Many-body physics with ultracold gasesdalibard/publications/2008_RMP.pdf · 2011. 10. 28. ·...

Many-body physics with ultracold gases

Immanuel Bloch*

Institut für Physik, Johannes Gutenberg–Universität, D-55099 Mainz, Germany

Jean Dalibard†

Laboratoire Kastler Brossel, CNRS, Ecole Normale Supérieure,24 rue Lhomond, F-75005 Paris, France

Wilhelm Zwerger‡

Physik-Department, Technische Universität München, D-85748 Garching, Germany

�Published 18 July 2008�

This paper reviews recent experimental and theoretical progress concerning many-body phenomenain dilute, ultracold gases. It focuses on effects beyond standard weak-coupling descriptions, such as theMott-Hubbard transition in optical lattices, strongly interacting gases in one and two dimensions, orlowest-Landau-level physics in quasi-two-dimensional gases in fast rotation. Strong correlations infermionic gases are discussed in optical lattices or near-Feshbach resonances in the BCS-BECcrossover.

DOI: 10.1103/RevModPhys.80.885 PACS number�s�: 03.75.Ss, 03.75.Hh, 74.20.Fg

CONTENTS

I. Introduction 885

A. Scattering of ultracold atoms 887

B. Weak interactions 888

C. Feshbach resonances 892

II. Optical Lattices 895

A. Optical potentials 895

B. Band structure 897

C. Time-of-flight and adiabatic mapping 898

D. Interactions and two-particle effects 899

III. Detection of Correlations 901

A. Time-of-flight versus noise correlations 901

B. Noise correlations in bosonic Mott and fermionic

band insulators 902

C. Statistics of interference amplitudes for

low-dimensional quantum gases 903

IV. Many-Body Effects in Optical Lattices 904

A. Bose-Hubbard model 904

B. Superfluid–Mott-insulator transition 905

C. Dynamics near quantum phase transitions 910

D. Bose-Hubbard model with finite current 911

E. Fermions in optical lattices 912

V. Cold Gases in One Dimension 913

A. Scattering and bound states 913

B. Bosonic Luttinger liquids; Tonks-Girardeau gas 916

C. Repulsive and attractive fermions 920

VI. Two-dimensional Bose Gases 921

A. The uniform Bose gas in two dimensions 922

B. The trapped Bose gas in 2D 924

VII. Bose Gases in Fast Rotation 929A. The lowest-Landau-level formalism 929B. Experiments with fast-rotating gases 931C. Beyond the mean-field regime 933D. Artificial gauge fields for atomic gases 935

VIII. BCS-BEC Crossover 936A. Molecular condensates and collisional stability 936B. Crossover theory and universality 938C. Experiments near the unitarity limit 945

IX. Perspectives 949A. Quantum magnetism 950B. Disorder 950C. Nonequilibrium dynamics 952

Acknowledgments 953Appendix: BEC and Superfluidity 953References 956

I. INTRODUCTION

The achievement of Bose-Einstein condensation�BEC� �Anderson et al., 1995; Bradley et al., 1995; Daviset al., 1995�, and of Fermi degeneracy �DeMarco and Jin,1999; Schreck et al., 2001; Truscott et al., 2001�, in ultra-cold, dilute gases has opened a new chapter in atomicand molecular physics, in which particle statistics andtheir interactions, rather than the study of single atomsor photons, are at center stage. For a number of years, amain focus in this field has been the exploration of thewealth of phenomena associated with the existence ofcoherent matter waves. Major examples include the ob-servation of interference of two overlapping conden-sates �Andrews et al., 1997�, of long-range phase coher-ence �Bloch et al., 2000�, and of quantized vortices andvortex lattices �Matthews, 1999; Madison et al., 2000;Abo-Shaeer et al., 2001� and molecular condensates withbound pairs of fermions �Greiner et al., 2003; Jochim et

*[email protected]†[email protected]‡[email protected]

REVIEWS OF MODERN PHYSICS, VOLUME 80, JULY–SEPTEMBER 2008

0034-6861/2008/80�3�/885�80� ©2008 The American Physical Society885

http://dx.doi.org/10.1103/RevModPhys.80.885

al., 2003b; Zwierlein et al., 2003b�. Common to all ofthese phenomena is the existence of a coherent, macro-scopic matter wave in an interacting many-body system,a concept familiar from the classic areas of superconduc-tivity and superfluidity. It was the basic insight ofGinzburg and Landau �1950� that, quite independentof a detailed microscopic understanding, an effectivedescription of the coherent many-body state is providedby a complex, macroscopic wave function ��x�= ��x� ��exp i��x�. Its magnitude squared gives the superfluiddensity, while the phase ��x� determines the superfluidvelocity via vs= �� /M��x� �see the Appendix for a dis-cussion of these concepts and their connection with themicroscopic criterion for BEC�. As emphasized by Cum-mings and Johnston �1966� and by Langer �1968�, thispicture is similar to the description of laser light as acoherent state �Glauber, 1963�. It applies both to thestandard condensates of bosonic atoms and to weaklybound fermion pairs, which are the building blocks ofthe BCS picture of superfluidity in Fermi systems. Incontrast to conventional superfluids like 4He or super-conductors, where the macroscopic wave function pro-vides only a phenomenological description of the super-fluid degrees of freedom, the situation in dilute gases isconsiderably simpler. In fact, as a result of the weak in-teractions, dilute BEC’s are essentially pure condensatessufficiently below the transition. The macroscopic wavefunction is thus directly connected with the microscopicdegrees of freedom, providing a complete and quantita-tive description of both static and time-dependent phe-nomena in terms of a reversible, nonlinear Schrödingerequation, the famous Gross-Pitaevskii equation �Gross,1961; Pitaevskii, 1961�. In dilute gases, therefore, themany-body aspect of a BEC is reduced to an effectivesingle-particle description, where interactions give riseto an additional potential proportional to the local par-ticle density. Adding small fluctuations around thiszeroth-order picture leads to the well-known Bogoliu-bov theory of weakly interacting Bose gases. Like theclosely related BCS superfluid of weakly interacting fer-mions, the many-body problem is then completelysoluble in terms of a set of noninteracting quasiparticles.Dilute, ultracold gases provide a concrete realization ofthese basic models of many-body physics, and many oftheir characteristic properties have been verified quanti-tatively. Excellent reviews of this remarkably rich areaof research have been given by Dalfovo et al. �1999� andby Leggett �2001� and, more recently, by Pethick andSmith �2002� and Pitaevskii and Stringari �2003�.

In the past several years, two major new develop-ments have considerably enlarged the range of physicsthat is accessible with ultracold gases. They are associ-ated with �i� the ability to tune the interaction strengthin cold gases by Feshbach resonances �Courteille et al.,1998; Inouye et al., 1998�; and �ii� the possibility ofchanging the dimensionality with optical potentials and,in particular, of generating strong periodic potentials forcold atoms through optical lattices �Greiner et al.,2002a�. The two developments, either individually or incombination, allow one to enter a regime in which the

interactions even in extremely dilute gases can no longerbe described by a picture based on noninteracting qua-siparticles. The appearance of such phenomena is char-acteristic for the physics of strongly correlated systems.For a long time, this area of research was confined to thedense and strongly interacting quantum liquids of con-densed matter or nuclear physics. By contrast, gases—almost by definition—were never thought to exhibitstrong correlations.

The use of Feshbach resonances and optical potentialsfor exploring strong correlations in ultracold gases wascrucially influenced by earlier ideas from theory. In par-ticular, Stoof et al. �1996� suggested that Feshbach reso-nances in a degenerate gas of 6Li, which exhibits a tun-able attractive interaction between two differenthyperfine states, may be used to realize BCS pairing offermions in ultracold gases. A remarkable idea in arather unusual direction in the context of atomic physicswas the proposal by Jaksch et al. �1998� to realize aquantum phase transition from a superfluid to a Mott-insulating state by loading a BEC into an optical latticeand increasing its depth. Further directions in the regimeof strong correlations were opened with the suggestionsby Olshanii �1998� and Petrov, Shlyapnikov, and Wal-raven �2000� to realize a Tonks-Girardeau gas withBEC’s confined in one dimension and by Wilkin andGunn �2000� to explore quantum Hall effect physics infast-rotating gases.

Experimentally, the strong-coupling regime in dilutegases was first reached by Cornish et al. �2000� usingFeshbach resonances for bosonic atoms. Unfortunately,in this case, increasing the scattering length a leads to astrong decrease in the condensate lifetime due to three-body losses, whose rate on average varies as a4

�Fedichev, Reynold, and Shlyapnikov, 1996; Petrov,2004�. A quite different approach to the regime ofstrong correlations, which does not suffer from problemswith the condensate lifetime, was taken by Greiner et al.�2002a�. Loading BEC’s into an optical lattice, they ob-served a quantum phase transition from a superfluid to aMott-insulating phase even in the standard regimewhere the average interparticle spacing is much largerthan the scattering length. Subsequently, the strong con-finement available with optical lattices made possiblethe achievement of low-dimensional systems where newphases can emerge. The observation of a �Tonks-Girardeau� hard-core Bose gas in one dimension by Ki-noshita et al. �2004� and Paredes et al. �2004� constituteda first example of a bosonic Luttinger liquid. In two di-mensions, a Kosterlitz-Thouless crossover between anormal phase and one with quasi-long-range order wasobserved by Hadzibabic et al. �2006�. The physics ofstrongly interacting bosons in the lowest Landau level isaccessible with fast-rotating BEC’s �Bretin et al., 2004;Schweikhard et al., 2004�, where the vortex lattice is pre-dicted to melt by quantum fluctuations. Using atoms like52Cr, which have a larger permanent magnetic moment,BEC’s with strong dipolar interactions have been real-ized by Griesmaier et al. �2005�. In combination withFeshbach resonances, this opens the way to tuning the

886 Bloch, Dalibard, and Zwerger: Many-body physics with ultracold gases

Rev. Mod. Phys., Vol. 80, No. 3, July–September 2008

nature and range of the interaction �Lahaye et al., 2007�,which might, for instance, be used to reach many-bodystates that are not accessible in the context of the frac-tional quantum Hall effect.

