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Expansion of Bose-Hubbard Mott insulators in optical lattices

Mark Jreissaty,1 Juan Carrasquilla,1 F. Alexander Wolf,1, 2 and Marcos Rigol1

1Department of Physics, Georgetown University, Washington, DC 20057, USA2Theoretical Physics III, Center for Electronic Correlations and Magnetism,

Institute of Physics, Augsburg University, 86135 Augsburg, Germany

We study the expansion of bosonic Mott insulators in the presence of an optical lattice afterswitching off a confining potential. We use the Gutzwiller mean-field approximation and considertwo different set ups. In the first one, the expansion is restricted to one direction. We show thatthis leads to the emergence of two condensates with well defined momenta, and argue that sucha construct can be used to create atom lasers in optical lattices. In the second set up, we studyMott insulators that are allowed to expand in all directions within the lattice. In this case, a singlecondensate develops in the system. However, its constituent bosons are found to populate manynonzero momentum modes. An analytic understanding of both phenomena in terms of the exactdispersion relation in the hard-core limit is presented.

PACS numbers: 03.75.Kk,03.75.-b,03.75.Hh,05.30.Jp

I. INTRODUCTION

In recent years, the study of strongly correlated ultra-cold gases in optical lattices has become an importantfocus of experimental and theoretical research [1, 2]. Ini-tially, many of the efforts were devoted to studying equi-librium quantum phase transitions, for which the obser-vation of the superfluid to Mott insulator phase transi-tions in three [3], two [4], and one [5] dimension was amajor experimental accomplishment. More recently, thestudy of the non-equilibrium dynamics of these systemshas started gaining attention [6–10]. The latter is possi-ble, and offers great insights into the coherent dynamicsof quantum systems, thanks to the nearly ideal isola-tion of the gas from the environment, and to the uniquecontrol that has been achieved experimentally over theparameters that determine the dynamics.The non-equilibrium dynamics of quantum systems is

a rich subject, see e.g, Refs. [11–13]. In this work, wewill be mainly interested in the expansion of a stronglyinteracting bosonic gas in an optical lattice. Experi-mentally, expansions (time-of-flight measurements) havebeen used to learn about the properties of equilibriumgases [14]. The idea behind those measurements is that,if the gas is allowed to expand in the absence of inter-actions, the initial momentum distribution function fullydetermines the density distribution after a long expan-sion time. Hence, by taking a picture of the latter, onecan determine the former by means of a simple rescaling[1, 2, 14]. On the other hand, if interactions are presentduring the expansion, unusual phenomena can occur [15],e.g., the transformation of the momentum distributionof an expanding gas of impenetrable bosons into the mo-mentum distribution function of a noninteracting Fermigas in equilibrium, which has a Fermi edge [16, 17]. Inaddition, it has been recently proposed that the expan-sion of two-component fermions in the presence of stronginteractions in an optical lattice can be used as a toolto cool the gas through a “quantum distillation” process[18].

There is another remarkable phenomenon that occursduring the expansion of a strongly interacting gas ofbosons in an optical lattice, and that is the emergence ofcoherence during the expansion of Mott insulating states.This phenomenon will be the main focus of this work. AMott insulator is an interaction-generated insulator thatexhibits a gap to one-particle excitations. The presenceof this gap produces an exponential decay of the one-particle correlations. Within the Bose-Hubbard modeldescription, the Mott insulator is the ground state oflattice bosons for commensurate fillings and sufficientlystrong interactions [19]. As mentioned before, Mott insu-lators have been created experimentally using ultracoldgases in optical lattices. We should stress that due tothe presence of a trapping potential, the Mott insulat-ing phases in experiments usually coexist with superfluiddomains. Surprisingly, when Mott insulators are allowedto expand by turning off the trap in the presence of alattice, coherence emerges between initially uncorrelatedparticles [20–22], a phenomenon that resembles a sort ofdynamical phase transition.

The study in Ref. [20] demonstrated that the expan-sion of pure Mott insulating states of hard-core bosons in1D leads to the development of quasi-long range correla-tions and to the emergence of quasi-condensates at finitemomenta k = ±π/2a (where a is the lattice parame-ter). In a related set-up, the onset of quasi-long rangecorrelations has been studied more recently (in the spin-1/2 language) using various analytic and approximate ap-proaches [23]. It is important to note that these resultswere obtained in the limit of infinite on-site repulsionsbetween the bosons (hard-core limit), and for pure Mottinsulating domains. It is, therefore, natural to wonderwhether the same results can be observed experimentallywith ultracold gases, for which the on-site interactionsare not infinite and the insulating domains are alwayssurrounded by superfluid ones. Systems containing hard-core bosons, in which superfluid and insulating domainscoexist, were studied in Ref. [21]. It was found there that,if most of the initial system is in a Mott insulating state,

2

quasi-condensates with momenta k ≈ ±π/2a are gener-ated. The superfluid wings surrounding the insulatingdomain in the initial density profiles appeared to haveno negative effect on the emergence of quasi-coherenceduring the expansion. Furthermore, the expansion of aMott insulator in a system with finite on-site interac-tions (soft-core bosons) was studied in Ref. [22]. Onceagain, quasi-long range coherence developed during theexpansion and lead to quasi-condensates with momenta|k| . π/2a. The differences between the position of thepeaks for soft-core bosons and hard-core bosons can beunderstood by considering their respective dispersion re-lations. Finally, a similar onset of power-law decayingcorrelations has been observed during the expansion of aMott insulator within the fermionic Hubbard model [24].

