Exploring the grand-canonical phase diagram of interacting bosons in optical lattices by trap...

PHYSICAL REVIEW A 82, 023601 (2010)

Exploring the grand-canonical phase diagram of interacting bosonsin optical lattices by trap squeezing

Tommaso RoscildeLaboratoire de Physique, CNRS UMR 5672, Ecole Normale Superieure de Lyon, Universite de Lyon, 46 Allee d’Italie, Lyon, F-69364, France

(Received 31 March 2010; published 3 August 2010)

In this article, we theoretically discuss how quantum simulators based on trapped cold bosons in opticallattices can explore the grand-canonical phase diagram of homogeneous lattice boson models, via control ofthe trapping potential independent of all other experimental parameters (trap squeezing). Based on quantumMonte Carlo, we establish the general scaling relation linking the global chemical potential to the Hamiltonianparameters for the Bose-Hubbard model in a parabolic trap, describing cold bosons in optical lattices; we findthat this scaling relation is well captured by a modified Thomas-Fermi scaling behavior—corrected for quantumfluctuations—in the case of sufficiently high density and/or sufficiently weak interactions, and by a mean-fieldGutzwiller ansatz over a much larger parameter range. This scaling relation allows us to control experimentallythe chemical potential, independent of all other Hamiltonian parameters, via trap squeezing; given that theglobal chemical potential coincides with the local chemical potential in the trap center, measurements of thecentral density as a function of the chemical potential give access to the information on the bulk compressibilityof the Bose-Hubbard model. Supplemented with time-of-flight measurements of the coherence properties, themeasurement of compressibility enables one to discern among the various possible phases realized by bosonsin an optical lattice with or without external (periodic or random) potentials—for example, superfluid, Mottinsulator, band insulator, and Bose glass. We theoretically demonstrate the trap-squeezing investigation of thesephases in the case of bosons in a one-dimensional optical lattice and in a one-dimensional incommensuratesuperlattice.

DOI: 10.1103/PhysRevA.82.023601 PACS number(s): 03.75.Lm, 71.23.Ft, 68.65.Cd, 72.15.Rn

I. INTRODUCTION

The impressive recent advances in the engineering ofinteracting Hamiltonians for cold trapped atoms suggest thepossibility of experimentally determining the equilibrium (andout-of-equilibrium) behavior of fundamental theoretical quan-tum many-body models. In particular, experiments on coldatoms in optical lattices have demonstrated the implementationof fundamental lattice models (of Hubbard type) with full tun-ability of all Hamiltonian parameters, as well as of the latticegeometry and dimensionality [1,2]. Remarkably, the under-standing of a large class of lattice many-body models (such asinteracting fermions, frustrated quantum magnets, etc.) keepsdefying standard theoretical and computational approaches.Hence, the perspective of realizing analog quantum simulators[3] based on cold atoms, literally implementing the physicsof the challenging models in question, represents a mostpromising and innovative route toward their understanding.

Nonetheless, several obstacles still separate the currentexperiments from giving original answers to long-standingquestions in quantum many-body physics on a lattice. Oneproblematic feature of cold-atom systems in a lattice is ther-mometry [4–11]: given that trapped cold atoms are not coupledto a thermal reservoir, their entropy can be controlled but nottheir temperature, and the knowledge of the temperature asa function of entropy is a problem which requires the priorknowledge of the equation of state of the many-body systemunder investigation [7,12,13]. A second puzzling feature ofcurrent cold-atom experiments, which is the main concern ofthis article, is the intrinsic inhomogeneity of these systems,imposed by the presence of an overall parabolic trappingpotential. While the size of the lattices realized in experimentscan easily beat the capability of current classical simulation

methods (at least in dimensions d > 1), the parabolic potentialsignificantly limits the size of the lattice over which a uniformphase of the Hamiltonian model is realized. Indeed, a parabolictrapping potential V (i) = Vt (ri − r0)2 (where ri is the positionof the ith lattice site and r0 is the center of the trap) imposes asite-dependent chemical potential and site-dependent energyjumps �E(i) = V (i + 1) − V (i) = Vt (ri+1 − ri)(ri+1 + ri −2r0) between nearest-neighboring sites. If the local many-bodyphase at a given filling n, realized around the ith site at thelocal chemical potential µ(i) = V (i), is not protected by aparticle-hole gap �g � �E(i), its existence is necessarilyrestricted to a narrow neighborhood of site i. This aspectintroduces significant limitations in the capability of realizingphases which are particularly sensitive to the filling andwhich display a small or no particle-hole gap over the groundstate (see, e.g., the supersolid phase of strongly correlatedbosons [14–17] or the Bose glass [18] for bosons in a randompotential).

In principle, one could consider the coexistence of severaldifferent phases in the trap as an advantage, given that onesingle experiment samples a significant portion of the phasediagram of the Hamiltonian implemented in the system via thechemical potential modulation imposed by the trap [19,20].Unfortunately, this form of parallelism is generally hard toexploit, as most measurement protocols for trapped cold atomsemployed to date give access to global observables, whichcollect a signal from all the different regions of the trap.This fact becomes particularly inconvenient when consideringthe measurement of the excitation spectrum, as sampled,for example, via lattice modulation and two-photon Braggspectroscopy [21–24]. In fact, it is quite difficult to associatethe different contributions to the excitation spectrum with

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http://dx.doi.org/10.1103/PhysRevA.82.023601

TOMMASO ROSCILDE PHYSICAL REVIEW A 82, 023601 (2010)

different spatial regions of the trap, a fact which makes itvery hard to extract precise information on the structure of theexcitations associated with a particular phase of interest. Inparticular, the presence of a trap imposes the existence of a haloof dilute particles at the cloud boundary, which in principlecan always host low-energy excitations. Consequently, thefundamental question of the presence or absence of an energygap over the ground state of a particular phase, realized locallyin the trap, becomes a formidable task for current cold-atomsimulators.

It is important to mention that single-site addressability inoptical lattices is becoming a reality in recent experiments onquantum gas microscopy [25–29]. This achievement definitelyallows one to enjoy the parallelism in the chemical potentialexhibited by trapped experiments [19] but only at the levelof local properties, defined on spatial regions over which thevariation of the local chemical potential can be consideredweak, and only as long as the so-called local-density approx-imation (LDA) can be fully trusted. Yet nonlocal propertiesof homogeneous phases, including correlation functions andcollective excitations, cannot be extracted from an inhomoge-neous sample, except for those associated with the region ofmaximal homogeneity, namely the trap center.

In this article, we propose an experimental protocolwhich aims at circumventing most of these difficulties whiletaking advantage of the trap as a low-energy probe for theproperties of the system (trap-squeezing spectroscopy [30]).The fundamental idea relies on the fact that trap effects areminimal in the center: both V (i) and �E(i) vanish, so that alocal, homogeneous phase can be established over a significantportion of the system, roughly of the order of ∼10 lattice sitesin each spatial direction. This sizable portion of the lattice canbe now regarded as the system of interest, while the rest of thelattice can be regarded as the environment, acting in particularas a particle reservoir (see Fig. 1 for a sketch). In light of

FIG. 1. (Color online) An atomic cloud in a smoothly varyingtrapping potential, such as a parabolic trap, can be virtually dividedinto two regions: a central C region over which the density isessentially uniform (within a tolerance ε) and whose local chemicalpotential µ(N,Vt ) essentially corresponds to the global chemicalpotential of the atomic cloud, and a periphery region C, in whichthe density varies rapidly in space due to the trapping potential. Thenearly homogeneous central region can be regarded as the “system”in a grand-canonical ensemble, while the periphery can be regarded asa particle “reservoir.” Increasing the trapping potential (as indicatedby the black arrows) increases the chemical potential of the reservoirC and, at equilibrium, that of the system C. Hence, the chemicalpotential of the system can be controlled continuously by the trappingpotential Vt .

this discussion, we can conclude that the trap center realizesthe textbook quantum simulation of the particular Hamiltonianimplemented in the system in the grand-canonical ensemble(namely at a nearly uniform local chemical potential). Whatremains to be shown is how to control the crucial parameterof the quantum simulation at the trap center, namely its localchemical potential, and how to retrieve selective informationon the local collective phase realized there.