In Fermi gases, the Pauli principle suppresses three-body losses, whose rate in fact decreases with increasingvalues of the scattering length �Petrov, Salomon, andShlyapnikov, 2004�. Feshbach resonances, therefore, al-low one to enter the strong-coupling regime kF�a��1 inultracold Fermi gases �O’Hara et al., 2002; Bourdel et al.,2003�. In particular, there exist stable molecular states ofweakly bound fermion pairs in highly excited rovibra-tional states �Cubizolles et al., 2003; Strecker et al., 2003�.The remarkable stability of fermions near Feshbachresonances allows one to explore the crossover from amolecular BEC to a BCS superfluid of weakly boundCooper pairs �Bartenstein et al., 2004a; Bourdel et al.,2004; Regal et al., 2004a; Zwierlein et al., 2004�. In par-ticular, the presence of pairing due to many-body effectshas been probed by rf spectroscopy �Chin et al., 2004� orthe measurement of the closed-channel fraction �Par-tridge et al., 2005�, while superfluidity has been verifiedby observation of quantized vortices �Zwierlein et al.,2005�. Recently, these studies have been extended toFermi gases with unequal densities for the spin-up andspin-down components �Partridge et al., 2006; Zwierleinet al., 2006�, where pairing is suppressed by the mis-match of the respective Fermi energies.

Repulsive fermions in an optical lattice allow one torealize an ideal and tunable version of the Hubbardmodel, a paradigm for the multitude of strong-correlation problems in condensed matter physics. Ex-perimentally, some basic properties of degenerate fermi-ons in periodic potentials, such as the existence of aFermi surface and the appearance of a band insulatorat unit filling, have been observed by Köhl et al. �2005a�.While it is difficult to cool fermions to temperaturesmuch below the bandwidth in a deep optical lattice,these experiments give hope that eventually magneti-cally ordered or unconventional superconducting phasesof the fermionic Hubbard model will be accessible withcold gases. The perfect control and tunability of the in-teractions in these systems provide a novel approach forstudying basic problems in many-body physics and, inparticular, for entering regimes that have never been ac-cessible in condensed matter or nuclear physics.

This review aims to give an overview of this rapidlyevolving field, covering both theoretical concepts andtheir experimental realization. It provides an introduc-tion to the strong-correlation aspects of cold gases, thatis, phenomena that are not captured by weak-couplingdescriptions like the Gross-Pitaevskii or Bogoliubovtheory. The focus of this review is on examples that havealready been realized experimentally. Even within thislimitation, however, the rapid development of the fieldin recent years makes it impossible to give a completesurvey. In particular, important subjects like spinorgases, Bose-Fermi mixtures, quantum spin systems inoptical lattices, or dipolar gases will not be discussed�see, e.g., Lewenstein et al. �2007��. Also, applications of

cold atoms in optical lattices for quantum informationare omitted completely; for an introduction, see Jakschand Zoller �2005�.

A. Scattering of ultracold atoms

For an understanding of the interactions between neu-tral atoms, first at the two-body level, it is instructive touse a toy model �Gribakin and Flambaum, 1993�, inwhich the van der Waals attraction at large distances iscut off by a hard core at some distance rc on the order ofan atomic dimension. The resulting spherically symmet-ric potential,

V�r� = �− C6/r6 if r � rc� if r � rc,

� �1�is, of course, not a realistic description of the short-rangeinteraction of atoms; however, it captures the main fea-tures of scattering at low energies. The asymptotic be-havior of the interaction potential is fixed by the van derWaals coefficient C6. It defines a characteristic length

ac = �2MrC6/�2�1/4 �2�

at which the kinetic energy of the relative motion of twoatoms with reduced mass Mr equals their interaction en-ergy. For alkali-metal atoms, this length is typically onthe order of several nanometers. It is much larger thanthe atomic scale rc because alkali-metal atoms arestrongly polarizable, resulting in a large C6 coefficient.The attractive well of the van der Waals potential thussupports many bound states �of order 100 in 87Rb�. Theirnumber Nb may be determined from the WKB phase

= �rc

�

dr2Mr�V�r��/� = ac2/2rc2 � 1 �3�

at zero energy, via Nb= � /+1/8�, where � � means tak-ing the integer part.1 The number of bound states in thismodel, therefore, depends crucially on the precise valueof the short-range scale rc. By contrast, the low-energyscattering properties are determined by the van derWaals length ac, which is sensitive only to the asymptoticbehavior of the potential. Consider the scattering instates with angular momentum l=0,1 ,2 , . . . in the rela-tive motion �for identical bosons or fermions, only evenor odd values of l are possible, respectively�. The effec-tive potential for states with l�0 contains a centrifugalbarrier whose height is of order Ec�2l3 /Mrac

2. Convert-ing this energy into an equivalent temperature, one ob-tains for small l temperatures around 1 mK for typicalatomic masses. At temperatures below that, the energy�2k2 /2Mr in the relative motion of two atoms is typicallybelow the centrifugal barrier. Scattering in states withl�0 is therefore frozen out, unless there exist so-called

1This result follows from Eq. �5� below by noting that a newbound state is pulled in from the continuum each time thescattering length diverges �Levinson’s theorem�.

887Bloch, Dalibard, and Zwerger: Many-body physics with ultracold gases


shape resonances, i.e., bound states with l�0 behind thecentrifugal barrier, which may be in resonance with theincoming energy; see Boesten et al. �1997� and Dürr et al.�2005�. For gases in the sub-millikelvin regime, there-fore, usually the lowest-angular-momentum collisionsdominate �s-wave for bosons, p-wave for fermions�,which in fact defines the regime of ultracold atoms. Inthe s-wave case, the scattering amplitude is determinedby the corresponding phase shift �0�k� via �Landau andLifshitz, 1987�

f�k� =1

k cot �0�k� − ik→

1

− 1/a + rek2/2 − ik

. �4�

At low energies, it is characterized by the scatteringlength a and the effective range re as the only two pa-rameters. For the truncated van der Waals potential �1�,the scattering length can be calculated analytically as�Gribakin and Flambaum, 1993�

a = ā�1 − tan� − 3/8�� , �5�

where, is the WKB phase �3� and ā=0.478ac is theso-called mean scattering length. Equation �5� showsthat the characteristic magnitude of the scattering lengthis the van der Waals length. Its detailed value, however,depends on the short-range physics via the WKB phase, which is sensitive to the hard-core scale rc. Since thedetailed behavior of the potential is typically not knownprecisely, in many cases neither the sign of the scatteringlength nor the number of bound states can be deter-mined from ab initio calculations. The toy-model result,however, is useful beyond the identification of ac as thecharacteristic scale for the scattering length. Indeed, ifignorance about the short-range physics is replaced bythe �maximum likelihood� assumption of a uniform dis-tribution of in the relevant interval �0,�, the prob-ability for finding a positive scattering length, i.e.,tan �1, is 3 /4. A repulsive interaction at low energy,which is connected with a positive scattering length, istherefore three times more likely than an attractive one,where a�0 �Pethick and Smith, 2002�. Concerning theeffective range re in Eq. �4�, it turns out that re is also onthe order of the van der Waals or the mean scatteringlength ā rather than the short-range scale rc, as mighthave been expected naively2 �Flambaum et al., 1999�.Since kac1 in the regime of ultracold collisions, thisimplies that the k2 contribution in the denominator ofthe scattering amplitude is negligible. In the low-energylimit, the two-body collision problem is thus completelyspecified by the scattering length a as the single param-eter, and the corresponding scattering amplitude

f�k� = − a/�1 + ika� . �6�

As noted by Fermi in the context of scattering of slowneutrons and by Lee, Huang, and Yang for the low-temperature thermodynamics of weakly interactingquantum gases, Eq. �6� is the exact scattering amplitudeat arbitrary values of k for the pseudopotential,3

V�x��¯� =4�2a

2Mr��x�

�

�r�r ¯ � . �7�

At temperatures such that kBT�Ec, two-body interac-tions in ultracold gases may be described by a pseudo-potential, with the scattering length usually taken as anexperimentally determined parameter. This approxima-tion is valid in a wide range of situations, provided nolonger-range contributions come into play as, e.g., in thecase of dipolar gases. The interaction is repulsive forpositive and attractive for negative scattering lengths.Now, as shown above, the true interaction potential hasmany bound states, irrespective of the sign of a. Forlow-energy scattering of atoms, however, these boundstates are irrelevant as long as no molecule formationoccurs via three-body collisions. The scattering ampli-tude in the limit k→0 is sensitive only to bound �orvirtual for a�0� states near zero energy. In particular,within the pseudopotential approximation, the ampli-tude �6� has a single pole k= i�, with �=1/a�0 if thescattering length is positive. Quite generally, poles of thescattering amplitude in the upper complex k plane areconnected with bound states with binding energy �b=�2�2 /2Mr �Landau and Lifshitz, 1987�. In the pseudo-potential approximation, only a single pole is captured;the energy of the associated bound state is just belowthe continuum threshold. A repulsive pseudopotentialthus describes a situation in which the full potential hasa bound state with a binding energy �b=�2 /2Mra2 on theorder of or smaller than the characteristic energy Ec in-troduced above. The associated positive scatteringlength is then identical with the decay length of the wavefunction �exp�−r /a� of the highest bound state. In theattractive case a�0, in turn, there is no bound statewithin a range Ec below the continuum threshold; how-ever, there is a virtual state just above it.

