trap of any dimension the system is characterized by a set off eigenenergies of an individual atom, {t v }. The (average) total number of particles N iss then related to the temperature and chemical potential by the equation ATT = ^iv(( e,-/i)/t), (2.1.1) V V wheree N(z) = l/(exp2 1) is the Bose occupation number (we use the convention ksks = 1)- The population of the ground state (c 0 = 0) is Woo =?, (2.1.2) JVoo k } exp(-/i/t)-l' andd in the thermodynamic limit can become macroscopic (comparable with N) only if \i\i 0. For a large but finite number of particles in a trap, N Q becomes comparable with NN and one has a crossover to the BEC regime at a small but finite /i. Wee now determine the temperature of the BEC crossover for a 2D Bose gas confined inn a circularly symmetric harmonic trap. In this case the index u corresponds to the pairr of quantum numbers {n x,n y } in such a way that e nx,n v = hw{n x + n y ), where u) iss the trap frequency. This particular level structure allows one to rewrite the sum in Eq.. (2.1.1) in the form of an integral NN = N 0 + N((e-n)/T)p(e)de. (2.1.3) Jo Jo 5 5

12 6 6 ChapterChapter 2. Overview Heree p(e) = e/{hu)) 2 is the density of states of the system and we have separated the groundd state population. The transformation from the sum to the integral is justified byy the inequality hw < T and by the fact that the relative contribution of several lowenergyy states is negligible (the density of states goes to zero). For a large population of thee ground state Eq. (2.1.2) gives -p/t «1/N 0 < 1. Then the first two leading terms inn the expansion of the integral in Eq. (2.1.3) are andd Eq. (2.1.3) reduces to the form N[l-N[l- (T/T c ) 2 ] =N 0 - {T/hwf (1 + lniv 0 )/Jvo, (2.1.5) where e TT cc =y/6n/7t=y/6n/7t 22 huj.huj. Fromm Eq. (2.1.4) one clearly sees that there is a sharp crossover to the BEC regime at TT &T C [32; 34]. Below T c we can neglect the last term in Eq. (2.1.5). Then we obtain thee occupation of the ground state iv 0 «N [l - (T/T c ) 2 ] similar to that in the 3D case. Abovee T c the first term in the rhs of Eq. (2.1.5) can be neglected compared to the second one.. Note that at T c the de Broglie wavelength of particles A ~ y/h 2 /mt c becomes comparablee with the mean interparticle separation ~ {Nmu) 2 /T c )~ 1 / 2. In the crossover regionn between the two regimes all terms in Eq. (2.1.5) are equally important and we estimatee the width of the crossover region as ATT /In N 1717 ~ V -IT' (2 - L7) Solutionss of Eq. (2.1.5) for various N are presented in Fig Forr a large number of particles the relative width of the crossover region is very small. Therefore,, one can speak of an ordinary BEC transition in an ideal harmonically trapped 2DD gas. Inn the ID case the spectrum is c n = Hun and the density of states is p(e) = l/hw. Heree the integral representation Eq. (2.1.3) fails as the integral diverges for -y/t 0 andd we should correctly take into account the lowest energy levels [33]. In the limit {-/x,, hw} «Twe rewrite Eq. (2.1.1) in the form TT.^. 1 i NN = N + ^ S n -^ + J +i exp(wt-a*/t)-r (2 ' 1-8) wheree the number M satisfies the inequalities 1 <C M < T/huj. The first sum is MM x ^2^2 n_ ti i hw = 1>(M+l-ii/hu>)-1>(l-n/hw) w -ip(l-n/?ko)+\n{m-n/hw), (2.1.9)

13 SectionSection 2.1. BEC of an ideal gas in ID and 2D harmonic traps N 0 /N N J - ^^ N=50 >vv N=200 >>> N=10 3 \-,,. M in" > o,, IN 111 \\ \ \.. s rr ' T i i T/T T Figuree 2.1.1: The ground state population in a 2D trap versus temperature. The curves aree calculated from Eq. (2.1.5) for various N. wheree V is the digamma function. The second sum in Eq. (2.1.8) can be transformed to ann integral I I i i dx dx hujr, hujr, ** - 1 n=m+ll exp " T fvjj JhuM/T exp(x - n/t) - 1 TT hu>{m - n/huj) 'hü'hü n T ' Finally,, eliminating the chemical potential by using the equality \i «T/N 0, we reduce Eq.. (2.1.8) to the form TT T TooToo hui TT f T huhu \ HUJNQ (2.1.10) ) Ass in the 2D case, we have two regimes, with the border between them at T c determined byy N w (T c /fkü)[ln(t c /tko) ]. Indeed, below this temperature the first term in thee rhs of Eq. (2.1.10) greatly exceeds the second one and the ground state population behavess as JV 0 ft) N - (T/huj)ln(T/huj). The crossover region is determined as the temperaturee interval where both terms are equally important: AT/T c ~l/lnat.. (2.1.11) Thee crossover temperature in a ID trap is T c ft) Nhu/lnN and, in contrast to 3D and 2DD cases, is much lower than the degeneracy temperature Tj. «Nfaj. In Fig we presentt N 0 (T) calculated from Eq. (2.1.10). Thee crossover region in the ID case is much wider than in 2D. This is not surprising ass the crossover itself is present only due to the discrete structure of the trap levels. The quasiclassicall calculation does not lead to any crossover [32]. Notee that an ideal 2D Bose gas in a finite box has the density of states independent off energy, just like a harmonically trapped ID gas. Therefore, it is characterized by a similarr crossover to the BEC regime [35; 33].

14 ChapterChapter 2. Overview ^ ' ' N 0 /N N N>*,, N N N' ' \ v.. V N=50 0 N=200 0 N=10 0 N= ^N.. ** * - *"""""-- ^** s *^>^^ "^ ""* *** T T ii ' i ' r i T/T T Figuree 2.1.2: The ground state population in a ID trap versus temperature. The curves aree calculated from Eq. (2.1.10) for various N Interacting Bose gas Wee now consider an interacting d-dimensional Bose gas in a trap with the potential U(r). Thee Hamiltonian of the system in the second quantization reads (see [36]) HH = &(r)(-^ + U(r)\*(r) + V(\r-v'\) (v) (r'mvmr'), (2.2.1) wheree V(r) is the potential of interaction between two atoms and ^(r), ^t(r) are the Bosonn field operators satisfying the commutation relations [tff (r), *(r')] = 0, [#(r), *V)] = 6(r - v'). (2.2.2) ) Inn the Heisenberg representation the time derivative of ^(r) is given by MM,*.»., / h 2 v 2 ih-ih- 4f] = f-^- + U{r) + J V{\v - r'\)&(r')4!(r')\ #. (2.2.3) dtdt L J " V 2 Usingg the commutation relations (2.2.2) we rewrite Eq. (2.2.3) in the form.0* * nn 22 vv 2 2 '' hh + lïtlït y^* = ' Z 2 L ~^! + U ^ + * ƒ F( ' r ~ r 'l)* + ( r ')*(r')- (2.2.4) ) Thee term V(0)^ gives only a trivial time dependence and can be eliminated by the substitutionn * > $?exp(iv(0)t/h). Let us now turn to the density-phase representation off the field operators ** = exp(i<j>)vü, * f = V^exp(-icé), (2.2.5)

15 SectionSection 2.2. Interacting Bose gas 9 9 wheree the density and phase operators are real and satisfy the commutation relation [n(r),<kr')]=i<s(r-r').. (2.2.6) Substitutingg Eqs. (2.2.5) into Eq. (2.2.4) and separating real and imaginary parts, we get thee coupled continuity and Euler hydrodynamic equations for the density and velocity vv = {h/m)v4>: otot m hh dt dt Eqs.. ( ) are obtained from the Hamiltonian (2.2.1) without any approximationss and, in principle, should describe any property of an interacting Bose gas. At thee same time, these equations are nonlinear and very complicated. However, in a dilute ultra-coldd weakly interacting gas we can use a number of simplifications. The short-range characterr of the interatomic potential 1 allows us to rewrite the integral in Eq. (2.2.8) as V( r'-r )n(r')«sn(r),, (2.2.9) L L wheree the mean-field coupling constant g can also depend on density or on average momentumm of particles (see Chapters 3 and 4). Wee further simplify Eqs. ( ) assuming small fluctuations of the density. In thiss case, Eq. (2.2.7) shows that fluctuations of the phase gradient are also small. Representingg the density operator as n(r) = no(r) -{- Sn(r) and shifting the phase by pt/h we thenn linearize Eqs. ( ) with respect to Sh, V< around the stationary solution nn = no, V<j> = 0. The zero order terms give the Gross-Pitaevskii equation for no: andd the chemical potential (i follows from the normalization condition n I I 0 (r)) = iv. (2.2.11) Thee first order terms provide equations for the density and phase fluctuations: hd(sh/y/n^)/dthd(sh/y/n^)/dt = (-h 2 V 2 /2m + U(r)+gno-iJ,)(2y/n^j>), ( hd{2y/rï^4>)/dt-hd{2y/rï^4>)/dt = (-h 2 V 2 /2m + U(r) + 3gno-ti)(ön/y/nï>). (2.2. Solutionss of Eqs. ( ) are obtained by representing Sh, V(j> in terms of elementaryy excitations: 6n(r)6n(r) = no(r) 1 / 2 5^t/-(r)c-* e "*/' l o l, + H.c. J (2.2.14) 4>{T)4>{T) = [4no(r)]- 1 / 2^/+(r)e-^t/,L a I/ + H.c. (2.2.15) *Inn this Thesis we do not consider dipolar or charged gases with long-range interatomic forces.

16 10 0 ChapterChapter 2. Overview Fromm Eqs. ( ) we then find equations for the eigenenergies e v and eigenfunctionss f^ of the excitations: (-h(-h 22 VV 22 /2m/2m + U(T) +9**-») ƒ+ = v f~, (-h(-h 22 VV 22 /2m/2m + U{v) + Zgno - /i) ƒ" = e ƒ +. (2.2.16) ) (2.2.17) ) Thee commutation relation (2.2.6) ensures that the functions f^ are normalized by the condition n \\ I (.tf(r) A7*(r) + /"(r) /+*(r)) = 1. (2.2.18) Equationss ( ) are exactly the same as the Bogolyubov-de Gennes equations 22 for elementary excitations of a Bose-Einstein condensate with the density profile no(r).. The functions f^ are related to the well-known Bogolyubov w, v functions by f^f^ = We thus see that the assumption of small density fluctuations is sufficient forr having the Bogolyubov spectrum of the excitations, irrespective of the presence or absencee of a true condensate. Thee spectrum of elementary excitations of a trapped Bose-condensed gas has been intensivelyy discussed in literature. In a vast majority of experiments the number of particless is very large and the chemical potential /A greatly exceeds the level spacing inn the trap. In this case the kinetic energy term in Eq. (2.2.10) is much smaller than thee nonlinear term and can be neglected. This approach is called the Thomas-Fermi (TF)) approximation, and in a harmonic trap the density profile no takes the well-known parabolicc shape nn 00 = [p - U(r)]/g. (2.2.19) Inn this case, the low-energy excitations (e v <C fi) can be found analytically [38; 39; 40].. The dependence of the excitation spectrum on the trapping geometry has been extensivelyy studied for 3D TF condensates (see [41] for review). For example, in very elongatedd cigar-shaped condensates the spectrum of low lying axial excitations reads _,, = fiw z^/j(j + 3)/2 [40; 42]. Stringari [42] has found the spectrum for very anisotropic pancake-shapedd condensates. The spectrum of low-energy excitations has also been found forr purely 2D and ID Thomas-Fermi clouds [43]. Inn a homogeneous Bose gas the chemical potential (i = griq and Eqs. ( ) givee the well-known spectrum and wavefunctions: c(fc)) = y/e(k) [E{k) + 2n], (2.2.20) t-mmt"'*'-t-mmt"' wheree E(k) = h 2 k 2 /2m is the free-particle spectrum, and V is the d-dimensional volume off the system. The Bogolyubov spectrum (2.2.20) is phonon-like, with e(k) «c a hk, for energiess of the order of or smaller than p. This corresponds to momenta k < l// c, where thee healing length l c ~ h/y/rnji. The speed of sound is c s = yfixfm. For larger momenta thee spectrum (2.2.20) is particle-like, with e(k) «E(k) + /x. 22 Similar equations have been found by de Gennes [37] for inhomogeneous superconductors.

17 SectionSection 2.2. Interacting Bose gas 11 1 Thee chemical potential /z is an approximate border between two classes of excitations. Inn order to analyze the role of these two classes let us separate the operator of the density fluctuationsfluctuations into two parts: Sn = 6h 8 + Sh pi where the indices s and p stand for the phononn (e < /i) and free-particle (e > //) parts respectively. We now use Eq. (2.2.14) and calculatee the density-density correlation function (<5n s (ri)<5n a (r2)), where the symbol (...) denotess the statistical average. A straightforward calculation yields K(nK(r 2 ))^^ %&[2N{ {k)/t) + l]co8k-{r 1 -T 3 ) e(k) e(k) (k)<h (k)<h (2.2.22) ) Heree we used the equality (d k a k ) = N(e(k)/T) for the equilibrium occupation number off excitations. Taking into account that this number is always smaller than T/e{k) we obtain n (ni(ri)n;(r2))/n22 < (/i/t d ) d / 2 max{(t/ M ), 1}, (2.2.23) wheree d is the dimension of the system, and the temperature of quantum degeneracy is TdTd = h 2 7iQ /m. We then see that well below Td in 2D and 3D weakly interacting gases thee phonon-induced density fluctuations are small and can be neglected. To ensure that theyy are small in the one-dimensional gas we require T <C (/itd) 1 / 2 in ID. Then, omitting fluctuationsfluctuations originating from the high-energy part (e > p) of the spectrum, we represent thee field operator Vf in the form ** = v/«4>exp (i<i> a ), (2.2.24) ) Inn order to make sure that fluctuations coming from high-energy excitations can be omitted,, we a priori assume that they are small and can be evaluated by linearizing the fieldfield operator 4( = exp(i<f))\/h with regard to the high-energy part. This is equivalent to writingg the preexponential factor in Eq. (2.2.24) as (y/nö + #') instead of just y/nö. The operatorr ty' accounts for both density and phase fluctuations and reads *'' = i ]T w k (r)a k e-* e(fc)t / ft - v k (r)a k e iê(fe)t/ft (2.2.25) ) e(fc)>m M Att energies significantly larger than \i the function u k (f / k ~)/2 > 0, and u k = (fk(fk + /k~)/ 2 ~* ^" 1/2 exp(ik r). The energy itself is e(k) «E(k) + fi and, hence, thee operator #' describes an ideal gas of Bose particles with chemical potential equal to /x.. One thus sees that high-energy Bogolyubov excitations correspond to the incoherent non-condensedd part of the gas. The density of this gas is exponentially small at T < /i andd for T > \i it is equal to (T/Td)3/22 j in 3D c(fc)>mm \ 2n ) (T/T d )ln(t//i),, Tf/xTd)- 1 / 2,, in 2D in ID. (2.2.26) )

18 12 2 ChapterChapter 2. Overview Att temperatures T <^T d {T <^ {pt d ) 1/2 in the ID case) the quantity (4>'i&) is really small,, which justifies Eq. (2.2.24) for the field operator. Equationn (2.2.24) allows us to calculate correlation functions at low temperatures too zero order in perturbation theory. For obtaining perturbative corrections, one should expandd Eqs. ( ) up to quadratic terms in On and (f>. This provides a correction forr the stationary solution and for the chemical potential as a function of density. One shouldd then include the low-energy density fluctuations and the high-energy fluctuations inn the expression for the field operator. This is equivalent to proceeding along the lines off the perturbation theory developed by Popov (see [12]) Phase fluctuations and quasicondensates in a uniformm gas Bose-Einsteinn condensation in a uniform gas is associated with the long-range order inn the system, i.e. with the finite value of the one-particle density matrix p(ri,r2) = (^,t (ri) 1 l r (r2))) at ri r^l» oo. Using Eq. (2.2.24) the density matrix at low temperaturess can be written in the form [12] p(n,r a )) = no (c- i [*-< ri >-*- (ra) ]) =noc-*<[*'< r '>-*'M a >. (2.3.1) Thee phase correlator is obtained from Eqs. (2.2.15) and (2.2.21). It reads // [0.(r) - <M0)] 2 \ = -L Y, i [ 2Ar ( c (*)/ T ) +!] C 1 " cosk * r )» ( ) Inn the 3D case this correlator is small even at temperatures larger than /x, where it is approximatelyy equal to T/(7r 2 Tdn 0 ' Z c )- Thee two-dimensional vacuum fluctuations of the phase are negligible and in 2D one hass a true condensate at zero temperature. At finite temperatures the correlator (2.3.2) takess the form (see [44]) /[^s(r)-0 s (O)] 2 \«T/(7rr d )ln(r// c ).. (2.3.3) Thiss result means that the long-wave fluctuations of the phase of the boson field provide aa power law decay of p(r) at r» oo in contrast to the 3D case. This was first found by Kanee and Kadanoff [44] and is consistent with the Bogolyubov theorem [10; 11] indicating thee absence of a true condensate at finite temperatures in 2D. The power law behavior of p(r)) is qualitatively different from the exponential decay at large distances in a classical gas,, which indicates a possibility of phase transition at sufficiently low T. The existence off a superfluid phase transition in 2D gases (liquids) has been proved by Berezinskii [45;; 46]. Kosterlitz and Thouless [47; 48] found that this transition is associated with the formationn of bound pairs of vortices below a critical temperature TKT which is of the orderr of T&. Recently, the exact value of this temperature has been found using Monte Carloo simulations [49; 50].

19 SectionSection 2.4. Realization of BEC's in lower dimensions 13 3 Earlierr theoretical studies of 2D systems have been reviewed in [12] and have led to the conclusionn that below the Kosterlitz-Thouless transition (KTT) temperature the Bose liquidd (gas) is characterized by the presence of a quasicondensate, which is a condensate withh fluctuating phase (see [13]). Indeed, from Eqs. (2.3.1) and (2.3.3) we see that the densityy matrix decays on a length scale R^ «l c exp(-nt d /T)» l c. In this case the system cann be divided into blocks with a characteristic size greatly exceeding the healing length ll cc but smaller than the radius of phase fluctuations R^. Then, there is a true condensate inn each block but the phases of different blocks are not correlated with each other. Thee situation is similar in the ID gas at temperatures T <C (//T^) 1 / 2, except for the presencee of a true condensate at T = 0. Eq. (2.3.2) gives (U s (r)) -< s (0)l 2 \ «-^=f + i«/f Inf, (2.3.4) wheree the first term in the rhs comes from the thermal part and the second one from the vacuumm part of the phase fluctuations. Thus, at zero temperature the density matrix undergoess a power-law decay [51; 52; 12], and there is noo true Bose-Einstein condensate. Thiss is consistent with a general analysis of ID Bose gases at T = 0 [53]. At finite temperaturess the long-range order is destroyed by long-wave fluctuations of the phase leadingg to an exponential decay of the one-particle density matrix at large distances [44;; 12]. Inn this Thesis we mostly concentrate on the properties of a trapped ID Bose gas in thee weakly interacting regime, where the healing length l c = h/^/mng greatly exceeds thee mean interparticle separation 1/n. This corresponds to small values of the parameter 77 = mg/h 2 n. Our previous discussions of the uniform ID interacting gas in this chapter weree related to this particular regime. In general, the ID Bose gas with repulsive shortrangee interactions characterized by the coupling constant g > 0 is integrable by using thee Bethe Ansatz at any g and n and has been a subject of extensive theoretical studies [54;; 55; 56; 57]. The equation of state and correlation functions depend crucially on thee parameter 7. The weakly interacting regime (7 «C 1) is realized at comparatively largee densities (or small g). For sufficiently small densities (or large 5), the parameter 77 ^> 1 and one has a gas of impenetrable bosons (Tonks gas), where the wavefunction stronglyy decreases when particle approach each other. In this respect the system acquires fermionicc properties Realization of BEC's in lower dimensions Inn commonly studied BEC's in 3D harmonic traps the mean-field interaction greatly exceedss the level spacings. In this case the Thomas-Fermi approximation can be used andd the shape of the condensate density profile is given by Eq. (2.2.19). For the Thomas- Fermii condensate the chemical potential equals 2/5 5 // Loivogm,-' ~u>~ \ p*s^)). (-) wheree u 3 = <jj x tv y cj z, No is the number of condensed particles, and g = 4nh 2 a/m is the 3DD coupling constant with a being the 3D scattering length.

20 14 4 ChapterChapter 2. Overview Inn order to reach quasi-id or quasi-2d regimes for BEC in harmonic traps one has too satisfy the condition fi «C hvo, where U)Q is the frequency of the tight confinement. In cylindricallyy symmetric traps (u> x =u) y = the approximate crossover to quasi-id or quasi-2d,, defined by /x 3 o = tkjo, occurs if the number of condensate particles becomes Görlitzz et al. [6] have explored the crossover from 3D to ID and 2D in a 23 Na BEC byy reducing the number of the condensed atoms. The scattering length for sodium is relativelyy small, a ~ 28A, and the traps in this experiment feature extreme aspect ratios resultingg in high numbers N\D > 10 4 and N 2 D > Forr the ID case, the condition fi = yields a linear density n\b «l/4a, implying thatt the linear density of a ID condensate is limited to less than one atom per scattering lengthh independent of the radial confinement. Therefore, tight transverse confinement, ass may be achievable in small magnetic waveguides [58; 59; 60; 61; 62; 63] or hollow laserr beam guides [64; 65], is by itself not helpful to increase the number of atoms in aa ID condensate. Large ID numbers may be achieved only at the expense of longer condensatess or if the scattering length is smaller. Inn anisotropic traps, a primary indicator of crossing the transition temperature for Bose-Einsteinn condensation is a sudden change of the aspect ratio of the ballistically expandingg cloud, and an abrupt change in its energy. The transition to lower dimensions iss a smooth cross-over, but has similar indicators. In the 3D Thomas-Fermi limit the degreee of anisotropy of a BEC is independent of the number of atoms iv 0, whereas in IDD and 2D the aspect ratio depends on NQ. Similarly, the release energy in 3D depends onn No [41] while in lower dimensions it saturates at the zero-point energy of the tightly confiningg dimension (s). AA trapped 3D condensate has a parabolic shape and its radius and half-length are givenn by = \/2^3 >/mü;^ and R z = ^/2/j, 3D /muj 2, resulting in an aspect ratio of = When the 2D regime is reached by reducing the atom number, the condensatee assumes a Gaussian shape with the width l z = y/h/muj z along the axial direction,, but retains the parabolic shape radially. The radius of a trapped 2D condensate decreasess with JV 0 as = (l28n^a 2 h 3 üü z /irm 3 uj\) 1/8 (see Section 3.1). Similarly, the half-lengthh of a trapped ID condensate is R Z ID 2 ) 1 / 3 (see Chapter 5). Inn a ballistically expanding cloud the size in the direction(s) of shallow confinement practicallyy does not change in time compared to a fast expansion in the direction(s) of tightt confinement. In the case of pancake Thomas-Fermi condensates the latter equals bb zz RR zz,, where the parameter b z is governed by the scaling equation [66; 67; 68] bb zz =uj=uj 22 JblJbl (2.4.2) Att long times of flight t» 1/<JJ Z Eq. (2.4.2) predicts b z «\[2u z t. Therefore, the aspect ratioo is b z (t)r z /Rj_ «Iff the cloud is in the quasi-2d regime, the scaling parameter equals b z (t) w u> z t and thee aspect ratio is b z (t)l z 2 D «(nhuj z /128mN 2 0 a 2 uj]_) 1 / 8 which is larger than in thee Thomas-Fermi regime. Görlitz et al. [6] have observed the crossover in the change off the aspect ratio by decreasing the number of particles in the condensate.

21 SectionSection 2.4. Realization of BEC's in lower dimensions bb timee off flight [ms] 10 0 Figuree 2.4.1: Signature of ID condensate. Radial size of expanding condensates with 10 4 atomss as a function of time of flight. The straight line is the expected behavior for the expansionn of the ground state radial harmonic oscillator with = 4 khz Inn cigar-shaped Thomas-Fermi condensates at t» 1/wj. the aspect ratio equals zz w uj z t. In the quasi-ld regime the radial size expands differently and the aspectt ratio reads {t)/r z \D * 2 muj^) 1 / 6 Lj z t and is larger than inn the Thomas-Fermi case. In both cases the scaling parameter is bx{t) «(see [66]).. The change in the aspect ratio with the decrease of the particle number has been observedd in Ref. [6]. Schreck et al. [8] have reached the quasi-ld regime in a 7 Li condensatee taking advantage of the small scattering length. Their measurements of the radiall size of the cloud agree with the time evolution of the radial ground state wave functionn (see Fig ). Ann efficient way to reach the quasi-2d and quasi-ld regimes in trapped condensates iss to apply a periodic potential of an optical lattice to a 3D condensate initially prepared inn a usual magnetic trap. In this way an array of quasi-2d Rb condensates has been obtainedd by Burger et al. [7]. Greiner et al. [9] realized an array of quasi-ld Rb condensatess in a two-dimensional periodic dipole force potential, formed by a pair of standingg wave laser fields. Advantagess of this method are obvious: First, an optical lattice can confine a large arrayy of 2D or ID systems, which allows measurements with a much higher number of involvedd atoms with respect to a single confining potential. Second, the macroscopic populationn of a single quantum state in the initial 3D trap naturally transfers the whole systemm into 2D or ID systems well below the degeneracy temperature.