Here we specialize our discussion to the case of bosons inoptical lattices, governed by the Bose-Hubbard Hamiltonianin an external potential, and leave the case of fermions tofuture work. In particular, we quantitatively discuss how tuningthe trap frequency independently of all other experimentalparameters gives direct access to the control of the chemicalpotential in the trap center. Remarkably, numerical simulationson the Bose-Hubbard model on the hypercubic lattice showthat there exists a simple relation between the strength of thetrapping potential and the chemical potential: This relationfollows approximately the predictions which can be obtainedboth from the atomic limit (AL) in a lattice and from theThomas-Fermi (TF) theory for weakly interacting gases incontinuum space, with deviations due to quantum and latticecorrections. Remarkably, such deviations are quantitativelycaptured at the mean-field level. This means that, unlike thecase of the temperature [12,13], the accurate knowledge of thechemical potential of a strongly correlated bosonic system inan optical lattice might not require in general an extensiveab initio calculation. This makes the trapping potential afundamental experimental knob whose effect on the system’sparameters can be readily assessed.

Nonetheless, the tuning of such a knob has to be done verycarefully, and in general it cannot be done after loading theatoms in the optical lattices. In fact, increasing the strengthof the trapping potential at equilibrium leads to the transferof particles from the wings to the center of the trap, but thetunneling amplitude associated with such a transfer can beextremely small in strongly interacting systems, requiring thenexceedingly slow ramps of the trapping potential to enforceadiabaticity in the protocol. A simple way to circumvent thisproblem is proposed, based on the loading of atoms in thetrap in the weakly interacting regime (namely without opticallattice) followed by the ramp of the optical lattice.

Once full control on the chemical potential at the trap centerhas been achieved, selective information can be retrieved onthis region of space by microscopy of the atomic cloud. Herewe propose a measurement scheme of the average centraldensity based on two tightly focused crossed beams resonantwith different optical transitions of the atoms; as alreadymentioned, other quantum gas microscopy schemes (withresolution as high as one lattice spacing) have been proposedor have even become experimentally available in the recentpast. The knowledge of the central density as a function ofthe chemical potential provides the fundamental informationon the bulk compressibility of the Hamiltonian model imple-mented in the system and hence on the particle-hole gap overthe ground state. We present the application of trap-squeezingspectroscopy to the measurement of the bulk-phase diagramof the Bose-Hubbard model in d = 1: In particular, we discussthe case without any applied external potential and the casein which an incommensurate potential with two wavelength

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EXPLORING THE GRAND-CANONICAL PHASE DIAGRAM . . . PHYSICAL REVIEW A 82, 023601 (2010)

components is applied to the system, giving rise to an extendedBose-glass phase whose compressible nature is perfectlycaptured via trap-squeezing spectroscopy.

The structure of the article is as follows: Sec. II introducesthe model investigated, the LDA, and the fundamental roleplayed by the average central density in the trap; Sec. IIIdiscusses the relationship between the chemical potential inthe trap center and the trap strength for the Bose-Hubbardmodel in dimensions d = 1, 2, and 3, with or without anexternal potential; Sec. IV discusses the issue of equilibriumpreparation of the system at a given trap strength, based ona two-step protocol for the ramp of the optical potentials;Sec. V discusses a proposal for the selective measurementof the average density in the trap center; applications to theBose-Hubbard model without and with external superlatticepotentials are presented in Secs. VI and VII respectively;Sec. VIII compares the central compressibility with theglobal compressibility of the atomic cloud, recently measuredin experiments [31]; and finally Sec. IX is devoted toconclusions.

II. LOCAL-DENSITY APPROXIMATION,AVERAGE CENTRAL DENSITY, AND CENTRAL

CHEMICAL POTENTIAL

In this section, we focus our attention on the general caseof the Bose-Hubbard model on the d-dimensional hypercubiclattice in an external potential, composed of a rapidly varyingpart, given by a superlattice potential created by an additionalstanding wave applied to the system [22,32,33], and a slowlyvarying parabolic part, coming from the overall Gaussianprofile of the lasers applied to the system.

H(J,U,V2,Vt ) = H0(J,U,V2) + Vt

∑i

(ri − r0)2ni, (1)

H0(J,U,V2) = −J∑〈ij〉

(b†i bj + H.c.) + U

2

∑i

ni(ni − 1)

+V2

∑i

gi({αl},{φl})ni, (2)

where

gi({αl},{φl}) =d∑

l=1

cos2(2παlil + φl) − d

2(3)

is a one-color superlattice potential. Here il (l = 1, . . . ,d)are the coordinates of the ith site, and 〈ij 〉 are the pairs ofnearest neighbors on the d-dimensional hypercubic lattice ofsize Ld .

Experiments on cold atoms in optical lattices are typicallyperformed with a fixed number of particles N . Nonetheless,given that the particle number is a good quantum number ofthe Hamiltonian, at T = 0 the system is in a definite N sectoreven in the grand-canonical ensemble. This allows us to regardthe canonical system with N particles as equivalent to a systemin the grand-canonical ensemble with Hamiltonian

Hµ = H − µ∑

i

ni, (4)

where µ = µ(Vt ,N,J,U,V2) is the chemical potential whichestablishes N particles in the ground state of the HamiltonianH(J,U,V2,Vt ). Moreover, we can rewrite Eq. (4) as

Hµ = H0 −∑

i

µ(i) ni, (5)

where we have introduced the site-dependent chemical po-tential µ(i) = µ − Vt (ri − r0)2. If µ(i) is a slowly varyingfunction in space around the reference site i∗ over the typicallength scale given by the correlation length of the HamiltonianHµ(i∗) = H0 − µ(i∗)

∑i ni , one can adopt the LDA: This

amounts to considering that the trapped system behaves aroundthe site i∗ in the same way as its bulk counterpart at thehomogeneous chemical potential µ = µ(i∗). In particular,in the absence of a superlattice potential, the local density〈ni〉 for i ≈ i∗ will be extremely close to the homogeneousdensity in the ground state of Hµ(i∗). In the presence ofa superlattice potential, the local density 〈ni〉 will be veryclose to the density of the bulk system at a point experi-encing the same superlattice potential; moreover, the averagedensity over a period of the superlattice (or quasiperiodfor incommensurate superlattices) centered around the sitei∗ will approximate very well the average density in theground state of the bulk system. All these expectations forthe behavior of the density are fully verified by numericallyexact calculations on the Bose-Hubbard model with or withouta superlattice [34–36]. The LDA typically breaks down whenthe local chemical potential µ(i) approaches a critical valuesitting at the boundary between two phases in the bulksystem, so that the correlation length of the bulk systemdiverges.