B. Weak interactions

For a qualitative discussion of what defines the weak-interaction regime in dilute, ultracold gases, it is usefulto start with the idealization of no interactions at all.Depending on the two fundamental possibilities for thestatistics of indistinguishable particles, Bose or Fermi,the ground state of a gas of N noninteracting particles iseither a perfect BEC or a Fermi sea. In the case of anideal BEC, all particles occupy the lowest availablesingle-particle level, consistent with a fully symmetricmany-body wave function. For fermions, in turn, the

2This is a general result for deep potentials with a power-lawdecay at large distances, as long as the scattering energy ismuch smaller than the depth of the potential well.

3Because of the � function, the last term involving the partialderivative with respect to r= �x� can be omitted when the po-tential acts on a function that is regular at r=0.



particles fill the N lowest single-particle levels up to theFermi energy �F�N�, as required by the Pauli principle.At finite temperatures, the discontinuity in the Fermi-Dirac distribution at T=0 is smeared out, giving riseto a continuous evolution from the degenerate gas atkBT�F to a classical gas at high temperatures kBT��F. By contrast, bosons exhibit in three dimensions�3D� a phase transition at finite temperature, where themacroscopic occupancy of the ground state is lost. In thehomogeneous gas, this transition occurs when the ther-mal de Broglie wavelength �T=h /2MkBT reaches theaverage interparticle distance n−1/3. The surprising factthat a phase transition appears even in an ideal Bose gasis a consequence of the correlations imposed by the par-ticle statistics alone, as noted already in Einstein’s fun-damental paper �Einstein, 1925�. For trapped gases, withgeometrical mean trap frequency �̄, the transition to aBEC is in principle smooth.4 Yet, for typical particlenumbers in the range N�104–107, there is a rathersharply defined temperature kBTc

�0�=��̄�N /��3��1/3,above which the occupation of the oscillator groundstate is no longer of order N. This temperature is againdetermined by the condition that the thermal de Brogliewavelength reaches the average interparticle distance atthe center of the trap �see Eq. �95� and below�.

As discussed above, interactions between ultracold at-oms are described by a pseudopotential �7�, whosestrength g=4�2a /2Mr is fixed by the exact s-wave scat-tering length a. Now, for identical fermions, there is nos-wave scattering due to the Pauli principle. In the re-gime kac1, where all higher momenta l�0 are frozenout, a single-component Fermi gas thus approaches anideal, noninteracting quantum gas. To reach the neces-sary temperatures, however, requires thermalization byelastic collisions. For identical fermions, p-wave colli-sions dominate at low temperatures, whose cross section�p�E2 leads to a vanishing of the scattering rates �T2�DeMarco et al., 1999�. Evaporative cooling, therefore,does not work for a single-component Fermi gas in thedegenerate regime. This problem may be circumventedby cooling in the presence of a different spin state that isthen removed, or by sympathetic cooling with a anotheratomic species. In this manner, an ideal Fermi gas, whichis one paradigm of statistical physics, has first been real-ized by DeMarco and Jin �1999�, Schreck et al. �2001�,and Truscott et al. �2001� �see Fig. 1�.

In the case of fermion mixtures in different internalstates, or for bosons, there is in general a finite scatter-ing length a�0, which is typically of the order of the vander Waals length Eq. �2�. By a simple dimensional argu-ment, interactions are expected to be weak when thescattering length is much smaller than the average inter-particle spacing. Since ultracold alkali-metal gases have

densities between 1012 and 1015 particles per cm3, theaverage interparticle spacing n−1/3 typically is in therange 0.1–1 �m. As shown above, the scattering length,in turn, is usually only in the few-nanometer range. In-teraction effects are thus expected to be very small, un-less the scattering length happens to be large near azero-energy resonance of Eq. �5�. In the attractive casea�0, however, even small interactions can lead to insta-bilities. In particular, attractive bosons are unstable to-ward collapse. However, in a trap, a metastable gaseousstate arises for sufficiently small atom numbers �Pethickand Smith, 2002�. For mixtures of fermions in differentinternal states, an arbitrary weak attraction leads to theBCS instability, where the ground state is essentially aBEC of Cooper pairs �see Sec. VIII�. In the case of re-pulsive interactions, in turn, perturbation theory worksin the limit n1/3a1.5 For fermions with two differentinternal states, an appropriate description is provided bythe dilute gas version of Landau’s theory of Fermi liq-uids. The associated ground-state chemical potential isgiven by �Lifshitz and Pitaevskii, 1980�

�F =�2kF

2

2M

1 + 4

3

kFa +

4�11 − 2 ln 2�152

�kFa�2 + ¯ � ,�8�

where the Fermi wave vector kF= �32n�1/3 is deter-mined by the total density n in precisely the same man-ner as in the noninteracting case. Weakly interactingBose gases, in turn, are described by the Bogoliubovtheory, which has na3 as the relevant small parameter.For example, the chemical potential at zero temperature

4A Bose gas in a trap exhibits a sharp transition only in thelimit N→� , �̄→0 with N�̄3=const, i.e., when the critical tem-perature approaches a finite value in the thermodynamic limit.

5We neglect the possibility of a Kohn-Luttinger instability�Kohn and Luttinger, 1965� of repulsive fermious to a �typi-cally� p-wave superfluid state, which usually only appears attemperatures very far below TF; see Baranov et al. �1996�.

810 nK

510 nK

240 nK7Li Bosons 6Li Fermions

FIG. 1. �Color online� Simultaneous cooling of a bosonic andfermionic quantum gas of 17Li and 6Li to quantum degeneracy.In the case of the Fermi gas, the Fermi pressure prevents theatom cloud from shrinking further in space as quantum degen-eracy is approached. From Truscott et al., 2001.



for a homogeneous gas is given by �Lifshitz and Pita-evskii, 1980�

�Bose =4�2a

Mn�1 + 32

3

na3

�1/2 + ¯ � . �9�

Moreover, interactions lead to a depletion

n0 = n�1 −83 �na

3/�1/2 + ¯ � �10�

of the density n0 of particles at zero momentum com-pared to the perfect condensate of an ideal Bose gas.The finite value of the chemical potential at zero tem-perature defines a characteristic length � by �2 /2M�2

=�Bose. This is the so-called healing length �Pitaevskiiand Stringari, 2003�, which is the scale over which themacroscopic wave function ��x� varies near a boundary�or a vortex core; see Sec. VII� where BEC is sup-pressed. To lowest order in na3, this length is given by�= �8na�−1/2. In the limit na31, the healing length istherefore much larger than the average interparticlespacing n−1/3. In practice, the dependence on the gas pa-rameter na3 is so weak that the ratio �n1/3��na3�−1/6 isnever very large. On a microscopic level, � is the lengthassociated with the ground-state energy per particle bythe uncertainty principle. It can thus be identified withthe scale over which bosons may be considered to belocalized spatially. For weak-coupling BEC’s, atoms aretherefore smeared out over distances much larger thanthe average interparticle spacing.

Interactions also shift the critical temperature forBEC away from its value Tc

�0� in the ideal Bose gas. Tolowest order in the interactions, the shift is positive andlinear in the scattering length �Baym et al., 1999�,

Tc/Tc�0� = 1 + cn1/3a + ¯ �11�

with a numerical constant c�1.32 �Arnold and Moore,2001; Kashurnikov et al., 2001�. The unexpected increaseof the BEC condensation temperature with interactionsis due to a reduction of the critical density. While aquantitative derivation of Eq. �11� requires quite sophis-ticated techniques �Holzmann et al., 2004�, the result canbe recovered by a simple argument. To leading order,the interaction induced change in Tc depends only onthe scattering length. Compared with the noninteractingcase, the finite scattering length may be thought of aseffectively increasing the quantum-mechanical uncer-tainty in the position of each atom due to thermal mo-

tion from �T to �̄T=�T+a. To lowest order in a, the

modified ideal gas criterion n�̄Tc3 =��3/2� then gives rise

to the linear and positive shift of the critical temperaturein Eq. �11� with a coefficient c̄�1.45, which is not farfrom the numerically exact value.

In the standard situation of a gas confined in a har-monic trap with characteristic frequency �̄, the influenceof weak interactions is quantitatively different for tem-peratures near T=0 or near the critical temperature Tc.At zero temperature, the noninteracting Bose gas has adensity distribution n�0��x�=N��0�x��2, which reflects the

harmonic-oscillator ground state wave function �0�x�. Itscharacteristic width is the oscillator length �0=� /M�̄,which is on the order of 1 �m for typical confinementfrequencies. Adding even small repulsive interactionschanges the distribution quite strongly. Indeed, in theexperimentally relevant limit Na��0, the density profilen�x� in the presence of an external trap potential U�x�can be obtained from the local-density approximation�LDA�

��n�x�� + U�x� = ��n�0�� . �12�

For weakly interacting bosons in an isotropic harmonictrap, the linear dependence �Bose=gn of the chemicalpotential on the density in the homogeneous case thenleads to a Thomas-Fermi profile n�x�=n�0��1− �r /RTF�2�.Using the condition �n�x�=N, the associated radiusRTF=��0 exceeds considerably the oscillator length sincethe dimensionless parameter �= �15Na /�0�1/5 is typicallymuch larger than 1 �Giorgini et al., 1997�.6 This broad-ening leads to a significant decrease in the density n�0�at the trap center by a factor �−3 compared with thenoninteracting case. The strong effect of even weak in-teractions on the ground state in a trap may be under-stood from the fact that the chemical potential �=��̄�2 /2 is much larger than the oscillator ground-stateenergy. Interactions are thus able to mix in many single-particle levels beyond the harmonic trap ground state.Near the critical temperature, in turn, the ratio � /kBTc

�n�0�a3�1/6 is small. Interaction corrections to the con-densation temperature, which dominate finite-size cor-rections for particle numbers much larger than N104,are therefore accessible perturbatively �Giorgini et al.,1997�. In contrast to the homogeneous case, where thedensity is fixed and Tc is shifted upward, the dominanteffect in a trap arises from the reduced density at thetrap center. The corresponding shift may be expressed as�Tc /Tc=−const�a /�Tc �Giorgini et al., 1997; Holzmannet al., 2004; Davis and Blakie, 2006�. A precise measure-ment of this shift has been performed by Gerbier et al.�2004�. Their results are in quantitative agreement withmean-field theory, with no observable contribution ofcritical fluctuations at their level of sensitivity. Quite re-cently, evidence for critical fluctuations has been in-ferred from measurements of the correlation length ��T−Tc�−� very close to Tc. The observed value �=0.67±0.13 �Donner et al., 2007� agrees well with theexpected critical exponent of the 3D XY model.