22

23 Chapterr 3 Quantum degenerate quasi-2dd trapped gases 3.11 Bose-Einstein condensation in quasi-2d trapped gases s WeWe discuss BEC in quasi-2d trapped gases and find that well below the transition temperatureature T c the equilibrium state is a true condensate, whereas at intermediate temperatures TT < T c one has a quasicondensate (condensate with fluctuating phase). The mean-field interactioninteraction in a quasi-2d gas is sensitive to the frequency UQ of the (tight) confinement inin the "frozen" direction, and one can switch the sign of the interaction by changing UJQ. VariationVariation of LÜQ can also reduce the rates of inelastic processes. This offers promising prospectsprospects for tunable BEC in trapped quasi-2d gases. Thee influence of dimensionality of the system of bosons on the presence and character off Bose-Einstein condensation (BEC) and superfluid phase transition has been a subject off extensive studies in spatially homogeneous systems. In 2D a true condensate can onlyy exist at T = 0, and its absence at finite temperatures follows from the Bogolyubov k~k~ 22 theorem and originates from long-wave fluctuations of the phase (see, e.g., [69; 12]).. However, as was first pointed out by Kane and Kadanoff [44] and then proved by Berezinskiii [45; 46], there is a superfluid phase transition at sufficiently low T. Kosterlitz andd Thouless [47; 48] found that this transition is associated with the formation of bound pairss of vortices below the critical temperature TKT = (nh 2 /2m)n s (m is the atom mass, andd n s the superfluid density just below TKT)- Earlier theoretical studies of 2D systems havee been reviewed in [12] and have led to the conclusion that below the Kosterlitz- Thoulesss Transition (KTT) temperature the Bose liquid (gas) is characterized by the presencee of a quasicondensate, that is a condensate with fluctuating phase (see [13]). In thiss case the system can be divided into blocks with a characteristic size greatly exceeding thee healing length but smaller than the radius of phase fluctuations. Then, there is a truee condensate in each block but the phases of different blocks are not correlated with eachh other. Thee KTT has been observed in monolayers of liquid helium [14; 15]. The only dilute atomicc system studied thus far was a 2D gas of spin-polarized atomic hydrogen on liquidheliumm surface (see [70] for review). Recently, the observation of KTT in this system has beenn reported [16]. BECC in trapped 2D gases is expected to be qualitatively different. The trapping potentiall introduces a finite size of the sample, which sets a lower bound for the momentum off excitations and reduces the phase fluctuations. Moreover, for an ideal 2D Bose gas inn a harmonic potential Bagnato and Kleppner [32] found a macroscopic occupation of thee ground state of the trap (ordinary BEC) at temperatures T < T c «JV 1 / 2 /^, where NN is the number of particles, and u the trap frequency. Thus, there is a question of 17 7

24 18 8 ChapterChapter 3. Quantum degenerate quasi-2d trapped gases whetherr an interacting trapped 2D gas supports the ordinary BEC or the KTT type off a cross-over to the BEC regime 1. However, the critical temperature will be always comparablee with T c of an ideal gas: On approaching T c from above, the gas density is nn cc ~ N/H^c, where RT C «y/t c /muj 2 is the thermal size of the cloud, and hence the KTTT temperature is ~ h 2 n c /m ~ N 1/2 hu «T c. Thee discovery of BEC in trapped alkali-atom clouds [3; 4; 5] stimulated a progress inn optical cooling and trapping of atoms. Present facilities allow one to tightly confine thee motion of trapped particles in one direction and to create a (quasi-)2d gas. In other words,, kinematically the gas is 2D, and in the "frozen" direction the particles undergo zeroo point oscillations. This requires thee frequency of the tight confinement OJQ to be much largerr than the gas temperature T and the mean-field interparticle interaction nog (no is thee gas density, and g the coupling constant). Recent experiments [72; 73; 27; 28; 74; 75] indicatee a realistic possibility of creating quasi-2d trapped gases and achieving the regime off quantum degeneracy in these systems. The character of BEC will be similar to that inn purely 2D trapped gases, and the main difference is related to the sign and value of thee coupling constant g. Inn this section we discuss BEC in quasi-2d trapped gases and arrive at two key conclusions.. First, well below T c the phase fluctuations are small, and the equilibrium state iss a true condensate. At intermediate temperatures T < T c the phase fluctuates on a distancee scale smaller than the Thomas-Fermi size of the gas, and one has a quasicondensatee (condensate with fluctuating phase). Second, in quasi-2d the coupling constant gg is sensitive to the frequency of the tight confinement UJQ and, for a negative 3D scatteringg length a, one can switch the mean-field interaction from attractive to repulsive byy increasing u> 0. Variation of UQ can also reduce the rates of inelastic processes. These findingsfindings are promising for tunable BEC. Inn a weakly interacting Bose-condensed gas the correlation (healing) length l c = h/y/mnogh/y/mnog (g > 0) should greatly exceed the mean interparticle separation. In (quasi-)2d thee latter is ~ l/-\/2~7mö, and we obtain a small parameter of the theory, (mg/2trh 2 ) <C 1 (seee [13]). Wee first analyze the character of BEC in a harmonically trapped 2D gas with repulsive interparticlee interaction, relying on the calculation of the one-particle density matrix. Similarlyy to the spatially homogeneous case [69; 12], at sufficiently low temperatures onlyy phase fluctuations are relevant. Then the field operator can be written as ^(R) = 11 /2 * " nn QQ '' (R)exp{i< (R)}, where <^>(R) is the operator of the phase fluctuations, and no(r) thee condensate density at T = 0. The one-particle density matrix takes the form [12] <*t(r)*(0)>> = VnoWMO) ex P {-((^(R)) 2 >/2}. (3.1.1) Heree rf^(r) = < (R) <^(0), and R = 0 at the trap center. Formula (3.1.1) is a generalizationn of Eq. (2.3.1) for the inhomogeneous case. For a trapped gas the operator 0(R) iss given by (we count only phonon excitations (see Overview)) 0(R)== Y, [4no(R)]" 1/2 /^ + h.c, (3.1.2) v.v. c v <nog 11 Indications of a BEC-type of phase transition in a weakly interacting trapped 2D Bose gas were obtainedd in [71].

25 SectionSection 3.1. Bose-Einstein condensation in quasi-2d trapped gases 19 wheree a is the annihilation operator of an elementary excitation with energy e, and ƒ =u v v v are the Bogolyubov u, v functions of the excitations. Inn the Thomas-Fermi regime the density no(r) has the parabolic shape, with the maximumm value no m = n 0 (0) ^ (Nm/irg) 1 / 2^, and the radius RTF W (ïfi/mu) 2 ) 1 / 2. Thee chemical potential is fi = no m g, and the ratio T c j\i w (irfp/mg) 1 / 2» 1. For calculatingg the mean square fluctuations of the phase, we explicitly found the (discrete) spectrumm and wavefunctions of excitations with energies e v -C fi by using the method developedd for 3D trapped condensates [38; 39; 40]. For excitations with higher energies wee used the WKB approach. At distances R greatly exceeding l c near the trap center, forr T» \i we obtain mt mt <(^(R)) 2 ))» ^ In (R/X T ), (3.1.3) wheree AT is the wavelength of thermal excitations (e «T). Eq. (3.1.3) holds at any TT for a homogeneous gas of density no m > where at T «C /i it reproduces the well-known resultt (see [12]). Strictly speaking, for T >> \i we should put the healing length l c instead off AT under the logarithm. However, this gives only a small correction at temperatures significantlyy lower than T c. In a trapped gas for T «C /J, due to the contribution of low-energyy excitations, Eq. (3.1.3) acquires a numerical coefficient ranging from 1 at RR -C RTF to approximately 3 at R «RTF- Thee character of the Bose-condensed state is determined by the phase fluctuations at RR ~ RTF- With logarithmic accuracy, from Eq. (3.1.3) we find Inn quasi-2d trapped alkali gases one can expect a value ~ 10-2 or larger for the small parameterr mg/2'nh 2, and the number of trapped atoms N ranging from 10 4 to For TT > \x the quantity no m A T «2 /m/g)(fi/t) and lniv is always significantly larger thann ln(no m A T ). Then,, from Eq. (3.1.4) we identify two BEC regimes. At temperatures well below TT cc the phase fluctuations are small, and there is a true condensate. For intermediate temperaturess T <T C the phase fluctuations are large and, as the density fluctuations are suppressed,, one has a quasicondensate (condensate with fluctuating phase). Thee characteristic radius of the phase fluctuations R<j> «Arexp (irh 2 /mt), following fromfrom Eq. (3.1.3) under the condition {(ö<j>(r)) 2 } ~ 1, greatly exceeds the healing length. Therefore,, the quasicondensate has the same Thomas-Fermi density profile as the true condensate.. Correlation properties at distances smaller than R^ and, in particular, local densityy correlators are also the same. Hence, one expects the same reduction of inelastic decayy rates as in 3D condensates [13]. However, the phase coherence properties of a quasicondensatee are drastically different. For example, in the MIT type [76] of experiment onn interference of two independently prepared quasicondensates the interference fringes cann significantly differ from the case of pure condensates. Wee now calculate the mean-field interparticle interaction in a quasi-2d Bose-condensed gas,, relying on the binary approximation. The coupling constant g is influenced by the trappingg field in the direction z of the tight confinement. For a harmonic tight confinement,, the motion of two atoms interacting with each other via the potential V(r) can be

26 20 0 ChapterChapter 3. Quantum degenerate quasi-2d trapped gases stilll separated into their relative and center of mass motion. The former is governed by V(r)V(r) together with the potential VH{Z) = muj^z 2 /4 originating from the tight harmonic confinement.. Then, similarly to the 3D case (see, e.g. [77]), to zero order in perturbation theoryy the coupling constant is equal to the vertex of interparticle interaction in vacuum att zero momenta and frequency 2 E = 2fi. For low E > 0 this vertex coincides with the amplitudee of scattering at energy E and, hence, is given by (see [36], pp ) gg = f(e) = J dr^(r)v(r)r f (r). (3.1.5) Thee wavefunction of the relative motion of a pair of atoms, 0(r), satisfies the Schrödinger equation n -^AA + VW + VHW-^' fjj{r)fjj{r) = ty(r). (3.1.6) mm I Thee wavefunction of the free x,y motion ipf(r) = <po(z) exp(iqp), with <fo(z) being the groundd state wavefunction for the potential VH{Z), p= {x,y}, and q = (me/h 2 ) 1 / 2. As thee vertex of interaction is an analytical function of E, the coupling constant for \L < 0 iss obtained by analytical continuation of ƒ (E) to E < 0. Thee possibility to omit higher orders in perturbation theory requires the above criterionn (mg/2'ïïh 2 ) < : 1. Thee solution of the quasi-2d scattering problem from Eq. (3.1.6) contains two distance scales:: the extension of ip(r) in the z direction, fo = (h/rruuo) 1 / 2, and the characteristic radiuss R e of the potential V(r). In alkalis it ranges from 20 A for Li to 100 A for Cs. Att low energies (qr e <^ 1) the amplitude f(e) is determined by the scattering of the s-wavee for the motion in the x,y plane. Wee first consider the limiting case lo ^ Re- Then the relative motion of atoms in the regionn of interatomic interaction is not influenced by the tight confinement, and ip(r) in Eq.. (3.1.5) differs only by a normalization coefficient from the 3D wavefunction: 1>(r)1>(r) = Wo{0)ihD(r). (3-1.7) Att r ;» R e we have ip^o = 1 a.fr. Hence, for R e <C r <C l 0, Eq. (3.1.7) takes the form 00 = ^asfo = Wo(0)(l a/r). This expression serves as a boundary condition at r * 0 forr the solution of Eq. (3.1.6) with V(r) = 0 (r»,re)- The latter can be expressed throughh the Green function G(r, r') of this equation: 0(r)) = ip 0 {z) exp(iq p) + AG(r, 0), (3.1.8) Thee coefficients A and n are obtained by matching the solution (3.1.8) at r > 0 with V>as( r )-- Similarlyy to the case of a purely ID harmonic oscillator (see, e.g., [78]), we have G(r,0)) = i ƒ 0000 exp{i(z 2 cot t/4zg - q 2 ïkt - t/2 + p 2 /Ull)\ at at fafa Jo iy^(47ri) 3 sin* * 22 For an inhomogeneous density profile in the x, y plane, /z should be replaced by no(r)p.

27 SectionSection 3.1. Bose-Einstein condensation in quasi-2d trapped gases 21 1 Underr the condition qlo <C 1 (p, <C huo) at r <C Zo we obtain c*^^ + * 47rrr 2(2TT) 3 / 2 ZO.""(A) ) ++ VK (3.1.9) ) Omittingg the imaginary part of G (3.1.9) and comparing Eq. (3.1.8) with V>as, we immediatelyy find AA r,, x-,-1 (3.1.10) ) V2nk V2nk T)=-T)=- 47Tfl9?o(0) ) 1+^fe In (^ ).. Inn Eq. (3.1.5) one can put ipf = ^o(o) = (l/27r/ ) 1/4. Then, using the well-known resultt ƒ dr^3 D (r)v(r) = 4nh 2 a/m, Eqs. (3.1.5), (3.1.7) and (3.1.10) lead to the coupling constant t ^2y/2^h^2y/2^h m l 0 /a+(l/v2^)\n(l/nq 2 l 2 )' Forr /i < 0, analytical continuation of Eqs. (3.1.9) and (3.1.11) to E h 2 q 2 /m < 0 leads too the replacement E > \E\ = 2\p\ in the definition of q. Thee coupling constant in quasi-2d depends on q (2m\p\/h 2 ) 1^2 and, hence, on the condensatee density. In the limit Zo ~> \a\ the logarithmic term in Eq. (3.1.11) is not important,, and g becomes density independent. In this case the quasi-2d gas can be treatedd as a 3D condensate with the density profile oc exp ( Z 2 /IQ) in the z direction. Ass follows from Eq. (3.1.11), for repulsive mean-field interaction in 3D (a > 0) the interactionn in quasi-2d is also repulsive. For a < 0 the dependence of g (and the scatteringg amplitude ƒ) on Zo has a resonance character (cf. Fig ): The couplingg constant changes sign from negative (attraction) at very large Zo to positive for lolo < I* = ( a /V^7r)ln(l/7r9 2 Zo)- The resonance originates from the fact that the moduluss of energy coincides with the binding energy of a pair of atoms in the quasi-2d geometry att Zo «I*- This should describe the case of Cs, where a < -600 A [79; 80; 81; 73; 27; 28] andd the condition Zo» R e assumed in Eq. (3.1.11) is satisfied at Zo < /*. Near the resonancee point Z* the quantity (m <7 /27r/i 2 ) becomes large, which violates the perturbation theoryy for a Bose-condensed gas and makes Eqs. (3.1.5) and (3.1.11) invalid. Forr Zo < Re (except for very small Zo) we used directly Eqs. (3.1.5) and (3.1.6) and calculatedd numerically the coupling constant g for Li, Na, Rb, and Cs. The potential V(r)V(r) was modeled by the Van der Waals tail, with a hard core at a distance RQ <SL R e selectedd to support many bound states and reproduce the scattering length a. The numericall results differ slightly from the predictions of Eq. (3.1.11). For Rb and Cs both aree presented in Fig Thee nature of the g(lo) dependence in quasi-2d can be understood just relying on the valuess of g in the purely 2D and 3D cases. In 2D at low energies the mean-field interaction iss always repulsive. This striking difference from the 3D case can be found from the solutionn of the 2D scattering problem in [36] and originates from the 2D kinematics: At distances,, where R e <C p <C q~ l {q * 0), the solution of the Schrödinger equation for the (free)) relative motion in a pair of atoms readsv> oc \n(p/d)/ \n(l/qd) (d > 0). We always havee ^ 2 increasing with p, unless we touch resonances corresponding to the presence of aa bound state with zero energy (d > oo). This means that it is favorable for particles to bee at larger p, i.e. they repel each other.

28 22 2 ChapterChapter 3. Quantum degenerate quasi-2d trapped gases JJJ 0.10 " " ' ---^^ -^\ *** O-v 0*v v "" N N / / / / / / / / / / Rb b R R =80A,, a=55a nn =10 8 cm" 2 ^^^ ^ RR =100A, a=-600a n o=10 8 cm' 22 ^ ^ " / / / / / / a) ) ---- b) ) Figuree 3.1.1: The parameter mg/2-ïïh 2 versus lo/r e at fixed n 0 for Rb (a) and Cs (b). Solidd curves correspond to the numerical results, dashed curves to Eq. (3.1.11). The dottedd curve in (b) shows the result of Eq. (3.1.11) in the region where m\g\/2nh 2 ~ 1. Inn quasi-2d for very large l 0 the sign of the interparticle interaction is the same ass in 3D. With decreasing l 0, the 2D features in the relative motion of atoms become pronounced,, which is described by the logarithmic term in Eq. (3.1.11). Hence, for a > 0 thee interaction remains repulsive, whereas for a < 0 the attraction turns to repulsion. Thee obtained results are promising for tunable BEC in quasi-2d gases, based on variationss of the tight confinement and, hence, l 0. However, as in the MIT studies of tunablee 3D BEC by using Feshbach resonances [82], an "underwater stone" concerns inelasticc losses: Variation of l 0 can change the rates of inelastic processes. For optically trappedd atoms in the lowest Zeeman state the most important decay process is 3-body recombination 3.. Thiss process occurs at interparticle distances r < max{i? e, o } [84; 85]. We will restrictt ourselves to the case where Zo ^ <x and is also significantly larger than R e. Then thee character of recombination collisions remains 3-dimensional, and one can treat them inn a similar way as in a 3D gas with the density profile (n 0 /^l 0 ) exp ( z 2 /IQ). However,, the normalization coefficient of the wavefunction in the incoming channel willl be influenced by the tight confinement. Relying on the Jastrow approximation, we writee this wavefunction as a product of the three wavefunctions ifrfak), each of them beingg a solution of the binary collision problem Eq. (3.1.6). In our limiting case the 33 Three-body recombination to a weakly bound quasi-2d molecular state was discussed in [83].

29 SectionSection 3.2. Superfluid transition in quasi-2d Fermi gases 23 3 solutionn is given by Eq. (3.1.7) divided by y>o(0) to reconstruct the density profile in thee z direction. The outgoing wavefunction remains the same as in 3D, since one has a moleculee and an atom with very large kinetic energies. Thus,, in the Jastrow approach we have an additional factor r/ 3 for the amplitude andd rf for the probability of recombination in a quasi-2d gas compared to the 3D case. Averagingg over the density profile in the z direction, we can relate the quasi-2d rate constantt a to the rate constant in 3D (see [85] for a table of azo in alkalis): aa = (v 6 /0<*3D. (3.1.12) Ass n is given by Eq. (3.1.10), for a > 0 the dependence a(fo) is smooth. For a < 0 thee rate constant a peaks at IQ «l* and decreases as IQ/(1* lo) 6 at smaller l 0. This indicatess a possibility to reduce recombination losses while maintaining a repulsive meanfieldfield interaction (g > 0). For Cs already at Z 0 «200 A (/ «500A) we have a ~ 10" 17 cm 4 /s,, and at densities 10 8 cm -2 the life-time r > 1 s. Thee predicted possibility to modify the mean-field interaction and reduce inelastic lossess by varying the frequency of the tight confinement opens new handles on tunable BECC in quasi-2d gases. These experiments can be combined with measurements of non-triviall phase coherence properties of condensates with fluctuating phase Superfluid transition in quasi-2d Fermi gases WeWe show that atomic Fermi gases in quasi-2d geometries are promising for achieving superfluidity.superfluidity. In the regime of BCS pairing for weak attraction, we calculate the critica temperaturetemperature T c and analyze possibilities of increasing the ratio oft c to the Fermi energy. InIn the opposite limit, where a strong coupling leads to the formation of weakly bound quasi-2dquasi-2d dimers, we find that their Bose-Einstein condensate will be stable on a long timetime scale. Recentt progress in trapping and cooling of Fermi isotopes of K [86; 87; 88; 89] and Lii [90; 8; 91; 92; 93; 94] has shown the ability to go far below the temperature of quantumm degeneracy and to manipulate independently the trapping geometry, density, temperaturee and interparticle interaction. The Duke experiment [95] presents intriguing resultss on the possibility of achieving a superfluid phase transition in the two-component Fermii gas of 6 Li. Two-dimensionall Fermi gases have striking features not encountered in 3D. In the superfluidd state, thermal fluctuations of the phase of the order parameter strongly modify thee phase coherence properties. The interaction strength depends logarithmically on the relativee energy of the colliding atoms. For degenerate Fermi gases this energy is of the orderr of the Fermi energy ep which is proportional to the 2D density n. Accordingly, the exponentiall dependence of the BCS transition temperature on the interaction strength transformss into a power law dependence on the density: T c oc n 1//2 [96; 97; 98; 99]. Thiss suggests a unique possibility to cross the critical point by adiabatically expanding a degeneratee Fermi gas. Since the ratio T/SF remains unchanged, the temperature scales ass n and decreases with density faster than T c.