The validity of the LDA for parabolic traps implies thatthe trapped system faithfully probes the density of the bulksystem at many different values of the chemical potentialaway from critical points. As mentioned in the introduction,this form of parallel sampling of the bulk-phase diagram isvalid only for local properties, and it cannot be exploitedexperimentally unless single-site addressability is achieved.On the contrary, imaging of the atomic cloud over a lengthscale R of 5–10 µm (corresponding to ∼10–20 lattice sitesof an optical lattice with λ ∼ 800 nm) can be achieved moreconventionally via large-aperture optics [37] (see Sec. V fora more detailed discussion). This means that the informationon the local density can be retrieved if 〈ni〉 does not changeappreciably over the length scale R. For a sufficently weaktrapping potential, such a condition can be easily met at thetrap center, where the variation in the local chemical potentialis the slowest. We hence introduce the average central density,

nC =:1

|C|∑i∈C

〈ni〉, (6)

where |C| is the size of the C region built around the trapcenter i0. The C region is then defined as verifying a conditionof quasi-homogeneity,

|nC − 〈ni0〉|〈ni0〉

� ε, (7)

with ε � 1. Another important observation singles out the trapcenter as the most interesting region of the trap. Indeed, even

023601-3


when it is possible to measure the density at the single-sitelevel, one can exclusively achieve the knowledge of a localobservable Ai which, via the LDA, can be associated with that(A) of the bulk system as a function of the chemical potential,A(µ) Ai(µi = µ). The complementary information on thenonlocal correlation functions of the bulk system is insteadcompletely missing, and indeed the trapped system cannotfaithfully reproduce the correlation properties of the bulksystem in general (as testified by the poor performance of theLDA at the level of correlations [36]). Nonetheless, the trappedsystem can faithfully reproduce the correlations around thetrap center over a length scale corresponding to the extent ofthe quasihomogeneous central region. This aspect is discussedin detail in Sec. VI, where it is shown in addition that theevolution of global correlation properties of the system (asprobed, e.g., by time-of-flight measurements) is dominatedby the evolution of correlations in the trap center, so thatcorrelations in the trap center can effectively be accessed in theexperiments.

The quasi-homogeneity condition, Eq. (7), allows us toidentify nC as a close approximation to the ground-statedensity of the bulk Hamiltonian, Eq. (4) (with Vt = 0), at achemical potential corresponding to the background chemicalpotential µ(Vt ,N,J,U,V2) of the trapped system. As furtherelaborated in Sec. V, nC is experimentally accessible withconventional methods. In order to convert the informationon nC into information on the bulk phase diagram of theHamiltonianH0 of Eq. (2), we need to know at which chemicalpotential the experiment is operating once the experimentalparameters Vt , N , J , U , and V2 are set. This is the goal of thenext section.

III. TRAP-SQUEEZING CONTROL OF THECHEMICAL POTENTIAL

In this section, we investigate the dependence of thebackground chemical potential µ(Vt ,N,J,U,V2), stabilizinga ground state with N particles, for the Bose-HubbardHamiltonian with parameters J , U in a trapping potentialof strength Vt and, generally, in a superlattice potential ofstrength V2. We review the conventional TF approximationand its generic prediction for the functional form of µ

in the general case of a d-dimensional hypercubic lattice.Remarkably, a numerical investigation based on numericallyexact quantum Monte Carlo shows that the TF prediction,modified by quantum fluctuations, is accurate at least in theregime of either sufficiently weak interaction or sufficientlyhigh density. As a result, the chemical potential turns outto be simply controlled in the experiments either via thecontrol on the particle population N , the trap strength Vt , orboth.

A. Atomic limit and Thomas-Fermi approximation

A simple theoretical approach giving the relationshipbetween the chemical potential and the other Hamiltonianparameters for the Bose-Hubbard model is the AL, which con-sists in discarding the quantum kinetic term in the Hamiltonianand in solving for the diagonal part. Such an approximationis reasonable in the strongly interacting and strongly trappedlimit U,VtR

2 � J , where R is the radius of the atomic cloud.

By minimizing the potential energy part with respect to thedensity

Hpot = U

2

∑i

ni(ni − 1) + V2

∑i

gini

+Vt

∑i

i2ni − µ∑

i

ni, (8)

one finds

ni = µ + U/2 − V2gi − Vt (ri − r0)2

U. (9)

By imposing the condition N = ∑i ni and passing from the

lattice to the continuum formulation, one readily obtains theAL chemical potential for a d-dimensional system:

µAL + U

2=

(d + 2

2

�(d/2 + 1)

πd/2

) 22+d

(UN )2

2+d Vd

2+d

t

=

⎧⎪⎨⎪⎩

0.82548 . . . (UN )2/3V1/3t (d = 1),

0.79788 . . . (UNVt )1/2(d = 2),

0.81346 . . . (UN )2/5V3/5t (d = 3).

(10)

Notice that the superlattice term does not enter in thisformula because it has been chosen so as to take symmetricvalues around zero, and therefore it vanishes upon spatialintegration.

On the other hand, a formula similar to Eq. (13) canbe obtained in the weakly interacting case via a standardGross-Pitaevskii (GP) approach plus a TF approximation [38].Taking the mean-field approximation ai ≈ �i , a

†i ≈ �∗

i onthe normally ordered Hamiltonian, Eq. (1) (where �i is thecondensate wave function), one obtains the lattice GP energyfunctional:

EGP = −J∑〈ij〉

(�∗i �j + c.c.) + U

2

∑i

|�i |4

+∑

i

(Vi − µ)|�i |2, (11)

where Vi = V2gi + Vt (ri − r0)2 − µ.The standard TF approximation consists in neglecting

completely the kinetic term in the GP functional. This isjustified in the continuum case because the kinetic energyis suppressed when the wave function is slowly varyingin space; on the other hand, in the lattice case, a slowlyvarying wave function is such that �∗

i �j ≈ |�i |2. Namely,the kinetic term does not cancel, but it effectively adds up tothe chemical potential term, µ → µ + 2dJ . Minimizing theGP functional leads then to the lattice TF equation for thedensity:

|�i |2 = µ + 2dJ − V2gi − Vt (ri − r0)2

U. (12)

Integrating over space in the continuum limit leads to the resultµTF = −2dJ + µAL + U/2.

The central prediction of both the AL and the TF approach isthe linear dependence of µ on the combination (N2V d

t )1/(2+d),with a slope dependent on U 2/(2+d): this prediction will beverified by a numerically exact calculation in a large-parameterregime. What is completely missing in this simple approach

023601-4


are (1) lattice commensuration effects, relevant in the caseof the appearance of band or Mott insulator (MI) states[given that the analytical expression Eq. (10) is obtainedin the continuum limit], and (2) a proper treatment of thequantum kinetic J term in the Hamiltonian. The lattice effectscan be readily restored by numerically performing the sumN = ∑

i ni instead of analytically integrating the continuumgeneralization of Eq. (9), at the expense of losing the closed-form prediction of Eq. (10). On the other hand, a full account ofthe quantum corrections requires a more extensive numericaltreatment. This is provided in the following, where we see thatsuch effects do not alter too drastically the TF/AL predictionsand they can be captured already at the mean-field level.

B. Quantum Monte Carlo results

Here we present the results of quantum Monte Carlosimulations of the d-dimensional Bose-Hubbard model on Ld

hypercubic lattices, based on the stochastic series expansionmethod with directed-loop updates [39]. We have performedsimulations at low temperatures T ∼ J/L in order to removesignificant thermal effects. We perform the simulation in thegrand-canonical ensemble, and we fine-tune the chemicalpotential µ which stabilizes a given particle number N forthe Hamiltonian, Eq. (1); this allows us to numerically samplethe function µ(Vt ,N,J,U,V2). In the following, we take J

as the energy scale and express all other quantities in unitsof J . We have performed the simulations for two values ofthe U/J ratio for each dimensionality d and both for zerosuperlattice potential (V2 = 0) and for an intense superlatticepotential (V2 = U ). We have considered both incommensuratesuperlattices (α = 830/1076 as in the experiment of Ref. [22])and commensurate ones (α = 3/4). Results for one spatialdimension have already been partially reported in Ref. [30].