A more challenging question is what happens in higherdimensions. This question poses greater conceptual andcomputational challenges. Since hard-core bosons in 1Dare described by an integrable model, one may expecttheir behavior to be unique. Hence, it may not extend tonon-integrable models far from any integrable limit, suchas the Bose-Hubbard model with finite on-site interactionin higher dimensions. In the particular case of one spa-tial dimension, since the boson’s on-site interaction hasto be strong enough for a Mott insulator to exist, the re-sults obtained for the 1D expansion [22] can be thought asstill affected by the infinite-repulsion integrable limit, andthus, may not generalize to higher dimensions. Further-more, while the 1D expansion can be studied by meansof an exact analytical/numerical approach in the hard-core limit [20, 21] and by means of an unbiased numericaltechnique for the soft-core case (the time-dependent den-sity matrix renormalization group approach [25]), no suchtool exists to study the expansion in higher dimensions.For the latter, no efficient and unbiased, analytical orcomputational approach is known that can deal with thedynamics in the presence of strong interactions. Approx-imations are therefore required to study these systems.

A first step towards understanding what happens inhigher dimensions was taken in Ref. [26]. In thatwork, which was a study in the hard-core limit utiliz-ing the Gutzwiller mean-field approximation, a three-dimensional (3D) Mott insulator was allowed to expandin a single direction in the lattice. The study showedthat condensates emerge at finite momenta following the“melting” of a Mott insulator. In addition, the momentaof the resulting condensates were found to be fully de-termined by the ratio between the hopping amplitudestransverse to and along the expansion. Hence, thesesystems can be used to create highly controllable atomlasers. Other nonequilibrium set-ups that allow one toobtain condensation at finite momentum [27, 28] do notseem suitable for the latter purpose.

In this work, we extend the results reported in Ref. [26]to the expansion of Mott insulators with finite on-siteinteraction in two-dimensions (2D). Our results in 2Dcan be straightforwardly extended to three dimensions.We first consider a setup similar to that in Ref. [26], in

which the soft-core bosons (SCB) are allowed to expandin a single direction. We show that similarly to the casein the hard-core limit, condensates emerge at finite mo-menta. We then proceed to study a system in which theMott insulator is allowed to expand in all directions of thelattice. For this particular case, we show that, at leastwithin the mean-field approximation, a unique conden-sate with many different momenta develops as the Mottinsulator expands. We also discuss how these results canbe understood in terms of the exact dispersion relationin the hard-core limit.The article is organized as follows: in Sec. II, we intro-

duce the model and the time-dependent mean-field ap-proach used to study the expansion. The explanation ofhow our initial states are prepared is presented in Sec. III.Section IV is devoted to the study of the expansion alongone direction. The case in which the Mott insulator isallowed to expand in all directions is discussed in Sec. V.Finally, the conclusions are presented in Sec. VI.

II. MODEL AND MEAN-FIELD APPROACH

Our study is relevant to ultracold bosons trapped indeep optical lattices. They can be well described by theBose-Hubbard Hamiltonian [29]

H = −∑

〈i,j〉

Jij

(

a†i aj +H.c.)

+U

2

∑

i

ni (ni − 1)

+∑

i

(Vx x2i + Vy y

2i ) ni , (1)

where a†i (ai) is the boson creation (annihilation) opera-

tor at a given site i, and ni = a†i ai is the particle numberoperator at the same site i. The first two terms of theexpression are the usual kinetic and interaction termsthat define the Hubbard model in the homogeneous case,where Jij is the hopping amplitude between neighbor-ing sites i and j (〈i, j〉), and U is the on-site interactionstrength. In experiments involving ultracold gases, a trapis required to confine the atoms. To a good approxima-tion, those traps are harmonic, as considered in the lastterm in Eq. (1). Here, Vx and Vy denote the strength ofthe trap in the x- and y-direction, while xi and yi arethe coordinates of site i with respect to the center of thetrap. In the remainder of the manuscript, positions willbe given in units of the lattice spacing a, which is takento be the same in all directions.The ground state phase diagram of the homogeneous

case has been studied extensively in all dimensions[19, 30–35]. In this situation Vx = Vy = 0 and Jij = J .As mentioned earlier, it contains both superfluid andMott insulating phases. The former occurs for incom-mensurate fillings, and for commensurate fillings for U/Jis below some critical value that depends on the fillingand dimensionality of the system. The insulating phasesare only present for commensurate fillings and sufficientlystrong interactions. This model is not exactly solvable in