In spite of the strong deviations in the density distri-bution compared to the noninteracting case, the one-and two-particle correlations of weakly interacting

6For fermions, the validity of the LDA, which is in fact just asemiclassical approximation �see, e.g., Brack and Bhaduri�1997��, does not require interactions. The leading term�Fermi�n2/3 of Eq. �8� leads to a density profile n�x�=n�0��1− �r /RTF�2�3/2 with a radius RTF=��0. Here �=kF�N��0= �24N�1/6�1 and the Fermi wave vector kF�N� in a trap is�F�N�=�2kF

2�N� /2M.



bosons are well described by approximating the many-body ground state of N bosons by a product

�GP�x1,x2, . . . ,xN� = �i=1

N

�1�xi� �13�

in which all atoms are in the identical single-particlestate �1�x�. Taking Eq. �13� as a variational ansatz, theoptimal macroscopic wave function �1�x� is found toobey the well-known Gross-Pitaevskii equation. Moregenerally, it turns out that for trapped BEC’s, the Gross-Pitaevskii theory can be derived mathematically by tak-ing the limits N→� and a→0 in such a way that theratio Na /�0 is fixed �Lieb et al., 2000�. A highly non-trivial aspect of these derivations is that they show ex-plicitly that, in the dilute limit, interactions enter onlyvia the scattering length. The Gross-Pitaevskii equationthus remains valid, e.g., for a dilute gas of hard spheres.Since the interaction energy is of kinetic origin in thiscase, the standard mean-field derivation of the Gross-Pitaevskii equation via the replacement of the field op-

erators by a classical c number �̂�x�→N�1�x� is thusincorrect in general. From a many-body point of view,the ansatz Eq. �13�, where the ground state is written asa product of optimized single-particle wave functions, isthe standard Hartree approximation. It is the simplestpossible approximation to account for interactions; how-ever, it contains no interaction-induced correlations be-tween different atoms at all. A first step beyond that isthe well-known Bogoliubov theory. This is usually intro-duced by considering small fluctuations around theGross-Pitaevskii equation in a systematic expansion inthe number of noncondensed particles �Castin and Dum,1998�. It is also instructive from a many-body point ofview to formulate Bogoliubov theory such that the bo-son ground state is approximated by an optimized prod-uct �Lieb, 1963b�

�Bog�x1,x2, . . . ,xN� = �i�j

�2�xi,xj� �14�

of identical, symmetric two-particle wave functions �2.This allows one to include interaction effects beyond theHartree potential of the Gross-Pitaevskii theory by sup-pressing configurations in which two particles are closetogether. The many-body state thus incorporates two-particle correlations that are important, e.g., to obtainthe standard sound modes and the related coherent su-perposition of “particle” and “hole” excitations. Thisstructure, which has been experimentally verified by Vo-gels et al. �2002�, is expected to apply in a qualitativeform even for strongly interacting BEC’s, whose low-energy excitations are exhausted by harmonic phonons�see the Appendix�.

Quantitatively, however, the Bogoliubov theory is re-stricted to the regime na31, where interactions leadonly to a small depletion �10� of the condensate at zerotemperature. Going beyond that requires one to specifythe detailed form of the interaction potential V�r� andnot only the associated scattering length a. The ground

state of a gas of hard-sphere bosons, for instance, losesBEC already for na3�0.24 by a first-order transition toa solid state �Kalos et al., 1974�. On a variational level,choosing the two-particle wave functions in Eq. �14� ofthe form �2�xi ,xj��exp−u��xi−xj�� with an effectivetwo-body potential u�r� describes so-called Jastrow wavefunctions. They allow taking into account strong short-range correlations; however, they still exhibit BEC evenin a regime in which the associated one-particle densitydescribes a periodic crystal rather than a uniform liquid,as shown by Chester �1970�. Crystalline order may thuscoexist with BEC. For a discussion of this issue in thecontext of a possible supersolid phase of 4He, see Clarkand Ceperley �2006�.

For weakly interacting fermions at kFa1, the varia-tional ground state, which is analogous to Eq. �13�, is aSlater determinant

�HF�x1,x2, . . . ,xN� = det��1,i�xj�� 15�

of optimized single-particle states �1,i�xj�. In the transla-tionally invariant case, they are plane waves �1,i�x�=V−1/2 exp�iki ·x�, where the momenta ki are filled up tothe Fermi momentum kF. Although both the Bose andFermi ground-state wave functions consist of symme-trized or antisymmetrized single-particle states, they de-scribe fundamentally different physics. In the Bose case,the one-particle density matrix g�1��=n0 /n approachesa finite constant at infinite separation, which is the basiccriterion for BEC �see the Appendix�. The many-bodywave function is thus sensitive to changes of the phase atpoints separated by distances r that are large comparedto the interparticle spacing. By contrast, the Hartree-Fock state �15� for fermions shows no long-range phasecoherence, and indeed the one-particle density matrixdecays exponentially g�1��r��exp�−�r� at any finite tem-perature �Ismail-Beigi and Arias, 1999�. The presence ofN distinct eigenstates in Eq. �15�, which is a necessaryconsequence of the Pauli principle, leads to a many-body wave function that may be characterized as near-sighted. The notion of nearsightedness depends on theobservable, however. As defined originally by Kohn�1996�, it means that a localized external potentialaround some point x� is not felt at a point x at a distancemuch larger than the average interparticle spacing. Thisrequires the density response function ��x ,x�� to beshort ranged in position space. In this respect, weaklyinteracting bosons, where ��x ,x��exp�−�x−x�� /�� / �x−x�� decays exponentially on the scale of the healinglength �, are more nearsighted than fermions at zerotemperature, where ��x ,x��sin�2kF�x−x�� / �x−x��3 ex-hibits an algebraic decay with Friedel oscillations attwice the Fermi wave vector 2kF. The characterizationof many-body wave functions in terms of the associatedcorrelation functions draws attention to another basicpoint emphasized by Kohn �1999�: in situations with alarge number of particles, the many-body wave functionitself is not a meaningful quantity because it cannot becalculated reliably for N�100. Moreover, physically ac-cessible observables are sensitive only to the resulting



one-or two-particle correlations. Cold gases provide aconcrete example for the latter statement: the standardtime-of-flight technique of measuring the absorption im-age after a given free-expansion time t provides the one-particle density matrix in Fourier space, while the two-particle density matrix is revealed in the noisecorrelations of absorption images �see Sec. III�.

C. Feshbach resonances

The most direct way of reaching the strong-interactionregime in dilute, ultracold gases is via Feshbach reso-nances, which allow the scattering length to be increasedto values beyond the average interparticle spacing. Inpractice, this method works best for fermions becausefor them the lifetime due to three-body collisions be-comes very large near a Feshbach resonance, in starkcontrast to bosons, where it goes to zero. The conceptwas first introduced in the context of reactions forming acompound nucleus �Feshbach, 1958� and, independently,for a description of configuration interactions in multi-electron atoms �Fano, 1961�. Quite generally, a Feshbachresonance in a two-particle collision appears whenever abound state in a closed channel is coupled resonantlywith the scattering continuum of an open channel. Thetwo channels may correspond, for example, to differentspin configurations for atoms. The scattered particles arethen temporarily captured in the quasibound state, andthe associated long time delay gives rise to a Breit-Wigner-type resonance in the scattering cross section.What makes Feshbach resonances in the scattering ofcold atoms particularly useful is the ability to tune thescattering length simply by changing the magnetic field�Tiesinga et al., 1993�. This tunability relies on the differ-ence in the magnetic moments of the closed and openchannels, which allows the position of closed-channelbound states relative to the open-channel threshold tobe changed by varying the external, uniform magneticfield. Note that Feshbach resonances can alternativelybe induced optically via one- or two-photon transitions�Fedichev, Kagan, Shlyapnikov, et al., 1996; Bohn andJulienne, 1999� as realized by Theis et al. �2004�. Thecontrol parameter is then the detuning of the light fromatomic resonance. Although more flexible in principle,this method suffers, however, from heating problems fortypical atomic transitions, associated with thespontaneous-emission processes created by the light ir-radiation.

On a phenomenological level, Feshbach resonancesare described by an effective pseudopotential betweenatoms in the open channel with scattering length

a�B� = abg�1 − �B/�B − B0�� . �16�

Here abg is the off-resonant background scatteringlength in the absence of the coupling to the closed chan-nel, while �B and B0 describe the width and position ofthe resonance expressed in magnetic field units �see Fig.2�. In this section, we outline the basic physics of mag-netically tunable Feshbach resonances, providing a con-nection of the parameters in Eq. �16� with the inter-

atomic potentials. Of course, our discussion covers onlythe basic background for understanding the origin oflarge and tunable scattering lengths. A more detailedpresentation of Feshbach resonances can be found in thereviews by Timmermans et al. �2001�; Duine and Stoof�2004�; and Köhler et al. �2006�.