30 24 4 ChapterChapter 3. Quantum degenerate quasi-2d trapped gases Experimentallyy it is possible to achieve the quasi-2d regime by confining the atoms inn one direction so tightly that the corresponding level spacing exceeds the Fermi energy. Underr this condition the degenerate Fermi gas is kinematically two-dimensional. Thus far,, this regime has been reached for Cs atoms [73; 27; 74; 75; 30; 31] and for Bose- Einsteinn condensates of Na [6] and Rb [7]. Inn the quasi-2d regime the mean-field interaction between particles exhibits a similar logarithmicc dependence on the particle energy as in the purely 2D case (see Section 3.1 andd Chapter 4). The amplitude of the s-wave scattering turns out to be sensitive to thee strength of the tight confinement. This opens new handles on manipulations of the interparticlee interaction and superfluid pairing. Inn this section we show that atomic Fermi gases in quasi-2d geometries can become strongg competitors of 3D gases in achieving superfluidity. The ability to increase the interparticlee interaction by tuning the trap frequencies gives an opportunity to realize a transitionn from the standard BCS pairing in the case of weak attraction to the limit of strongg interactions and pairing in coordinate space. In the latter case one eventually gets aa dilute system of weakly bound quasi-2d dimers of fermionic atoms, which can undergo Bose-Einsteinn condensation. For the BCS case, we calculate the critical temperature T c too second order in perturbation theory and discuss possibilities of increasing the ratio Tc/ep-Tc/ep- In the other extreme, we find that the interaction between the quasi-2d dimers is repulsive,, and their collisional relaxation and decay are strongly suppressed. This allows uss to conclude that BEC of these composite bosons will be stable on a long time scale. Wee consider an ultracold two-component Fermi gas in the quasi-2d regime and confine ourselvess to the s-wave interaction and superfluid pairing between atoms of different components.. We assume that the characteristic radius of the interaction potential is muchh smaller than the harmonic oscillator length in the tightly confined direction, /o = (ft/mwo) 1^2,, where m is the atom mass, and UJQ is the confinement frequency. Then the interactionn problem involves two length scales: IQ and the 3D scattering length a. For aa < 0 and \a\ «C IQ, there is a peculiar quasi-2d weakly bound s-state of two particles, withh the binding energy (see Chapter 4) soso = 0.915(huo/ir)exp(->/2irlo/\a\) < HUQ. (3.2.1) Inn this case the coupling constant for the intercomponent interaction takes the form gg = (47rft 2 /m)ln -1 ( o/ ), where the relative collision energy e is assumed to be either muchh smaller or much larger than SQ (see Chapter 4). As in degenerate Fermi gases one hass e ~ p, the interaction is attractive (g < 0) if the density is sufficiently high and one satisfiess the inequality ee 00 /e/e FF < 1. (3.2.2) Thus,, the inequality (3.2.2) is the necessary condition for the BCS pairing. For findingfinding the critical temperature T c below which the formation of Cooper pairs becomes favorable,, we go beyond the simple BCS approach and proceed along the lines of the theoryy developed by Gor'kov and Melik-Barkhudarov for the 3D case [100]. Thee critical temperature T c is determined as the highest temperature for which the linearizedd equation for the order parameter (gap) A = (g^ü) has a nontrivial solution [101].. Assuming that the quasi-2d gas is uniform in two in-plane directions, the gap

31 SectionSection 3.2. Superüuid transition in quasi-2d Fermi gases 25 >-- -q -q q'' q» \ /> q' -q-> > ^ -q' - q _ ^!s_>_ -q' Figuree 3.2.1: The leading contributions to 5V(q, q') equationn in the momentum space takes the 2D form HH A(q) ) -/{ { 2 2 Z(*,q,q') ) K(q')K(q') + zz - (tf) + «0 ++ '>} } SV(q,q d 2 L')Ktf)\A(q L ') g' ' (2TT)2' ' (3.2.3) ) wheree JK"(g) = (l/2 (g)) tanh( (g)/2t), (g) = h 2 q 2 /2m - fj,, \i «e F = irh 2 n/m is the chemicall potential, and n is the total density of the two equally populated components. Thee first term in the rhs of Eq. (3.2.3) results from the direct interaction between particles,, and we renormalized the interaction potential in terms of the scattering amplitude ƒƒ (vertex function). The latter is a solution of the quasi-2d scattering problem. The parameterr z has a meaning of the total energy of colliding particles in their center of masss reference frame. It is of the order of ep and drops out of the final answer. The term 6V(ci,6V(ci, q') describes the modification of the interparticle interaction due to the presence off other particles (many-body effects). The leading contributions to this term are second orderr in the scattering amplitude. They are shown in Fig and correspond to an indirectt interaction between two particles when one of them interacts with a particleholee pair virtually created from the ground state (filled Fermi sea) by the other particle. Thesee second order contributions are important for the absolute value of the critical temperaturee (preexponential factor in the 3D case), whereas higher order terms involving moree interaction events can be neglected. Thee amplitude ƒ is independent of the momenta q, q' and we will use the 2D relation (seee [12]) // = 47rln- 1 { 0 /(-*)}. (3-2.4) Thee last term in the rhs of Eq. (3.2.3) is a small correction, since 5V ~ f 2. Accordingly, thee quantity / represents a small parameter of the theory. In the quasi-2d regime (z(z <?C fkjo) the motion of particles in the tightly confined direction provides a correction too Eq. (3.2.4), which is ~ (z/fuv 0 )f 2 (see Chapter 4). It is much smaller than 5V and willl be omitted. Equationss (3.2.3) and (3.2.4) show that the momentum dependence of the order parameterr appears only due to the second order term that contains many-body contributionss to the interparticle interaction SV. The latter is a function of p = q + q' and 2 ' '

32 26 6 ChapterChapter 3. Quantum degenerate quasi-2d trapped gases rapidlyy decays for p > 2qp, where qp = \J2m\xfK is the Fermi momentum. For p < 2qp thee quantity SV is almost constant. Therefore, one has A(q') «A(qp) in a wide momentumm range near the Fermi surface. Then, for q = qp a direct integration of Eq. (3.2.3) yields s A{qF)A{qF) = -&> A( ) ln (-J^) - m<*,1r)j5e " (c ) A(, F ). wheree 7 « is the Euler constant, and C is a numerical factor determined by the momentumm dependence of A and SV. The calculation of V(<7F,<ZF) is straightforward andd gives SV{qp,qp) = {h 2 /2irm)f 2 (2ii). Then, using Eq. (3.2.4) we obtain the critical temperature e TT cc «(2M/7r)exp ( rre/- 1 (2/i) ). Thee exponent in this equation should be large and the quantity Re/ -1 (2/i) should be negative.. As the chemical potential is ji «E F, from Eq. (3.2.4) one sees that these requirementss are reached under the condition (3.2.2). Using Eqs. (3.2.1) and (3.2.4) the criticall temperature takes the form TcTc = l ^ l = o.ifi^s^exp i r ^ ~ \ «*F- (3.2.5) Thee relative correction to this result is of the order of 1/ ln( o/ p)\ «C 1. Notee that Eq. (3.2.5) predicts by a factor of e smaller value for the critical temperature thann a simple BCS calculation 4. This means that the attractive interaction between particless becomes weaker once we take into account the polarization of the medium. Thee ratio T c JEp is not necessarily very small. For example, using Feshbach resonances thee scattering length is tunable over a wide interval of negative values [86; 87; 91; 92; 93;; 94]. Keeping the exponential term equal to 0.05 in Eq. (3.2.5), with UJQ ~ 100 khz wee obtain T C /E F ~ 0.1 for 2D densities n ~ 10 9 cm" 2 (T c ~ 40 nk). Ass in the purely 2D case [96; 97; 98; 99], the transition temperature T c oc n 1 / 2 and thee ratio TJEF increases with decreasing density as n -1 / 2. This is a striking difference fromm the 3D case, where this ratio decreases exponentially with density. In the presence off the in-plane confinement, one can approach the BCS transition in a degenerate Fermi gass by adiabatically expanding the quasi-2d trap in the in-plane direction(s). As the degeneracyy parameter T/EF is conserved in the course of the adiabatic expansion, the ratioo T/T c will decrease as n 1 / 2. Equations (3.2.1) and (3.2.5) also show that one can increasee 0 and T C /EQ by tuning \a\ to larger values or by making the tight confinement strongerr and thus decreasing IQ. Whatt happens if EQ and EF become comparable with each other, i.e. one reaches thee quasi-2d resonance for two-body collisions? Then Eq. (3.2.5) leads to T c ~ p andd is no longer valid. In fact, for 0 > F the formation of bound quasi-2d dimers of distinguishablee fermions becomes energetically favorable and one encounters the problem off Bose-Einstein condensation of these bosonic molecules. Thus, an increase of the ratio EQ/EFEQ/EF from small to large values is expected to provide a transformation of the BCS pairingg to molecular BEC. This type of crossover has been discussed in literature in 4 Forr the purely 2D case see this calculation in [96; 102; 103].

33 SectionSection 3.2. Superüuid transition in quasi-2d Fermi gases 27 7 thee context of superconductivity [104; 105; 106; 102; 97; 98; 99] and in relation to superfluidityy in 2D films of 3 He [96; 103]. The idea of using a Feshbach resonance for achievingg a superfluid transition in the BCS-BEC crossover regime in ultracold 3D Fermi gasess has been proposed in refe. [107; 108]. Wee will not consider the crossover regime and confine ourselves to the limiting case of molecularr BEC (eo» EF). A subtle question is related to the stability of the expected condensate,, which depends on the interaction between the molecules. For the repulsive interactionn one will have a stable molecular BEC, and the attractive interaction should causee a collapse. Thee molecule-molecule scattering is a 4-body problem described by the Schrödinger equation n *(r 1,r 2,R)) = 0. (3.2.6) ) Heree ri is the distance between two given distinguishable fermions, r 2 is the distance betweenn the other two, R is the distance between the centers of mass of these pairs, andd U is the interatomic potential. The total energy is E = 2e 0 + e, with e being the relativee molecule-molecule kinetic energy. Thee interaction between molecules is present only at intermolecular distances of the orderr of or smaller than the size of a molecule d* = h/y/mëö. Therefore, at energies ee -C Q the scattering between molecules is dominated by the s-wave channel and can bee analyzed on the basis of the solution of Eq. (3.2.6) for e = 0. For large R the correspondingg wavefunction is *(ri,r 2,R) «K 0 (ri/d*)ko(r2/d+)\n(ar/d*), where the decayingg Bessel function Kofa^/d*) represents the 2-body bound state. The parameter aa is a universal constant which can be found by matching the quantity \n(ar/d*) with thee solution of Eq. (3.2.6) at short distances. Finally, matching ïn(ar/d*) with the wavefunctionn of free relative motion of two molecules at distances d* -C R <C A e, where A ee = h/yjme is their de Broglie wavelength, we obtain the coupling constant (scattering amplitude)) for the interaction between molecules: gg mm = {27rh 2 /m) ln- 1 (2a 2 e" 27 oa) >! e «<*) (3-2.7) AA precise value of a is not important as it gives rise to higher order corrections in Eq.. (3.2.7). However, in order to make sure that this constant is neither anomalously largee nor anomalously small we have integrated Eq. (3.2.6) numerically. For this purpose, itt is convenient to transform Eq. (3.2.6) into an integral equation for a function which dependss only on three independent coordinates. This has been done by using the method off ref. [109]. Our calculations lead to a «1.6. They show the absence of 4-body weaklyy bound states and confirm an intuitive picture that the interaction between two moleculess can be qualitatively represented by means of a purely repulsive potential with thee range ~ d*. For the interaction between Bose-condensed dimers, in Eq. (3.2.7) one hass e = 2n m g Tn -C EQ, where n m is the density of the dimers (see Section 3.1). We thus concludee that a Bose condensate of these weakly bound dimers is stable with respect to collapse.. Thee 2D gas of bosons becomes Bose-condensed below the Kosterlitz-Thouless transitionn temperature TKT [47; 48] which depends on the interaction between particles.

34 28 8 ChapterChapter 3. Quantum degenerate quasi-2d trapped gases Accordingg to the recent quantum Monte Carlo simulations [49; 50], for the 2D gas with thee coupling constant (3.2.7) the Kosterlitz-Thouless temperature is given by TKTTKT = (nh 2 n m /m) In" 1 [fa/4*r) In (l/n m d 2 )], (3.2.8) wheree the numerical factor 77 «380. For Ep <C Q, the density of dimers n m fa n/2 and the parameterr (l/n m c^) «2TT Q/ F- Then, for Q/EF = 10, Eq. (3.2.8) gives TKT/^F ~ 0.1 andd at densities 10 8 cm -2 the transition temperature in the case of 6 Li is TKT ~ 30 nk. Thee weakly bound dimers that we are considering are molecules in the highest rovibrationall state and they can undergo collisional relaxation and decay. The relaxation process occurss in pair dimer-dimer or dimer-atom collisions. It produces diatomic molecules in deepp bound states and is accompanied by a release of the kinetic energy. The size of these deeplyy bound molecules is of the order of the characteristic radius of the interatomic potentiall R e <C IQ, and their internal properties are not influenced by the tight confinement. Therefore,, the relaxation can be treated as a 3D process and it requires the presence of att least three fermionic atoms at distances ~ R e between them. Since at least two of themm are identical, the relaxation probability acquires a small factor (kr e ) 2 compared too the case of bosons, where k ~ l/d* = y/ëöj^ö/lo is a characteristic momentum of atoms.. The 3D density of atoms in the quasi-2d geometry is ~ TI/IQ. Thus, qualitatively, thee inverse relaxation time can be written as r^1 ~ a re \n(r e /lo) 2 (eo/fk»jo)/lo 1 where a Te \ iss the relaxation rate constant for the highest rovibrational states of 3D molecules of twoo bosonic atoms. We estimate r re \ keeping in mind the recent measurements for Rb2 moleculess [110] which give a re \ «3x cm 3 /s. For lo in the interval from 10-5 to cm, the suppression factor (R e /lo) 2 ( o/fou>o) ranges from 10-3 to 10-5 and at 2D densitiess n ~ 10 8 cm -2 we find the relaxation time r re i of the order of a second or larger. Dimer-dimerr pair collisions can lead to the formation of bound trimers, accompanied byy a release of one of the atoms. The formation of deeply bound trimer states will be suppressedd at least in the same way as the relaxation process discussed above. Therefore, itt is important that there are no weakly bound trimers in (quasi-)2d. Just as in 3D [111; 112]] (see also ref. [109]), this can be established by using the zero-range approach (R e 0).. We have performed this analysis along the lines of the 3D work [109]. Qualitatively, thee symmetry of the 3-fermion system containing two identical fermions provides a strong centrifugall repulsion that does not allow the presence of 3-body bound states. This is in contrastt to 2D bosons where one has two fully symmetric trimer bound states [113]. Thus,, the life-time of quasi-2d dimers of fermionic atoms is rather long and one easily estimatess that it greatly exceeds the characteristic time of elastic collisions. One can even thinkk of achieving BEC in the initially non-condensed gas of dimers produced out of a non-superfluidd atomic Fermi gas under a decrease of n or /o- Inn conclusion, we have found the temperature of superfluid phase transition in twocomponentt quasi-2d Fermi gases. Our results are promising for achieving this transition inn both the regime of BCS pairing and the regime of BEC of weakly bound dimers.

35 Chapterr 4 Interatomic collisions in a tightlyy confined Bose gas WeWe discuss pair interatomic collisions in a Bose gas tightly confined in one (axial) directionrection and identify two regimes of scattering. In the quasi-2d regime, where the confinementfinement frequency UJQ greatly exceeds the gas temperature T, the scattering rates exhibit 2D2D features of the particle motion. At temperatures T ~ hwq one has a confinementdominateddominated 3D regime, where the confinement can change the momentum dependence of thethe scattering amplitudes. We describe the collision-induced energy exchange between the axialaxial and radial degrees of freedom and analyze recent experiments on thermalization and spinspin relaxation rates in a tightly (axially) confined gas of Cs atoms Introduction Collisionall properties of ultra-cold gases strongly confined in one direction attract a great deall of interest since the start of active studies of spin-polarized atomic hydrogen. In thee latter case the interest was related to recombination and spin relaxation collisions andd to elastic scattering in the (quasi-)2d gas of atomic hydrogen adsorbed on liquid Hee surface (see [70] for review). The discovery of Bose-Einstein condensation in trapped alkali-atomm clouds [3; 4; 5] stimulated a progress in evaporative and optical cooling and inn trapping of neutral atoms. Present facilities make it possible to (tightly) confine the motionn of particles in one direction to zero point oscillations. Then, kinematically the gass is 2D, and the only difference from the purely 2D case is related to the value of the interparticlee interaction which now depends on the tight confinement. Thus, one now hass many more opportunities to create (quasi-) 2D gases. In the recent experiments with opticallyy trapped Cs [73; 27; 74; 75; 29; 30] about 90% of atoms are accumulated in the groundd state of the harmonic oscillator potential in the direction of the tight confinement. Inn this chapter we consider a Bose gas tightly confined in one (axial) direction and discusss how the axial confinement manifests itself in pair elastic and inelastic collisions. Wee identify two regimes of scattering. At temperatures T -C TUVQ (^o is the axial frequency)) only the ground state of the axial harmonic oscillator is occupied, and one has aa quasi-2d regime. In this case, the 2D character of the relative motion of particles at largee separation between them, manifests itself in a logarithmic energy dependence of thee scattering amplitude. For a negative 3D scattering length a, we observe resonances inn the dependence of the elastic scattering rate on a. This is quite different from the 3DD case where the scattering rate always increases with a 2. The presence of these resonancess in quasi-2d follows from the analysis given in Section 3.1 and finds its origin in increasingg role of the 2D kinematics of the particle motion with increasing ratio a //o, wheree IQ = (h/mujo) 1^2 is the axial extension of the atom wavefunction, and m the atom mass

36 30 0 ChapterChapter 4. Interatomic collisions in a tightly confined Bose gas Att temperatures T ~ HLÜQ we have a confinement-dominated 3D regime of scattering,, where the 2D character of the particle motion is no longer pronounced in the scatteringg process, but the axial confinement can strongly influence the energy (temperature)) dependence of the scattering rate. Treating collisions as three-dimensional, thee wavevector p of the relative motion of colliding atoms does not decrease with T. Thee atoms undergo zero-point oscillations in the axial direction and this corresponds too p ~ 1/Zo- If the 3D scattering amplitude is momentum-dependent at these p, which iss the case for \a\ > Zo, then the temperature dependence of the elastic collisional rate becomess much weaker. This means that for a large 3D scattering length the tight axial confinementt suppresses a resonant enhancement of the collisional rate at low energies. Inn many of the current experiments with ultra-cold gases one tunes o to large positivee or negative values by varying the magnetic field and achieving Feshbach resonances [114;; 115; 116; 82; 117; 28; 118; 119; 120]. In the unitarity limit (\a\ -> oo) the 3D elastic crosss section is a = Sir/p 2 and the rate of 3D elastic collisions strongly increases with decreasingg temperature. The tight confinement of the axial motion makes the scattering ratee practically temperature independent at T ~ fajq. We obtain a similar suppression off resonances for inelastic collisions, where the resonant temperature dependence in 3D iss related to the energy dependence of the initial wavefunction of colliding atoms. We analyzee the Stanford and ENS experiments on elastic [27; 29; 30] and spin relaxation [27]] collisions in a tightly axially confined gas of cesium atoms and discuss the origin of significantt deviations of the observed collisional rates from the 3D behavior. Wee develop a theory to describe the collision-induced energy exchange between axiall and radial degrees of freedom of the particle motion. We establish selection rules forr transitions between particle states in the axial harmonic potential and calculate the correspondingg transition amplitudes. This allows us to consider temperatures T > hu>o andd analyze thermalization rates in non-equilibrium clouds. In the Stanford and ENS experimentss these clouds were created by means of degenerate Raman sideband cooling [73;; 27; 74; 75; 29; 30] which effectively leads to a gas with different axial (T z ) and radial (X p )) temperatures. After the cooling is switched off, the temperatures T z and T p start too approach each other, and ultimately the gas reaches the equilibrium temperature. At sufficientlyy low T only a few axial states are occupied and the temperature dependence off the corresponding thermalization rates should deviate from the 3D behavior, thus exhibitingg the influence of the axial confinement on the scattering process. We calculatee the thermalization rates and establish the conditions under which this influence is pronounced.. Thee minimum energy exchange between the radial and axial degrees of freedom of twoo colliding atoms is equal to 2hu>o- This follows from the symmetry of the interatomic potentiall with respect to simultaneous inversion of the axial coordinates of the two atoms, whichh ensures the conservation of parity of their wavefunction under this operation. Accordingly,, the sum of two (axial) vibrational quantum numbers can be changed only by ann even value. The rate of energy transfer from the radial to axial motion is proportional too the difference between the radial and axial temperatures AT = T p T z, if they are closee to each other. As the total energy of colliding particles should exceed 2hwo in order too enable the energy transfer, the rate of this process at temperatures T < hujq becomes exponentiallyy small: AE oc ATexp( 2fkoo/T). Due to the presence of the energy gap

37 SectionSection D scattering problem 31 1 HWQHWQ in the excitation spectrum of the axial harmonic oscillator, the heat capacity of the axiall degree of freedom is de z /dt z ~ exp( hu)o/t), which leads to a thermalization ratee AT/AT oc exp( TULOQ/T). This exponential temperature dependence shows that the thermalizationn is suppressed at very low temperatures. One can deeply cool the axial motion,, but radially the cloud remains "hot" on a very long time scale D scattering problem First,, we discuss the purely 2D elastic scattering in pair collisions of ultra-cold atoms interactingg via a short-range potential U{p). At interparticle distances p -* oo the wavefunctionn of colliding atoms is represented as a superposition of the incident plane wavee and scattered circular wave [36]: iftp)iftp) «e*t> - f(q, 0) J^-/ qp - (4-2.1) Thee quantity f(q,<p) is the scattering amplitude, q is the relative momentum of the atoms,, and <j> the scattering angle. Note that f(q,<j>) in Eq. (4.2.1) differs by a factor of y/8irq y/8irq from the 2D scattering amplitude defined in [36]. Similarlyy to the 3D case, the scattering amplitude is governed by the contribution of thee s-wave scattering if the relative momentum q satisfies the inequality qr e «C 1, where ReRe is the characteristic radius of interaction. In the case of alkali atoms, the radius R e iss determined by the Van der Waals tail of the potential U(p) and ranges from 20 A for Lii to 100 A for Cs. The s-wave scattering amplitude is independent of the scattering anglee <f>. The probability a(q) for a scattered particle to pass through a circle of radius pp per unit time is equal to the intensity of the scattered wave multiplied by 2-Kpv, where vv = 2Hq/m is the relative velocity of colliding atoms. From Eq. (4.2.1) we have "(S)) = ^!/(<?) 2. (4.2.2) Thee velocity v is equal to the current density in the incident wave of Eq. (4.2.1). The ratioo of ct(q) to this quantity is the 2D cross section which has the dimension of length: {q) {q) = I/(9)I74<7. (4.2.3) Forr the case of identical bosons Eqs. (4.2.2) and (4.2.3) have an extra factor 2 in the rhs. Thee quantity a(q) is nothing else than the rate constant of elastic collisions at a givenn q. The average of ct(q) over the momentum distribution of atoms, multiplied by thee number of pairs of atoms in a unit area, gives the number of scattering events in this areaa per unit time. Forr finding the s-wave scattering amplitude one has to solve the Schrödinger equation forr the s-wave of the relative motion of colliding atoms at energy e = h 2 q 2 /m: *\*\.,J.w. ^»V A,, + U(p)] j,,(q,p) = ^-^.(9,P)- (4-2.4) mm J m

38 32 2 Chapterr 4. Interatomic collisions in a tightly confined Bose gas Att distances p^> R e the relative motion is free and one can omit the interaction between atoms.. Then the solution of Eq. (4.2.4), which for qp» 1 gives the partial s-wave of ip(p)ip(p) (4.2.1), takes the form Mq,P)Mq,P) = Mqp)-^Ho(qp), p<# e, (4.2.5) wheree Jo and HQ are the Bessel and Hankel functions. Onn the other hand, at distances p <SC 1/q one can omit the relative energy of particles inn Eq. (4.2.4). The resulting (zero energy) solution depends on the momentum q only throughh a normalization coefficient. In the interval of distances where R e <C p -C 1/q, thee motion is free and this solution becomes ip a oc \n(p/d), where d > 0 is a characteristic lengthh that depends on a detailed shape of the potential U(p) and has to be found fromfrom the exact solution of Eq. (4.2.4) with q = 0. This logarithmic expression serves ass a boundary condition for ij> s (q,p) (4.2.5) at qp -C 1, which immediately leads to the scatteringg amplitude [36] 2TT 2TT f{q)f{q) = ln(l/ g d.) + W2' (4-2 ' 6) wheree cf* = (d/2) expc and C «0.577 is the Euler constant. Itt is important to mention that the condition qr e <C 1 is sufficient for the validity off Eq. (4.2.6). This equation also holds for the case of resonance scattering, where the potentiall U(p) supports a weakly bound s-level. In this case the spatial shape of i> a (q } p) att distances where R e <C p <C 1/g, is the same as the shape of the wavefunction of the weaklyy bound state. This gives d* = h/^m Q, where o is the binding energy. We thus havee the inequality d*» R e, and the quantity qd* in Eq. (4.2.6) can be both small and large.. The rate constant a(q) peaks at q = 1/d* and decreases as 1/[1 + (4/7T 2 ) In (qd*)] withh increasing or decreasing q. Note that the 2D resonance is actually a resonance in thee logarithmic scale of energies. The decrease of a by factor 2 from its maximum value requiress a change of energy e = h 2 q 2 /m by factor 20. Forr qd* <?; 1 one may omit the imaginary part in Eq. (4.2.6), and the scattering amplitudee becomes real and positive 1. The positive sign of ƒ (q) has a crucial consequence forr the mean-field interparticle interaction in purely 2D Bose gases. In the ultra-cold limit wheree qr e <C 1, the scattering amplitude is related to the energy of interaction in a pair off particles (coupling constant g). For a short-range potential U(p), the energy of the mean-fieldd interaction in a weakly interacting gas is the sum of all pair interactions. In a uniformm Bose-condensed gas the coupling constant g for condensate atoms is equal to the amplitudee of scattering (with an extra factor h 2 /m for our definition of ƒ) at the energy off the relative motion e = h 2 q 2 /m = 2//, where p is the chemical potential 2. Hence, we ^orr interatomic potentials which have a shallow well (spin-polarized atomic hydrogen), the bound statee with an exponentially small binding energy eo, characteristic for shallow 2D potential wells [36], doess not exist due to the presence of a strong repulsive core [121]. For potentials with a deep well (alkali atoms)) the situation is similar to that in 3D: There are many bound states and only accidentally one can encounterr a very weakly bound s-level (d m oo). Thus, for realistic momenta of particles in ultracold gasess we always have the inequality qd* < ; 1. 2 Thiss is easily established by comparing the relation for the vertex of elastic interaction, obtained to zeroo order in perturbation theory [77], with the expression for the scattering amplitude [36]. In the case off an inhomogeneous density profile one should replace /x by nog, where no is the coordinate-dependent condensatee density.