3

any dimension. Unbiased results for the phase diagramhave been obtained using quantum Monte Carlo simula-tions [30, 34, 35], density-matrix renormalization group(1D) [33], and strong coupling expansions [31, 32]. Inter-estingly, a simple approximation such as the Gutzwillermean-field theory, which is exact only in the limit of infi-nite dimensions, can still describe qualitatively the com-plete phase diagram of this model for finite dimensions[19].In the presence of a trap, superfluid and Mott insulat-

ing phases coexist and form a “wedding cake” structure[36–38]. The particular shape of the density distribu-tion function can be understood within the local densityapproximation [1], given the phase diagram in the ho-mogeneous case (see Fig. 1, and the accompanying ex-planation). Experimentally, such a structure has beenobserved by several groups [39–43]. As mentioned in theintroduction, the unique opportunities provided by thestudy of these systems, as opposed to real materials, comefrom the experimentalist’s high degree of control over thedepth and structure of the optical lattice and the confin-ing potential. Essentially, all the microscopic parametersin Eq. (1) can be manipulated and made time-dependent.For example, J , U , and Vx, Vy can be modified by chang-ing the intensity of the laser beams that produce the lat-tice; Vx, Vy can be modified by introducing an additionaltrapping (or anti trapping) magnetic fields; and U canbe modified using Feshbach resonances. Together withthe high degree of isolation from the environment, thislevel of control makes these experiments ideal to studythe effects of strong correlations and dimensionality inthe non-equilibrium dynamics of the expanding gases.In this work, we are interested in studying the expan-

sion of Mott insulating states in the lattice. For this, weprepare the system in its ground state, for a particularnumber of particles, U , J , and Vx, Vy. At time t = 0,we turn off the trap and allow the system to expand in

the presence of the optical lattice. During the expansion,particles interact strongly and we compute observablessuch as the density, momentum distribution function,and condensate fraction, as a function of time. All ourcalculations are performed utilizing the time-dependentGutzwiller mean-field theory. This is a good approxima-tion in high dimensions, and, in Ref. [26], it was shown toqualitative reproduce the behavior seen in an exact diag-onalization study of the expansion in (small) 2D lattices.This mean-field theory is based on the restriction that

the ground-state wave function is of the Gutzwiller-typeproduct state

|ΨMF〉 =L∏

i=1

nc∑

n=0

γin|n〉i , (2)

where L is the number of lattice sites, nc is a cut-off takento be large enough such that all our results are indepen-dent of nc (nc = 3 in our calculations), |n〉i are the Fockstates for lattice site i, and the complex coefficients γinare variational parameters determined by minimizing the

expectation value

〈ΨMF|H − µN |ΨMF〉. (3)

In Eq. (3), µ is the chemical potential, i.e., we work on

the grand canonical ensemble, and N the total numberof particle operator.The energy minimization leads to the set of equations

∑

〈j〉i

−Jij[√n+ 1 γi(n+1)Φ

∗j +

√nγi(n−1)Φj

]

+ n

[

U

2(n+ 1) + Vi − µ

]

γin = 0, (4)

where∑

〈j〉idenotes a summation over all j that are

nearest neighbors of i, Vi = Vx x2i + Vy y

2i , and γi(−1) =

γi(nc+1) = 0. The mean-field Φj is defined as

Φj = 〈aj〉 =nc∑

n=1

√n γ∗j(n−1)γjn. (5)

By solving Eq. (4), one obtains all the coefficients γinin Eq. (2), and hence determines the initial state for thetime evolution.An alternative approach, which is equivalent to that

of solving Eq. (4), can be used. It consists in findingthe ground state of the following mean-field decoupledHamiltonian [44],

Hmf = −∑

〈i,j〉

Jij

(

a†iΦj +Φ∗i aj − Φ∗

iΦj

)

+H.c.

+U

2

∑

i

ni(ni − 1) +∑

i

(Vi − µ) ni. (6)

This Hamiltonian is obtained by a decoupling of the hop-ping terms in Eq. (1) as,

a†i aj ≃ a†iΦj + ajΦ∗i − Φ∗

iΦj . (7)

Equation (1) can then be written as a sum over single-site

Hamiltonians Hi, coupled only through constant neigh-boring terms Φj . This implies that the ground state ofthe decoupled Hamiltonian can be written as the prod-uct state in Eq. (2). The optimal set of coefficients γinis related to the ground-state eigenvector components ofthe Hamiltonian in Eq. (6) subject to the condition inEq. (5), which has to be self-consistently satisfied. Inpractice, this condition is reached as follows: an arbi-trary set of nonzero Φ0

i is used to construct a set of de-

coupled Hamiltonians Hi which are diagonalized for eachsite in order to obtain a set of γin’s. Given γin, a set ofΦnew

i is computed according to Eq. (5). If the elementson this new set are equal to those of the previous set Φ0

i

within a certain desired precision, then self consistencyhas been reached. If not, we set Φ0

i = Φnewi and construct

and a new set of decoupled Hamiltonians and the processof diagonalizing and finding Φnew

i is iteratively repeated

4

until self consistency is satisfied. We have verified thatboth approaches based on Eq. (4) and Eq. (6) providequantitatively consistent results. However, at least in ourimplementations, the one based on the mean-field decou-pled Hamiltonian is more stable and considerably fasterthan the one based on the direct solution of Eq. (4).Following the release of the potential trap, the time-

evolution of |ψMF〉 is computed using the time-dependentvariational principle [45–48], which minimizes the expec-tation value

〈ΨMF|i∂t − H + µN |ΨMF〉 , (8)

and leads to the following set of differential equations,

iγin = −∑

〈j〉i

Jij[√n+ 1 γi(n+1)Φ

∗j +

√nγi(n−1)Φj

]

+ n

[

U

2(n− 1) + Vi − µ

]

γin . (9)

Here, we use the same notation as in Eq. (4). It is impor-tant to note that Eq. (9) determines the time evolutionwhile preserving the normalization of the wave-functionand the total number of particles N in the system. Wesolve this set of L×(nc + 1) differential equations numer-ically, using a fourth-order Runge-Kutta method. Self-consistency is enforced by monitoring the total energy,particle number, and the normalization.