Open and closed channels. We start with the specificexample of fermionic 6Li atoms, which have electronicspin S=1/2 and nuclear spin I=1. In the presence of amagnetic field B along the z direction, the hyperfinecoupling and Zeeman energy lead for each atom to theHamiltonian

Ĥ� = ahfŜ · Î + �2�BŜz − �nÎz�B . �17�

Here �B�0 is the standard Bohr magneton and �n��B� is the magnetic moment of the nucleus. This hy-perfine Zeeman Hamiltonian actually holds for anyalkali-metal atom, with a single valence electron withzero orbital angular momentum. If B→0, the eigen-states of this Hamiltonian are labeled by the quantumnumbers f and mf, giving the total spin angular momen-tum and its projection along the z axis, respectively. Inthe opposite Paschen-Back regime of large magneticfields �B�ahf /�B30 G in lithium�, the eigenstates arelabeled by the quantum numbers ms and mI, giving theprojection on the z axis of the electron and nuclearspins, respectively. The projection mf=ms+mI of the to-tal spin along the z axis remains a good quantum num-ber for any value of the magnetic field.

Consider a collision between two lithium atoms, pre-pared in the two lowest eigenstates �a� and �b� of theHamiltonian �17� in a large magnetic field. The loweststate �a� �with mfa=1/2� is ��ms=−1/2 ,mI=1� with asmall admixture of �ms=1/2 ,mI=0�, whereas �b� �withmfb=−1/2� is ��ms=−1/2 ,mI=0� with a small admixtureof �ms=1/2 ,mI=−1�. Two atoms in these two lowest

FIG. 2. Magnetic field dependence of the scattering length be-tween the two lowest magnetic substates of 6Li with a Fesh-bach resonance at B0=834 G and a zero crossing at B0+�B=534 G. The background scattering length abg=−1405aB is ex-ceptionally large in this case �aB the Bohr radius�.



states thus predominantly scatter into their triplet state.7

Quite generally, the interaction potential during the col-lision can be written as a sum

V�r� = 14 �3Vt�r� + Vs�r�� + Ŝ1 · Ŝ2�Vt�r� − Vs�r�� 18�

of projections onto the singlet Vs�r� and triplet Vt�r� mo-lecular potentials, where the Ŝi’s �i=1,2� are the spinoperators for the valence electron of each atom. Thesepotentials have the same van der Waals attractive behav-ior at long distances, but they differ considerably atshort distances, with a much deeper attractive well forthe singlet than for the triplet potential. Now, in a largebut finite magnetic field, the initial state �a ,b� is not apurely triplet state. Because of the tensorial nature ofV�r�, this spin state will thus evolve during the collision.More precisely, since the second term in Eq. �18� is notdiagonal in the basis �a ,b�, the spin state �a ,b� may becoupled to other scattering channels �c ,d�, provided thez projection of the total spin is conserved �mfc+mfd=mfa+mfb�. When the atoms are far apart, the Zeeman+hyperfine energy of �c ,d� exceeds the initial kinetic en-ergy of the pair of atoms prepared in �a ,b� by an energyon the order of the hyperfine energy. Since the thermalenergy is much smaller than that for ultracold collisions,the channel �c ,d� is closed and the atoms always emergefrom the collision in the open-channel state �a ,b�. How-ever, due to the strong coupling of �a ,b� to �c ,d� via thesecond term in Eq. �18�, which is typically on the orderof eV, the effective scattering amplitude in the openchannel can be strongly modified.

Two-channel model. We now present a simple two-channel model that captures the main features of a Fes-hbach resonance �see Fig. 3�. Consider a collision be-tween two atoms with reduced mass Mr, and model thesystem in the vicinity of the resonance by the Hamil-tonian �Nygaard et al., 2006�

Ĥ =�−�2

2Mr�2 + Vop�r� W�r�

W�r� −�2

2Mr�2 + Vcl�r� � . �19�

Before collision, the atoms are prepared in the openchannel, whose potential Vop�r� gives rise to the back-ground scattering length abg. Here the zero of energy ischosen such that Vop��=0. In the course of the colli-sion, a coupling to the closed channel with potentialVcl�r� �Vcl��0� occurs via the matrix element W�r�,whose range is on the order of the atomic scale rc. Forsimplicity, we consider here only a single closed channel,which is appropriate for an isolated resonance. We alsoassume that the value of abg is on the order of the vander Waals length �2�. If abg is anomalously large, as oc-curs, e.g., for the 6Li resonance shown in Fig. 2, an ad-

ditional open-channel resonance has to be included inthe model, as discussed by Marcelis et al. �2004�.

We assume that the magnetic moments of the collid-ing states differ for the open and closed channels, anddenote their difference by �. Varying the magnetic fieldby �B, therefore, amounts to shifting the closed-channelenergy by ��B with respect to the open channel. In thefollowing, we are interested in the magnetic field regionclose to Bres such that one �normalized� bound state�res�r� of the closed-channel potential Vcl�r� has an en-ergy Eres�B�=��B−Bres� close to 0. It can thus be reso-nantly coupled to the collision state where two atoms inthe open channel have a small positive kinetic energy. Inthe vicinity of the Feshbach resonance, the situation isnow similar to the well-known Breit-Wigner problem�see, e.g., Landau and Lifshitz �1987�, Sec. 134�. A par-ticle undergoes a scattering process in a �single-channel�potential with a quasi- or true bound state at an energy�, which is nearly resonant with the incoming energyE�k�=�2k2 /2Mr. According to Breit and Wigner, thisleads to a resonant contribution

�res�k� = − arctan„��k�/2�E�k� − ��… �20�

to the scattering phase shift, where �=��B−B0� is con-ventionally called the detuning in this context �for thedifference between Bres and B0, see below�.The associ-ated resonance width ��k� vanishes near zero energy,with a threshold behavior linear in k=2MrE /� due tothe free-particle density of states. It is convenient to de-fine a characteristic length r��0 by

7The fact that there is a nonvanishing s-wave scatteringlength for these states is connected with the different nuclearand not electronic spin in this case.

closedchannel

openchannel

incidentenergy

boundstate

Energy

Interactomicdistance

FIG. 3. �Color online� The two-channel model for a Feshbachresonance. Atoms prepared in the open channel, correspond-ing to the interaction potential Vop�r�, undergo a collision atlow incident energy. In the course of the collision, the openchannel is coupled to the closed channel Vcl�r�. When a boundstate of the closed channel has an energy close to zero, a scat-tering resonance occurs. The position of the closed channel canbe tuned with respect to the open one, e.g., by varying themagnetic field B.



��k → 0�/2 = �2k/2Mrr�. �21�The scattering length a=−limk→0 tan��bg+�res� /k thenhas the simple form

a = abg − �2/2Mrr

�� . �22�

This agrees precisely with Eq. �16� provided the widthparameter �B is identified with the combination��Babg=�2 /2Mrr� of the two characteristic lengths abgand r�.

On a microscopic level, these parameters may be ob-tained from the two-channel Hamiltonian �19� by thestandard Green’s-function formalism. In the absence ofcoupling W�r�, the scattering properties of the openchannel are characterized by Gop�E�= �E−Hop�−1, withHop=P2 /2Mr+Vop�r�. We denote by ��0� the eigenstateof Hop associated with the energy 0, which behaves as�0�r��1−abg/r for large r. In the vicinity of the reso-nance, the closed channel contributes through the state�res, and its Green’s function reads

Gcl�E,B� ��res��res�/�E − Eres�B�� . �23�

With this approximation, one can project the eigenvalue

equation for the Hamiltonian Ĥ onto the backgroundand closed channels. One can then derive the scatteringlength a�B� of the coupled-channel problem and write itin the form of Eq. �16�. The position of the zero-energyresonance B0 is shifted with respect to the “bare” reso-nance value Bres by

��B0 − Bres� = − ��res�WGop�0�W��res� . �24�

The physical origin of this resonance shift is that an in-finite scattering length requires that the contributions tok cot ��k� in the total scattering amplitude from theopen and closed channels precisely cancel. In a situationin which the background scattering length deviates con-siderably from its typical value ā and where the off-diagonal coupling measured by �B is strong, this cancel-lation already appears when the bare closed-channelbound state is far away from the continuum threshold. Asimple analytical estimate for this shift has been given byJulienne et al. �2004�,

B0 = Bres + �Bx�1 − x�/�1 + �1 − x�2� , �25�

where x=abg/ ā. The characteristic length r* defined inEq. �21� is determined by the off-diagonal coupling via

��res�W��0� = ��2/2Mr�4/r�. �26�Its inverse 1/r* is therefore a measure of how strongly

the open and closed channels are coupled. In the experi-mentally most relevant case of wide resonances, thelength r* is much smaller than the background scatteringlength. Specifically, this applies to the Feshbach reso-nances in fermionic 6Li and 40K at B0=834 and 202 G,respectively, which have been used to study the BCS-BEC crossover with cold atoms �see Sec. VIII�. Theyare characterized by the experimentally determinedparameters abg=−1405aB, �B=−300 G, �=2�B and abg=174aB, �B=7.8 G, �=1.68�B, respectively, where aB

and �B are the Bohr radius and Bohr magneton. Fromthese parameters, the characteristic length associatedwith the two resonances turns out to be r�=0.5aB andr�=28aB, both obeying the wide-resonance conditionr� �abg�.