39 SectionSection 4.3. Scattering in axially confined geometries. General approach 33 have e hh,.. 2wh 1 n,,. _. A<1, (427) ^m^^-^-rït^^11 «wheree <jr c = y/2mp,fh is the inverse healing length. In a dilute thermal 2D gas, due to thee logarithmic dependence of ƒ on q, the thermal average of the mean-field interaction leadss to the coupling constant g = (h 2 /m)f(qt), where qr = y/mt/h is the thermal momentumm of particles. At sufficiently low temperatures, where qrd* < [ 1, we again havee g > 0. Thus,, in an ultra-cold purely 2D gas the coupling constant for the mean-field interactionn is always positive in the dilute limit and, hence, the interaction is repulsive. This strikingg difference from the 3D case is a consequence of the 2D kinematics. For low energies,, at interparticle distances p^> R e, the (free) relative motion of a pair of atoms is governedd by the wavefunction ip a oc In (p/d). The probability density \ip 9 \ 2 of finding two atomss at a given separation increases with p as the condition p > d is always reached, unlesss the atoms have a bound state with energy e 0 (d oo). This means that it is favorablee for particles to be at larger p, i.e. they repel each other Scattering in axially confined geometries. Generall approach Inn this section we discuss elastic scattering of atoms (tightly) confined in the axial (z) direction,, assuming that the motion in two other (x, y) directions is free. We analyze how thee scattering is influenced by the confinement and calculate a complete set of scattering amplitudess corresponding to collision-induced transitions between particle states in the confiningg potential. We still call this scattering elastic as the internal states of atoms are nott changing. Forr a harmonic axial confinement, the motion of two atoms interacting with each otherr via the potential V(r) can be still separated into their relative and center-of-mass motion.. The latter drops out of the scattering problem. The relative motion is governed byy the potential V(r) and by the potential VH(Z) = rmc\z 2 fa. originating from the axial confinementt with frequency wo- For the incident wave characterized by the wavevector q off the motion in the x, y plane and by the quantum number v of the state in the potential VH(Z),VH(Z), the wavefunction of the relative motion satisfies the Schrödinger equation 7711 Z </>(r)) = ev(r), (4-3.1) wheree e = h 2 q 2 /m + UHUJQ. Thee scattering depends crucially on the relation between the radius of interatomic interactionn R e and the characteristic de Broglie wavelength of particles A e. The latter is introducedd qualitatively, as the motion along the z axis is tightly confined. Accounting forr the zero point axial oscillations one can write A e ~ h/vmi, with ë = + hu)o/2. We willl consider the ultra-cold limit where AA > R e - (4.3.2) )

40 34 4 ChapterChapter 4. Interatomic collisions in a tightly confined Bose gas Eq.. (4.3.2) immediately leads to the inequality qr e <C 1, as the de Broglie wavelength forr the motion in the x,y plane is ~ 1/q. For small v the harmonic oscillator length Joo = (fr/mojo) 1 / 2 plays the role of the axial de Broglie wavelength of atoms. Therefore, thee ultra-cold limit (4.3.2) also requires the condition IQ» R e. For large u, the axial dee Broglie wavelength is ~ l l\fv and, according to Eq. (4.3.2), this quantity should be muchh larger than R e. Underr the condition qr e <C 1, the scattering amplitudes are determined by the contributionn of the s-wave for the motion in the x, y plane. In the case of identical bosons, the s-wavee scattering requires even values of v and u' as the wavefunction ip should conserve itss sign under the transformation z + z. The quantum numbers v and v' should be evenn also for distinguishable particles. Otherwise at distances of interatomic interaction, rr < R e, the wavefunction ip will be small at least as R e /lo, ensuring the presence of this smalll parameter in the expressions for the scattering amplitudes. Thee scattering amplitudes corresponding to transitions from the initial state v (of thee relative motion in the potential VH{Z)) to final states v' are defined through the asymptoticc form of the wavefunction ip at an infinite separation p in the x, y plane: ikr)) «<p v {z)e^p - f»ae)vaz)\hr^ e iq ''' P, (4-3.3) wheree <Pv(z) and <p u >{z) are the (real) eigenfunctions of the states v and v''. For each off the scattered circular waves the value of the momentum q v > follows from the energy conservationn law fr 2 ql,/m = e tiwqv' > 0. Relyingg on the condition (4.3.2), we develop a method that allows us to express thee scattering amplitudes through the 3D scattering length. At interparticle distances rr» R e the relative motion in the x, y plane is free, and the motion along the z axis is governedd only by the harmonic oscillator potential VH(Z). Then, the solution of Eq. (4.3.1) withh V(r) = 0 can be expressed through the Green function G (r, r') of this equation. Retainingg only the s-wave for the motion in the x,y plane, we have V>(r)) = tpmmw) + A,G E (r,0), (4.3.4) andd the expression for the Green function G e (r, 0) reads GG (r,(r, 0) = > yv (z)<p V i (0) x I (4.3.5) [[ K 0 (\q v '\p)/27r;ql,<0 Heree the summation is also performed over closed scattering channels for which q 2,, < 0. Thee function KQ{X) = (i7r/2)ho(ix) and it decays as y/n/2x exp( x) at x» 1. Thus, forr p * oo the terms corresponding to the closed channels vanish. Then, comparing Eq.. (4.3.4) at p * oo with Eq. (4.3.3), we find a relation between the scattering amplitudess and the coefficients A v \ wheree 0 is the step function. jj vvvv >> = -A v ip v, (O)0(e - hw 0 v f ), (4.3.6)

41 Sectionn 4.3. Scattering in axially confined geometries. General approach 35 Thee condition IQ ^> R e ensures that the relative motion of atoms in the region of interatomicc interaction is not influenced by the axial (tight) confinement. Therefore, thee wavefunction V'(r) in the interval of distances where Re <^ r <C A e, differs only by a normalizationn coefficient from the 3D wavefunction of free motion at zero energy. Writing thiss coefficient as yv(0)i7, we have i){r)i){r)««mom««mom 11 - a M- ( Eq.. (4.3.7) contains the 3D scattering length a and serves as a boundary condition for ip(r)ip(r) (4.3.4) at r - 0. Forr r» 0, a straightforward calculation of the sum in Eq. (4.3.5) yields wheree the complex function w(x) is given by w(x)w(x) = Um JV V /ÏV,, JV A(2j-1)!!,,. _ && W (4.3.9) ) Withh the Green function (4.3.8), the wavefunction (4.3.4) at r 0 should coincide with ip(r)ip(r) (4.3.7). This gives the coefficient 777 = =i (4.3.10) 11 + (a/v2^l 0 )w{ /2Hw 0 ) andd provides us with the values of the coefficients A v. Then, using Eq. (4.3.6) and explicitt expressions < v(0) = (1/27T/Q) 1 / 4 (I/ l)\\/y/ïa, we immediately obtain the scattering amplitudee /oo(^) and express all other scattering amplitudes through this quantity: foo(e)foo(e) = 47r^(0)ar? = 2^j* (4.3.11) where e ff vvvv.{e).{e) = P >foo( W( - hjj Q v)6{e - huov'), (4.3.12) jmj^=(y -W-i)ii > > V?(0)) vwl Onee can see from Eqs. (4.3.11) and (4.3.12) that for any transition v i/ the scatteringg amplitude is a universal function of the parameters a/lo and e/hwo- The quantityy P vv > in Eq. (4.3.12) is nothing else than the relative probability amplitude off having an axial interparticle separation \z\ <C Zo ( m particular, \z\ < R e ) for both incomingg (i/) and outgoing (i/) channels of the scattering process. It is thus sufficient to studyy only the behavior of foo(e). Wee emphasize the presence of two distinct regimes of scattering. The first one, which wee call quasi-2d, requires relative energies e <CfojjQ. In this case, the relative motion of particless is confined to zero point oscillations in the axial direction, and the 2D kinematics

42 366 Chapter 4. Interatomic collisions in a tightly confined Bose gas off the relative motion at interatomic distances p> IQ should manifest itself in the dependencee of the scattering amplitude on e/2hwo and CL/IQ. In the other regime, at energies alreadyy comparable with HLJQ, the 2D kinematics is no longer pronounced in the scatteringg process. Nevertheless, the latter is still influenced by the (tight) axial confinement. Qualitatively,, the scattering amplitudes become three-dimensional, with a momentum ~~ 1/Zo related to the quantum character of the axial motion. Thus, we can say that this iss a confinement-dominated 3D regime of scattering. With increasing the relative energy too e» hwo, the momentum is increasing to y/me/h and the confinement-dominated 3D regimee continuously transforms to ordinary 3D scattering Quasi-2D regime Inn the quasi-2d regime, due to the condition e «C huo, the incident and scattered waves havee quantum numbers u = v' = 0 for the motion in the axial harmonic potential VH(Z). Thee relative energy = h 2 q 2 /m and the inequality qlo <C 1 is satisfied. In this case Eq.. (4.3.9) gives W(E/2FUJQ)W(E/2FUJQ) = In (Bhuo/ne) + wr, (4-4. wheree B « Then our equation (4.3.11) recovers Eq. (3.1.11) of Section 3.1, obtainedd in this limit (In Section 3.1 the coefficient B was put equal to unity). Usingg Eq. (4.4.1) we can represent /oo(e) (4.3.11) in the 2D form (4.2.6), with dd mm = (d/2)expc = ^/r^jbloexp{-y/^^/2lo/a). (4.4.2) Thiss fact has a physical explanation. Relying on the same arguments as in the purely 2D case,, one finds that in the interval of distances where l 0 <C p <C 1/q, the wavefunction ipip ex <po(z) In (p/d). On the other hand, for p > IQ we have i/>(r) = <po(z)ip a (p), where ip s iss given by the 2D expression (4.2.5) with f(q) = foo{ ). This follows from Eqs. (4.3.4)- (4.3.6),, as all closed scattering channels [y' ^ 0) in Eq. (4.3.5) for the Green function G e (r,0)) have momenta \q v '\ ^ l/^o and will be exponentially suppressed at p ^$> IQ. Matchingg the two expressions for the wavefunction ip one immediately obtains the 2D equationn (4.2.6). However, the parameter d* (4.4.2) can be found only from the solution off the quasi-2d scattering problem. Wee thus conclude that the scattering problem in the quasi-2d regime is equivalent too the scattering in an effective purely 2D potential which leads to the same value of d*.. For positive a <C Zo, this potential can be viewed as a (low) barrier, with a height Voo ~ h 2 a/mlq and radius IQ. Hence, in the case of positive a we have a small (positive) scatteringg amplitude, in accordance with Eqs. (4.2.6) and (4.4.2). For a negative a satisfyingg the condition \a\ -C IQ, the effective potential is a shallow well which has a depthh Vol and radius fo- This shallow well supports a weakly bound state with an exponentiallyy small binding energy EQ, which leads to an exponentially large d* as follows fromfrom Eq. (4.4.2). As a result, we have a resonance energy dependence of the scattering amplitudee /bo at a fixed ratio a/fo, and a resonance behavior of /oo as a function of CL/IQ att a fixed /hu)q. Thee resonance in the energy dependence of /oo is quite similar to the logarithmic-scale resonancee in the purely 2D case, discussed in Section 4.2. The quasi-2d resonance is also

43 SectionSection 4.4. Quasi-2D regime a"<r^/ a"<r^/ 10-- // all =~ y \L \L <r<r o * *.'*.' al\=\ ^^ * TTTTll 1 r " " 10* * e/2ft<o 10" 1 1 Figuree 4.4.1: The function / 0 o 2 versus energy for a/lo (dashedd curve) and a/lo = 1 (dotted curve). -11 (solid curve), a/lo = oo describedd by Eq. (4.2.6), where the length d* is now given by Eq. (4.4.2). As expected, thee dependence of /oo on e is smooth. Thee resonance in the dependence of the quasi-2d scattering amplitude on a/lo nas beenn found and discussed in Section 3.1. Relying on the above introduced effective 2D potentiall for the quasi-2d scattering, we can now explain this resonance on the same groundss as the resonance in the energy dependence of /oo- We will do this in terms of thee relative energy e and the binding energy in the effective potential, e 0 = h 2 /mdl oc exp(- N /7r72i 0 / a )-- For e/e 0 = (qd*) 2» 1, the scattering amplitude in Eq. (4.2.6) is real andd negative. It increases in magnitude with decreasing ratio e/e 0, that is with decreasing qq or /o- In the opposite limit, where e/eo <C 1, the scattering amplitude is real and positive andd it increases with the ratio e/eo. The region of energies e/e 0 ~ 1 corresponds to the resonance,, where both the real and imaginary parts of /oo are important. The real part reachess its maximum at e/e 0 = exp(-7r), drops to zero at e/e 0 = 1, and acquires the maximumm negative value for e/e 0 = expir. The dependence of Im/ 00 on e/e 0 is the samee as that of the quantity /oo 2 - Both of them peak at e/e 0 = 1 and decrease with increasingg or decreasing e/eo- Qualitatively,, the picture remains the same for \a\ ~ lo- In Fig we present the dependencee of / 0 o 2 on e/2hu>o at a/lo equal to -1, 1, and oo. In the two last cases we alwayss have e/e 0 <T 1, and / 0 o 2 increases with e at e <T hu> 0 - For a/l 0 = -1 we have thee above described logarithmic-scale resonance in the behavior of /oo 2 - Thee quasi-2d resonance is much more pronounced in the dependence of the scattering amplitudee on the parameter a/l 0. The reason is that / 0 o logarithmically depends on the particlee energy, whereas the dependence on a/l 0 is a power law. For e <?C hu>o, Eq. (4.3.11) yields s 16TT 2 2 oo o (y/2wlo/a(y/2wlo/a + In (Bhujo/ne)) 2 + TT 2 ' l/< < (4.4.3) )

44 38 8 ChapterChapter 4. Interatomic collisions in a tightly confined Bose gas IIII >< r! r! a a s.. ^ ^ 0.06 e/fico 0 = i a<0 0 a) ) lal// / Figuree 4.4.2: The function / 00 2 versus \a\/l 0 at various energies for a < 0 (a) and a > 0 (b).. Thee quantity / 0 o 2 differs only by a factor of fi/m from the rate constant of elastic collisionss (see Eq. (4.2.2)), and one can think of observing the resonance dependence off /oo 2 on a/lo in an experiment. For example, one can keep e (temperature) and uu 00 constant and vary a by using Feshbach resonances. The resonance is achieved at aa = v2nl 0 /ln(bhujo/ire). This is a striking difference from the 3D case, where the crosss section and rate constant of elastic collisions monotonically increase with a 2. In Fig we present / 00 2 versus a/l 0 at a fixed e/hcoo- In order to extend the results too the region of energies where the validity of the quasi-2d approach is questionable, the quantityy / 00 2 was calculated by using Eq. (4.3.11) for the scattering amplitude. The resonancee is still visible at e/fouo = 0.06 and it disappears for e/hu) 0 = 0.2. Thee obtained results allow us to conclude that for \a\ > l 0 the approximate border line betweenn the quasi-2d and confinement-dominated 3D regimes is e sa e» = 0.1fko 0. For \a\\a\ <C /o, as we will see below, the confinement-dominated regime is practically absent. Thee output of kinetic studies in thermal gases is usually related to the mean collisional frequencyy (the rate of interatomic collisions) Q = an, where a is the mean rate constant off elastic collisions, and n the gas density. In the quasi-2d regime, the rate constant a

45 SectionSection 4.5. Confinement-dominated 3D regime 39 9 followss directly from Eq. (4.2.2), with twice as large rhs for identical bosons: öö = -< /oo 2 >, (4-4-4) wheree the symbol {) stands for the thermal average. Our numerical calculations show thatt the average of /oo 2 over the Boltzmann distribution of particles only slightly broadenss the resonances in Fig and Fig Due to the logarithmic dependence of /ooo on the relative energy, the thermal average is obtained with a good accuracy if one simplyy replaces e by the gas temperature T. Thus, in order to observe the manifestation off the 2D features of the particle motion in their collisional rates one has to achieve very loww temperatures T < O.lhujQ Confinement-dominated 3D regime Inn the confinement-dominated 3D regime, where e ~ hw 0, the axial confinement influences thee scattering process through the confined character of the axial motion. In order to analyzee this influence, we first examine the function w(e/2hu)o) which determines the energyy dependence of the scattering amplitudes. The imaginary part of w(x), following fromm Eq. (4.3.9), is equal to TT ^ V ^ ~ 1 )" n /-r([x]+3/2) lmw{x)lmw{x) =TT^ {2j)}} = Isfn ^, (4.5.1) wheree [x] is the integer part of x. The function lmw(x) has a step-wise behavior as shown inn Fig It is constant at non-integer x and undergoes a jump at each integer x, takingg a larger value for larger x. With increasing x, the jumps become smaller and for xx» 1 we have Imu;(x) «2y/nx. The real part of w(x) was calculated numerically from Eq.. (4.3.9) and is also given in Fig At any x we have Reiu(x) < 1, except for narroww intervals in the vicinity of integer x. In each of these intervals the function Reiy (x) logarithmicallyy goes to infinity as x approaches the corresponding integer value. This is consistentt with the step-wise behavior of lmw(x): As one can see directly from Eq. (4.3.9), forr x approaching an integer j the analytical complex function w oc In (j x io). Thee described behavior of the function w(e/2hu;o) has a direct influence on the scatteringg amplitudes. For s/2hujo close to an integer j, the amplitude is small and it is equall to zero for = 2hu>oj. This phenomenon originates from the fact that for e close too 2hu>oj, a new scattering channel opens (really or virtually). For this channel the momentumm \q v r\ = yjm\ - 2hu 0 j\/h is very small. Hence, at distances p < \l/q*'\ the wavefunctionn ip (4.3.4) will be determined by the contribution of this low-momentum termm if p > l 0. This is clearly seen from Eqs. (4.3.5) and (4.3.4) and makes the situation somewhatt similar to that in the quasi-2d regime of scattering. In the latter case, the wavefunctionn if) (4.3.4) at distances p <C \jq is also determined by the contribution of thee low-momentum channel as long as p > / 0 - Then, as follows from the analysis in Sectionn 4.4, this wavefunction and the scattering amplitude /oo behave as l/ln(^j 0 /e) inn the limit e 0. In the present case, the wavefunction if; and the scattering amplitudes aree small as 1/ In (huo/\ - 2hu 0 j\) for e > 2hw 0 j.

46 40 0 ChapterChapter 4. Interatomic collisions in a tightly confined Bose gas Figuree 4.5.1: The functions Reu;(:r) (solid curve) and Imw(x) (dashed lines). The dotted curvee shows the function 2^pnx corresponding to the asymptotic behavior of Imw at large x. x. Thee energy dependence of /oo 2 on s/2hcj 0 for a/l 0 equal to 1, 1, and oo is displayedd in Fig Outside narrow energy intervals in the vicinity of integer e/2hu>o, thee quantity /oo 2 is a smooth function of e. One can also see a sort of a step-wise decreasee of /oo 2 with increasing e, originating from the step-wise increase of the function lmw(e/2hujo).lmw(e/2hujo). For e/2fku 0 close to integer values j > 0 we find a fine structure si inn nature to the behavior of /oo 2 at e < Ku> a. For a/l 0 = 1 and a/l 0 = oo there are narroww dips corresponding to the logarithmic decrease of / 00 2 as > 2hLj 0 j, and for aa /lo/lo = 1 these dips are accompanied by resonances. Note that the thermal distribution off particles averages out this fine structure, and the latter will not be pronounced in kineticc properties. Thee difference between the confinement-dominated 3D regime and the ordinary 3D regimee of scattering will manifest itself in the rate of elastic collisions (mean collisional frequencyy 0). For the Boltzmann distribution of particles, one can find this quantity by turningg to the thermal distribution for the relative motion of colliding partners. Collisioninducedd transitions between the states of the relative motion in the axial potential VH(Z) aree described by the rate constants aa vvvv,{e),{e) = (h/m)\f uv,(e)\ 2, (4.5.2) wheree the scattering amplitudes f uv * are given by Eqs. (4.3.11) and (4.3.12), and an extra factorr 2 for identical bosons is taken into account. The collisional frequency Q. = an, wheree n is the (2D) density, and the mean rate constant of elastic collisions, a, is obtained byy averaging a. vv > (4.5.2) over the thermal distribution of relative energies e and by makingg the summation over all possible scattering channels. We thus have nn = ön = ^/^^a^( ) J 4exp(-^). (4.5.3) vv' vv' Heree A T = (27rh 2 /mt) 1 / 2 is the thermal de Broglie wavelength, s = h 2 q 2 /m + HU}QV,

47 SectionSection 4.5. Confinement-dominated 3D regime a// 0 =-1 1 a) ) 5--,, // *" ~ ~ ~" x a/i =~ jj a// =1, i V V ## * b) ) zl2hcozl2hco n n Figuree 4.5.2: The function /oo 2 versus energy. In (a) the parameter a/lo = 1. In (b) thee dashed curve corresponds to the unitarity limit (a/lo = oo), and the dotted curve to a/la/l 00 = 1- andd the quantum numbers v and v' take only even values. The distribution function overr v and q is normalized to unity. The normalization coefficient A is obtained by the summationn over both even and odd v and is equal to AA = 2(1 - exp (-fiwo/ï 1 )). (4.5.4) ) Notee that the number of collisions per unit time and unit surface area in the x, y plane iss equal to an 2 /2 = ftn/2. Thee manifestation of the tight axial confinement of the particle motion in collisional ratess depends on the relation between the scattering length a and the characteristic dee Broglie wavelength A ~ h/y/m(e + hwo/2) accounting for the zero point axial oscillations.. For the scattering length satisfying the condition \a\ <C A e, the scattering amplitudess are energy independent at any e, except for extremely small energies in thee quasi-2d regime. This follows directly from Eqs. (4.3.10)-(4.3.12). The condition \a\\a\ < A automatically leads to the inequalities \a\ <C h and \a\ < h/sfrnz. Hence, thee function w(e/2hujo) is much smaller than lo/\a\, unless e < /k^oexp(-z 0 / a ) (see Eq.. (4.4.1) and Fig ). Accordingly, Eq. (4.3.10) gives r\ = 1 and Eqs. (4.3.11), (4.3.12)) lead to the scattering amplitudes fn,.fn,. = 4^wp v {H)tp v >{0)e{e - hu o v)0(e - fiwoi/). (4.5.5)

48 42 2 ChapterChapter 4. Interatomic collisions in a tightly congned Bose gas Thee amplitudes (4.5.5) are nothing else than the 3D scattering amplitude averaged over thee axial distribution of particles in the incoming {v) and outgoing (v') scattering channels.. Prom Eqs. (4.5.2) and (4.5.5) one obtains the same rate of transitions v v' as inn the case of 3D scattering of particles harmonically confined in the axial direction and interactingg with each other via the potential V(r). This is what one should expect, since underr the condition \a\ <C A e the amplitude of 3D scattering is momentum independent. Thee integration over d 2 q in Eq. (4.5.3) leads to the mean collisional frequency = j(* )^Erf<0)( (0)«P (-53W{ y}y Thus,, in the case where \a\ < AT, the tight confinement in the axial direction can manifestt itself in the collisional rates only through the axial distribution of particles and thee discrete structure of quantum levels in the axial confining potential. The expression forr the collisional frequency ft can be reduced to the form 87rhn 87rhn m m (Q\,(Q\, (4.5.6) wheree the coefficient ranges from 1 at T <C hw$ to 2/n for T > huj 0. The condition \a\\a\ <C A T is equivalent to \a\ < Z 0 and q T \a\ < 1, where q T = y/mt/h is the thermal momentumm of particles. For T > hw 0, Eq. (4.5.6) gives the collisional frequency which coincidess with the three-dimensional result averaged over the classical Boltzmann profile off the 3D density in the axial direction, UB{Z)'. fifi 33 DD = (CTSDV) f ÏB&dz. (4.5.7) Heree a is the 3D elastic cross section, and v is the relative velocity of colliding particles. In otherr words, the quantity (l/2)an 2 (l/2)firc coincides with the number of 3D collisions perr unit time and unit surface area in the x, y plane, given by (l/2)(asdv) Jn 2 B{z)dz. Fromm Eq. (4.5.6) we conclude that for \a\ -C l 0 the confinement-dominated 3D regime off scattering is not pronounced. At temperatures T < HUJQ the collisional rate only slightlyy deviates from the ordinary 3D behavior. This has a simple physical explanation. Forr \a\ "C AT, treating collisions as three-dimensional we have 2 ~ 87ra 2 i;n3 >- At low temperaturess T < hwo the velocity v ~ h/mlo and the 3D density is n^d ~ n/fo- For TT > fot>o we have v ~ [T/m) 1 / 2 and n 3 > ~ n(rrio; /r) 1/2. In both cases the "flux" VTISDVTISD ~ won, and there is only a small numerical difference between the low-t and high-tt collisional frequencies. Thee ultra-cold limit (4.3.2) assumes that the characteristic radius of interatomic interactionn R e < ; IQ. Therefore, the condition \a\ <tc IQ is always satisfied, unless the scatteringg length is anomalously large (\a\» R e ). Below we will focus our attention on thiss case, assuming that \a\ > IQ. Lett us first show how the 3D result follows from our analysis at T» hwq, irrespective off the relation between a and A e. At these temperatures the main contribution to the sum inn Eq. (4.5.3) comes from e» hwo and large u and u'. Accordingly, we can replace the summationn over v and v' by integration. At energies much larger than HWQ the quantity