III. THE INITIAL STATE

We are interested in the expansion of Mott-insulatingstates, though we know a priory that in presence of atrapping potential such Mott states are inevitably sur-rounded by superfluid regions. Nevertheless, one can aimat finding a set up for which those Mott domains are aslarge as possible for a given value of J/U . Our approachto achieving large Mott domains is based on the local-density approximation (LDA). The LDA assumes thatthe local properties of the confined system can be mappedto those properties of the homogeneous system with thesame value of J/U and an effective chemical potentialcorresponding to the spatially varying µi = µ−Vi at sitei in the trap. One can, therefore, get an approximatepicture of what the trapped system looks like based onthe phase diagram of the homogeneous system. Withinthe LDA, in order to get the largest Mott domain, itis enough to set the chemical potential at the center ofthe trap to be the largest chemical potential of the ho-mogeneous system that allows for a Mott insulator withdensity n = 1. That is just the value of the chemical po-tential µ+ right at the transition from the Mott insulatorwith density n = 1 to the superfluid with n > 1. Thisis based on the observation that, as the distance fromthe center of the trap is increased, the effective chemicalpotential of the homogeneous system decreases. In 2D,the critical value µ+ can be determined within the mean-field theory by finding the largest solution for µ from the

-0.5

0

0.5

1

1.5

2

0 0.02 0.04

µ/U

J/U

(a)

MI (n=2)

MI (n=1)

MI (n=0)

SF

µ+/U

0

0.2

0.4

0.6

0.8

1

1.2

0 20 40 60 80 100-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

n i µ i/U

i

(b)

niµi/U

FIG. 1. (Color online) (a) Mean-field phase diagram of theBose-Hubbard model in 2D. The red arrow follows the dif-ferent phases occurring in the trapped system as one movesaway from the center of the trap. (b) The density profile asobtained from the LDA with the largest Mott domain plottedtogether with the spatially varying chemical potential µi/U .

following relation that determines the boundaries of theMott lobes [49],

2 (Jx + Jy)

U=

(n− µ/U) (µ/U − n+ 1)

1 + µ/U, (10)

where Jx and Jy are the hopping parameters along the xand y directions respectively. These ideas are illustratedin Fig. 1, where the mean-field phase diagram of the ho-mogeneous 2D system is depicted [Fig. 1(a)], as well asthe density profile across the center of a trapped systemthat contains the largest Mott domain as obtained fromthe LDA [Fig. 1(b)].We should stress that once µ+ has been identified, we

select a value of µ close to µ+ and solve Eq. (4) and/orEq. (6) to obtain the initial state. Hence, our initialstate is always the exact ground state within the mean-field approximation, i.e., it is not the LDA ground state.This is the way the initial wave function in Eq. (2) isconstructed for all our expansions.

IV. EXPANSION IN ONE DIRECTION

Let us first consider the expansion of a Mott insula-tor along one direction in the lattice. We study the casein which the initial state is confined only along the xdirection, i.e., Vy = 0 while Vx is kept finite, but thesystem has open boundary conditions in the y direction(a box trap). We allow for the hopping amplitudes tobe anisotropic, namely, that the hopping along the x di-rection (Jx) differs from the one along the y direction(Jy). We then study the expansion for several values ofU/Jx, large enough so that the Mott-insulating phase ex-ists, as well as different values of η = Jy/Jx. In all cases

5

−100 0 100x

−20

0

20

y

0 1

−20

0

20

y

0 1

−20

0

20

y

0 1

−20

0

20

y

0 1

nU/J=25

−20

0

20

y

0 1

−100 0 100x

−20

0

20

0 1

−20

0

20

0 1

−20

0

20

0 1

−20

0

20

0 1

nU/J=60

−20

0

20 0 1

FIG. 2. (Color online) Comparison between the time evo-lution of the density profile across the trap of systems withU/Jx = 25 (left) and U/Jx = 60 (right), at t = 0, 10, 20, 30and 40 (from top to bottom) following the release from trapswith Vx = 0.03 and Vx = 0.08, respectively. η = 0.3 and thetime is reported in units of ~/Jx.

considered in this section, the expansion takes place in alattice with 200×40-sites. As stated in the introduction,this study extends the results presented in Ref. [26] onthe expansion of hard-core bosons in three-dimensionallattices to the expansion of soft-core bosons confined intwo-dimensional lattices. The initial state is selected suchthat the Mott-insulating domain is the largest possiblefor given values of U/Jx, η, and Vx/Jx, as explained thethe previous section.In Fig. 2, we show the time evolution of the density

profiles of systems with U/Jx = 25 and 60, for η = 0.3. Inthe initial state (t = 0), both systems exhibit large Mottinsulating domains in the center of the lattice (x = 0),surrounded by small superfluid regions. The latter beinglarger for the smallest interaction strength, i.e., U/Jx =25. For t > 0, the particles are allowed to expand byswitching off the confining potential along the x direction.As a result, the Mott-insulating domains “melt”. Thisprocess is seen to be qualitatively similar for the twovalues of U/Jx considered in Fig. 2. Note that during thetime evolution, due to the symmetry of the initial state,the system expands only along the x direction, and thedensity profile maintains the reflection symmetry withrespect to both x and y axes.The corresponding momentum distributions nk, of the

systems we have just described, are presented in Fig. 3.At t = 0, although barely observable in the plots (firstrow in Fig. 3), there is a sizable population of low momen-tum modes, because of the presence of the inevitable su-perfluid regions surrounding the initial Mott insulators.Those modes are embedded in a roughly homogeneousbackground of low-populated momenta. The latter are