1. d = 1 without superlattice

We begin our discussion with the case of the absence of asuperlattice (V2 = 0). Figures 2 and 3 show results for the twocomplementary regimes of U/J = 5 and U/J = 20. Judgingfrom the phase diagram of the 1d Bose-Hubbard model [40],for U/J = 5 the kinetic and the potential part of the bulkHubbard Hamiltonian are in strong competition for most fillingvalues n <∼ 3, so that a variation of the chemical potentialbrings the system through an alternation of correlated super-fluid (SF) phases and MI phases separated by quantum criticalpoints. Despite the important quantum effects taking place inthe system, it is remarkable to observe in Fig. 2 that all µ valuesobtained by various combinations of Vt and N collapse ontothe same universal curve, which is a homogeneous functionof N2Vt/J , in agreement with the TF/AL prediction Eq. (10).References [41,42] have introduced the so-called character-istic density, ρ = N (Vt/J )d/2, as the relevant parameter tocharacterize the density of a trapped system in d dimensions.Our data suggest that the chemical potential is a homogeneousfunction of the characteristic density. This result is furtherconfirmed in the case d = 2,3; see also Sec. III D for a generaldiscussion.

Even more remarkably, for sufficiently large N and/or Vt/J

values, the universal curve obeys a modified TF (mTF) scaling

0 2 4 6 8 10 12

( N2 V

t /J )

1/3

0

5

10

15

20

25

30

µ / J

TF fit

0 1 2 3 4-2

0

2

4

FIG. 2. (Color online) Chemical potential for the 1d Bose-Hubbard model with U = 5J . The universal curve µ = µ(N2Vt/J )has been obtained via the collapse of many data sets, with N =20, . . . ,120 and Vt = 0.004, . . . ,0.140J . The solid curve is a linearfit to the ansatz, Eq. (13). Inset: zoom on the crossover from the mTFbehavior to the low-N and low-Vt behavior.

of the kind

µmTF(Vt ,N,J,U,V2)/J = Cd (U/J )2

2+d

+ γd (U/JN)2

2+d (Vt/J )d

2+d , (13)

where Cd = Cd (U/J,V2/J ). Here, γd is a d-dependent slope:Numerical evidence shows that γd does not seem to dependon the other Hamiltonian parameters (see Ref. [30] and thefollowing discussion). A fit to the d = 1 data for U/J = 5gives us γd=1 = 0.817(2), which compares surprisingly wellwith the TF/AL prediction of Eq. (10). Moreover, Cd is ad-dependent offset term which is also found to depend onU/J and on V2/J and which contains the most relevantquantum corrections to the TF/AL result. It is important tostress that in general Cd (U/J )

22+d is not the correct value

of µ/J for Vt/J,N → 0. In fact, in this limit, the systembecomes extremely dilute and the dependence on U/J shoulddrop out. Indeed, in the limit of low N and Vt , the µ curvecrosses over to the correct dilute limit (see inset of Fig. 2),which gives the well-known result µ(N = 0,Vt = 0) = −2dJ

as obtained from the solution of the tight-binding model. Asseen in Sec. III A, this result coincides with the N = 0,Vt = 0value predicted by the lattice TF theory [Eqs. (10) and (12)],as should be expected, given that in the dilute limit the Bosegas becomes ideal. This also means that the quantum offsetterm (−2dJ ) of the chemical potential from lattice TF theorydoes not correspond at all to the quantum correction appearingin the mTF scaling, Eq. (13). Hence, the form of Eq. (13) isfar from being trivial.

The results for U = 20J are instead more elaborate. In thiscase, the inspection of the 1d Bose-Hubbard phase diagramreveals that the system with a filling n <∼ 3 is an MI formost of the chemical potential values. Hence, the latticestabilizes commensurate insulating regions in the trap whichare completely missed in the continuum limit leading to theTF/AL formula, Eq. (10). Nonetheless, it is remarkable toobserve that all the data obtained for different N and Vt

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0 50 100

N2 V

t /J

0

5

10

15

20

µ / J

hardcore bosons

0 500 1000 1500

N2 V

t/J

010203040506070

µ / J

0 2 4 6 8 10 12

(N2 V

t /J)

1/3

0

20

40

60

mTF fit

-50 0 500

1

2

3

4

-50 0 50 -50 0 50 -50 0 50

a

b

c

d

(a) (c) (d)(b)

(f) (g)

(e)

FIG. 3. (Color online) Chemical potential for the 1d Bose-Hubbard model with U = 20J . (a)–(e) Universal curve µ =µ(N 2Vt/J ) [panel (e)] and representative density profiles (a)–(d). Theuniversal curve is obtained via the collapse of many data sets (sameparameters as in Fig. 2). The differently colored regions correspond tothe number of shells populated in the “wedding cake” structure, from1 up to 4. The four representative density profiles in (a)–(d) show fourdifferent shell occupations (N = 100 and Vt = 0.004, 0.014, 0.070,and 0.135J ). (f) Zoom on the low-N and low-Vt region of panel (e)(first shell occupied only), showing agreement with the correspondingdata for a hardcore-boson system. (g) Universal curve presented as afunction of N 2/3(Vt/J )1/3: The solid curve represents a linear fit tothe ansatz, Eq. (13).

values collapse onto the same universal curve, which is againa homogeneous function of N2Vt , consistent with the TF/ALprediction. This curve shows a succession of kinks separatingvarious regimes of filling and trapping: An inspection in themicroscopic structure of the states corresponding to the variousregions shows that each kink marks the appearance of particlesin a new shell of the trapped cake structure of the densityprofile [see Fig. 3(a)]. When the system has filling 〈ni〉 < 1throughout the trap, the large U/J value suppresses multipleoccupation and the system behaves effectively as a hardcore-boson system. This is shown by the excellent agreementbetween the low-N /low-Vt data for the softcore-boson systemand the same data for a hardcore-boson system, which can beobtained exactly by standard Jordan-Wigner diagonalization[43]. By increasing the filling and/or the trapping, the conditionof single occupancy is eventually violated, and the second shellof the cake starts to be filled. In the AL, the condition for thepresence of a site with occupation n = 2 in the center of thetrap in a 1d system is Vt (N/2)2 � U , which gives the criticalvalue (N2Vt/J )n=2 = 4U/J . Assuming that the population offurther shells appears upon depletion of the shell with singleoccupation, we find that the n = 3 shell starts being occupied

when

Vt (N1/2)2 = Vt (N2/2)2 + U = 2U, (14)

which means that the n = 2 shell stops growing upon increas-ing Vt and the n = 3 shell starts building up. Here, Ni indicatesthe number of particles in the ith shell. With the conditionN1 + N2 = N , Eq. (14) gives

(N2Vt/J )n=3 = 4(1 +√

2)2U/J. (15)

A similar reasoning leads to the critical value for the populationof the n = 4 shell,

(N2Vt/J )n=4 = 2(1 +√

2)2(1 +√

3)2U/J, (16)

and so on. Hence, for U = 20J we obtain (N2Vt/J )n = 80,466.27, and 1740.16 for n = 2,3, and 4; the corresponding nu-merical estimates from the quantum Monte Carlo simulationsare (N2Vt/J )n ≈ 66, 400, and 1200, which, not surprisingly,are lower than the AL prediction due to quantum effects thatallow the particles to tunnel from the wings of the cloud intothe center at a lower trapping strength or particle number.