Weakly bound states close to the resonance. In additionto the control of scattering properties, an important fea-ture of Feshbach resonances concerns the possibility toform weakly bound dimers in the regime of small nega-tive detuning �=��B−B0�→0−, where the scatteringlength approaches +�. We briefly present below somekey properties of these dimers, restricting for simplicityto the vicinity of the resonance �B−B0� ��B�.

To determine the bound state for the two-channelHamiltonian �19�, one considers the Green’s functionG�E�= �E−Ĥ�−1 and looks for the low-energy pole atE=−�b�0 of this function. The corresponding boundstate can be written

�x��b�� = 1 − Z�bg�r�Z�res�r� � , �27�where the coefficient Z characterizes the closed-channeladmixture. The values of �b and Z can be calculated

explicitly by projecting the eigenvalue equation for Ĥ oneach channel. Close to resonance, where the scatteringlength is dominated by its resonant contribution, Eq.�22� and the standard relation �b=�2 /2Mra2 for a�0show that the binding energy

�b = ��B − B0��2/�� 28�

of the weakly bound state vanishes quadratically, withcharacteristic energy ��=�2 /2Mr�r��2. In an experimen-tal situation which starts from the atom continuum, it isprecisely this weakly bound state which is reached uponvarying the detuning by an adiabatic change in the mag-netic field around B0. The associated closed-channel ad-mixture Z can be obtained from the binding energy as

Z = −��b��

2��*

= 2r*

�abg��B − B0�

��B� . �29�

For a wide resonance, where r� �abg�, this admixtureremains much smaller than 1 over the magnetic fieldrange �B−B0�� B�.

The bound state ��b�� just presented should not beconfused with the bound state �op

�b��, that exists for abg�0 in the open channel, for a vanishing coupling W�r�.The bound state �op

�b�� has a binding energy of order�2 / �2Mrabg

2 �, that is much larger than that of Eq. �28�when �B−B0� ��B�. For �B−B0��B� the states ��b��and �op

�b�� have comparable energies and undergo anavoided crossing. The universal character of the aboveresults is then lost and one has to turn to a specific studyof the eigenvalue problem.

To conclude, Feshbach resonances provide a flexibletool to change the interaction strength between ultra-cold atoms over a wide range. To realize a proper many-body Hamiltonian with tunable two-body interactions,



however, an additional requirement is that the relax-ation rate into deep bound states due to three-body col-lisions must be negligible. As discussed in Sec. VIII.A,this is possible for fermions, where the relaxation rate issmall near Feshbach resonances �Petrov, Salomon, andShlyapnikov, 2004, 2005�.

II. OPTICAL LATTICES

In the following, we discuss how to confine cold atomsby laser light into configurations of a reduced dimen-sionality or in periodic lattices, thus generating situa-tions in which the effects of interactions are enhanced.

A. Optical potentials

The physical origin of the confinement of cold atomswith laser light is the dipole force

F = 12��L� � ��E�r��2� �30�

due to a spatially varying ac Stark shift that atomsexperience in an off-resonant light field �Grimm et al.,2000�. Since the time scale for the center-of-mass motionof atoms is much slower than the inverse laser frequency�L, only the time-averaged intensity �E�r��2 enters.The direction of the force depends on the sign ofthe polarizability ��L�. In the vicinity of an atomicresonance from the ground �g� to an excited state �e�at frequency �0, the polarizability has the form ��L��e�d̂E�g��2 /��0−�L�, with d̂E the dipole operator inthe direction of the field. Atoms are thus attracted to thenodes or to the antinodes of the laser intensity for blue-��L��0� or red-detuned ��L��0� laser light, respec-tively. A spatially dependent intensity profile I�r�, there-fore, creates a trapping potential for neutral atoms.Within a two-level model, an explicit form of the dipolepotential may be derived using the rotating-wave ap-proximation, which is a reasonable approximation pro-vided that the detuning �=�L−�0 of the laser field issmall compared to the transition frequency itself ��

�0. With � as the decay rate of the excited state, oneobtains for �� Grimm et al., 2000�

Vdip�r� =3c2

2�03

�

�I�r� , �31�

which is attractive or repulsive for red ��0� or blue��0� detuning, respectively. Atoms are thus attractedor repelled from an intensity maximum in space. It isimportant to note that, in contrast to the form suggestedin Eq. �30�, the light force is not fully conservative. In-deed, spontaneous emission gives rise to an imaginarypart of the polarizability. Within a two-level approxima-tion, the related scattering rate �sc�r� leads to an absorp-tive contribution ��sc�r� to the conservative dipole po-tential �31�, which can be estimated as �Grimm et al.,2000�

�sc�r� =3c2

2��03��

2

I�r� . �32�

As Eqs. �31� and �32� show, the ratio of scattering rate tothe optical potential depth vanishes in the limit ��.A strictly conservative potential can thus be reached inprinciple by increasing the detuning of the laser field. Inpractice, however, such an approach is limited by themaximum available laser power. For experiments withultracold quantum gases of alkali-metal atoms, the de-tuning is typically chosen to be large compared to theexcited-state hyperfine structure splitting and in mostcases even large compared to the fine-structure splittingin order to sufficiently suppress spontaneous scatteringevents.

The intensity profile I�r ,z� of a Gaussian laser beampropagating along the z direction has the form

I�r,z� = �2P/w2�z��e−2r2/w2�z�. �33�

Here P is the total power of the laser beam, r is thedistance from the center, and w�z�=w01+z2 /zR2 is the1/e2 radius. This radius is characterized by a beam waistw0 that is typically around 100 �m. Due to the finitebeam divergence, the beam width increases linearly withz on a scale zR=w0

2 /�, which is called the Rayleighlength. Typical values for zR are in the millimeter tocentimeter range. Around the intensity maximum, a po-tential depth minimum occurs for a red-detuned laserbeam, leading to an approximately harmonic potential

Vdip�r,z� � − Vtrap�1 − 2�r/w0�2 − �z/zR�2� . �34�

The trap depth Vtrap is linearly proportional to the laserpower and typically ranges from a few kilohertz up to1 MHz �from the nanokelvin to the microkelvin regime�.The harmonic confinement is characterized by radial �rand axial �z trapping frequencies �r= �4Vtrap/Mw0

2�1/2

and �z=2Vtrap/MzR2 . Optical traps for neutral atoms

have a wide range of applications �Grimm et al., 2000�.In particular, they are inevitable in situations in whichmagnetic trapping does not work for the atomic statesunder consideration. This is often the case when interac-tions are manipulated via Feshbach resonances, involv-ing high-field-seeking atomic states.

Optical lattices. A periodic potential is generated byoverlapping two counterpropagating laser beams. Dueto the interference between the two laser beams, an op-tical standing wave with period � /2 is formed, in whichatoms can be trapped. More generally, by choosing thetwo laser beams to interfere under an angle less than180°, one can also realize periodic potentials with alarger period �Peil et al., 2003; Hadzibabic et al., 2004�.The simplest possible periodic optical potential isformed by overlapping two counterpropagating beams.For a Gaussian profile, this results in a trapping poten-tial of the form

V�r,z� − V0e−2r2/w2�z� sin2�kz� , �35�

where k=2 /� is the wave vector of the laser light andV0 is the maximum depth of the lattice potential. Note



that due to the interference of the two laser beams, V0 isfour times larger than Vtrap if the laser power and beamparameters of the two interfering lasers are equal.

Periodic potentials in two dimensions can be formedby overlapping two optical standing waves along differ-ent, usually orthogonal, directions. For orthogonal po-larization vectors of the two laser fields, no interferenceterms appear. The resulting optical potential in the cen-ter of the trap is then a simple sum of a purely sinusoidalpotential in both directions.

In such a two-dimensional optical lattice potential, at-oms are confined to arrays of tightly confining one-dimensional tubes �see Fig. 4�a��. For typical experimen-tal parameters, the harmonic trapping frequencies alongthe tube are very weak �on the order of 10–200 Hz�,while in the radial direction the trapping frequencies canbecome as high as up to 100 kHz. For sufficiently deeplattice depths, atoms can move only axially along thetube. In this manner, it is possible to realize quantumwires with neutral atoms, which allows one to studystrongly correlated gases in one dimension, as discussedin Sec. V. Arrays of such quantum wires have been real-ized �Greiner et al., 2001; Moritz et al., 2003; Kinoshita etal., 2004; Paredes et al., 2004; Tolra et al., 2004�.

For the creation of a three-dimensional lattice poten-tial, three orthogonal optical standing waves have to beoverlapped. The simplest case of independent standingwaves, with no cross interference between laser beamsof different standing waves, can be realized by choosingorthogonal polarization vectors and by using slightly dif-ferent wavelengths for the three standing waves. The

resulting optical potential is then given by the sum ofthree standing waves. In the center of the trap, for dis-tances much smaller than the beam waist, the trappingpotential can be approximated as the sum of a homoge-neous periodic lattice potential

Vp�x,y,z� = V0�sin2 kx + sin2 ky + sin2 kz� �36�

and an additional external harmonic confinement due tothe Gaussian laser beam profiles. In addition to this, aconfinement due to the magnetic trapping is often used.

For deep optical lattice potentials, the confinement ona single lattice site is approximately harmonic. Atomsare then tightly confined at a single lattice site, with trap-ping frequencies �0 of up to 100 kHz. The energy ��0=2Er�V0 /Er�1/2 of local oscillations in the well is on theorder of several recoil energies Er=�2k2 /2m, which is anatural measure of energy scales in optical lattice poten-tials. Typical values of Er are in the range of severalkilohertz for 87Rb.