49 SectionSection 4.5. Confinement-dominated 3D regime 43 3 y/vfhy/vfh is nothing else than the axial momentum k z and we have e = h 2 (q 2 + k 2 )/m. Forr these energies the function w(e/2huj 0 ) in Eq. (4.3.11) takes its asymptotic form ww w iy/2-ire/hujo. Using Eq. (4.3.12), this immediately allows us to write 87rq22 _ P2 ^., \fvv'\fvv' I P vv > i2/i i ^2. 2\ ~ ^» wheree p = yjq 2 + A; 2 is the 3D momentum of the relative motion, and <J 3 D = 87ra 2 /(l + pp 22 aa 22 )) is the cross section for the 3D elastic scattering. For large v and i/, Eq. (4.3.13) givess P vv > = (4/7r 2 i/i/ / ) 1 / 4, and the integration over v' in Eq. (4.5.3) multiplies <T 3 D by thee relative 3D velocity v. Then, turning from the integration over v to the integration overr the axial momentum, we reduce Eq. (4.5.3) to 0-/^ ( w *-p(-3).. (4.,8) andd one can easily check that Eq. (4.5.8) coincides with the three-dimensional result 1$D (4.5.7).. Inn the limiting case, where the thermal momentum of particles satisfies the inequality qr\o\qr\o\ > 1, we obtain nn 3D3D = - \~-\; «ZTM»l. (4.5.9) Inn the opposite limit, where qr\o\ -C 1, at temperatures T» ÜLOQ we automatically have \a\\a\ < AT and, accordingly, recover Eq. (4.5.6) with = 2/TT: nn 3D3D = (?-) 2 -, «r a <l. (4.5.10) Ass mentioned in Section 4.4, for \a\ > IQ the approximate border line between the quasi-2dd and confinement-dominated 3D regimes is e* «O.lhuJo- In the temperature intervall e* < T < tuuo, the leading scattering channel will be the same as in the quasi- 2DD case, that is v = i/ = 0. However, the expression for the scattering amplitude /oo iss different. From Fig and Eq. (4.3.11) one concludes that the real part of the functionn w can be neglected and the scattering amplitude takes the form.. 2v / 2lr /ooo = lo/alo/a + iy/ir/2 Then,, retaining only the scattering channel v v' = 0, Eqs. (4.5.2) and (4.5.3) yield nn 8nhn fa\ 2 1-exp (-frdp/t) (AK^\ UU ~~ m \lo) l + 7ra 2 /2l 2 ' * Thee difference of Eq. (4.5.11) from the quasi-2d result of Eqs. (4.4.3) and (4.4.4) is relatedd to the absence of the logarithmic term in the denominator. This follows from the factt that now we omitted the real part of the function w, which is logarithmically large inn the quasi-2d regime. J

50 44 4 ChapterChapter 4. Interatomic collisions in a tightly confined Bose gas T/hm T/hm Figuree 4.5.3: The quantity mvt/hn versus temperature. In (a) the parameter a/lo = 1 (dashedd curve), and a/l 0 = 1 (dotted curve). In (b) a/l 0 = oo (unitarity limit). The solidd curves in (a) and (b) show the 3D result (4.5.7). The arrow in (b) indicates the lowestt ratio T/HLÜQ in the Stanford and ENS experiments. Itt is worth to note that for IQ ;» \a\, Eq. (4.5.11) is only slightly different from the 3D resultt (4.5.10). This is consistent with the above given analysis leading to Eq. (4.5.6). Onn the other hand, for large \a\/lo the difference between Eq. (4.5.11) and the 3D resultt (4.5.9) is significant. This originates from the fact that for a large scattering lengthh a the 3D amplitude of scattering in the ultra-cold limit depends on the particle momenta.. For a tight axial confinement, treating collisions as three-dimensional, the relativee momentum of colliding particles at temperatures T < HUJO is ~ l/7 0 and it no longerr depends on temperature. Hence, the scattering rate is quite different from that in 3D.. Given these arguments, one expects a strongly pronounced confinement-dominated 3DD regime of scattering if the ratio a /Zn ~> 1. Thiss is confirmed by our numerical calculations for the temperature dependence of CtCt from Eq. (4.5.3). In Fig we present the results for a/lo equal to 1, 1, and oo. Thee largest deviation from the 3D regime is observed in the unitarity limit (a * oo). Fromm Fig we see that in the Stanford [27] and ENS [29; 30] experiments performed inn this limit 3 one should have significant deviations of collisional rates from the ordinary 3 Inn the Stanford experiment [27] the unitarity limit for collisions between Cs atoms in the hyperfine

51 SectionSection 4.6. Thermalization rates DD behavior Thermalization rates Wee will now discuss the collision-induced energy exchange between axial and radial degreess of freedom of the particle motion in an ultra-cold Bose gas tightly confined in the axiall direction of a pancake-shaped trap. It is assumed that the radial confinement is shalloww and it does not influence the scattering amplitudes. In this geometry, using degeneratee Raman sideband cooling, the Stanford [73; 27] and ENS [74; 75; 29; 30] groupss created Cs gas clouds with different axial (T z ) and radial (T p ) temperatures. Afterr switching off the cooling, interatomic collisions lead to energy exchange between the axiall and radial particle motion and the temperatures T z and T p start to approach each other.. Ultimately, the gas reaches a new equilibrium state, with a temperature in betweenn the initial T z and T p. The corresponding (thermalization) rate has been measured att Stanford [27] and ENS [29; 30] and it provides us with the information on the regimes off interatomic collisions in the gas. Thee radial motion of particles is classical. Therefore, we will calculate the rate of energyy exchange between the radial and axial degrees of freedom for a given value of the radiall coordinate p and then average the result over the Boltzmann density profile in the radiall direction. The latter is given by n(p)=n(0)exp(-^^),, (4.6.1) wheree n(0) = muj 2 N/2'ïïT is the 2D density for p = 0, <x> the radial frequency, and N the totall number of particles. Collision-induced transitions u * u' change the energy of the axiall motion by fou)o(v' v). We will assume that in the course of evolution the axial andd radial distribution of particles remain Boltzmann, with instantaneous values of T z andd T p. Then the rate of energy transfer from the radial to axial motion can be written onn the same grounds as Eq. (4.5.3) and reads (4.6.2) ) wheree e = H?q 2 /m + HUQU, and the normalization coefficient A depends now on both T z andd T p. Thee radial energy of the gas is E p = 2NT P, and the axial energy is given by E z = JV/kt>o[expp (hwo/t) l] -1. The time derivatives of these energies take the form NhNh 22 ww 22 tt z z Êz=Êz= Jm0, 4T4T 22 2 *_,; Èp = 2Nt smhsmh 22 p. (4.6.3) (u(u 00 /2T/2T zz ) ) statee F = 3, mp = 3 has been achieved by tuning the magnetic field to the Feshbach resonance at 30 G. Inn the ENS experiment [29; 30] the magnetic field was very small. According to the recent calculations [122],, this corresponds to a large and negative scattering length (a ~ 1500 A) for collisions between Cs atomss in F = 3 states. For the ENS axial frequency u»o = 80 khz we have IQ «200 A. Hence, the ratio a /ioo in this experiment was approximately equal to 7, which is already rather close to the unitarity limit..

52 46 6 ChapterChapter 4. Interatomic collisions in a tightly confined Bose gas Givenn the initial values of T z and T p, Eqs. (4.6.2) and (4.6.3) provide us with the necessary informationn on the evolution of T z {t) and T x (t). Forr a small difference ST = T p T z, these equations can be linearized with respect too ST. As the total energy is conserved, Eqs. (4.6.3) reduce to STST -- Ir { ) (4 ' 64) Inn Eq. (4.6.2) we represent the exponent as {ejt + 8Tfou)Qv/T 2 ) and turn from the integrationn over dq to integration over de. The zero order term of the expansion in powerss of ST vanishes. The (leading) linear term, being substituted into Eq. (4.6.4), leadss to the differential equation for 5T(t): wheree the thermalization rate Q t h(t) is given by üthüth = ~1MT [ M*- + 4smh (ir J J J>-" ) STST = -n th (T)ST, (4.6.5) LJ ]UA^ exp ("r) (4.6.6) ) Thee degeneracy parameter is n(0)af, = N(hw/T) 2 and it is small as the gas obeys the Boltzmannn statistics. The normalization coefficient A is again given by Eq. (4.5.4). Thee quantum numbers u and v' take only even values and, hence, in order to change thee state of the axial motion one should have a relative energy e > 2hjjQ. Therefore, at temperaturess lower than HUJO the rate of transitions changing the axial and radial energy is occ exp ( 2hu}o/T). On the other hand, the axial energy E z oc exp ( hujo/t) and thus the thermalizationn rate Cl t h ex exp ( hu> 0 /T). This can be easily found from Eq. (4.6.6) and showss that the quantum character of the axially confined particle motion exponentially suppressess the thermalization process at temperatures T <C ^o- In particular, this is thee case for the quasi-2d regime. Inn the most interesting part of the confinement-dominated 3D regime, where e* < TT < hu, the energy exchange between the axial and radial motion of particles is mostly relatedd to transitions between the states with v 1 = 0 and v = 2. The relative energy e shouldd be larger than 2huJo and, at the same time, this energy is well below AfoujQ. Hence, thee scattering amplitude /20(e) is determined by Eqs. ( and (4.3.12) in which one cann put w(e/2hu>o) «z37r/2 (see Eq. (4.5.1) and Fig ). This gives andd from Eq. (4.6.6) we obtain ^ - ^ T ^ ^ 1 2V/2V V ƒ200 = ll00 /a/a + i{3/2)- Thee characteristic frequency X1Q is given by - - * - " ^ -- ^ ÜQÜQ = OJ 2 N/U>O- (4.6.8) )

53 SectionSection 4.6. Thermalization rates 47 7 Q ** * Ï Ï o> > 55 "VJi 10 0 Figuree 4.6.1: Thermalization rate versus temperature in the unitarity limit (a = oo). The solidd curve shows the result of our calculations for Q, t h/ >o, and the dotted line the 3D resultt n^p/qo- Squares and circles show the data of the Stanford and ENS experiments. Att temperatures T» rate: : icic th th HLJO, Eq. (4.6.6) leads to the 3D result for the thermalization oo 3D 3D 166 (o_ ifcc ( > n» : «- 1 " 1 * l Ï57rr Uo -^--^- ) Q 0 - q T \a\ < 1. (4.6.9) ) (4.6.10) ) Comparingg Eqs. (4.6.9) and (4.6.10) with Eq. (4.6.7) one sees that iï t h should acquire its maximumm value at T ~ fio; 0 - For \a\ > IQ this maximum value is on the order of n 0 /2ir. Ass one expects from the discussion in Section 4.5, the difference of the thermalization ratee from Q 3 is pronounced for large values of a. For example, in the unitarity limit Eq.. (4.6.9) gives O^P a 1/T 2, whereas in the confinement-dominated regime we have fua(l/t)exp(-wt).. Itt should be emphasized that for any T, w 0, and a the ratio fl t h/^o depends only onn the parameters T/huj 0 and a/l 0. This can be found directly from Eq. (4.6.6). In Fig we present the temperature dependence of Ct t h in the unitarity limit, obtained numericallyy from Eq. (4.6.6), and compare our results with the data of the Stanford [27] andd ENS [29; 30] experiments. With the current error bars, the ENS results do not show significantt deviations from the classical 3D behavior. These results agree fairly well with ourr calculations. The Stanford experiment gives somewhat lower values of Q t /j/oo at the lowestt temperatures of the experiment. Inn the hydrodynamic regime for the gas cloud, where the characteristic collisional frequencyy greatly exceeds the radial frequency w, our assumption of quasiequilibrium att instantaneous (time-dependent) values of T z and T p may not be valid. Nevertheless, thee shape of the curve fl t h(t) qualitatively remains the same, including the exponential decreasee with temperature at T < hw^ and power law decrease with increasing T at

54 488 Chapter 4. Interatomic collisions in a tightly confined Bose gas temperaturess larger than fiwo- However, the maximum value of ft t h will be somewhat lowerr (in particular, of the order of u; [27]). Thee number of particles (per "2D" sheet of atoms) in the Stanford experiment [27] wass N ~ 10 4 [123], which is by a factor of 20 higher than at ENS for T «hu 0 [29; 30]. We thenn estimate the 2D density of atoms for these temperatures to be n ~ 2.5 x 10 8 cm -2 at Stanfordd (u> «90 Hz), and n ~ 0.5 x 10 8 cm -2 at ENS (u> «180 Hz). For these densities, thee ratio of the collisional frequency fi in Fig to the radial frequency is Q/UJ ~ 0.3 inn the ENS experiment, and Q/UJ ~ 3 in the experiment at Stanford. At temperatures TT > HLJQ the density n and the ratio Q/UJ are smaller in both experiments. We thus seee that the ENS experiment [29; 30] was in the collisionless regime, although rather closee to the hydrodynamic regime at temperatures T «hbjq. For these temperatures, the Stanfordd experiment [27] has already achieved the hydrodynamic regime, and this can explainn the discrepancy between our calculations and the Stanford results in Fig Inelastic 2-body processes Inelasticc scattering of atoms is also influenced by the tight axial confinement of the particlee motion. In this section we will consider the inelastic 2-body processes, such as spinn relaxation, in which the internal states of colliding atoms are changing, and the releasedd internal-state energy of the atoms is transferred to their kinetic energy. Our goall is to establish a relation between the inelastic rates in 3D and those in the (tightly) axiallyy confined geometry. The analysis given below relies on two important conditions widelyy met for the 2-body spin relaxation [124]: i)thee energy release per collision greatly exceeds the gas temperature and the frequency off the axial confinement. Accordingly, the inelastic transitions occur at comparatively shortt interparticle distances ~ Ri n which are much smaller than the characteristic de Brogliee wavelength of particles. ii)thee inelastic transitions are caused by weak (spin-dipole, spin-orbit, etc.) interatomic interactionss and can be treated with perturbation theory. Too first order in perturbation theory the amplitude of inelastic scattering, defined in thee same way as in the previous sections, is given by a general expression [125] fin(e)fin(e) = ƒ Mr)U int (r)1> f (T)<Pr. (4.7.1) Heree fa(r) and tpf (r) are the true wavefunctions of the initial and final states of the relativee motion of colliding atoms, and Ui n t (r) is the (weak) interatomic potential responsible forr inelastic transitions. This potential is the same as in the 3D case. The function ipj iss also the same as in 3D, since the relative energy in the final state is much larger than hu)q.hu)q. Thus, the only difference of the amplitude fi n (4.7.1) from the amplitude of inelastic scatteringg in the 3D case is related to the form of the wavefunction fa. Thee characteristic interatomic distance Ri n at which the inelastic transitions occur, satisfiess the inequality Ri n <C A (see item ii). Therefore, we are in the ultra-cold limitt similar to that determined by Eq. (4.3.2) in the case of elastic scattering, and thee conditions qri n <C 1 and i^ «C IQ are satisfied. The former ensures a dominant contributionn of the s-wave (of the initial wavefunction fa) to the scattering amplitude

55 SectionSection 4.7. Inelastic 2-body processes 49 9 finfin (4.7.1). Due to the condition Ri n -C Zo> at distances r ~ Ri n the wavefunction fa hass a three-dimensional character: ipi(r) <x V'3D('") 1 where V, 3Z?(Ï') is the wavefunction of thee 3D relative motion at zero energy. For r» R e we have ^3 >(0 = (1 a / r ), and m orderr to be consistent with Eq. (4.3.7) we should write fa(r)fa(r) = ri( )<pa0)ï>3d(r), (4.7.2) wheree the coefficient 7](E) is given by Eq. (4.3.10), and v is the quantum number of the initiall state of the relative motion in the axial harmonic potential VH(Z). Inn the 3D case, the amplitude /o of inelastic scattering at zero initial energy is determinedd by Eq. (4.7.1) with fa replaced by ^D- Hence, Eq. (4.7.2) directly gives a relation betweenn the two scattering amplitudes: finfin =»?(e) V(0)/o. (4-7.3) Duee to the high relative kinetic energy of particles in the final state of the inelastic channel,, the density of final states in this channel is independent of the axial confinement. Therefore,, relying on Eq. (4.7.3) the mean rate constant ö in of inelastic collisions in the axiallyy confined geometry and the corresponding collisional frequency can be represented inn the form ctinctin = {\r){e)\ 2 <pl{0)0(e - hw 0 v))a 0 ; ü in = a in n, (4.7.4) wheree OLQ is the 3D inelastic rate constant at zero energy. Notee that in the ultra-cold limit the 3D inelastic rate constant is temperature independentt and equal to OLQ if the scattering length \a\ < R e. For \a\» R e, the wavefunctionn of the relative motion in the region of interatomic interaction takes the form fa(r)fa(r) = rjzd'tpzdi'f), where I^DI 2 = (1 +p 2 a 2 )~ l and p is the 3D relative momentum off colliding particles (see, e.g. [126]). Hence, for the inelastic rate constant we have {QQ(1{QQ(1 +p 2 a 2 ) -1 ). In the presence of axial confinement, averaging the frequency of inelasticc collisions over the (quantum) axial density profile TI^D{Z), we obtain Thee density profile nzoiz) accounts for the discrete structure of quantum levels in the axiall confining potential and for the quantum spatial distribution of particles. Therefore, Eq.. (4.7.5) gives the ordinary 3D result only at temperatures T» hwq, where TLZD(Z) becomess the Boltzmann distribution TIB(Z). Wee first analyze the influence of axial confinement on Qi n (4.7.4) for the case where \a\\a\ < A T or, equivalently, \a\ < / 0 and qr\a\ < 1. In this case we may put rf = 1 at any T,, except for extremely low temperatures in the quasi-2d regime. Then Eq. (4.7.4) gives nn inin = <rf(0))oo» = ^fjr ^h 1/2 (**). (4.7.6) Onee can easily check that Eq. (4.7.6) coincides with Eq. (4.7.5) in which p\a\ <C 1. The reasonn for this coincidence is that, similarly to the case of elastic scattering described by Eq.. (4.5.5), for 77 = 1 the scattering amplitude fi n (4.7.3) is independent of the relative

56 50 0 ChapterChapter 4. Interatomic collisions in a tightly congned Bose gas energyy e. Hence, the inelastic rate is influenced by the axial confinement only through thee axial distribution of particles. However, this influence is significant, in contrast to thee case of elastic scattering under the same conditions (see Eq. (4.5.6)). Qualitatively, forr qr\o\ *C 1 we have Q,i n ~ OCQUZD- At temperatures T -C ÏUUQ, a characteristic valuee of the 3D density is n 3 > ~ n/fo and we obtain Sli n ~ aon/lo. For T» hu>o, thee 3D density UZD ~ n^muj^/t) 1 ' 2 and hence the frequency of inelastic collisions is «i»» - (aon/zoxwt) 1 / 2. Wee now discuss the temperature dependence of the inelastic rate for the case where l a ll ^fo, which in the ultra-cold limit (4.3.2) assumes that \a\» R e. In the quasi-2d regimee and in the temperature interval e* < T < hwo of the confinement-dominated 3D regime,, the most important contribution to the inelastic rate in Eq. (4.7.4) comes from collisionss with the axial quantum number u = 0. Then, using Eq. (4.3.11) we express the parameterr j] through the elastic amplitude /oo and obtain a relation between Q,i n and thee mean frequency of elastic collisions Q(T): a (r)) = ( /M ( )P) I3^F=n ( r)/ J, (4.7.7) wheree 0 = (l/12sir 3 ) 1 ' 2 (7nloao/ha 2 ) is a dimensionless parameter independent of temperature.. The temperature dependence of il is displayed in Fig and Fig and wass discussed in Sections 4.4 and 4.5. Note that the parameter 0 is not equal to zero for \a\\a\ oo. In this case, since the amplitude /o is calculated with the wavefunction tp^o whichh behaves as a/r for r oo, we have OCQ OC a 2 and 0 = const. Att temperatures T» hwo, using the same method as in Section 4.5 for the case of elasticc scattering, from Eq. (4.7.4) we recover the 3D result Qf^ given by Eq. (4.7.5) withh TI3D(Z) = ns{z). In the limiting case, where qr\o\ ^> 1, we find ^^i^f^mt)^)^^i^f^mt)^) Comparingg Eq. (4.7.8) with Eq. (4.7.7), we see that in the confinement-dominated 3D regimee the deviation of the inelastic rate from the ordinary 3D behavior should be larger thann that in the case of elastic scattering. Ass follows from Eq. (4.7.7) and Fig , for \a\ > lo the inelasticfrequencyüi n reachess its maximum at temperatures near the border between the quasi-2d and confinement-dominatedd 3D regimes. The maximum value of X7j n is close to tltl inin = 0. (4.7.9) m m Fromm Eq. (4.7.4) one finds that at any T the ratio li n /&in depends only on two parameters:: T/HUJQ and a/lo. In Fig we present our numerical results for 0,^/0,^ as a functionn of T/foujQ for a/lo equal to 1, 1, and oo. As expected, the deviations from the 3DD behavior are the largest in the unitarity limit. Thee inelastic rate of spin relaxation in a tightly (axially) confined gas of atomic cesiumm has been measured for the unitarity limit in the Stanford experiment [27]. Due to aa shallow radial confinement of the cloud in this experiment, the 2D density n ~ \/T

57 SectionSection 4.8. Concluding remarks ~~ 0.8- '-«? =-11 = \ QQ IQ..,n,n,n 0 g. ^ ^ 'all.-'all =11 "*- -«r^m^ \n 3D /S S a) ) ~~ 1-5- nn /n inin in f f \\ ü 3D /n \\ in in ^ ^ - ^ b) ) t\t\ n ii i i ' T T Figuree 4.7.1: The quantity fij n /Öj versus temperature. In (a) the parameter a/lo = 1 (dashedd curve), and a/l 0 = 1 (dotted curve). In (b) a/l 0 = oo (unitarity limit). The solidd curves in (a) and (b) show the 3D result (4.7.7). (seee Eq. (4.6.1)). Then, Eq. (4.7.8) gives the 3D inelastic frequency fi?f ~ 1/T 5 / 2, whereass in the temperature interval e* < T < hujo of the confinement-dominated regime Eqs.. (4.5.11) and (4.7.7) lead to Sl in ~ (1/T)(1 - exp(-hu 0 /T)). In order to compare ourr calculations with the data of the Stanford experiment on spin relaxation, in Fig wee display the ratio of Qi n (T/fkJo) to ili n at T = 3fkoo which was the highest temperaturee in the experiment. The temperature dependence of the inelastic rate, following fromm the Stanford results, agrees fairly well with the calculations and shows significant deviationss from the 3D behavior. It should be noted that, in contrast to thermalization rates,, the inelastic decay rate is not sensitive to whether the gas is in the collisionless or hydrodynamicc regime [27] Concluding remarks Inn conclusion, we have developed a theory which describes the influence of a tight axial confinementt of the particle motion on the processes of elastic and inelastic scattering. Thee most interesting case turns out to be the one in which the 3D scattering length aa exceeds the extension of the wavefunction in the axial direction, /Q. In the ultra-

58 522 Chapter 4. Interatomic collisions in a tightly confined Bose gas Figuree 4.7.2: The ratio of Q,i n (T/hu>o) to fij n at T = 3hu>o versus temperature in the unitarityy limit (a = oo). The solid curve shows the result of our numerical calculations, andd the dotted line the 3D limit. Circles show the data of the Stanford experiment. coldd limit defined by Eq. (4.3.2), the condition \a\ > IQ automatically requires large \a\\a\ compared to the radius of interatomic interaction R e. Then we have a pronounced confinement-dominatedd 3D regime of scattering at temperatures on the order of HUJQ. Treatingg interatomic collisions as three-dimensional, the relative momentum of colliding atomss is related to the quantum character of the axial motion in the confining potential andd becomes of the order of l/lo- As a result, the scattering rate can strongly deviate fromm the ordinary 3D behavior. The axial extension of the wavefunction, achieved in thee experiments at Stanford and ENS [73; 27; 74; 75; 29; 30], is l 0 w 200 A. The requiredd value of the scattering length, o > IQ and \a\ ~3> R e, is characteristic for Cs atomss (R e «100A) and can also be achieved for other alkali atoms by using Feshbach resonances.. Inn order to observe the 2D features of the particle motion in the rates of interatomic collisionss one has to reach the quasi-2d regime of scattering, which requires much lower temperatures,, at least by an order of magnitude smaller than HLOQ. For wo «80kHz (hujq(hujq fa 4 zzk) as in the Stanford [73; 27] and ENS [74; 75; 29; 30] experiments, these are temperaturess below 400nK. As one can see from Fig , the rate of elastic collisions iss still rather large for these temperatures and, hence, one can think of achieving them byy evaporative cooling. Moreover, for realistic radial frequencies u> ~ 100Hz there is a hopee to achieve quantum degeneracy and observe a cross-over to the BEC regime. The cross-overr temperature is T c «N x f 2 fuü (see [32] and the discussion in Section 3.1), and forr N ~ 1000 particles in a quasi-2d layer we find T c w loonk. Anotherr approach to reach BEC in the quasi-2d regime will be to prepare initially aa 3D trapped condensate and then adiabatically slowly turn on the tight axial confinement.. Manipulating the obtained (quasi-) 2D condensate and inducing the appearance of thermall clouds with temperatures T < T c, one can observe interesting phase coherence phenomenaa originating from the phase fluctuations of the condensate in quasi-2d (see Sectionn 3.1). Interestingly,, at temperatures T ~ T c the collisional frequency ft can be on the

59 SectionSection 4.8. Concluding remarks 53 orderr of the cross-over temperature if \a\ > fo- This follows directly from Fig andd Eq. (4.4.4) which give Q ~ -nhnfm even at temperatures by two orders of magnitude smallerr than hwo. As the 2D density of thermal particles is n ~ Nmuj 2 /T, we immediately obtainn Ml(T c ) ~ N l / 2 hu) ta T c. This condition means that the trapped gas becomes a stronglyy interacting system. The mean free path of a particle, v/q(t c ), is already on thee order of its de Broglie wavelength h/y/mt c. At the same time, the system remains dilute,, since the mean interparticle separation is still much larger than the radius of interatomicc interaction R e. In this respect, the situation is similar to the 3D case with aa large scattering length a» R e at densities where na? ~ 1. The investigation of the cross-overr to the BEC regime in such strongly interacting quasi-2d gases should bring inn analogies with condensed matter systems or dense 2D gases. Well-known examples off dense 2D systems in which the Kosterlitz-Thouless phase transition [47; 48] has been foundd experimentally, are monolayers of liquid helium [14; 15] and the quasi-2d gas of atomicc hydrogen on liquid helium surface [16]. Onn the other hand, for \a\ <C l 0 the collisional frequency near the BEC cross-over, f2(t c )) < T c /h, and the (quasi-)2d gas remains weakly interacting. Then, the nature of thee cross-over is questionable (see Section 3.1). Generally speaking, one can have both thee ordinary BEC cross-over like in an ideal trapped gas [32] and the Kosterlitz-Thouless typee [47; 48] of a cross-over. We thus see that axially confined Bose gases in the quasi-2d regimee are remarkable systems where by tuning a or IQ one can modify the nature of the BECC cross-over.