It is evident from Fig. 3 that lattice commensuration effectsbecome weaker and weaker on the µ curve for increasingfilling. In particular, when plotting the curve as a function ofthe natural TF parameter (N2Vt/J )1/3, as done in Fig. 3(c), weobserve that the mTF scaling of Eq. (13) sets in approximatelyonce the n = 3 shell starts to be filled. A fit of the large-N andlarge-Vt data with Eq. (13) delivers a value of γd=1 consistentwith the value obtained from the U/J = 5 data, confirmingthat γd only depends on dimensionality.

2. d = 1 with superlattice

Figure 4 shows the µ curve for the 1d Bose-Hubbard modelin a strong superlattice with α = 3/4, φ = 0, and V2 = U andfor three values of U/J = 10, 20, and 30. Again, the chemicalpotential is clearly a homogeneous function of N2Vt/J , asshown by the collapse of all the µ values onto the sameuniversal curve dependent on U/J and V2/J . For large U/J ,V2/J , and/or for small N2Vt/J , we observe kinks in the µ

curve characteristic of lattice commensuration effects: In the

0 5 10

N 2/3

(Vt / J)

1/3

0

20

40

60

80

µ / J

N=20N=30N=40N=50N=60N=70N=80N=90N=100N=110N=120

0 5 10

N 2/3

(Vt / J)

1/3

0

2

4

6

8

µ / J

(U

/ J)

-2/3

V2 = U = 10

20

30

FIG. 4. (Color online) Chemical potential for the trapped 1d

Bose-Hubbard model with a commensurate superlattice potential(V2 = U , α = 3/4). The right panel shows that all curves have auniversal scaling of the slope with the ratio U/J . Dashed lines arefits to the ansatz, Eq. (13).

023601-6


case of a commensurate superlattice as the one considered here,these effects are clearly related to the fractional filling shells[30] appearing in the AL of the system at fillings (2n + 1)/4with n = 0,1,2, . . . . Yet, the overall trend of the curve forsufficiently high filling follows the mTF behavior [Eq. (13)],and a fit to that equation gives again a γd consistent with thatfound in the previous section in the absence of the superlattice.This confirms the TF/AL prediction that the presence ofa superlattice is not influencing the overall slope of theµ curve. Similar curves for the incommensurate superlatticecase α = 1076/830 have already appeared in Ref. [30].

It is important to stress that lattice commensuration effectsin the presence of a superlattice are strongly related to thepossibility of maintaining the spatial phase φ of the superlatticefixed in the experiments. If this phase is allowed to fluctuate (asgenerally happens from shot to shot of the same experiment[44], unless phase locking is explicitly enforced [32]), thechemical potential in the experiment is going to changeaccordingly, assuming that all other experimental parametersremain unchanged. In this case, it is then more convenient tointroduce a phase-averaged chemical potential 〈µ〉φ , namely,averaged over random fluctuations of the phase φ. Figure 5shows 〈µ〉φ for two superlattices: for the commensurate onediscussed previously and for an incommensurate one whichhas one additional color component, namely

gi(α,α′,φ,φ′) = cos2(2παi + φ) + cos2(2πα′i + φ′) − 1,

(17)

with α = 1076/830 and α′ = 1473/830. This particular super-lattice is further discussed in Sec. VII. The chemical potentialvalues are typically averaged over a sample of ∼100–200 phasevalues. When comparing the average results with typical ones,we notice that fluctuations are small around the average, so thatfor each different realization of the superlattice, the averagechemical potential 〈µ〉φ stabilizes in the system a number ofparticles N which is close to the desired one.

As a summary of the study of one-dimensional systems,Table I shows the result of the fit of the various cases considered

3 4 5 6 7 8 9 10 11

N2/3

(Vt / J)

1/3

0

10

20

30

40

50

60

< µ

>φ

/ J

commensurate superlatticeincommensurate 2-color superlattice

FIG. 5. (Color online) Phase-averaged chemical potential for thetrapped 1d Bose-Hubbard model with a commensurate superlatticepotential and with a two-color incommensurate potential (see text).For both cases, V2 = U = 20J and N = 100.

TABLE I. Results of the fit of the chemical potential data tothe scaling ansatz, Eq. (13), for the 1d trapped Bose-Hubbard modelwithout any superlattice and with a commensurate or incommensuratesuperlattice.

U/J V2 α γd=1 Cd=1

5 0 – 0.818(1) −1.23(1)10 – 0.818(2) −1.21(2)20 – 0.815(5) −1.32(4)10 U 3/4 0.817(2) −1.23(2)20 3/4 0.820(2) −1.40(1)30 3/4 0.822(3) −1.58(1)10 U 830/1076 0.816(3) −1.22(3)20 830/1076 0.822(4) −1.42(4)30 830/1076 0.819(5) −1.54(5)

(no superlattice, commensurate and incommensurate superlat-tice) to the ansatz of Eq. (13). We observe that the coefficientsγd=1 are essentially all consistent within error bars, as theTF/AL theory would simply predict; more quantitatively, theyare all close to the TF/AL prediction γd=1 = 0.825 48, which isa remarkable fact given the simplicity of that theory. Moreover,the constant terms Cd=1(U/J,V2/J ) appear to be only weaklydependent on the Hamiltonian parameters; in particular, withinerror bars they appear not to depend on the parameter α of thesuperlattice but only on the superlattice strength.

3. d = 2, 3

Figure 6 shows the chemical potential for the d = 2 Bose-Hubbard model as numerically determined via quantum MonteCarlo. We have considered the two values U/J = 15 and 40for the interaction: According to the phase diagram of thehomogeneous 2d Bose-Hubbard model [45], for the first valuethe system is always in an SF state at all fillings, while forthe second value the system experiences MI phases at fillingsn = 1,2, . . . upon changing the chemical potential. From Fig. 6we see that, no matter the phase the system is in, data sets fordifferent particle numbers and varying trapping potentials allcollapse onto the same curve when plotted as a function of

0 10 20 30 40

( N Vt / J)

1/2

0

50

100

150

200

µ / J

N=250 300 350 400 450 500 550 600 650 700 750 800

0 10 20 30 40 50 60

( N Vt / J)

1/2

0

50

100

150

200

250

300 N=150 200 250 300 350 400 450 500 550 600 650 700

U/J=40

15

U/J=40

15

V2=0 V

2=U


Bose-Hubbard model without (left panel) and with (right panel)a superlattice potential (α = 830/1076 in all spatial dimensions).Dashed lines are fits to the ansatz, Eq. (13).

023601-7


TABLE II. Results of the fit of the chemical potential data tothe scaling ansatz, Eq. (13), for the 2d trapped Bose-Hubbard modelwithout any superlattice and with an incommensurate superlattice.