−π 0 πkx

−π/2

0

π/2

k y

0 55

−π/2

0

π/2

k y

0 55

−π/2

0

π/2

k y

0 55

−π/2

0

π/2

k y

0 55

nk

U/J=25

−π/2

0

π/2

k y

0 55

−π 0 πkx

−π/2

0

π/2

0 100

−π/2

0

π/2

0 100

−π/2

0

π/2

0 100

−π/2

0

π/2

0 100

nk

U/J=60

−π/2

0

π/2 0 100

FIG. 3. (Color online) Comparison between the time evolu-tion of the momentum distribution function of systems withU/Jx = 25 (left) and U/Jx = 60 (right), at t = 0, 10, 20, 30and 40 (from top to bottom) following the release from trapswith Vx = 0.03 and Vx = 0.08, respectively. η = 0.3 and thetime is reported in units of ~/Jx.

the result of the large Mott-insulating domains presentin the initial states (top panels in Fig. 2). Remarkably,shortly after the expansion has started, particles get re-distributed in momentum space, and they tend to fill upstates surrounding two well defined momentum valuesk = (±k∗x, 0). This effect, which can be clearly observedin all panels with t > 0 in Fig. 3, is already present atvery early times t < 10. As the expansion goes on, thevalues of the momentum of the highly populated modesbecome time independent. Their finite momenta reflectsthe motion of the two groups of particles moving awayfrom the center of the systems in Fig. 2. We find thatthe structure around the momentum peaks, better seenin the left column in Fig. 3 around k, disappears as thevalue of U/Jx is increased, and only sharp peaks are left.This is apparent by comparing the momentum distribu-tion functions between t = 20 and t = 40 in Fig. 2. There,the peaks for U/Jx = 60 are much sharper than those forU/Jx = 25.To a very good approximation, on can predict the val-

ues of k seen in Fig. 3 using energy and momentum con-servation arguments. In order to do that, we first realizethat if U/Jx is large enough, double or higher occupanciesof the lattice sites become energetically too costly andcan be, to a first approximation, ignored. The systemcan then be thought as composed of hard-core bosons.Following Ref. [26], the kinetic energy can then be writ-ten as,

Ekin =∑

k

ǫknk, (11)

where nk is the momentum distribution function at any

6

given time, k = (kx, ky), and ǫk is the dispersion relation

ǫk = −2Jx cos kx − 2Jy cos ky. (12)

In order to further simplify the calculations, one canassume that the system before the expansion is a pureMott insulator, thus neglecting the small superfluid re-gions that are always present in the trap, and implyingthat the initial kinetic energy is Ekin ≈ 0. (Note thatif double occupancies are neglected, the kinetic energy isessentially the total energy of the system after the traphas been turned off.) We further recall the observationthat, in Fig. 3, many particles tend to populate modeswith k = (±k∗x, 0), from which we understand that they component of k is zero due to the symmetry of theexpansion, which occurs only along the x direction. Wecan now ask, within our simplified picture of the systemspresented in Figs. 2 and 3, which modes [with (k∗x, 0)]could be populated by all the particles, after the expan-sion, while still conserving the energy. (We are assumingthat interactions during the expansion redistribute theenergy in such a way that all particles condense into asingle mode.) From Eq. (11), it follows that

cos k∗x = −η. (13)

If, in addition to energy conservation, we enforce con-servation of the total lattice momentum, or the symme-try of the expansion for that matter, we conclude thatk = (±k∗x, 0), where k∗x = arccos(−η).From Eq. (13), it follows that, in the limit of U/Jx ≫ 1,

the location of the kmodes that are populated during theexpansion depends only on the ratio η. Remarkably, wefind that the peaks in Fig. 3 agree very well with thatprediction [Eq. (13)]. This occurs despite the fact that inFig. 3 the results were obtained for finite values of U/Jxand in the presence of small superfluid domains aroundthe Mott insulator. In Fig. 4(a), we report a completestudy of the dependence of the location of the peaks inthe mean-field calculations as a function of η, for threevalues of U/Jx (points), together with the predictions ofEq. (13) (solid line). For the smallest values of U/Jx thatallow for the realization of a Mott insulator, one expectssome dependence of the location of the peaks on U/Jx,because the dispersion relation is expected to exhibit thelargest differences from Eq. (12). The deviations of themean-field results from Eq. (13) are, however, very smallfor small values η. The largest deviations, for all valuesof U/Jx, are observed for the largest values of η. Thisoccurs because as η is increased, the superfluid wingssurrounding the Mott insulator increase their size, as aresult, the total kinetic energy is lower than that of theMott insulator, and one of our assumptions (Ekin ≈ 0)becomes less and less valid. Nevertheless, as Fig. 4(a)shows, for the largest values of η, the differences are stillrelatively small even for the smallest values of U/Jx thatsupport a Mott insulator. They also can be seen to de-crease when increasing U/Jx. For U/Jx = 100, the re-sults are barely distinguishable from the analytic resultsfor the hard-core limit. Overall, the fact that only for