60

61 Chapterr 5 Regimes of quantum degeneracyy in trapped ID gases WeWe discuss the regimes of quantum degeneracy in a trapped ID gas and obtain the diagram ofof states. Three regimes have been identified: the BEC regimes of a true condensate and quasicondensate,quasicondensate, and the regime of a trapped Tonks gas (gas of impenetrable bosons). presencepresence of a sharp cross-over to the BEC regime requires extremely small interaction betweenbetween particles. We discuss how to distinguish between true and quasicondensates in phasephase coherence experiments. Low-temperaturee ID Bose systems attract a great deal of interest as they show a remarkablee physics not encountered in 2D and 3D. In particular, the ID Bose gas with repulsivee interparticle interaction (the coupling constant g > 0) becomes more non-ideal withh decreasing ID density n [56; 57]. The regime of a weakly interacting gas requires thee correlation length l c = h/y/mng (m is the atom mass) to be much larger than the meann interparticle separation 1/n. For small n or large interaction, where this condition iss violated, the gas acquires Fermi properties as the wavefunction strongly decreases at shortt interparticle distances [54; 56; 57]. In this case it is called a gas of impenetrable bosonss or Tonks gas (cf. [127]). Spatiallyy homogeneous ID Bose gases with repulsive interparticle interaction have beenn extensively studied in the last decades. For the delta-functional interaction, Lieb andd Liniger [56; 57] have calculated the ground state energy and the spectrum of elementaryy excitations which at low momenta turns out to be phonon-like. Generalizing thee Lieb-Liniger approach, Yang and Yang [128] have proved the analyticity of thermodynamicall functions at any finite temperature T, which indicates the absence of a phase transition.. However, at sufficiently low T the correlation properties of a ID Bose gas are qualitativelyy different from classical high-t properties. In the regime of a weakly interactingg gas (nl c» 1) the density fluctuations are suppressed [44; 129], whereas at finite TT the long-wave fluctuations of the phase lead to exponential decay of the one-particle densityy matrix at large distances [44; 129; 130]. A similar picture, with a power-law decayy of the density matrix, was found at T = 0 [51; 52]. Therefore, the Bose-Einstein condensatee is absent at any T, including T = 0. From a general point of view, the absencee of a true condensate in ID at finite T follows from the Bogolyubov k~ 2 theorem ass was expounded in [10; 11] (for the T = 0 case see [53]). Earlier studies of ID Bose systemss are reviewed in [12]. They allow us to conclude that in ID gases the decrease of temperaturee leads to a continuous transformation of correlation properties from ideal-gas classicall to interaction/statistics dominated. A ID classical field model for calculating correlationn functions in the conditions, where both the density and phase fluctuations aree important, was developed in [131] and for Bose gases in [132]. Interestingly, ID gases cann posses the property of superfluidity at T = 0 [133; 12]. Moreover, at finite T one cann have metastable supercurrent states which decay on a time scale independent of the sizee of the system [134]. 55 5

62 566 Chapter 5. Regimes of quantum degeneracy in trapped ID gases Thee earlier discussion of ID Bose gases was mostly academic as there was no possible realizationn of such a system. Fast progress in evaporative and optical cooling of trapped atomss and the observation of Bose-Einstein condensation (BEC) in trapped clouds of alkalii atoms [3; 4; 5] stimulated a search for non-trivial trapping geometries. At present, theree are significant efforts to create (quasi-)id trapped gases [135; 62], where the radial motionn of atoms in a cylindrical trap is (tightly) confined to zero point oscillations. Then, kinematicallyy the gas is ID, and the difference from purely ID gases is only related to thee value of the interparticle interaction which now depends on the radial confinement. Thee presence of the axial confinement allows one to speak of a trapped ID gas. Ketterlee and van Druten [33] considered a trapped ID ideal gas and have revealed aa BEC-like behavior of the cloud. They have established that at temperatures T < Nfojj/Nfojj/ In 2iV, where N is the number of particles and ui the trap frequency (we use the Boltzmannn constant fc^ = 1), the population of the ground state rapidly grows with decreasingg T and becomes macroscopic. AA fundamental question concerns the influence of interparticle interaction on the presencee and character of BEC. In this chapter we discuss the regimes of quantum degeneracy inn a trapped ID gas with repulsive interparticle interaction. We find that the presence off a sharp cross-over to the BEC regime, predicted in [33], requires extremely small interactionn between particles. Otherwise, the decrease of temperature leads to a continuouss transformation of a classical gas to quantum degenerate. We identify 3 regimes at TT <C Td, where Td «Nhu; is the degeneracy temperature. For a sufficiently large interparticlee interaction and the number of particles much smaller than a characteristic value iv*,, at any T < Td one has a trapped Tonks gas characterized by a Fermi-gas density profile.. For N ^> N* we have a weakly interacting gas. The presence of the trapping potentiall introduces a finite size of the sample and drastically changes the picture of longwavee fluctuations of the phase compared to the uniform case. We calculate the density andd phase fluctuations and find that well below Td there is a quasicondensate, i.e. the BECC state where the density fluctuations are suppressed but the phase still fluctuates. Att very low T the long-wave fluctuations of the phase are suppressed due to a finite size off the system, and we have a true condensate. The true condensate and the quasicondensatee have the same Thomas-Fermi density profile and local correlation properties, and wee analyze how to distinguish between these BEC states in an experiment. Wee first discuss the coupling constant g for possible realizations of ID gases. These realizationss imply particles in a cylindrical trap, which are tightly confined in the radial (p)(p) direction, with the confinement frequency LÜQ greatly exceeding the mean-field interaction.. Then, at sufficiently low T the radial motion of particles is essentially "frozen" andd is governed by the ground-state wavefunction of the radial harmonic oscillator. If thee radial extension of the wavefunction, IQ = (h/mwo) 1^2» R e, where R e is the characteristicc radius of the interatomic potential, the interaction between particles acquires aa 3D character and will be characterized by the 3D scattering length a. In this case, assumingg lo ~S> \a\, we have gg = 2h 2 ajmll. (5.0.1) Thiss result follows from the analysis in [136] and can also be obtained by averaging the 3D interactionn over the radial density profile. Thus, statistical properties of the sample are thee same as those of a purely ID system with the coupling constant given by Eq. (5.0.1).

63 ChapterChapter 5. Regimes of quantum degeneracy in trapped ID gases 57 Inn the regime of a weakly interacting gas, where nl c» 1, we have a small parameter 77 = l/(n/ c ) 2 = mg/h 2 n < 1. (5.0.2) Forr particles trapped in a harmonic (axial) potential V(z) = mu) 2 z 2 /2, one can introduce aa complementary dimensionless quantity a = mgl/h 2 which provides a relation between thee interaction strength g and the trap frequency u (Z = n/h/mw is the amplitude of axiall zero point oscillations). Att T = 0 one has a true condensate: In the Thomas-Fermi (TF) regime the mean squaree fluctuations of the phase do not exceed ~ 7 1 / 2 and, hence, they are small underr the condition (5.0.2) (see [43]). The condensate wavefunction is determined by thee Gross-Pitaevskii equation which gives the TF parabolic density profile no(z) = n 0m(ll Z 2 /I$T F ). The maximum density no m = fjt/g, the TF size of the condensate RTFRTF = (Zfi/mu 2 ) 1 / 2, and the chemical potential \i = hw{zna/ay/2) 2 t z. For a» 1 we aree always in the TF regime (/i ^> hw). In this case, Eq. (5.0.2) requires a sufficiently largee number of particles: NN» N* = a 2. (5.0.3) Notee that under this condition the ratio fi/td ~ (a 2 /JV) 1 / 3 -C 1. For a«l the criterion (5.0.2)) of a weakly interacting gas is satisfied at any N, and the condensate is in the TF regimee if N» a -1. In the opposite limit the mean-field interaction is much smaller than thee level spacing in the trap HUJ. Hence, one has a macroscopic occupation of the ground statee of the trap, i.e. there is an ideal gas condensate with a Gaussian density profile. Att this point, we briefly discuss the cross-over to the BEC regime, predicted by Ketterlee and van Druten for a trapped ID ideal gas [33]. They found that the decrease off T to below T c = Nhw/ In 2N strongly increases the population of the ground state, whichh rapidly becomes macroscopic. This sharp cross-over originates from the discrete structuree of the trap levels and is not observed in quasiclassical calculations [32]. We arguee that the presence of the interparticle interaction changes the picture drastically. Onee can distinguish between the (lowest) trap levels only if the interaction between particless occupying a particular level is much smaller than the level spacing. Otherwise thee interparticle interaction smears out the discrete structure of the levels. For T close too T c the occupation of the ground state is ~ T c /hw «N/ In 2N [33] and, hence, the mean-fieldd interaction between the particles in this state (per particle) is Ng/l In 2N. The sharpp BEC cross-over requires this quantity to be much smaller than hw, and we arrive att the condition N/\n2N -C a -1. For a realistic number of particles (N ~ ) thiss is practically equivalent to the condition at which one has the ideal gas Gaussian condensatee (see above). Ass we see, the sharp BEC cross-over requires small a. For possible realizations of ID gases,, using the coupling constant g (5.0.1), we obtain a = 2ÜI/IQ. Then, even for the ratioo I/IQ ~ 10 and moderate radial confinement with IQ ~ 1/xm, we have a ~ 0.1 for Rbb atoms (a w 50 A). Clearly, for a reasonably large number of particles the cross-over conditionn JV < Q" 1 can only be fulfilled at extremely small interparticle interaction. Onee can think of reducing a to below la and achieving a < 10-3 by using Feshbach resonancess as in the MIT and JILA experiments [82; 120]. In this case one can expect thee sharp BEC cross-over already for N ~ 10 3.

64 58 8 ChapterChapter 5. Regimes of quantum degeneracy in trapped ID gases Wee now turn to the case of large a. For 7» 1 one has a Tonks gas [56; 57] (thee realization of the axially uniform Tonks gas in a square-well (axial) potential was discussedd in [136]). The one-to-one mapping of this system to a gas of free fermions [54] ensuress the fermionic spectrum and density profile of a trapped Tonks gas. For (axial) harmonicc trapping the condition 7» 1 requires N <C AT*. The chemical potential is equal too Nhuj, and the density distribution is n(z) = (y/2n/7rl)y/l (z/r) 2, where the size off the cloud R = y/2nl The density profile n(z) is different from both the profile of the zero-temperaturee condensate and the classical distribution of particles, which provides a roott for identifying the trapped Tonks gas in experiments. The interference effects and dynamicall properties of this system are now a subject of theoretical studies [137; 138]. Inn Rb and Na this regime can be achieved for N < 10 3 by the Feshbach increase of a to ~~ 500 A and using UJ ~ 1Hz and U)Q ~ 10kHz (a ~ 50). Largee a and N satisfying Eq. (5.0.3), or small a and N 3> a -1, seem most feasible inn experiments with trapped ID gases. In this case, at any T <C Td one has a weakly interactingg gas in the TF regime. Similarly to the uniform ID case, the decrease of temperaturee to below Td continuously transforms a classical ID gas to the regime of quantum degeneracy.. At T = 0 this weakly interacting gas turns to the true TF condensate (see above).. It is then subtle to understand how the correlation properties change with temperaturee at T <C Td. For this purpose, we analyze the behavior of the one-particle density matrixx by calculating the fluctuations of the density and phase. We a priori assume small densityy fluctuations and prove this statement relying on the zero-temperature equations forr the mean density no (z) and excitations. The operator of the density fluctuations is (see,, e.g. [71]) 00 0 ti(z)ti(z) = n^iz^if-waj + h.c, (5.0.4) 3=1 3=1 wheree dj is the annihilation operator of the excitation with quantum number j and energy j j,, f^ = Uj Vj, and the w, v functions of the excitations are determined by the same Bogolyubov-dee Gennes equations (generalization of the Bogolyubov method for spatially inhomogeneouss systems see in [37]) as in the presence of the TF condensate. Thee solution of these equations gives the spectrum j = fujy/j(j + l)/2 [43; 42] and thee wavefunctions 1/2 2 '?«=ft^) ) ^(l-x 2 ) ) 2 2 Pfa),Pfa), (5.0.5) wheree j is a positive integer, Pj are Legendre polynomials, and x = Z/RTFmeann square fluctuations of the density, (Sh 2.^) = ((h'(z) - n'(z')) 2 ), we have zz' zz' ;2 2 "Omm ~[ 2/in 0m KTF For the withh Nj = [exp(ej/t) 1-1 1] 1 being the occupation numbers for the excitations. At TT» fko the main contribution to the density fluctuations comes from quasiclassical excitationss (j > 1). The vacuum fluctuations are small: (Sh 2 zl)o ~ ^om7 1//2 - For the thermall fluctuations on a distance scale \z z'\ ^> Z c, we obtain {ön{ön 22 zzzz,),) TT «n 2 0rn(T/T d ) min{(t/,i), 1}. (5.0.6)

65 ChapterChapter 5. Regimes of quantum degeneracy in trapped ID gases 59 9 Wee see that the density fluctuations are strongly suppressed at temperatures T «C T d. Then,, one can write the total field operator as \j){z) = y/no(z) exp(i<f>(z)), where <j>(z) iss the operator of the phase fluctuations, and the one-particle density matrix takes the formm (see, e.g. [12]) which is a generalization of Eq. (2.3.1) for the inhomogeneous case {ft{z)j>{z')){ft{z)j>{z')) = Vno(*)M*')«p{-<*&')/2}, (5.0.7) withh 5<j> zz ' = <f>{z) <j>{z'). The operator <j>(z) is given by (we count only phonon part (seee Overview)) 4>{z)4>{z) = [4no(z)]" 1/2 Yl tf(*)&j + *-c, (5.0.8) andd for the mean square fluctuations we have,, *-^ ZejTlomKTF J 3-3- Cj</x Forr the vacuum fluctuations we find (c.f. [139; 43]) (^>o«(7 1/2 /T)ln(k-z' //c),, andd they are small for any realistic size of the gas cloud. The thermal fluctuations of thee phase are mostly provided by the contribution of the lowest excitations and the restrictionn 6j < fj, is not important. A direct calculation, with Nj = T/ej, yields 1212 \ _ 47> (Wi(Wi zz >)T>)T = log g (l-x')(l(l-x')(l + x)] 3T3T dd hu hu (l(l + x')(l-x) (5.0.9) ) Inn the inner part of the gas sample the logarithm in Eq. (5.0.9) is of order unity. Thus,, we can introduce a characteristic temperature TT phph = T d hw/fi (5.0.10) att which the quantity {ó<f% z,) «lona distance scale \z z'\ ~ RTF- The characteristicc radius of phase fluctuations is R<f> «RrFiTph/T) oc iv 2 / 3 /T, and for T < T p h it exceedss the sample size RTF- This means that at T < T p h both the density and phase fluctuationsfluctuations are suppressed, and there is a true condensate. The condition (5.0.3) always providess the ratio T ph /hw «(4iV/a 2 ) 1/3 > 1. Inn the temperature range, where T d» T ; > T p h, the density fluctuations are suppressed,, but the phase fluctuates on a distance scale R<f, <C RTF- Thus, similarly to thee quasi-2d case (see Section 3.1), we have a quasicondensate. The radius of the phase fluctuationsfluctuations greatly exceeds the correlation length: R<j> fn l c {T d /T) :» l c. Hence, the qua sicondensatee has the same density profile as the true condensate. Correlation properties att distances smaller than R^ are also the same. However, the phase fluctuations lead to aa drastic difference in the phase coherence properties. Inn Fig we present the state diagram of the trapped ID gas for a = 10 (JV* = 100).. For N» N+, the decrease of temperature to below T d leads to the appearance

66 60 0 ChapterChapter 5. Regimes of quantum degeneracy in trapped ID gases a=10,, N=100 quasicondensate e truee condensate Thomas-Fermii profile degeneracyy limit ii i i i i 111 n i i i T/fico o Figuree 5.0.1: Diagram of states for a trapped ID gas. off a quasicondensate which at T < T p h turns to the true condensate. In the T N planee the approximate border line between the two BEC regimes is determined by the equationn (T/fkj) = (32./V/9./V,,) 1 / 3. For N < N* the system can be regarded as a trapped Tonkss gas (for an infinitely long atom waveguide, the approximate border line between thee Tonks and weakly interacting regimes is given in terms of density and temperature inn [140]). Phasee coherence properties of trapped ID gases can be studied in 'juggling' experimentss similar to those with 3D condensates at NIST and Munich [141; 142]. Small clouds off atoms are ejected from the main cloud by stimulated Raman or RF transitions. Observingg the interference between two clouds, simultaneously ejected from different parts off the sample, allows the reconstruction of the spatial phase correlation properties. Similarly,, temporal correlations of the phase can be studied by overlapping clouds ejected att different times from the same part of the sample. In this way juggling experiments providee a direct measurement of the one-particle density matrix. Repeatedly juggling cloudss of a small volume fl from points z and z' of the sample, for equal time of flight to thee detector we have the averaged detection signal I = fl[no(z)+no(z') + 2(^(z)'ip(z'))]. Att T <C T p h the phase fluctuations are small and one has a true condensate. In thiss case, for z' = z we have = UQ(Z) and I = 4ilno(z), and there is aa pronounced interference effect: The detected signal is twice as large as the number off atoms in the ejected clouds. The phase fluctuations grow with T and for T > T p h, wheree the true condensate turns to a quasicondensate, the detection signal decreases ass described by (ft (z)4>(z')) from Eqs. (5.0.7) and (5.0.9). For T» T ph the phase fluctuationss completely destroy the interference between the two ejected clouds, and II = 2Qn 0 {z). Inn conclusion, we have identified 3 regimes of quantum degeneracy in a trapped ID gas:: the BEC regimes of a quasicondensate and true condensate, and the regime of a trappedd Tonks gas. The creation of ID gases will open handles on interesting phase coherencee studies and the studies of "fermionization" in Bose systems.

67 Chapterr 6 Phase-fluctuating 3D Bose-Einsteinn condensates 6.11 Phase-fluctuating 3D condensates in elongated traps s WeWe find that in very elongated 3D trapped Bose gases, even at temperatures far below thethe BEC transition temperature T c, the equilibrium state will be a 3D condensate with fluctuatingfluctuating phase (quasicondensate). At sufficiently low temperatures the phase fluctuationstions are suppressed and the quasicondensate turns into a true condensate. The presence ofof the phase fluctuations allows for extending thermometry of Bose-condensed gases well belowbelow those established in current experiments. Phasee coherence properties are among the most interesting aspects of Bose-condensed gases.. Since the discovery of Bose-Einstein condensation (BEC) in trapped ultra-cold cloudss of alkali atoms [3; 4; 5], various experiments have proved the presence of phase coherencee in trapped condensates. The MIT group [76] has found the interference of two independentlyy prepared condensates, once they expand and overlap after switching off thee traps. The MIT [143], NIST [141] and Munich [142] experiments provide evidence forr the phase coherence of trapped condensates through the measurement of the phase coherencee length and/or single particle correlations. Thesee results support the usual picture of BEC in 3D gases. In equilibrium, the fluctuationsfluctuations of density and phase are important only in a narrow temperature range near thee BEC transition temperature T c. Outside this region, the fluctuations are suppressed andd the condensate is phase coherent. This picture precludes the interesting physics of phase-fluctuatingg condensates, which is present in 2D and ID systems (see Section 3.1, Chapterr 5 and refe. therein). Inn this section we show that the phase coherence properties of 3D Thomas-Fermi (TF) condensatess depend on their shape. In very elongated 3D condensates, the axial phase fluctuationsfluctuations are found to manifest themselves even at temperatures far below T c. Then ass the density fluctuations are suppressed, the equilibrium state will be a condensate with fluctuatingfluctuating phase (quasicondensate) similar to that in ID trapped gases (see Chapter 5). Decreasingg T below a sufficiently low temperature, the 3D quasicondensate gradually turnss into a true condensate. Thee presence and the temperature dependence of axial phase fluctuations in sufficientlyy elongated 3D condensates suggests a principle of thermometry for Bose-condensed gasess with indiscernible thermal clouds. The idea is to extract the temperature from a measurementt of the axial phase fluctuations, for example by measuring the single-particle correlationn function. This principle works for quasicondensates or for any condensate that cann be elongated adiabatically until the phase fluctuations become observable. Soo far, axial phase fluctuations have not been measured in experiments with cigar- 61 1

68 622 Chapter 6. Phase-fluctuating 3D Bose-Einstein condensates shapedd condensates. We discuss the current experimental situation and suggest how one shouldd select the parameters of the cloud in order to observe the phase-fluctuating 3D condensates.. Wee first consider a 3D Bose gas in an elongated cylindrical harmonic trap and analyzee the behavior of the single-particle correlation function. The natural assumption of thee existence of a true condensate at T = 0 automatically comes out of these calculations.. In the TF regime, where the mean-field (repulsive) interparticle interaction greatly exceedss the radial (u p ) and axial (UJ Z ) trap frequencies, the density profile of the zerotemperaturee condensate has the well-known shape no(p, z) = no m (l p 2 /R 2 z 2 /L 2 ), wheree no m = fi/g is the maximum condensate density, with // being the chemical potential,, g = knk 2 a/m, m the atom mass, and a > 0 the scattering length. Under the conditionn u p» w z, the radial size of the condensate, R = (2^/mu; 2,) 1 / 2, is much smaller thann the axial size L = (2^/ma; 2 ) 1 / 2. Fluctuationss of the density and phase of the condensate, in particular at finite T, are relatedd to elementary excitations of the cloud. The density fluctuations are dominated byy the excitations with energies of the order of \i. The wavelength of these excitations is muchh smaller than the radial size of the condensate. Hence, the density fluctuations have thee ordinary 3D character and are small. Therefore, one can write the total field operator off atoms as ^(r) = ^/no(r)exp(i0(r)), where < (r) is the operator of the phase. The single-particlee correlation function is then expressed through the mean square fluctuations off the phase (see, e.g. [12]): (Vit( r )^( r '))) = v/n 0 (r)n 0 (r')exp{-<[^(r, r')] 2 )/2}, (6.1.1) withh <S< (r,r') = < (r) - < (r'). The operator < (r) is given by (see, e.g., [71]) 0(p)) = ^(r)]" 1 / 2 f+ {r)&i/ + hmcn (6. L2) V V wheree a v is the annihilation operator of the excitation with quantum number (s) v and energyy, ƒ+ = u v + v v, and the w, v functions of the excitations are determined by the Bogolyubov-dee Gennes equations. Thee excitations of elongated condensates can be divided into two groups: "low energy" axiall excitations with energies e v < hw p, and "high energy" excitations with e > huj p. Thee latter have 3D character as their wavelengths are smaller than the radial size R. Therefore,, as in ordinary 3D condensates, these excitations can only provide small phase fluctuations.fluctuations. The low-energy axial excitations have wavelengths larger than R and exhi aa pronounced ID behavior. Hence, one expects that these excitations give the most importantt contribution to the long-wave axial fluctuations of the phase. Thee solution of the Bogolyubov-de Gennes equations for the low-energy axial modes givess the spectrum j = hw z^jj{j + 3)/4 [42], where j is a positive integer. The wavefunctionss fj~ of these modes have the form f+(r)f+(r) _ /(7 + 2)(2j+3)jno(r) (ltl),z\

69 SectionSection 6.1. Phase-üuctuating 3D condensates in elongated traps 63 4-r r 3-- ff UU "T " i i!i Izl/L L Figuree 6.1.1: The function f(z/l). The solid curve shows the result of our numerical calculation,, and the dotted line is f(z) = 2\z\/L following from Eq. (6.1.5). wheree Pj are Jacobi polynomials. Note that the contribution of the low-energy axial excitationss to the phase operator (6.1.2) is independent of the radial coordinate p. Relyingg on Eqs. (6.1.2) and (6.1.3), we now calculate the mean square axial fluctuationss of the phase at distances \z - z'\ < R. As in ID trapped gases (see Chapter 5), the vacuumm fluctuations are small for any realistic axial size L. The thermal fluctuations are determinedd by the equation (we count only the "low energy" excitations) J.J. Cj <ÜrUüp withh N 0 = (87r/15)n 0 m-r 2 i being the number of Bose-condensed particles, and Nj the equilibriumm occupation numbers for the excitations. The main contribution to the sum overr j in Eq. (6.1.4) comes from several lowest excitation modes, and at temperatures TT» hu z we may put Nj = T/tj. Then, in the central part of the cloud ( z, z' 4C L) a straightforwardd calculation yields ([04>{z,z')]([04>{z,z')] 22 )) TT = 5l\z-z'\/L, (6.1.5 wheree the quantity b\ represents the phase fluctuations on a distance scale \z z'\ ~ L andd is given by SS 22 LL(T)(T) = 32fiT/15N 0 (huj z ) 2. (6.1.6) Notee that at any z and z' the ratio of the phase correlator (6.1.4) to 5 2 L is a universal functionn of z/l and z'/'l: ([64>(z,([64>(z, z')] 2 ) T = 6 2 L(T)f(z/L, z'/l). (6.1.7) Inn Fig we present the function f (z/l) = f {z/l, -z/l) Eq.. (6.1.4). calculated numerically from