15 0 – 0.795(3) −2.1(1)40 – 0.790(4) −3.1(1)15 U 830/1076 0.803(6) −2.6(2)40 830/1076 0.799(3) −3.53(8)

(NVt )1/2, in agreement with the TF/AL prediction, both inthe absence and in the presence of a superlattice potential.For sufficiently high N and/or Vt , the dependence of µ on(NVt )1/2 is linear and can be well fitted to the ansatz, Eq. (13),giving the results summarized in Table II. It is remarkablethat the numerical results for the slope γd=2 are very closeto being all numerically consistent with each other, and withthe TF/AL prediction γd=2 = 0.797 88. On the opposite end,for small N and/or low Vt , the local filling can drop to values〈ni〉 � 1, which, in the absence of a superlattice, amounts tothe onset of a n = 1 Mott plateau for U/J = 40: This stronglattice commensurability effect, completely neglected in theTF/AL approach in continuum space, is responsible for thedeviation of the U/J = 40 results from the linear behavior.As already observed for the d = 1 case [30], the µ curveexhibits a crossover to a low-density regime which loses thedependence on the U/J value, as shown by the merging of theU/J = 40 curve with the U/J = 15 one. In general, one doesexpect all µ curves at arbitrary U/J values to merge onto thesame universal curve for a diluted system when N,Vt → 0.Yet a large enough U/J suppresses double occupancy in asystem with filling smaller than unity, so that the actual valueof U/J becomes irrelevant and the system enters the so-calledTonks or hardcore regime: This is the effect responsible for themerging of the µ curves at U/J = 40 and U/J = 15 in Fig. 6,well before the regime of extreme dilution is attained [the twocurves come close to each other for (NVt )1/2 values whichcorrespond to a filling n ∼ 1 in the trap center]. In contrast,for the NVt values we considered, the hardcore regime is notobserved in presence of a strong superlattice V2 = U , whichcompetes with the repulsion and maintains doubly occupiedsites down to the lowest trap fillings we have explored.

To conclude, Fig. 7 shows analogous results for the d = 3case. We again choose two values of repulsion: U/J = 20,for which the homogeneous system is SF at all fillings [46],and U/J = 50, for which the system has MI regions at fillingsn = 1,2, . . . . In agreement with the TF/AL prediction, thechemical potential is a homogeneous function of the productN2/5V

3/5t , and it becomes linear for sufficiently large filling

in the trap center (n >∼ 1); the results of fits to Eq. (13) aresummarized in Table III and show numerical consistencybetween the estimated slopes γd=3 and the proximity to theTF/AL prediction γd=3 = 0.813 46, both in the absence andin the presence of a superlattice [47]. In the opposite limit oflow filling n <∼ 1, and in the absence of the superlattice, weobserve again that the µ curves at different U/J values tend tomerge together, which corresponds to the onset of the hardcoreregime with suppression of double occupancy.

0 10 20 30 40

N 2/5

(Vt / J)

3/5

0

50

100

µ / J

N=250 300 350 400 450 500 550 600 650 700 750 800

0 10 20 30 40 50

N 2/5

(Vt / J)

3/5

0

50

100

150 N=250 300 350 400 450 500 550 600 650 700 750 800

U/J=20

50

U/J=20

50

V2=0 V

2=U


Bose-Hubbard model without (left panel) and with (right panel)a superlattice potential (α = 830/1076 in all spatial dimensions).Dashed lines are fits to the ansatz, Eq. (13).

C. Quantum Monte Carlo versus Gutzwiller mean-field results

In the previous section, we have seen that the chemicalpotential of strongly correlated bosonic systems can bedescribed via a modified TF scaling for sufficiently highdensities or sufficiently strong trapping potentials, at whichlattice commensuration effects become very weak. On theother hand, lattice effects can be easily taken into accountat the classical level via a numerical calculation in the AL,neglecting intersite hopping. Yet such a drastic approximationis going to be unreliable when the hopping energy becomescomparable with the repulsion one, typically when 2dnJ ∼Un2. Nonetheless, quantum effects can be restored at anapproximate level within Gutzwiller mean-field theory, whichkeeps the cost of numerical calculations at a minimum whileproducing predictions which turn out to be in suprisingly goodagreement with those of quantum Monte Carlo.

Gutzwiller mean-field theory assumes a factorized ansatzfor the wave function of the many-body system,

|�〉 = ⊗Ni=1

(nmax∑n=0

f (i)n |n〉i

), (18)

where |n〉i is a Fock state with n particles at site i and nmax

is a suitable truncation imposed on the local Hilbert spacefor numerical purposes. Assuming all f (i)

n coefficients to bereal, the expectation value of the Hamiltonian, Eq. (1), on theGutzwiller ansatz (GA) wave function takes the form

〈H〉 =∑

i

εi , (19)

TABLE III. Same as in Table II bur for the 3d trapped Bose-Hubbard model.


20 0 – 0.801(3) −3.38(8)50 – 0.795(5) −4.55(11)20 U 830/1076 0.812(14) −4.4(2)50 830/1076 0.81(5) −6.3(1.7)

023601-8


where

εi = −Jγi

∑n

√n + 1f (i)

n f(i)n+1

+∑

n

∣∣f (i)n

∣∣2[U

2n(n − 1) + Vin

](20)

and

γi = 1

2

∑l

∑n

√n + 1f (i+l)

n f(i+l)n+1

(21)Vi = V2gi + Vt (ri − r0)2 − µ.

Here,∑

l runs on the z = 2d nearest neighbors in a hypercubiclattice.

The ground state corresponds then to the minimization ofthe Hamiltonian expectation value with respect to the Nnmax

coefficients. Instead of proceeding with a full minimization,the usual procedure is to minimize with respect to the localvariables f (i)

n at a site i while holding fixed the variablesat all the other sites. We perform random sweeps over thelattice sites, touching each site once per sweep on average,until convergence is reached—typically less than a hundredsweeps are necessary for convergence. For the Hubbard modelunder investigation, this leads to a very efficient location of theabsolute energy minimum, which can be verified by repeatingthe minimization starting from a different initial condition.

Equation (18) generally describes a state which doesnot have a well-defined particle number. Given that theminimization of the energy is done in an unconstrained wayat each site, we need to adjust the chemical potential µ aposteriori in order to achieve a desired average particle numberN = ∑

i,n |f (i)n |2n, with an analogous procedure to that used

with quantum Monte Carlo. Hence, this procedure samples thefunction µ = µ(Vt ,N,J,U,V2) of interest. We use differentparticle numbers N and trapping potentials, ranging in thesame intervals as those explored in quantum Monte Carlocalculations.

Figures 8 and 9 compare the results of the GA withthose of quantum Monte Carlo calculations for the trappedBose-Hubbard model in dimensions d = 1, 2, and 3. Dataobtained in the AL are also shown as a reference. All dataare normalized to the mTF behavior, Eq. (13), attained in thelarge density or strong confinement limit. The results in theAL are seen to generally overestimate the chemical potential;this is not surprising, given that there is no kinetic energygain in adding particles to the system in the AL and henceall the energy gain to balance repulsion has to be providedby the chemical potential. Quite surprisingly, instead, weobserve that the mean-field GA accounts very well for thequantum corrections to the AL for all system dimensions, andthat the chemical potential predicted via the GA is typicallyless than 4% off with respect to the quantum Monte Carloprediction. Indeed, the GA, while successfully describing themain features of the phase diagram of the Bose-Hubbardmodel, is not particularly accurate in determining the phaseboundaries; given its mean-field nature, it is not quantitativelytrustworthy in low dimensions, and its predictions on thelocation of phase boundaries can be up to 100% off in d = 1.We find that the GA works equally well even in the presence of

0 1 2 3 4 5 6

N 2/3

(Vt / U)

1/3

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

µ / µ

mT

F

QMCGutzwiller Ansatzatomic limit

0 1 2 3 4 5 6

N 2/3

(Vt / U)

1/3

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

µ / µ

mT

F


FIG. 8. (Color online) Chemical potential for the trapped 1d BoseHubbard model with U = 10J , as obtained via quantum Monte Carlo,Gutzwiller ansatz (GA), and AL calculations. µmTF is the modifiedTF prediction of Eq. (13) with fitting parameters from Table I. Upperpanel: V2 = 0. Lower panel: V2 = U , α = 830/1076.

an incommensurate superlattice potential, despite its inabilityto describe Anderson localization and Bose-glass physics (datafor the case d = 1 are shown in Fig. 8).