0

π/4

π/2

3π/4

π

0 10 20 30 40

k x* (t

)

t

(b)

η=0.1

η=0.3

η=0.5

η=0.7

η=0.9π/2

5π/8

3π/4

7π/8

0 0.2 0.4 0.6 0.8

k x* (t

) t→

∞

η

(a)

arccos(−η)

U/Jx=100

U/Jx=60

U/Jx=25

FIG. 4. (Color online) (a) Location of the momentum peak k∗

x

after a long expansion time obtained from the mean-field cal-culation as function of the hopping ratio η, for U/Jx = 25, 60,and 100 (points). The prediction of Eq. (13) is depicted asa solid line. (b) The time evolution of the location of themomentum peak for U/Jx = 60 and several values of η. Thehorizontal dashed-dotted lines show the values predicted inEq. (13).

large values of U/Jx one can realize a Mott insulator inthe initial state allows for the hard-core description to bea good approximation in all cases analyzed here.We conclude this section by showing, in Fig. 4(b), re-

sults for the location of the momentum peak as functionof time, for U/Jx = 60. That figure makes clear that, astime passes, the location of the peak rapidly approachesthe values predicted by Eq. (13) (dotted horizontal lines).It also shows that it takes a shorter time to reach thosevalues for smaller values of η. In our calculations, thevalue of k∗x is calculated as the local center of mass of themomentum distribution function around the position ofthe brightest peak.

V. EXPANSION IN ALL DIRECTIONS

We now examine the situation in which the initial Mottinsulator is allowed to expand in all directions. For sucha case, we consider Vx = Vy = V and Jx = Jy = J . Weturn off the trap at t = 0, and study the time evolution ofthe density and momentum distribution for systems withU/J = 25, 30, 40, 50, 60, 80 and 100. Note that theMott insulator can only be achieved in systems with largeenough values of U/J (U/J & 23.2 within the mean-fieldapproach in 2D).Figure 5 depicts the time evolution of the density pro-

file of the expanding bosons with U/J = 25 (left) andU/J = 50 (right). Interestingly, although the rightmostcase has an interaction energy that is twice that of theleftmost one, the expansion is somehow similar in bothcases. At the moment the trap is turned off (t = 0), theMott insulator domain is easily discernible from the su-

7

0

40y

0 1

0

40

y

0 1

0

40

y

0 1

−40 0 40x

−40

0

40

y

0 1 0 1 0 1 0 1n

U/J=25 U/J=50

−40 0 40x

FIG. 5. (Color online) Comparison between the time evo-lution of the density profile across the trap of systems withU/J = 25 (left) and U/J = 50 (right), at t = 0, 4, 8, and12 (from top to bottom) following the release from traps withV = 0.053 and V = 0.161, respectively. The time is reportedin units of ~/J .

perfluid one surrounding it. The lattice sites in the vicin-ity of the center of the system have unit occupancies andare surrounded by a thin ring with density smaller thanone. Note that, as expected from the way we constructthe initial state (see Sec. III), the superfluid ring in thesystem with U/J = 50 is smaller than the one in thesystem with U/J = 25.

After turning off the trap, the bosonic cloud expandsand, as in the previous section, the Mott insulator“melts”. In Fig. 5, it is remarkable that the expansionalong the diagonals is always faster than that along the xand y axes. This leads to a square like density profile fort > 0. Such a behavior is accentuated as U/J increases,as seen in the right panels in Fig. 5, where the expansionoccur almost entirely along the diagonals.

The evolution of the momentum distribution function,corresponding to the density profiles depicted in Fig. 5,is shown in Fig. 6. At t = 0, peaks can be seen in thecenter of the momentum distribution function. They cor-respond to the bosons that are in the superfluid ringssurrounding the Mott insulator. The difference betweenthe size of those rings for U/J = 25 (Fig. 5, left) and

0

π

k y

0 2

0

π

k y

0 2

0

π

k y

0 2

−π 0 πkx

−π

0

π

k y

0 2 0 2 0 2 0 2nk

U/J=25 U/J=50

−π 0 πkx

FIG. 6. (Color online) Comparison between the time evolu-tion of the momentum distribution function of systems withU/J = 25 (left) and U/J = 50 (right), at t = 0, 4, 8, and12 (from top to bottom) following the release from traps withV = 0.053 and V = 0.161, respectively. The time is reportedin units of ~/J .