70 64 4 ChapterChapter 6. Phase-fluctuating 3D Bose-Einstein condensate Thee phase fluctuations decrease with temperature. As the TF chemical potential is fifi = {l5n 0 g/n) 2 / 5 (mü 2 /8) 3^ (ü = Wp /3 u;] /3 ), Eq. (6.1.6) can be rewritten in the form S\S\ = {T/T C ){N/N 0 )VH 2 C, (6.1.8) wheree T c «N l / Z hw is the BEC transition temperature, and N the total number of particles.. The presence of the 3D BEC transition in elongated traps requires the inequality TT cc» hujp and, hence, limits the aspect ratio to u) p /uj z <C N. The parameter S 2 is given by y 32/i(AT 00 = N) fu p \ 4/3 _ gwmwj?' 16 Exceptt for a narrow interval of temperatures just below T c, the fraction of non-condensed atomss is small and Eq. (6.1.8) reduces to <$ = (T/T C )S 2. Thus, the phase fluctuations can bee important at large values of the parameter S 2, whereas for 5 2 <C 1 they are small on anyy distance scale and one has a true Bose-Einstein condensate. In Fig we present thee quantity S 2 for the parameters of various experiments with elongated condensates. Inn the Konstanz [144] and Hannover [145] experiments the ratio T/T c was smaller thann 0.5. In the recent experiment [65] the value S 2 C «3 has been reached, but the temperaturee was very low. Hence, the axial phase fluctuations were rather small in these experimentss and they were dealing with true condensates. The last statement also holds forr the ENS experiment [146] where T was close to T c and the Bose-condensed fraction wass N 0 /N «0.75. Thee single-particle correlation function is determined by Eq. (6.1.1) only if the condensatee density no is much larger than the density of non-condensed atoms, n'. Otherwise, thiss equation should be completed by terms describing correlations in the thermal cloud. However,, irrespective of the relation between no and n', Eq. (6.1.1) and Eqs. (6.1.4)- (6.1.6)) correctly describe phase correlations in the condensate as long as the fluctuations off the condensate density are suppressed. This is still the case for T close to T c and NN 00 <C N, if we do not enter the region of critical fluctuations. Then, Eq. (6.1.8) gives S\S\ (N/NQ) 3/5. At the highest temperatures of the Bose-condensed cloud in the MIT sodiumm experiment [147], the condensed fraction was No/N ~ 0.1 and the phase fluctuationss were still small. Onn the contrary, for N 0 /N «0.06 in the hydrogen experiment [148], with S 2 from Fig we estimate 8\ «1. The same or even larger value of 8\ was reached in the Munichh Rb experiment [142] where the gas temperature was varying in a wide interval aroundd T c. In the Rb experiment at AMOLF [149], the smallest observed Bose-condensed fractionn was NQ/N «0.03, which corresponds to Si «5. However, axial fluctuations of thee phase have not been measured in these experiments. Wee will focus our attention on the case where NQ «N and the presence of the axial phasee fluctuations is governed by the parameter S 2. For 5 2» 1, the nature of the Bosecondensedd state depends on temperature. In this case we can introduce a characteristic temperature e T 00 = 15(^) 2 AT/32/i (6.1.10) att which the quantity Si «1 (for N 0» N). In the temperature interval T<f, < T < T c, thee phase fluctuates on a distance scale smaller than L. Thus, as the density fluctuationss are suppressed, the Bose-condensed state is a condensate with fluctuating phase or

71 SectionSection 6.1. Phase-Buctuating 3D condensates in elongated traps // / % % * * + + r-fh r-fh ii (o/co (o/co PP z Figuree 6.1.2: The parameter 6% for experiments with elongated condensates. The up and downn triangles stand for 6% in the sodium [147] and hydrogen [148] MIT experiments, respectively.. Square, cross, diamond, circle, and star show &\ for the rubidium experiments att Konstanz [144], Munich [142], Hannover [145], ENS [146], and AMOLF [149]. quasicondensate.. The expression for the radius of phase fluctuations (phase coherence length)) follows from Eq. (6.1.5) and is given by 1$1$» L(T^/T). (6.1.11) ) Thee phase coherence length 1$ greatly exceeds the correlation length l c = h/^/mji. Eqs.. (6.1.11) and (6.1.10) give the ratio l^/l c «(T c /T)(T c /hw p ) 2 > 1. Therefore, the quasicondensatee has the same density profile and local correlation properties as the true condensate.. However, the phase coherence properties of quasicondensates will be drasticallyy different (see below). Thee decrease of temperature to well below T$ makes the phase fluctuations small [b\[b\ -C 1) and continuously transforms the quasicondensate into a true condensate. Itt is interesting to compare the described behavior of the interacting gas for 6% 3> 1, withh the two-step BEC predicted for the ideal Bose gas in elongated traps [150]. In bothh cases, at T c the particles Bose-condense in the ground state of the radial motion.. However, the ideal gas remains non-condensed (thermal) in the axial direction for TT > TID = Nhu> z /\n2n (assuming T\D < T c ), and there is a sharp cross-over to the axiall BEC regime at T w Tip. The interacting Bose gas below T c forms the 3D TF (non-fluctuating)) density profile, and the spatial correlations become non-classical in all directions.. For 8\ S> 1, the axial phase fluctuations at T ~ T c are still large, and one has aa quasicondensate which continuously transforms into a true condensate at T below T^. Notee that T^ is quite different from T\D of the ideal gas. Lett us now demonstrate that 3D elongated quasicondensates can be achieved for realisticc parameters of trapped gases. As found above, the existence of a quasicondensate requiress large values of the parameter 8% given by Eq. (6.1.9). Most important is the dependencee of 6% on the aspect ratio of the cloud UJ P /LJ Z, whereas the dependence on thee number of atoms and on the scattering length is comparatively weak. Fig

72 66 6 ChapterChapter 6. Phase-üuctuating 3D Bose-Einstein condensate lij.u f/t.f/t. / \X\X // a) *ZL--" i ' r r co/co co/co PP z Figuree 6.1.3: The ratios T c /T$ = 5% and /i,/t$ in (a) and the temperature T$ in (b), versuss the aspect ratio IJÜ P /CÜ Z for trapped Rb condensates with N = 10 5 and to p = 500 Hz.. showss Tc/Ttj, = S 2, fji/t,),, and the temperature T$ as functions of uj p /to z for rubidium condensatess a,t N = 10 5 and UJ P = 500 Hz. Comparing the results for 5% in Fig with thee data in Fig , we see that 3D quasicondensates can be obtained by transforming thee presently achieved BEC's to more elongated geometries corresponding to UJ P /UJ Z > 50. Onee can distinguish between quasicondensates and true BEC's in various types of experiments.. By using the Bragg spectroscopy method developed at MIT one can measuree the momentum distribution of particles in the trapped gas and extract the coherence lengthh /^[143]. The use of two (axially) counter-propagating laser beams to absorb a photonn from one beam and emit it into the other one, results in axial momentum transfer too the atoms which have momenta at Doppler shifted resonance with the beams. These atomss form a small cloud which will axially separate from the rest of the sample provided thee mean free path greatly exceeds the axial size L. The latter condition can be assured byy applying the Bragg excitation after abruptly switching off the radial confinement of thee trap. The axial momentum distribution is then conserved if the dynamic evolution off the cloud does not induce axial velocities. According to the scaling approach [67; 68], thiss is the case for the axial frequency decreasing as uj z {t) = w 2 (0)[l + ujph 2 ]^1/ 2.

73 SectionSection 6.2. Observation of phase fluctuations in Bose-Einstein condensates 67 Inn "juggling" experiments described in Chapter 5 and similar to those at NIST and Munichh [141; 142], one can directly measure the single-particle correlation function. The latterr is obtained by repeatedly ejecting small clouds of atoms from the parts z and z' off the sample and averaging the pattern of interference between them in the detection regionn over a large set of measurements. As follows from Eqs. (6.1.6) and (6.1.7), for z'z' = z the correlation function depends on temperature as exp{ Sj j (T)f(z/L)/2}, wheree f{z/l) is given in Fig Thee phase fluctuations are very sensitive to temperature. From Fig we see that onee can have T^/T c < 0.1, and the phase fluctuations are still significant at T < /J, where onlyy a tiny indiscernible thermal cloud is present. Thiss suggests a principle for thermometry of 3D Bose-condensed gases with indiscerniblee thermal clouds. If the sample is not an elongated quasicondensate by itself, itt is first transformed to this state by adiabatically increasing the aspect ratio UJ P /UJ Z. Thiss does not change the ratio T/T c as long as the condensate remains in the 3D TF regime.. Second, the phase coherence length 1$ or the single-particle correlation function aree measured. These quantities depend on temperature if the latter is of the order of T$ orr larger. One thus can measure the ratio T/T c for the initial cloud, which is as small as thee ratio T$/T c for the elongated cloud. Wee believe that the studies of phase coherence in elongated condensates will reveal manyy new interesting phenomena. The measurement of phase correlators will allow one too study the evolution of phase coherence in the course of the formation of a condensate outt of a non-equilibrium thermal cloud Observation of phase fluctuations in Bose-Einstein condensates s WeWe describe a method of observing phase fluctuations through the measurement of the densitydensity profile after releasing the gas from the trap. It is shown how the phase fluctuationstions of an elongated BEC transform into the density modulations in the course of free expansion.expansion. The average value of the modulations is found as a function of the number of particles,particles, temperature, time of flight, and trap frequencies. Fluctuationss of the phase of a Bose condensate are related to thermal excitations and alwayss appear at finite temperature. However, as shown in Section 6.1, the fluctuations dependd not only on temperature but also on the trap geometry and on the particle number.. Typically, fluctuations in spherical traps are strongly suppressed as the wavelengths off excitations are smaller than the size of the atomic cloud. In contrast, wavelengths of thee excitations in strongly elongated traps can be larger than the radial size of the cloud. Inn this case, the low-energy axial excitations acquire a ID character and can lead to more pronouncedd phase fluctuations, although the density fluctuations of the equilibrium state aree still suppressed. Because of the pronounced phase fluctuations, the coherence propertiess of elongated condensates can be significantly altered as compared with previous observations.. In particular, the axial coherence length can be much smaller than the size off the condensate, which can have dramatic consequences for practical applications.

74 ChapterChapter 6. Phase-fluctuating 3D Bose-Einstein condensates (a) ) i l T~0.6T C C r r J J T~0.9T a a x[\im] x[\im] x[(im] ] Figuree 6.2.1: Absorption images and corresponding density profiles of BECs after 25 ms time-of-flightt taken for various temperatures T and aspect ratios [A = 10 (a), 26 (b), 51 (c)].. Thiss section provides a theoretical basis for the experiment on observation of phase fluctuationss carried out at the University of Hannover (see [151; 152]). The experiment is performedd with Bose-Einstein condensates of up to N 0 = 5 x 10 5 rubidium atoms in the \F\F = 2, rrif = +2} state of a cloverleaf-type magnetic trap. The fundamental frequencies off the magnetic trap are cu x = 2n x 14 Hz and ui p = 2ir x 365 Hz along the axial and radial direction,, respectively. Due to the highly anisotropic confining potential with an aspect ratioo A = LO P /UJ X of 26, the condensate is already elongated along the horizontal x axis. Inn addition, further radial compression of the ensemble by means of a superimposed blue detunedd optical dipole trap is possible [65]. The measurements are performed for radial trapp frequencies UJ P between 2TT x 138 Hz and 2n x 715 Hz corresponding to aspect ratios AA between 10 and 51. After rf evaporative cooling to the desired temperature, the system iss allowed to reach an equilibrium state by waiting for lsec (with rf "shielding"). Then, withinn 200 /xs the trapping potential is switched off and the density profile of the cloud iss detected after a variable time-of-flight. Figuree shows examples of experimental data for various temperatures T < T c andd aspect ratios A. The usual anisotropic expansion of the condensate related to the anisotropyy of the confining potential is clearly visible in the absorption images. The

75 SectionSection 6.2. Observation of phase Buctuations in Bose-Einstein condensates 69 linee density profiles below reflect the parabolic shape of the BEC density distribution. Remarkably,, pronounced stripes in the density distribution are observed in some cases. Onn average these stripes are more pronounced at high aspect ratios of the trapping potentiall [Fig (c)], high temperatures (bottom row of Fig ), and low atom numbers.. Thee appearance of stripes in the process of ballistic expansion can be understood qualitativelyy as follows. As mentioned in Section 6.1, in the equilibrium state of a BEC inn a trap the density distribution remains largely unaffected even if the phase fluctuates. Thee reason is that the mean-field interparticle interaction prevents the transformation of locall velocity fields provided by the phase fluctuations into modulations of the density. However,, after switching off the trap, the mean-field interaction rapidly decreases and thee axial velocity field is then converted into a particular density distribution. Thee stripes and their statistical properties can be described analytically using the scalingg theory (see e.g. [66; 67; 68]). The condensate without initial fluctuations evolve accordingg to the scaling solution *=^gr e '' (6.2.D wheree <f>o = (m/2h)(b p /b p )p 2, p is the radial coordinate, and no is the initial Thomas- Fermii density profile. The scaling parameter b p (t) is the ratio of the radial size at time tt to the initial size. This parameter is determined by the scaling equations [66; 67; 68]). Forr initially elongated condensate, on a time scale t <C <JJ 2 /<JJ Z we have b 2 p{t) = 1 + u) 2 t 2, whereass the axial size of the released cloud remains unchanged. Equationss determining the ballistic expansion in the presence of the density and phase fluctuations,fluctuations, Sn and <f>, are obtained from the Gross-Pitaevskii equation linearized wi regardd to On, V<f> around the solution (6.2.1). This gives two coupled equations: *,, = ^-«SÖSfi, (6.2.2) gnpsn 1 - fsn\ hv 2 Jn -*** " "Mf + 4t HW""ST' (623) wheree f = -(h/m)(v p >no V p ' + nov 2,,), and p' p/b p. Combining Eqs. (6.2.2) and (6.2.3),, we find: Inn the Thomas-Fermi limit {\ijhjj p» 1) the last term in the rhs of Eq. (6.2.4) can be neglected.. For the axial excitations the quantities 0 and Sn are independent of the radial coordinatee (see [42], and Eq. (6.1.2) in the previous section). This allows us to average Eq.. (6.2.4) over the radial coordinate. The averaging procedure nullifies the first term in thee rhs and yields

76 70 0 ChapterChapter 6. Phase-fluctuating 3D Bose-Einstein condensates withh n(x) = /i(l - x 2 /L 2 ), and L being the axial size. At t = 0 we have b p = 1, andd Eq. (6.2.5) is nothing else than the eigenmode equation for the axial excitations of trappedd elongated condensates. For low-energy excitations the third term in the lhs of thee eigenmode equation (quantum pressure term) can be omitted and one has the energy spectrumm ej - hw x y/j(j + 3)/4, where j is a positive integer (see [42] and the previous section).. Thee main contribution to the density modulations in the expanding cloud comes from quasiclassicall excitations (j > 1), at least on a time scale t» l/u p. Therefore, we will relyy on the local density approximation and neglect spatial derivatives of fi(x). Then Eq.. (6.2.5) transforms to the equation for (local) Bogolyubov eigenmodes characterized byy the axial momentum k: hh 22 SnSn kk + e%(t)sn k =0, (6.2.6) ) where e ëfc(t)) = 'fi{x)h'fi{x)h22 kk 22 h*k 4 2mb2mb 2 2 4m 2 2 (6.2.7) ) iss the "local Bogolyubov spectrum". For short times, namely t < 1/UJ P, the phonon partt of this spectrum is dominant and ë k (t) w hc(x)k/b p (t), where c(x) = y/fi(x)/2m is thee local velocity of axially propagating phonons in the initial cloud. Then Eq. (6.2.6) is solvedd in terms of hypergeometric functions of the variable -.22 _ -u# 2 -- Forr larger times wee have b p m u) p t and Eq. (6.2.6) can be solved in terms of Bessel functions of the variable tt = è\{qi)t/hiji{x). The two regimes may be matched by using the asymptotic expression forr the hypergeometric function at r > 1, and for the Bessel function at t < 1. In this wayy we obtain an analytic expression for the density fluctuations ön(x, t) = ]T fc 5n k {x, t). Sincee the momentum k and the quantum number j are related to each other as k ww xx j/2c(x),j/2c(x), the relative density fluctuations for r» 1 can be represented in the form: Sn(x,t) Sn(x,t) == 2 2_j (f>j exp { (ej/hu> no(x) no(x) p ) 2 In r } sin ^ ^ ^(x)^ ^ (6.2.8) ) wheree <pj is the initial (t=0) contribution of the j-th mode to the phase operator in Eq.. (6.1.2) (withi/ = j). Fromm Eq. (6.2.8) one obtains a closed relation for the mean square density fluctuationss <J 2 by averaging (<5n/n 0 ) 2 over different realizations of the initial phase. Taking intoo account that ((&j + a])(cbj> + a],))/4 = NJSJJI/2, using the quasiclassical average (P.(P. ' (x)) 2 ~ 4(1 - :r 2 ) -3 / 2 /7rj, and transforming the sum over j into an integral we j j obtain n uu xx jj } Inr \n\n 00 (x(x }} t))t)) ) TT(1 - (x/l)*)w T* J j=0 j 2 4fi(x)uj4fi(x)uj *** U^OuJ p p ^ ( 4> 2 2 ) (6.2.9) ) Inn the central part of the cloud (x «0) the integration of Eq. (6.2.9) gives '(on(0,t)\'(on(0,t)\ 22 \\ = (<j BEC \ 2 _ UJ X T /h77 /,, (hw p r \no(q,t)j\no(q,t)j / \ no J WpT^V n M 1 + V + V2V2. (6.2.10) Ulnr

77 SectionSection 6.2. Observation of phase fluctuations in Bose-Einstein condensates 71 I I ^^ O.OTT h rf rf & & x/l L Figuree 6.2.2: (a) A typical initial phase distribution calculated for co x = 2-K X 14 HZ, uipuip = 2n x 508 Hz, N 0 = 2 x 10 5, and T = 0.5 T c. (b) Corresponding density profile after 255 ms time-of-flight from a direct solution of the Gross-Pitaevskii equation (solid line) andd analytical theory (dashed line). Notee that Eq. (6.2.10) provides a direct relation between the observed density fluctuations andd temperature, and thus can be used for thermometry at very low T. Inn the experiment [151; 152] Eq. (6.2.10) has been used to compare directly the averagedd measured value of [(CBEC/^O) 2 ]^ * tne predicted value of (cr BEC /no) t heor y for variouss values of T,NQ,LO X,LJ P, and t (see Fig ). The theoretical value takes into accountt the limited experimental imaging resolution. Thee experimental results follow the expected general dependence very well. With the directt link of the phase fluctuations in the magnetic trap to the observed density modulationn given by Eq. (6.2.10), the data therefore confirm the predicted general behavior of phasee fluctuations in elongated BECs. However, the measured values are approximately byy a factor of 2 smaller than those predicted by theory. This discrepancy could be due to, forr example, a reduction of the observed contrast caused by a small tilt in the detection laserr beam with respect to the radial stripes. Most of the observed experimental data, whichh exhibit fluctuations well above the noise limit, correspond to T > T$. This implies thatt the measurements were performed in the regime of quasicondensation; i.e., the phase coherencee length 1$ = LT$/T of the initial condensate was smaller than the axial size L. Forr instance, for A = 51, T = 0.5 T c, and N 0 = 3 x 10 4, one obtains 1$ w L/3. Recently,, theoretical predictions of this chapter have been experimentally tested by Gerbierr et al. [153]. They have measured the temperature-dependent coherence length of

78 722 Chapter 6. Phase-fluctuating 3D Bose-Einstein condensates u.zu C3 3 "" a a ti ti ^^ a a o o A A X=10 0 >.. = 22 XX = 26 -'* A.. = 36 ^ XX = nm-- % %, A aa -a /n o)to«y y Figuree 6.2.3: Average standard deviation of the measured line densities [ (er BEC /no) 2 ]*/ p 2 comparedd to the theoretical value of (o" BEC /7io) t heory obtained from Eq. (6.2.10). The dashedd line is a fit to the experimental data. ann elongated Bose condensate by using Bragg spectroscopy and have found quantitative agreementt between their measurements and our theoretical results. The measurements aree consistent with the absence of density fluctuations, even when the phase fluctuationss are strong, and confirm the existence of the regime of quasicondensate - novel phenomenonn in the physics of elongated 3D BECs.

79 Bibliography y [1]] S. Bose, Z. Phys. 26, 178 (1924). [2]] A. Einstein, Sitzber. Kgl. Preuss. Akad. Wiss. 261 (1924). [3]] M. H. Anderson et a/., Science 269, 198 (1995). [4]] K. B. Davis et a/., Phys. Rev. Lett. 75, 3969 (1995). [5]] C. C. Bradley, C. A. Sackett, J. J. Tolett, and R. G. Hulet, Phys. Rev. Lett. 75, (1995). [6]] A. Görlitz et al, Phys. Rev. Lett. 87, (2001). [7]] S. Burger et al, Europhys. Lett. 57, 1 (2002). [8]] F. Schreck et al., Phys. Rev. Lett. 87, (2001). [9]] M. Greiner et a/., Phys. Rev. Lett. 87, (2001). 10]] N. D. Mermin and H. Wagner, Phys. Rev. Lett. 17, 1133 (1966). 11]] P. C. Hohenberg, Phys. Rev. 158, 383 (1967). 12]] V. N. Popov, Functional Integrals in Quantum Field Theory and Statistical Physics (D.. Reidel Pub., Dordrecht, 1983). 13]] Y. Kagan, B. V. Svistunov, and G. V. Shlyapnikov, Sov. Phys. JETP 66, 480 (1987).. 14]] D. J. Bishop and J. D. Reppy, Phys. Rev. Lett. 40, 1727 (1978). 15]] D. J. Bishop and J. D. Reppy, Phys. Rev. B 22, 5171 (1980). 16]] A. I. Safonov et a/., Phys. Rev. Lett. 81, 4545 (1998). 17]] E. Hanamura and H. Huag, Phys. Rep. 33, 209 (1977). 18]] C. Comte and P. Nozières, J. Phys. (Paris) 43, 1069 (1982). 19]] A. Mysyrowicz, J. Phys. (Paris) 41, 281 (1980). 20]] A. Griffin, D. W. Snoke, and S. Stringari, Bose-Einstein Condensation of Excitons andand Biexcitons (Cambridge Univ. Pess, Cambridge, 1995). 21]] D. W. Snoke, J. P. Wolfe, and A. Mysyrowicz, Phys. Rev. Lett. 64, 2543 (1990). 22]] D. Snoke et al, Nature 418, 754 (2002). 23]] L. V. Butov et al, Nature 417, 47 (2002). 24]] L. V. Butov, A. C. Gossard, and D. S. Chemla, Nature 418, 751 (2002). 25]] Proceedings of the International Conference on High-T c Superconductors and Materialsterials and Mechanisms of Superconductivity, Interlaken, Switzerland, edited by J. Mullerr and J. L. Olsen, Physica (Amsterdam) C, Pts. I and II (1988). [26]] R. Brusetti et a/., Physica C , 459 (1988). 73 3

80 74 4 Bibliography Bibliogra V.. Vuletic, A. J. Kerman, C. Chin, and S. Chu, Phys. Rev. Lett. 82, 1406 (1999). V.. Vuletic, C. Chin, A. J. Kerman, and S. Chu, Phys. Rev. Lett. 83, 943 (1999). I.. Bouchoule, Ph.D. thesis, LKB ENS, Paris, I.. Bouchoule, M. Morinaga, C. Salomon, and D. S. Petrov, Phys. Rev. A 65, (2002).. M.. Hammes et al, physics/ V.. Bagnato and D. Kleppner, Phys. Rev. A 44, 7439 (1991). W.. Ketterle and N. J. van Druten, Phys. Rev. A 54, 656 (1996). W.. J. Mullin, J. Low Temp. Phys. 106, 615 (1997). E.. B. Sonin, Sov. Phys. JETP 29, 520 (1969). L.. D. Landau and E. M. Lifshitz, Quantum Mechanics, Non-relativistic (Butterworth-Heinemann,, Oxford, 1999). Theory P.. G. de Gennes, Superconductivity of Metals and Alloys (Benjamin, New York, 1966).. S.. Stringari, Phys. Rev. Lett. 77, 2360 (1996). P.. Öhberg et al, Phys. Rev. A 56, R3346 (1997). M.. Fliesser, A. Csordas, P. Szepfalusy, and R. Graham, Phys. Rev. A 56, R2533 (1997).. F.. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari, Rev. Mod. Phys. 71, 463 (1999).. S.. Stringari, Phys. Rev. A 58, 2385 (1998). T.-L.. Ho and M. Ma, J. Low Temp. Phys. 115, 61 (1999). W.. Kane and L. Kadanoff, Phys. Rev. 155, 80 (1967). V.. L. Berezinskii, Sov. Phys. JETP 32, 493 (1971). V.. L. Berezinskii, Sov. Phys. JETP 34, 610 (1972). J.. M. Kosterlitz and D. J. Thouless, J. Phys. C 6, 1181 (1973). J.. M. Kosterlitz, J. Phys. C 7, 1046 (1974). N.. Prokof'ev, O. Ruebenacker, and B. Svistunov, Phys. Rev. Lett. 87, (2001).. N.. Prokof'ev and B. Svistunov, Phys. Rev. A 66, (2002). M.. Schwartz, Phys. Rev. B 15, 1399 (1977). F.. D. M. Haldane, Phys. Rev. Lett. 47, 1840 (1981). L.. P. Pitaevskii and S. Stringari, J. Low Temp. Phys. 85, 377 (1991). M.. Girardeau, J. Math. Phys. 1, 516 (1960). M.. Girardeau, Phys. Rev. 139, B500 (1965). E.. H. Lieb and W. Liniger, Phys. Rev. 130, 1605 (1963).