A semiquantitative justification of the success of the GA inpredicting the chemical potential of the trapped Bose-Hubbardmodel can be provided based on LDA. LDA predicts that thetotal density of the system can be written as

N =∑

i

n(µi), (22)

where µi = µ − Vt (ri − r0)2 is the local chemical potentialand n(µ) is the density as a function of the chemical potentialfor the bulk system. The GA might provide a very poorprediction nGA(µ) for the n(µ) function of the bulk system,especially in d = 1, and this might suggest that the globalchemical potential µ which stabilizes N particles in the systemshould be also poorly estimated. Yet what matters in thedetermination of N is not the whole n(µ) curve, but only itsintegral. If the difference �n(µ) = n(µ) − nGA(µ) oscillatesfrom positive to negative and back, it will be averaged tozero upon integration. This is most likely the case for theHamiltonian parameters we have considered. As observed, forexample in Fig. 8, the most serious problems arise for lowN , which, in a trapped system, implies a chemical potentialexcursion (from the tails to the center) over which the �n

function has completed only a few oscillations (if any).

023601-9


0 2 4 6 8

(N Vt / U)

1/2

0.9

1

1.1

1.2

1.3

1.4

1.5

µ / µ

mT

F


1 2 3 4

N 2/5

(Vt / U)

3/5

0.9

1

1.1

1.2

1.3

1.4

1.5

µ / µ

mT

F


FIG. 9. (Color online) Chemical potential for the trapped 2d and3d Bose Hubbard model. µmTF is the modified TF prediction ofEq. (13) with fitting parameters from Tables II and III. Upper panel:d = 2, U = 15J , V2 = 0. Lower panel: d = 3, U = 20J , V2 = 0.

D. Discussion

From these results, we can conclude that the chemicalpotential for the Bose-Hubbard model in a trap and in asuperlattice potential has a simple scaling form as a functionof the experimental parameters. On the one hand, an accuratedetermination relies in general on numerics due to the stronginteractions in the system; yet we find that the chemical poten-tial appears as a homogeneous function of N2/(2+d)V

d/(2+d)t ,

and in particular a linear function thereof for sufficiently highfilling in the trap center (typically n >∼ 1 in absence of asuperlattice and even lower in presence of a superlattice),and/or sufficiently weak interaction, verifying this way aquantum-modified version of the TF/AL prediction, Eq. (13).Reference [48] has shown that, under the assumption thatLDA holds, one can prove the relationship ρ = Id (µ/J ; U/J ),where ρ is the characteristic density, already introduced, andId is an unknown function. By inverting the previous relation,this amounts to saying that µ is a homogeneous functionof ρ, in agreement with our findings. Therefore, our resultsallow us to explicitly reconstruct the relation between µ andρ. This correspondence of our results with the LDA predictionsuggests that µ is exactly a homogeneous function of ρ if (andpossibly only if) LDA is exact. Indeed, deviations from perfecthomogeneity are seen in our numerical data, for instancein Fig. 8, and they can be attributed in part to numericaluncertainty; in part to the fact that in any finite system different

µ values correspond to the same N value (which is discretized),so that there is a natural uncertainty on µ; and in part to possiblesystematic deviations, enhanced in the presence of a rapidlyoscillating superlattice. Yet we observe that the homogeneityproperty is an important simplifying assumption, and it isverified with an accuracy of a few percentage points in theworst case.

Moreover, in the case of the Bose-Hubbard model withoutsuperlattices, Ref. [49] has numerically shown that densityprofiles (and profiles of other local observables) for trappedbosons having the same characteristic density ρ and the sameHamiltonian parameters U and J , appear to be invariant upto a rescaling of the space coordinates with the characteristiclength η = √

J/Vt . This implies that under rescaling of thecoordinates with η, the density at site i, 〈ni〉 = 〈n(r i/η)〉, isa unique function of ρ. At the same time, the local chemicalpotential, µi , expressed in terms of rescaled variables, turnsout as well to be a unique function of ρ:

µi = µri /η = µ(ρ; U,J ) + Vt (ri − r0)2

= µ(ρ; U,J ) + J [(ri − r0)/η]2, (23)

thanks to the result that µ = µ(ρ; U,J ). As a consequence,the results of Ref. [49] suggest that 〈n(r i/η)〉 is a uniquefunction of µr i /η. In the limit of an infinitely shallow trap,Vt → 0 and N → ∞ at fixed ρ, the dependence of 〈n(ri/η)〉on µri /η must reproduce the dependence of the bulk densityon the chemical potential, n(µ). Hence the unique functionrelating 〈n(r i/η)〉 to µr i /η must be identical with that of thebulk limit. As a consequence, the exact invariance of thedensity profiles (and, with the same reasoning, of any otherlocal observable in the trap) under rescaling with η wouldimply that the LDA is exact. It is trivial to show that theexactness of the LDA implies the invariance of local quantitiesunder rescaling with η. Hence, we can conclude that the scaleinvariance of trapped systems at fixed characteristic densityis exact if and only if LDA is exact. Yet, even when LDAis not exact, the system might display scale invariance witha better accuracy than that of LDA, as shown numericallyin Ref. [49] in the case of density fluctuations and localcompressibility.

At the experimental level, our results reveal the possibilityof controlling the chemical potential in the trapped systemby (1) controlling the trapping frequency at fixed particlenumber, or, vice versa, (2) controlling the particle numberat fixed trap strength, or (3) controlling both simultaneously.Hence, the present experimental setups have direct access tothe measurement of phase diagrams of strongly correlatedbosons in the grand-canonical ensemble at variable chemicalpotential. Most noticeably, an accurate knowledge (withina few percentage points) of the zero-temperature chemicalpotential as a function of the Hubbard Hamiltonian parameterscan be obtained with little numerical effort via the mean-fieldGA, namely without fully solving numerically the many-bodyproblem. This aspect is to be contrasted with the case ofaccurate thermometry of strongly correlated bosons, whichat the moment can be reliably assessed only via exhaustive abinitio simulations [12,13].

023601-10


IV. TRAP-SQUEEZING PROTOCOL

The previous section has shown how the chemical potentialof a trapped system is simply controlled by the number ofparticles N and by the trapping potential Vt . Indeed, recent ex-periments have probed the compressibility of fermions loadedin optical lattices by varying the trapping potential [31] or theatom number [50,51], although the explicit relation betweenthe experimental parameters and the chemical potential wasnot known for that system. In the experiments, the atomnumber is typically subject to significant fluctuations due toshot noise of order

√N , as well as to other systematic effects.

On the contrary, the trapping potential Vt can be controlled ona much finer scale via the application of a dipolar trap along thespatial direction of interest, namely by shining d red-detunedrunning laser waves onto the sample to control the confinementin d spatial dimensions. (This is sketched in Fig. 10 for thecase of variable one-dimensional confinement.) The dipolartrap adds up to the confining (anticonfining) potential givenby the overall Gaussian intensity profile of the red-detuned(blue-detuned) optical lattice, and it gives the possibility ofvarying the trap strength Vt independently of the Hamiltonianparameters U , J , V2, offering in this way the possibilityof simulating the Bose-Hubbard Hamiltonian in a variabletrapping potential [31,52].

In the remainder of the article, we hence focus on theproposal of extracting the phase diagram of the Bose-Hubbardmodel in the grand-canonical ensemble by controlling thechemical potential of the system via trap squeezing and atfixed particle number. Two fundamental issues are addressedin the next subsections concerning the experimental feasibilityof this proposal: (1) we discuss the fundamental difficultyin achieving adiabatic trap squeezing once the particles areloaded in the optical lattice and (2) we propose a simple loadingsequence which overcomes this difficulty and which allows usto control the particle number in the central tubes or layers inthe case of 1d/2d confinement.