U/J = 50 (Fig. 5, right) is apparent in the initial mo-mentum distribution, as the height of the peak for theU/J = 25 case is significantly larger than that in theU/J = 50 case. Surprisingly, for t > 0, these peaksrapidly evolve into circular like structures, which in turn,become square-like as time passes [Fig. 6]. The initialring structure is less evident in the U/J = 50 case att = 4 because the formation of the square-like struc-ture occurs faster as the value of U/J is increased. Also,the structure discernible in the momentum distributionfunction for small values of kx and ky (bottom panelsfor U/J = 25), disappears as the value of U/J is in-creased (see how much weaker it is in bottom panels forU/J = 50) and most of the bosons have momenta thatlay within the edges of the square. The existence of thisstructure, in which bosons remain with small values of kxand ky, can be directly related to the size of the initialsuperfluid domain, which, as said before, decreases as thesystem approaches the hard-core limit.

A comparison between the density and momentumdistributions of systems with different values of U/J

8

−40 0 40

x

−40

0

40

y

−40 0 40

x−40 0 40

x−40 0 40

x−40 0 40

x

0

1

n

U/J=25

−π 0 πkx

−π

0

π

k y

U/J=30

−π 0 πkx

U/J=50

−π 0 πkx

U/J=80

−π 0 πkx

U/J=100

−π 0 πkx

0

2

n k

FIG. 7. (Color online) Top row: Comparison between the density profiles of systems with U/J = 25, 30, 50, 80, and 100 (fromleft to right), at t = 12 following the release from traps with V = 0.053, 0.0764, 0.161, 0.278 and 0.356, respective. Bottomrow: Corresponding momentum distribution functions.

(U/J = 25, 30, 50, 80, and 100), at the time the bosonsreach the boundaries of our 80×80 lattice (t = 12), ispresented in Fig. 7. These plots make evident that theexpansion mainly occurs along the diagonals as U/J in-creases and, at the same time, the momentum distribu-tion function acquires a clearer square shape.An understanding of the behavior exhibited by these

systems, when allowed to expand in all directions, canbe gained following a similar line of reasoning as donein the previous section. If U ≫ 1, a good description isprovided by the hard-core boson picture. For hard-corebosons, after the trapping potential is turned off, thereis only kinetic energy, just as in the expansion in onedirection. The dispersion relation ǫk in the isotropic casereduces to

ǫk = −2J(coskx + cos ky). (14)

We now assume, once again to simplify the argument,that the system before the expansion is a perfect Mott in-sulator, i.e., all the particles are localized in their respec-tive sites, and occupy all possible momenta in momentumspace. Therefore, the kinetic energy is again Ekin = 0,and it is conserved during the expansion. Furthermore,let us assume that after long time, the majority of bosonswill occupy (“condense” to) modes with particular k vec-tors (as seen in the mean-field calculations). Energy con-servation is guaranteed by the assumption that the en-ergy of each occupied mode vanishes. Under those as-sumptions, the k vector of the modes populated by thebosons can be calculated from the dispersion relation as,

ǫk = −2J(coskx + cos ky) = 0. (15)

The solution to this equation is given by the square-likecontour plot seen in Fig. 8, for the case with w = 0.Analytically, such contour plot is described by the ex-pressions

ky = π ± kx, ky = −π ± kx. (16)

Remarkably, the momentum distribution of all systemsin Fig. 7 has converged to a distinctive square-like struc-ture. It closely resembles the prediction of Eq. (16), asrepresented in Fig. 8. The deviations from the exact re-sults in Eq. (16) can be understood to be the result of (i)the finite values of U/J in all our calculations, and (ii)the fact that the Mott insulating domains with n = 1 arealways surrounded by superfluid ones. Once again, thefirst modifies the dispersion relation in Eq. (14), and thesecond one reduces the energy from the value assumed inEq. (15). As clearly seen in Fig. 7, when U/J increasesand the superfluid domains in the initial state reducetheir size, the momentum distribution becomes closer tothe square-like contour defined in Eq. (16). The robust-ness of the results, despite points (i) and (ii) above, is in-deed remarkable. Hence, we expect that they should bereproducible in experiments with ultracold bosons evenat finite, but low, temperatures.The analysis above, also allows one to understand the

expansion pattern seen in the density profiles in Figs. 5and 7. The density profiles for t > 0 are determined bythe velocities with which bosons expand across lattice.In the hard-core limit, the group velocity is

vg = ∇ǫk = 2J(sinkxx+ sin ky y), (17)

which produces the vector field plot seen in Fig. 8. Itis clear that the group velocities, which are perpendicu-lar to the contour plots, increase in magnitude for thosek-modes along the diagonal. (The maximum group ve-locity occurs when ǫk = 0 along the diagonal.) Therefore,if such modes are highly populated, the net effect is thatthe system will expand preferably along the directions in-dicated by those group velocity vectors. Indeed, the highpopulation of such modes is apparent in the momentumdistribution functions depicted in Fig. 7. The low popu-lation of the modes in the vertices of the square indicatesthe absence of particles expanding with momenta closeto (±π, 0) and (0,±π), which have vanishing group ve-

9

-4

-2

0

2

4

-3 -2 -1 0 1 2 3-3

-2

-1

0

1

2

3

kx

k y

FIG. 8. (Color online) Depiction of the group velocity vectorfield, given by Eq. (17) in k-space, along with a series of con-tour plots satisfying the equation −2Jx cos kx − 2Jy cos ky =w, where w = −2, −1.5, −1, −0.5, 0, 0.5, 1, and 2.