Spin-dependent optical lattice potentials. For large de-tunings of the laser light forming the optical latticescompared to the fine-structure splitting of a typicalalkali-metal atom, the resulting optical lattice potentialsare almost the same for all magnetic sublevels in theground-state manifold of the atom. However, for morenear-resonant light fields, situations can be created inwhich different magnetic sublevels can be exposed tovastly different optical potentials �Jessen and Deutsch,1996�. Such spin-dependent lattice potentials can, e.g.,be created in a standing wave configuration formed bytwo counterpropagating laser beams with linear polar-ization vectors enclosing an angle �Jessen and Deutsch,1996; Brennen et al., 1999; Jaksch et al., 1999; Mandel etal., 2003a�. The resulting standing wave light field can bedecomposed into a superposition of a �+- and a�−-polarized standing wave laser field, giving rise to lat-tice potentials V+�x , �=V0 cos2�kx+ /2� and V−�x , �=V0 cos2�kx− /2�. By changing the polarization angle ,one can control the relative separation between the twopotentials �x= � /��x /2. When is increased, both po-tentials shift in opposite directions and overlap againwhen =n, with n an integer. Such a configuration hasbeen used to coherently move atoms across lattices andrealize quantum gates between them �Jaksch et al., 1999;Mandel et al., 2003a, 2003b�. Spin-dependent lattice po-tentials furthermore offer a convenient way to tune in-teractions between two atoms in different spin states. Byshifting the spin-dependent lattices relative to eachother, the overlap of the on-site spatial wave functioncan be tuned between zero and its maximum value, thuscontrolling the interspecies interaction strength within arestricted range. Recently, Sebby-Strabley et al. �2006�have also demonstrated a novel spin-dependent latticegeometry, in which 2D arrays of double-well potentialscould be realized. Such “superlattice” structures allowfor versatile intrawell and interwell manipulation possi-bilities �Fölling et al., 2007; Lee et al., 2007; Sebby-Strabley et al., 2007�. A variety of lattice structures canbe obtained by interfering laser beams under different

(a)

(b)

FIG. 4. �Color online� Optical lattices. �a� Two- and �b� three-dimensional optical lattice potentials formed by superimposingtwo or three orthogonal standing waves. For a two-dimensional optical lattice, the atoms are confined to an arrayof tightly confining one-dimensional potential tubes, whereasin the three-dimensional case the optical lattice can be ap-proximated by a three-dimensional simple cubic array oftightly confining harmonic-oscillator potentials at each latticesite.



angles; see, e.g., Jessen and Deutsch �1996� and Gryn-berg and Robillard �2001�.

B. Band structure

We now consider single-particle eigenstates in an infi-nite periodic potential. Any additional potential fromthe intensity profile of the laser beams or from somemagnetic confinement is neglected �for the single-particle spectrum in the presence of an additional har-monic confinement, see Hooley and Quintanilla �2004��.In a simple cubic lattice, the potential is given by Eq.�36�, with a tunable amplitude V0 and lattice constantd= /k. In the limit V0�Er, each well supports a num-ber of vibrational levels, separated by an energy ��0�Er. At low temperatures, atoms are restricted to thelowest vibrational level at each site. Their kinetic energyis then frozen, except for the small tunneling amplitudeto neighboring sites. The associated single-particleeigenstates in the lowest band are Bloch waves with qua-simomentum q and energy

�0�q� =32��0 − 2J�cos qxd + cos qyd + cos qzd� + ¯ .

�37�

The parameter J�0 is the gain in kinetic energy due tonearest-neighbor tunneling. In the limit V0�Er, it canbe obtained from the width W→4J of the lowest band inthe 1D Mathieu equation

J

4

ErV0Er�

3/4

exp�− 2V0Er

�1/2� . �38�For lattice depths larger than V0�15Er, this approxima-tion agrees with the exact values of J given in Table I tobetter than 10% accuracy. More generally, for any peri-odic potential Vp�r+R�=Vp�r� which is not necessarilydeep and separable, the exact eigenstates are Blochfunctions �n,q�r�. They are characterized by a discreteband index n and a quasimomentum q within the firstBrillouin zone of the reciprocal lattice �Ashcroft andMermin, 1976�. Since Bloch functions are multiplied bya pure phase factor exp�iq ·R�, upon translation by oneof the lattice vectors R, they are extended over thewhole lattice. An alternative single-particle basis, whichis useful for describing the hopping of particles amongdiscrete lattice sites R, is provided by Wannier functionswn,R�r�. They are connected with the Bloch functions bya Fourier transform

�n,q�r� = �R

wn,R�r�eiq·R �39�

on the lattice. The Wannier functions depend only onthe relative distance r−R, and, at least for the lowestbands, they are centered around the lattice sites R �seebelow�. By choosing a convenient normalization, theyobey the orthonormality relation

� d3r wn*�r − R�wn��r − R�� = �n,n��R,R� �40�for different bands n and sites R. Since the Wannierfunctions for all bands n and sites R form a complete

basis, the operator �̂�r�, which destroys a particle at anarbitrary point r, can be expanded in the form

�̂�r� = �R,n

wn�r − R�âR,n. �41�

Here âR,n is the annihilation operator for particles in thecorresponding Wannier states, which are not necessarilywell localized at site R. The Hamiltonian for free motionon a periodic lattice is then

Ĥ0 = �R,R�,n

Jn�R − R��âR,n† âR�,n. �42�

It describes the hopping in a given band n with matrixelements Jn�R�, which in general connect lattice sites atarbitrary distance R. The diagonalization of this Hamil-tonian by Bloch states �39� shows that the hopping ma-trix elements Jn�R� are uniquely determined by theBloch band energies �n�q� via

�R

Jn�R�exp�iq · R� = �n�q� . �43�

In the case of separable periodic potentials Vp�r�=V�x�+V�y�+V�z�, generated by three orthogonal opticallattices, the single-particle problem is one dimensional.A complete analysis of Wannier functions in this casehas been given by Kohn �1959�. Choosing appropri-ate phases for the Bloch functions, there is a uniqueWannier function for each band, which is real and expo-nentially localized. The asymptotic decay �exp�−hn�x��is characterized by a decay constant hn, which is a de-creasing function of the band index n. For the lowestband n=0, where the Bloch function at q=0 is finite atthe origin, the Wannier function w�x� can be chosen tobe symmetric around x=0 �and correspondingly it isantisymmetric for the first excited band�. More pre-cisely, the asymptotic behavior of the 1D Wannier func-tions and the hopping matrix elements is �wn�x��x�−3/4 exp�−hn�x�� and Jn�R��R�−3/2 exp�−hn�R��, re-spectively �He and Vanderbilt, 2001�. In the particular

TABLE I. Hopping matrix elements to nearest J and next-nearest neighbors J�2d�, bandwidth W, and overlap betweenthe Wannier function and the local Gaussian ground state in1D optical lattices. Table courtesy of M. Holthaus.

V0 /Er 4J /Er W /Er J�2d� /J ��w ��2

3 0.444 109 0.451 894 0.101 075 0.97195 0.263 069 0.264 211 0.051 641 0.9836

10 0.076 730 0.076 747 0.011 846 0.993815 0.026 075 0.026 076 0.003 459 0.996420 0.009 965 0.009 965 0.001 184 0.9975



case of a purely sinusoidal potential V0 sin2�kx� withlattice constant d=� /2, the decay constant h0 increasesmonotonically with V0 /Er. In the deep-lattice limitV0�Er, it approaches h0d=V0 /Er /2. It is importantto realize that, even in this limit, the Wannier functiondoes not uniformly converge to the local harmonic-oscillator ground state � of each well: wn�x� decays ex-ponentially rather than in a Gaussian manner and al-ways has nodes in order to guarantee the orthogonalityrelation �40�. Yet, as shown in Table I, the overlap isnear 1 even for shallow optical lattices.

C. Time-of-flight and adiabatic mapping

Sudden release. When releasing ultracold quantumgases from an optical lattice, two possible release meth-ods can be chosen. If the lattice potential is turned offabruptly and interaction effects can be neglected, agiven Bloch state with quasimomentum q will expandaccording to its momentum distribution as a superposi-tion of plane waves with momenta pn=�q±n2�k, with nan arbitrary integer. This is a direct consequence ofthe fact that Bloch waves can be expressed as a super-position of plane-wave states exp�i�q+G� ·r� with mo-menta q+G, which include arbitrary reciprocal-latticevectors G. In a simple cubic lattice with lattice spacingd= /k, the vectors G are integer multiples of the fun-damental reciprocal-lattice vector 2k. After a certain

time of flight, this momentum distribution can be im-aged using standard absorption imaging methods. If onlya single Bloch state is populated, as is the case for aBose-Einstein condensate with quasimomentum q=0,this results in a series of interference maxima that can beobserved after a time-of-flight period t �see Fig. 5�. Asshown in Sec. III.A, the density distribution observedafter a fixed time of flight at position x is the momentumdistribution of the particles trapped in the lattice,

n�x� = �M/�t�3�w̃�k��2G�k� . �44�

Here k is related to x by k=Mx /�t due to the assump-tion of ballistic expansion, while w̃�k� is the Fouriertransform of the Wannier function. The coherence prop-erties of the many-body state are characterized by theFourier transform

G�k� = �R,R�

eik·�R−R��G�1��R,R�� 45�

of the one-particle density matrix G�1��R ,R��= �âR† âR��.

In a BEC, the long-range order in the amplitudesleads to a constant value of the first order coherencefunction G�1��R ,R�� at large separations �R−R�� see theAppendix�. The resulting momentum distribution coin-cides with the standard multiple wave interference pat-tern obtained with light diffracting off a material grating�see Fig. 5�c� and Sec. IV.B�. The atomic density distri-bution observed after a fixed time-of-flight time thusyields information on the coherence properties of themany-body system. It should be noted, however, that theobserved density distribution after time of flight can de-viate from the in-trap momentum distribution if interac-tion effects during the expansion occur, or the expansiontime is not so long that the initial size of the atom cloudcan be neglected �far-field approximation� �Pedri et al.,2001; Gerbier et al., 2007�.