81 Bibliography Bibliography 75 5 [57]] E. H. Lieb, Phys. Rev. 130, 1616 (1963). [58]] J. Fortagh, A. Grossmann, C. Zimmermann, and T. W. Hansen, Phys. Rev. Lett. 81,, 5310 (1998). [59]] J. Reichel, W. Hansel, and T. W. Hansen, Phys. Rev. Lett. 83, 3398 (1999). [60]] D. Muller et al., Phys. Rev. Lett. 83, 5194 (1999). [61]] N. H. Dekker et al, Phys. Rev. Lett. 84, 1124 (2000). [62]] M. Key et al., Phys. Rev. Lett. 84, 1371 (2000). [63]] R. Folman et al, Phys. Rev. Lett. 84, 4749 (2000). [64]] T. Kuga et al, Phys. Rev. Lett. 78, 4713 (1997). [65]] K. Bongs et al, Phys. Rev. A 63, (R) (2001). [66]] Y. Kagan, E. L. Surkov, and G. V. Shlyapnikov, Phys. Rev. A 54, R1753 (1996). [67]] Y. Castin and R. Dim, Phys. Rev. Lett. 77, 4315 (1996). [68]] Y. Kagan, E. L. Surkov, and G. V. Shlyapnikov, Phys. Rev. A 55, R18 (1997). [69]] E. M. Lifshitz and L. P. Pitaevskii, Statistical Physics, Part 2 (Pergamon Press, Oxford,, 1980). [70]] J. T. M. Walraven, in Atomic Hydrogen on Liquid Helium Surfaces, Fundamentaltal Systems in Quantum Optics, Proceedings of the Les Houches Summer School, SessionSession LIII, edited by J. Dalibard, J. M. Raimond, and J. Zinn-Justin (Elsevier Sciencee Publishers, Amsterdam, 1992). [71]] S. I. Shevchenko, Sov. J. Low Temp. Phys. 18, 223 (1992). [72]] H. Gauck et al, Phys. Rev. Lett. 81, 5298 (1998). [73]] V. Vuletic, C. Chin, A. J. Kerman, and S. Chu, Phys. Rev. Lett. 81, 5768 (1998). [74]] I. Bouchoule et al, Phys. Rev. A 59, R8 (1999). [75]] M. Morinaga, I. Bouchoule, J.-C. Karam, and C. Salomon, Phys. Rev. Lett. 83, (1999). [76]] M. R. Andrews et al, Science 275, 637 (1997). [77]] A. A. Abrikosov, L. P. Gorkov, and I. E. Dzyaloshinski, Methods of Quantum Field TheoryTheory in Statistical Physics (Dover Publications Inc., New York, 1975). [78]] R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals (McGraw- Hill,, New York, 1965). [79]] S. J. J. M. F. Kokkelmans, B. J. Verhaar, and K. Gibble, Phys. Rev. Lett. 81, 951 (1998).. [80]] D. Guery-Odelin, J. Soding, P. Desbiolles, and J. Dalibard, Optics Express 2, 323 (1998).. [81]] S. A. Hopkins et al, Phys. Rev. A 61, (2000). [82]] J. Stenger et al, Phys. Rev. Lett. 82, 2422 (1999). [83]] L. Pricoupenko and M. Olshanii, cond-mat/

82 76 6 Bibliography Bibliogr [84]] P. O. Fedichev, M. W. Reynolds, and G. V. Shlyapnikov, Phys. Rev. Lett. 77, 2921 (1996).. [85]] B. D. Esry, C. H. Greene, and J. P. B. Jr., Phys. Rev. Lett. 83, 1751 (1999). [86]] B. DeMarco and D. Jin, Science 285, 1703 (1999). [87]] T. Loftus et ai, Phys. Rev. Lett. 88, (2002). [88]] G. Roati, F. Riboli, G. Modugno, and M. Inguscio, Phys. Rev. Lett. 89, (2002).. [89]] G. Modugno et ai, Science 297, 2240 (2002). [90]] A. Truscott et al, Science 291, 2570 (2001). [91]] S. Granade, M. E. Gehm, K. M. O'Hara, and J. E. Thomas, Phys. Rev. Lett. 88, (2002). [92]] K. M. O'Hara et al, Phys. Rev. A 66, (R) (2002). [93]] Z. Hadzibabic et ai, Phys. Rev. Lett. 88, (2002). [94]] K. Dieckmann et al., Phys. Rev. Lett. 89, (2002). [95]] K. O'Hara et al, Science 298, 2179 (2002). [96]] K. Miyake, Progr. Theor. Phys. 69, 1794 (1983). [97]] M. Randeria, J.-M. Duan, and L.-Y. Shieh, Phys. Rev. Lett. 62, 981 (1989). [98]] S. Schmitt-Rink, C. M. Varma, and A. E. Ruckenstein, Phys. Rev. Lett. 63, 445 (1989).. [99]] M. Drechsler and W. Zwerger, Ann. Phys. 1, 15 (1992). [100]] L. P. Gor'kov and T. K. Melik-Barkhudarov, Sov. Phys. JETP 13, 1018 (1961). 101]] A. J. Leggett, Rev. Mod. Phys. 47, 331 (1975). 102]] M. Randeria, Bose-Einstein Condensation, edited by A. Griffin, D. Snoke, and S. Stringarii (Cambridge University Press, Cambridge, 1995). 103]] M. Y. Kagan, Sov. Phys. Uspekhi 37, 69 (1994). 104]] D. M. Eagles, Phys. Rev. 186, 456 (1969). 105]] A. J. Leggett, Modern Trends in the Theory of Condensed Matter, edited by A. Pekalskii and J. Przystawa (Springer, Berlin, 1980). 106]] P. Nozieres and S. Schmitt-Rink, J. Low Temp. Phys. 59, 195 (1985). 107]] M. Holland, S. J. J. M. F. Kokkelmans, M. L. Chiofalo, and R. Walser, Phys. Rev. Lett.. 87, (2001). 108]] E. Timmermans, K. Furuya, P. W. Milonni, and A. K. Kerman, Phys. Lett. A 285, 2288 (2001). 109]] D. S. Petrov, Phys. Rev. A 67, (2003). 110]] D. J. Heinzen et ai, workshop on Cold Molecules, Les Houches, Ill]] V. Efimov, Sov. J. Nucl. Phys. 12, 589 (1971).

83 Bibliography Bibliography 77 7 [112]] V. Efimov, Nucl. Phys. A 210, 157 (1973). [113]] E. Nielsen, D. V. Fedorov, and A. S. Jensen, Phys. Rev. A 56, 3287 (1997). [114]] E. Tiesinga, A. J. Moerdijk, B. J. Verhaar, and H. T. C. Stoof, Phys. Rev. A 46, R11677 (1992). [115]] E. Tiesinga, B. J. Verhaar, and H. T. C. Stoof, Phys. Rev. A 47, 4114 (1993). [116]] S. Inouye et a/., Nature 392, 151 (1998). [117]] P. Courteille et al, Phys. Rev. Lett. 81, 69 (1998). [118]] C. Chin, V. Vuletic, A. J. Kerman, and S. Chu, Phys. Rev. Lett. 85, 2717 (2000). [119]] J. L. Roberts, N. R. Claussen, S. L. Cornish, and C. E. Wieman, Phys. Rev. Lett. 85,, 728 (2000). [120]] S. L. Cornish et ai, Phys. Rev. Lett. 85, 1795 (2000). [121]] Y. Kagan, G. V. Shlyapnikov, I. A. Vartan'yantz, g^ N. A. Glukhov, JETP Lett. 35,, 477 (1982). [122]] P. J. Leo, C. J. Williams, and P. S. Julienne, Phys. Rev. Lett. 85, 2721 (2000). [123]] V. Vuletic, private communication. [124]] J. Weiner, V. S. Bagnato, S. Zilio, and P. S. Julienne, Rev. Mod. Phys. 71, 1 (1999).. [125]] A. S. Davydov, Quantum mechanics (Pergamon, Oxford, 1976). [126]] P. O. Fedichev, M. W. Reynolds, U. M. Rahmanov, and G. V. Shlyapnikov, Phys. Rev.. A 53, 1447 (1996). [127]] L. Tonks, Phys. Rev. 50, 955 (1936). [128]] C. N. Yang and C. P. Yang, J. Math. Phys. 10, 1115 (1969). [129]] J. W. Kane and L. P. Kadanoff, J. Math. Phys. 6, 1902 (1965). [130]] L. Reatto and G. V. Chester, Phys. Rev. 155, 88 (1967). [131]] D. J. Scolapino, M. Sears, and R. A. Ferrell, Phys. Rev. B 6, 3409 (1972). [132]] Y. Castin et al, J. Mod. Opt. 47, 2671 (2000). [133]] E. B. Sonin, Sov. Phys. JETP 32, 773 (1971). [134]] Y. Kagan, N. V. Prokof'ev, and B. V. Svistunov, Phys. Rev. A 61, (2000). [135]] J. Denschlag, D. Cassettari, and J. Schmiedmayer, Phys. Rev. Lett. 82, 2014 (1999).. [136]] M. Olshanii, Phys. Rev. Lett. 81, 938 (1998). [137]] E. B. Kolomeisky, T. J. Newman, J. P. Straley, and X. Y. Qi, Phys. Rev. Lett. 85, (2000). [138]] M. D. Girardeau and E. M. Wright, Phys. Rev. Lett. 84, 5239 (2000). [139]] H. Monien, M. Linn, and N. Elstner, Phys. Rev. A 58, R3395 (1998). [140]] J. H. Thywissen et o/., Eur. Phys. J. D 7, 361 (1999).

84 78 8 Bibliography Bibliogr [141]] E. W. Hagley et al., Science 283, 1706 (1999). [142]] I. Bloch, T. W. Hansen, and T. Esslinger, Nature (London) 403, 166 (2000). [143]] J. Stenger et al, Phys. Rev. Lett. 82, 4569 (1999). [144]] U. Ernst et al, Europhys. Lett. 41, 1 (1998). [145]] K. Bongs et al, Phys. Rev. Lett. 83, 3577 (1999). [146]] J. Söding et al., Appl. Phys. B 69, 257 (1999). [147]] D. M. Stamper-Kurn et al, Phys. Rev. Lett. 81, 500 (1998). [148]] D. G. Fried et al, Phys. Rev. Lett. 81, 3811 (1998). [149]] K. Dieckmann, Ph.D. thesis, FOM Institute for Atomic and Molecular Physics, Amsterdam,, [150]] N. J. van Druten and W. Ketterle, Phys. Rev. Lett. 79, 549 (1997). [151]] S. Dettmer et al, Phys. Rev. Lett. 87, (2001). [152]] D. Hellweg et al, Appl. Phys. B 73, 781 (2001). [153]] F. Gerbier et al, cond-mat/

85 Summary y Thiss Thesis is devoted to theoretical investigation of trapped low-dimensional quantum gases.. Inn Chapter 2 we review theoretical grounds of the physics of degenerate low-dimensionall Bose gases. We discuss Bose-Einstein condensation (BEC) in ideal low-dimensional gases,, macroscopic description of an interacting quantum Bose gas, elementary excitationss and their description in terms of Bogolyubov-de Gennes equations, phase fluctuationss and coherence, and the phenomenon of quasicondenstaion. Then,, in Chapter 3 we investigate quasi-2d Bose and Fermi gases. In the first section wee discuss BEC in quasi-2d trapped gases and arrive at two key conclusions. First, welll below the temperature of quantum degeneracy Td the phase fluctuations are small, andd the equilibrium state is a true condensate. At intermediate temperatures the phase fluctuatesfluctuates on a distance scale smaller than the Thomas-Fermi size of the gas, and one hass a quasicondensate. Second, in quasi-2d the coupling constant is sensitive to the frequencyy of the tight confinement OJQ and, for negative 3D scattering length a, one can switchh the mean-field interaction from attractive to repulsive by increasing UÜQ. Inn the second section of Chapter 3 we show that quasi-2d atomic Fermi gases can stronglyy compete with 3D gases in achieving superfluidity. In the regime of BCS pairingg for weak attraction we calculate the critical temperature T c to the second order in perturbationn theory and analyze possibilities of increasing the ratio T C / F, where e F is thee Fermi energy. In the opposite limit, where a strong coupling leads to the formation off weakly bound quasi-2d dimers, we find that the interaction between the dimers is repulsivee and their collisional relaxation and decay are strongly suppressed. We thus concludee that a Bose-Einstein condensate of these composite bosons will be stable on a ratherr long time scale. Inn Chapter 4 we consider a Bose gas tightly confined in one (axial) direction and discusss how the axial confinement manifests itself in elastic and inelastic pair collisions. Wee identify two regimes of scattering. At temperatures T < : hu>o one has a quasi-2d regimee in which the 2D character of the relative motion of particles manifests itself in a logarithmicc energy dependence of the scattering amplitude. At temperatures T ~ HUJQ wee have a confinement-dominated 3D regime of scattering, where the 2D character of thee particle motion is no longer pronounced in the scattering process, but the axial confinementt can strongly influence the energy (temperature) dependence of the scattering rate.. For a large 3D scattering length, the tight confinement provides a much weaker temperaturee dependence of the elastic collisional rate than in the 3D case and suppresses itss resonant enhancement at low energies. We have employed our theory for explaining thee Stanford and ENS experiments with tightly confined Cs clouds. Inn Chapter 5 we discuss the regimes of quantum degeneracy in a trapped ID gas with repulsivee interparticle interaction. We find that the presence of a sharp crossover to the BECC regime requires extremely small interaction between particles. Otherwise, the decreasee of temperature leads to a continuous transformation of a classical gas to a quantum degenerate.. We identify three regimes at T < Td- For a sufficiently large interparticle 79 9

86 80 0 Summary y interactionn and the number of particles much smaller than a characteristic value N+, at anyy T <C T^ one has a trapped Tonks gas characterized by a Fermi-gas density profile. Forr N :» N* we have a weakly interacting gas. In this case we calculate the density and phasee fluctuations and find that even well below Ta there is a quasicondensate. At very loww T the long-wave fluctuations of the phase are suppressed due to a finite size of the system,, and we have a true condensate. The true condensate and the quasicondensate havee the same Thomas-Fermi density profile and local correlation properties, and we analyzee how to distinguish between these BEC states in an experiment. Finally,, in Chapter 6, we show that phase coherence properties of 3D Thomas-Fermi condensatess depend on their shape. In very elongated 3D condensates, the axial longwavee fluctuations of the phase manifest themselves even at temperatures far below the BECC transition temperature. Then the equilibrium state is a quasicondensate similar too that in ID trapped gases. At sufficiently low temperatures, the 3D quasicondensate graduallyy transforms into a true condensate with decreasing T. In the second section wee investigate the influence of equilibrium phase fluctuations on the density profile of an elongatedd 3D Thomas-Fermi condensate after switching off the trap. We find a relation betweenn the phase fluctuations in the trap and density modulations in the expanding cloud.. The results of this chapter have been used for explaining experiments with elongatedd condensates in Hannover and have been confirmed by experiments in Orsay.

87 Samenvatting g Ditt proefschrift is gewijd aan de theoretische onderzoek van opgesloten laag-dimensionale kwantumgassen.. Inn hoofdstuk 2 geven we een overzicht van de theoretische basis van de natuurkunde vann laag-dimensionale Bose gassen in het ontaarde kwantumgebied. We bespreken de Bose-Einstein-condensatiee (BEC) van een ideaal gas in lage dimensies, de makroscopische beschrijvingg van een wisselwerkend kwantum Bose gas, de elementaire excitaties en hun beschrijvingg in termen van Bogolyubov-de-Gennes-vergelijkingen, de fase fluctuaties en dee coherente eigenschappen van het condensaat, en het fenomeen van quasicondensatie. Vervolgens,, in hoofdstuk 3 onderzoeken we quasi-twee-dimensionale Bose en Fermi gassen.. In het eerste deel bespreken we BEC in quasi-2d opgesloten gassen en komen tot tweee sleutelconclusies. Ten eerste, de fase fluctueert weinig bij temperaturen veel lager dann de temperatuur Td van de kwantum ontaarding. Bij tussenliggende temperaturen fluctueertfluctueert de fase op een schaal kleiner dan de Thomas-Fermi lengte van het gas: men heeftt een quasicondensaat. De tweede conclusie is dat in het quasi-2d geval de interactiekonstantee gevoelig is voor de frequentie van de sterke opsluiting wo- Daarnaast kan menn bij een systeem met negatieve 3D verstrooiingslengte a, de mean-field wisselwerking veranderenn van aantrekkend tot afstotend door UJQ te verhogen. Inn het tweede deel van hoofdstuk 3 laten we zien dat atomaire quasi-2d Fermi gassen sterkk met 3D gassen kunnen concureren als het gaat om het bereiken van superfluïditeit. Inn het geval van BCS paring bij zwakke aantrekking berekenen we de kritieke temperatuur TT cc tot de tweede orde in storingsrekening en analyseren we de mogelijkheden om de ratioo T c /ep te vergroten; hier is p de Fermi energie. In het tegenovergesteld geval, waarinn sterke interactie tot vorming van zwak-gebonden quasi-2d dimeren leidt, vinden wee dat de wisselwerking tussen de dimeren afstotend is en dat hun verval en relaxatie doorr botsingen sterk zijn onderdrukt. Wij concluderen daarom dat een Bose-Einstein condensaatt van deze samengestelde bosonen over langere tijd stabiel zal zijn. Inn hoofdstuk 4 beschouwen we een Bose gas, dat in één (axiale) richting sterk opgeslotenn is. Ook beschrijven wij hoe de axiale opsluiting zich in elastische en inelastische paarbotsingenn uit. We identificeren twee verschillende verstrooiingsgebieden. Bij temperaturenn van T <s: hwq vindt men een quasi-2d gebied waarbij het 2D karakter van de relatievee beweging van deeltjes zichzelf manifesteert in een logaritmische energie afhankelijkheidd van de verstrooiingsamplitude. Bij temperaturen T ~ hwo vinden we een 3D regimee waarbij de opsluiting het verstrooiingsgedrag domineert. Het 2D karakter van de deeltjesbewegingg is dan niet meer belangrijk in het verstrooiingsproces, maar de energie (temperatuur)) afhankelijkheid van de verstrooingsfrequentie zal sterk door de axial opsluitingg worden beïnvloed. De nauwe opsluiting maakt de temperatuurafhankelijkheid vann de elastische botsingsfrequentie voor grote 3D verstrooiingslengtes veel zwakker dan inn het 3D geval. Ook wordt de resonante verstrooiing bij lage energieën onderdrukt. Wijj hebben onze theorie toegepast op experimenten die op Stanford en bij de ENS zijn gedaann met sterk opgesloten Cs. Inn hoofdstuk 5 bespreken we de gebieden van de kwantumontaarding in een opgeslo- 81 1

88 82 2 Samenvatting Samenva tenn ID gas met afstotende wisselwerking tussen de deeltjes. Wij vinden dat voor een scherpee BEC-overgang de wisselwerking tussen de atomen bijzonder zwak moet zijn. In hett tegenovergesteld geval lijdt de verlaging van de temperatuur tot een onafgebroken vervormingg van een klassiek tot een kwantum ontaard gas. We onderscheiden drie gebiedenn voor T <CT(j. Als de wisselwerking tussen de deeltjes sterk genoeg is en het aantal atomenn veel kleiner is dan de karakteristieke waarde JV*, dan heeft men een opgesloten Tonkss gas voor willekeurige T <C Tj. Het Tonks gas heeft een dichtheidprofiel gelijk aann die van een ideaal Fermi gas. Voor N > N+ hebben wij een zwak wisselwerkende gas.. Voor dit geval berekenen we de dichtheid en fase fluctuaties. Wij vinden dat men zelfss voor temperaturen ver onder Td een quasicondensaat krijgt. Voor zeer lage temperaturenn T zijn de langegolf fluctuaties van de fase onderdrukt wegens de eindige lengte vann het systeem. Dit resulteert in een echt condensaat. Het echte condensaat en het quasicondensaatt hebben dezelfde Thomas-Fermi dichtheidsprofielen en lokale correlatie eigenschappen.. Wij analyseren hoe men het verschil tussen deze BEC-toestanden in een experimentt zichtbaar kan maken. Tenslotte,, in hoofdstuk 6 laten we zien dat fase coherente eigenschappen van een 3D Thomas-Fermii condensaat afhankelijk zijn van zijn vorm. In zeer lange 3D condensatenn manifesteren de axiale langegolf fase fluctuaties zich bij temperaturen veel lager dan dee kritieke temperatuur voor BEC. Het thermische evenwicht van het systeem is dan eenn quasicondensaat vergelijkbaar met ID opgesloten gassen. Voor voldoende lage temperaturenn vervormt het 3D condensaat geleidelijk tot een echt condensaat met kleiner wordendee T. In het tweede deel onderzoeken we hoe de fase fluctuaties van de evenwichtstoestandd het dichtheidsprofiel van een lang 3D Thomas-Fermi condensaat beïnvloeden naa het uitzetten van de val. Wij vinden een relatie tussen de fase fluctuaties die de wolk hadd tijdens de opsluiting in de val en de dichtheidsmodulaties van de expanderende wolk. Dee resultaten van dit hoofdstuk zijn gebruikt om experimenten met lange condensaten in Hannoverr te verklaren. Ook zijn deze resultaten door experimenten in Orsay bevestigd.

89 Acknowledgments s Theree is quite a number of people that contributed to this Thesis either explicitly or implicitly.. II was lucky to have been doing physics under the guidance of two outstanding scientistss - Gora Shlyapnikov and Jook Walraven. Thanks to them I gained an invaluable experiencee from which I will benefit in the years to come. Itt was a gift to share an office with a very good friend Igor Shvarchuck. I much appreciatedd his company and support. Thank you, Igor. II will never forget a friendly atmosphere of the Quantum Gases group at AMOLF. Kai Dieckmann,, Martin Zielonkowski, Wolf von Klitzing, Christian Buggle, Lorenzo Vichi, Markk Kemmann, Tobias Tiecke and Imogen Colton greatly helped my research. I wish themm all good luck. Colleaguess from the neighboring office were always open for my questions. Ruediger Langg helped me with TeX and provided me with a useful information which affected thee style of this book. I am grateful to Annemieke Petrignani for the translation of the summaryy of this Thesis into Dutch. Infrastructuree of AMOLF and people working there made the institute a perfect place too do research. I thank our travel and accommodation guru Roudy van der Wijk for her kindnesss and for a wonderful apartment that she found for me. Thanks to personnel officerss Wouter Harmsen and Danielle de Vries I was never lost in the labyrinth of dutch bureaucracy.. Jan van Elst helped me a lot with Linux configuration. Duringg my PhD period I enjoyed the good company of many people all over the world.. In particular, I would like to thank colleagues with whom I worked on various projectss or just discussed physics. These are Isabelle Bouchoule, Christophe Salomon, Yvann Castin, Subhasis Sinha, Jean Dalibard, David Guery-Odélin, Maciej Lewenstein, Luiss Santos, Jason Ho, Yuri Kagan, Boris Svistunov, Maxim Olshanii, Allard Mosk, Markuss Holzmann, Klaus Sengstock, Jan Arlt and many others. Conversations with myy friends Peter Fedichev, Misha Baranov and Andrey Muryshev always brought fresh wavess of motivation. Withh great pleasure I recollect the times of fishing, playing tennis, and cycling to the seasidee with Pavel Bouchev, Peter Fedichev and Yaroslav Usenko. It was a real pity that Pavell and Peter moved to Innsbruck. However, thanks to them we spent a marvelous skiingg vacation in Austria. II am infinitely grateful to my parents Sergei and Tatiana and my wife Anna for their enormouss tolerance and encouragement. It would not be an exaggeration to say that withoutt their help this Thesis would have never been written. 83 3

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