locities. In Fig. 7, the accuracy of this description canbe seen to improve with increasing U/J . It is interestingto note that the system expands at the greatest speedpossible by populating the modes with the highest groupvelocity that are consistent with the conservation of theenergy, as can be concluded from the momentum distri-butions and comparing them with the analytical results.We have already shown that, during the expansion

from a mostly Mott insulating state, bosons tend topopulate well defined momentum modes. Results havebeen presented for mean-field calculations and supportedby analytic arguments. One can now wonder whetherall these highly populated modes are coherently coupledforming a simple Bose-Einstein condensate, or, whetherthey are part of fragmented condensates. In stronglycorrelated systems, it is possible that modes with differ-ent momenta couple to each other and still form a singleBose-Einstein condensate.In the presence of interactions, condensation can be

understood as effective single particle states (natural or-bitals) being highly populated. Following Penrose andOnsager [50], one needs to diagonalize the one-particledensity matrix ρi,j and study its eigenvalues λη

∑

j

ρi,jϕηj = ληϕ

ηi . (18)

If only one of them scales proportionally to the totalnumber of bosons in the system (Nb), one has simpleBose-Einstein condensation, otherwise, if several of themscale proportionally to Nb, this means that fragmenta-tion occurs [51]. (And, of course, if none of them scalesproportionally to Nb, the system does not exhibit Bose-Einstein condensation.) Interestingly, exact results forexpanding hard-core bosons in one dimension [16] havealready shown that the largest eigenvalue of the one-

particle density matrix in a system out of equilibriumcan diverge (∝

√Nb in that case) while still being com-

posed by bosons with many different momenta. In thosesystems, there were as many populated momenta as thoseavailable in the Fermi sea of the noninteracting fermionsto which hard-core bosons were mapped.Within the mean-field approximation, the off-diagonal

elements of the one particle density matrix reduce to

ρi,j = Φ∗iΦj , for i 6= j, (19)

and the diagonal ones are just the site occupations.In Fig. 9, we show the time evolution of the largest

eigenvalue λ0 of ρi,j for three different values of U/J(U/J = 25, 50 and 100) in our systems. Due to thepresence of the superfluid rings in the initial state, λ0 isnot always small at t = 0. (As expected, its value att = 0 decreases as U/J is increased.) For all the sys-tems studied in this work, we find that the occupation ofthe highest populated natural orbital always grows dur-ing the expansion [Fig. 9], while the occupation of theother natural orbitals remains negligible at all times. Inaddition, we have verified that λ0 always increases pro-portionally to the total number of particles. Hence, atleast within the mean-field approximation, these systemsexhibit simple Bose-Einstein condensation. That meansthat all the different occupied nk modes in Figs. 6 and 7are coupled into a single coherent state.

0 5 10t

0

200

400

600

λ 0

U/J=25U/J=50U/J=100

FIG. 9. (Color online) Comparison between the time evo-lution of the occupation of the highest populated naturalorbital λ0 for systems with U/J = 25 (black), U/J = 50(red), and U/J = 100 (green), after turning off traps withV = 0.053, 0.162, and 0.356, respectively.

VI. CONCLUSIONS

We have presented a detailed study of the expansion ofn = 1 bosonic Mott insulators in two-dimensional opticallattices. In all cases studied here, the initial states wereprepared in such a way that the largest fraction possibleof the bosons would be part of the Mott insulating do-main. Two setups were considered: the expansion alongone direction of an anisotropic system, which was initiallytrapped only along the x direction, and the expansion of

10

an isotropic system which was allowed to expand in alldirections. In the first case, we have shown that the ex-pansion of the Mott insulator leads to a high populationof two well defined momentum states. Those momentawere found to be controlled by the ratio of the hoppingamplitudes along the x and y directions. Similarly, theexpansion of a Mott insulator in all directions leads tothe generation of a single condensate being composedby bosons with many different momenta, for which wefound an analytical argument that explains which modesare populated. The prediction that all those momentummodes are coupled into a single condensate is a remark-able one. However, since this is not required by our en-ergy arguments, and has only been seen within the mean-field calculations reported here, it needs to be verified inexperiments or by unbiased calculations.It is important to notice that the phenomena we have

observed here arise because of a subtle interplay betweenquantum tunneling and strong correlations. Both effectsplay a crucial role in the generation of the condensatesas, on the one hand, tunneling is necessary for the sys-tem to expand, and on the other hand, interactions arenecessary for the bosons to redistribute the energy andcouple coherently. In absence of interactions, the systemstill expands, but the momentum distribution remains

unchanged from its initial value, i.e., it remains feature-less if one starts from purely localized particles in thelattice sites (a Fock state), and no condensation occurs.

Finally, our study supports the conclusion in Ref. [26],and generalizes it to the soft-core boson case, thatstrongly correlated atom lasers can be generated from theexpansion of Mott insulating states. Furthermore, afterthe results in Sec. IV, one can see that the momenta ofthose matter waves can be controlled by changing theratio between the hopping parameters in the different di-rections in the lattice. In addition, if one cannot changethose, in an isotropic lattice (Sec. V), the momenta ofthe lasers can be controlled by changing the directionalong which the Mott insulator is allowed to expand. Inthe latter set up, the best results would be obtained byconstraining the Mott insulator to expand along the di-agonals in the lattice.

ACKNOWLEDGMENTS

This work was supported by the US Office of NavalResearch.

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