Adiabatic mapping. One advantage of using opticallattice potentials is that the lattice depth can be dynami-cally controlled by simply tuning the laser power. Thisopens another possibility for releasing atoms from thelattice potential, e.g., by adiabatically converting a deepoptical lattice into a shallow one and eventually com-pletely turning off the lattice potential. Under adiabatictransformation of the lattice depth the quasimomentum

BEC CCD Chip

Imaging laser

g

(a) (b) (c)

FIG. 5. �Color online� Absorption imaging. �a� Schematicsetup for absorption imaging after a time-of-flight period. �b�Absorption image for a BEC released from a harmonic trap.�c� Absorption image for a BEC released from a shallow opti-cal lattice �V0=6Er�. Note the clearly visible interference peaksin the image.

20 Er 4 Er Free particle(a) (b)

E

-3�k -2�k -�k-�k

�k�k

2�k 3�kp

�ω

q-�k �k

q-�k �k

q

FIG. 6. �Color online� Adiabatic band mapping. �a� Bloch bands for different potential depths. During an adiabatic ramp down thequasimomentum is conserved and �b� a Bloch wave with quasimomentum q in the nth energy band is mapped onto a free particlewith momentum p in the nth Brillouin zone of the lattice. From Greiner et al., 2001.



q is preserved, and during the turn-off process a Blochwave in the nth energy band is mapped onto a corre-sponding free-particle momentum p in the nth Brillouin-zone �see Fig. 6� �Kastberg et al., 1995; Greiner et al.,2001; Köhl et al., 2005a�.

The adiabatic mapping technique has been used withboth bosonic �Greiner et al., 2001� and fermionic �Köhlet al., 2005a� atoms. For a homogeneously filled lowest-energy band, an adiabatic ramp down of the lattice po-tential leaves the central Brillouin zone—a square ofwidth 2�k—fully occupied �see Fig. 7�b��. If, on theother hand, higher-energy bands are populated, one alsoobserves populations in higher Brillouin zones �see Fig.7�c��. As in this method each Bloch wave is mapped ontoa specific free-particle momentum state, it can be usedto efficiently probe the distribution of particles overBloch states in different energy bands.

D. Interactions and two-particle effects

So far we have only discussed single-particle behaviorof ultracold atoms in optical lattices. However, short-ranged s-wave interactions between particles give rise toan on-site interaction energy, when two or more atomsoccupy a single lattice site. Within the pseudopotentialapproximation, the interaction between bosons has theform

Ĥ� = �g/2� � d3r �̂†�r��̂†�r��̂�r��̂�r� . �46�Inserting the expansion Eq. �41� leads to interactions in-volving Wannier states in both different bands and dif-ferent lattice sites. The situation simplifies, however, fora deep optical lattice and with the assumption that onlythe lowest band is occupied. The overlap integrals arethen dominated by the on-site term Un̂R�n̂R−1� /2,which is nonzero if two or more atoms are in the sameWannier state. At the two-particle level, the interactionbetween atoms in Wannier states localized around Rand R� is thus reduced to a local form U�R,R� with

U = g� d3r�w�r��4 = 8/kaEr�V0/Er�3/4 �47��for simplicity, the band index n=0 is omitted for thelowest band�. The explicit result for the on-site interac-tion U is obtained by taking w�r� as the Gaussian groundstate in the local oscillator potential. As mentionedabove, this is not the exact Wannier wave function of thelowest band. In the deep-lattice limit V0�Er, however,the result �47� provides the asymptotically correct be-havior. Note that the strength �U� of the on-site interac-tion increases with V0, which is due to the squeezing ofthe Wannier wave function w�r�.

Repulsively bound pairs. Consider now an optical lat-tice at very low filling. An occasional pair of atoms atthe same site has an energy U above or below the centerof the lowest band. In the attractive case U�0, a two-particle bound state will form for sufficiently large val-ues of �U�. In the repulsive case, in turn, the pair is ex-pected to be unstable with respect to breakup into twoseparate atoms at different lattice sites to minimize therepulsive interaction. This process, however, is forbid-den if the repulsion is above a critical value U�Uc. Thephysical origin for this result is that momentum and en-ergy conservation do not allow the two particles to sepa-rate. There are simply no free states available if the en-ergy lies more than zJ above the band center, which isthe upper edge of the tight-binding band. Here z de-notes the number of nearest neighbors on a lattice. Twobosons at the same lattice site will stay together if theirinteraction is sufficiently repulsive. In fact, the two-particle bound state above the band for a repulsive in-teraction is the precise analog of the standard boundstate below the band for attractive interactions, andthere is a perfect symmetry around the band center.

Such “repulsively bound pairs” have been observed inan experiment by Winkler et al. �2006�. A dilute gas of87Rb2 Feshbach molecules was prepared in the vibra-tional ground state of an optical lattice. Ramping themagnetic field across a Feshbach resonance to negativea, these molecules can be adiabatically dissociated andthen brought back again to positive a as repulsive pairs.Since the bound state above the lowest band is built

2 hk

(b)4 4

44

3

3

3

3

33

3 3

1

2

2

2

2

(a)

2hk

2 hk

(c)

FIG. 7. Brillouin zones. �a� Brillouin zones of a 2D simple cubic optical lattice. �b� For a homogeneously filled lowest Bloch band,an adiabatic shutoff of the lattice potential leads to a homogeneously populated first Brillouin zone, which can be observedthrough absorption imaging after a time-of-flight expansion. �c� If in addition higher Bloch bands are populated, higher Brillouinzones become populated as well. From Greiner et al., 2001.



from states in which the relative momentum of the twoparticles is near the edge of the Brillouin zone, the pres-ence of repulsively bound pairs can be inferred fromcorresponding peaks in the quasimomentum distributionobserved in a time-of-flight experiment �see Fig. 8� �Win-kler et al., 2006�. The energy and dispersion relation ofthese pairs follows from solving UGK�E ,0�=1 for abound state of two particles with center-of-mass momen-tum K. In close analogy to Eq. �79� below, GK�E ,0� isthe local Green’s function for free motion on the latticewith hopping matrix element 2J. Experimentally, the op-tical lattice was strongly anisotropic such that tunnelingis possible only along one direction. The correspondingbound-state equation in one dimension can be solvedexplicitly, giving �Winkler et al., 2006�

E�K,U1� = 2J�2 cos Kd/2�2 + �U1/2J�2 − 4J �48�for the energy with respect to the upper band edge.Since E�K=0,U1��0 for arbitrarily small values of U1�0, there is always a bound state in one dimension. Bycontrast, in 3D there is a finite critical value, which isUc=7.9136 J for a simple cubic lattice. The relevant on-site interaction U1 in one dimension is obtained fromEq. �78� for the associated pseudopotential. With �0 theoscillator length for motion along the direction of hop-ping, it is given by

U1 = g1� dx�w�x��4 = 2/��a/�0. �49�Evidently, U1 has the transverse confinement energy�� as the characteristic scale, rather than the recoilenergy Er of Eq. �47� in the 3D case.

Tightly confined atom pairs. The truncation to the low-est band requires that both the thermal and on-site in-teraction energy U are much smaller than ��0. In thedeep-lattice limit V0�Er, the condition U��0 leads toka�V0 /Er�1/41 using Eq. �47�. This is equivalent to a

�0, where �0=� /M�0= �Er /V0�1/4d / is the oscillatorlength associated with the local harmonic motion in thedeep optical lattice wells. The assumption of staying inthe lowest band in the presence of a repulsive interac-tion thus requires the scattering length to be much

smaller than �0, which is itself smaller than, but of thesame order as, the lattice spacing d. For typical valuesa5 nm and d=0.5 �m, this condition is well justified.In the vicinity of Feshbach resonances, however, thescattering lengths become comparable to the latticespacing. A solution of the two-particle problem in thepresence of an optical lattice for arbitrary values of theratio a /�0 has been given by Fedichev et al. �2004�. Ne-glecting interaction-induced couplings to higher bands,they have shown that the effective interaction at ener-gies smaller than the bandwidth is described by apseudopotential. For repulsive interactions a�0, the as-sociated effective scattering length reaches a bound aeff�d on the order of the lattice spacing even if a→� neara Feshbach resonance. In the case in which the free-space scattering length is negative, aeff exhibits a geo-metric resonance that precisely describes the formationof a two-particle bound state at �U�=7.9136 J as dis-cussed above.

This analysis is based on the assumption that particlesremain in a given band even in the presence of stronginteractions. Near Feshbach resonances, however, this isusually not the case. In order to address the question ofinteraction-induced transitions between different bands,it is useful to consider two interacting particles in a har-monic well �Busch et al., 1998�. Provided the range ofinteraction is much smaller than the oscillator length �0,the interaction of two particles in a single well is de-scribed by a pseudopotential. The ratio of the scatteringlength a to �0, however, may be arbitrary. The corre-sponding energy levels E=��0�3/2−!� as a function ofthe ratio �0 /a follow from the transcendental equation

�0/a = 2��!/2�/�„�! − 1�/2… = f3�!� , �50�where ��z� is the standard gamma function. In fact, thisis the analytical continuation to an arbitrary sign of thedimensionless binding energy ! in Eq. �82� for n=3,since harmonic confinement is present in all three spatialdirections.

As shown in Fig. 9, the discrete levels for the relative

-2

2

4

6

-4 -2

2 4 l0/a

E/h�0-3/2

FIG. 9. �Color online� Energy spectrum of two interacting par-ticles in a 3D harmonic-oscillator p

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