A. The adiabaticity issue

Optical lattice experiments have revealed the unique pos-sibility of tuning the Hamiltonian parameters in real time,for example, via changing the intensity of the standing wavewhich controls the U/J parameter [53]. A fundamental issue

FIG. 10. (Color online) Schematic view of the confinementcontrol on the longitudinal direction in a set of one-dimensional tubes.

raised by such real-time control is the possible generationof excitations in the system by a nonadiabatic change of theHamiltonian parameters. This becomes particularly dramaticwhen the parameter change implies the crossing of a quantumcritical point, at which the gap over the ground state vanishes,leading to efficient Landau-Zener tunneling [54]. Similarly tothe change in the optical lattice strength, a change in the dipolartrapping strength can also lead to quantum phase transitions,driven this time by the chemical potential, and hence it issubject to adiabaticity issues. In fact, adiabaticity is evenmore problematic in the case of trap squeezing: Regardingthe center of the trap as the system and the trap periphery asthe particle reservoir (as mentioned in the introduction), theprocess of transfer of particles between system and reservoirunder trap squeezing might be pathologically slow if theparticles are strongly localized in one part or the other, dueto strong interactions in the presence of an optical lattice or(quasi-)disorder potentials.

We exemplify the previous discussion in the case of latticehardcore bosons, for which the exact many-body spectrum canbe calculated exactly via Jordan-Wigner diagonalization [43],in the case of density n � 1 on each site. The fermionizationapproach can be extended to the case n > 1 [55] under thecondition of having a well-developed layer-cake structure ofthe density profile (see Appendix A for a detailed discussion).Figure 11 shows the evolution of the density profile upontrap squeezing for the trapped Bose-Hubbard model with U =20J and with N = 100 particles, as obtained via extendedfermionization. For the chosen range of trap strengths, weobserve that the system goes from an MI phase with n = 1particles in the trap center to an SF phase with 1 < n < 2 andfinally to an MI phase with n = 2 particles. In particular, thelocal transition from MI to SF in the trap center at fixed particlenumber is exclusively due to the redistribution of particles fromstates in the wings of the trap to states in the center. The statesin the trap periphery are localized by the joint effect of thestrong repulsion exerted by the particles in the trap center andof the confining potential. On the other hand, the states inthe trap center are localized by the confining potential over alength scale which is well below the width of the atomic cloud.Hence, the spatial overlap between such states is negligible,

FIG. 11. (Color online) Density profile as a function of trapstrength for the Bose-Hubbard model with N = 100 particles andwith U = 20/J , resulting from an extended fermionization calcula-tion (see text and Appendix A).

023601-11


0.005 0.01 0.015 0.02 0.025

Vt / J

0

0.2

0.4

0.6

0.8

∆ / J

FIG. 12. Lowest energy gap � as a function of trap strength forthe Bose-Hubbard model. Parameters as in Fig. 11.

and they are separated by a high potential barrier, which isrepresented by the atoms forming the n = 1 Mott plateaus.This implies that the tunnel splitting between the states in thecenter and the states in the wings can be extremely small,and hence the ground state can become nearly degeneratewith the first excited state. This is indeed revealed by thedirect investigation of the lowest energy gap, which, within theextended fermionization approach, is estimated as the lowestparticle-hole excitation energy for the spinless fermions, and itis shown in Fig. 12. We observe that such a gap goes to a valuevery close to zero [56] when the first particles move from thetrap wings to the trap center, corresponding to the occurrenceof an MI-to-SF transition in the trap core. The gap shows asmany dips as particles transferred to the center, revealing thetightly avoided level crossings corresponding to the particleredistribution. More precisely, each deep minimum of the� versus Vt curve corresponds to the successive migrationof two particles from the wings to the center, because suchparticles occupy states which are nearly degenerate symmetricor antisymmetric superpositions of localized states on thetwo opposite wings of the trap. Hence, this succession oftightly avoided level crossings makes adiabatic trap squeezingessentially impossible in the presence of MI regions, assqueezing times which are several orders of magnitudebigger than the typical tunneling time (∼ms) would berequired.

As a second example, we consider the case of N = 50hardcore bosons in an incommensurate superlattice potential,Eq. (3), with α = 830/1076 (as experimentally realized inRef. [22]) and V2 = 20J . Such a potential is known to leadto Anderson localization of all single-particle states for V2 >

4J [57,58]. In this case, standard fermionization provides theexact spectrum, and in particular the lowest energy gap as afunction of the trap strength, as shown in Fig. 13. We observethat the gap can become extremely small, this time due tothe quasidegeneracy of Anderson localized states, which arespatially well separated, so that the tunnel splitting betweenthem is very small.

Hence, from these two examples, we can conclude thateither strong repulsion induced by the optical lattice or

0 0.005 0.01 0.015 0.02

Vt / J

0

0.2

0.4

0.6

0.8

1

1.2

∆ / J

FIG. 13. Lowest energy gap � as a function of trap strength forN = 50 hardcore bosons in an incommensurate superlattice (α =830/1076) of strength V2 = 10J .

a (pseudo-)disorder potential give rise to localized single-particle states, which become quasidegenerate upon changingthe trap frequency. This leads to a very efficient Landau-Zenertunneling, which essentially makes adiabatic trap squeezingimpossible. Although the results shown here are obtained for aone-dimensional system, their underlying mechanism is quitegeneral and applies to higher dimensions as well. Hence, wecan generally conclude that adiabatic trap squeezing for asystem of particles loaded in a strong optical (super)latticeis virtually impossible. Luckily, the statement of the problemalready contains in itself the solution: Trap squeezing hasto be performed before loading the particles in the optical(super)lattice, namely when the particles still enjoy their fullmobility, so that they can adiabatically follow the variationin the trapping potential. In the case of spatially anisotropicoptical lattices, discussed in the next subsection, this consider-ation applies to the loading of the particles in the optical latticealong the spatial dimensions in which the particles are moreweakly confined.

B. Trap-squeezing protocol for low-dimensional systems

In the case of spatially anisotropic optical lattices, allowingus to simulate the 1d or 2d Bose-Hubbard model [21,59],the control of the chemical potential can be achieved by trapsqueezing along the spatial dimensions of interest, namelyalong the tubes in 1d-anisotropic optical lattices and alongthe layers in 2d-anisotropic ones. Yet, if the strength ofthe trapping potential is changed before confining the atomsin tubes or in layers, the final atomic population present inthe tubes or layers will be significantly affected, so that oneviolates the condition of working at fixed particle number in thelow-dimensional elements (tubes or layers) of the system, andin particular in the central ones. The problem of guaranteeing atthe same time the control on the particle number in the tubesor layer and adiabatic trap squeezing can be easily solvedby considering the following loading protocol, sketched inFig. 14:

(a) Particles are initially trapped in a dipolar trap or amagneto-optical trap (MOT) at fixed strength.

023601-12


(a) (b)

(c) (d)

FIG. 14. (Color online) Sketch of the proposed loading andsqueezing protocol of atoms in a spatially anisotropic optical lattice.The description of the various stages is presented in Sec. IV B

(b) A deep one-dimensional (two-dimensional) opticallattice is then ramped up, defining layers (tubes) in which theatoms are still moving in continuum space. For a given initialatom number N , the overall trapping potential (dipolar trapor MOT plus the optical lattice beam profile) uniquely definesthe populations in each tube (layer). These populations remainessentially fixed for the rest of the experiment, thanks to thedeep optical lattice. Given that the atoms are still in the weaklyinteracting regime, we can determine these populations via theTF approximation. The results are given by Eqs. (B6) and (B9)in Appendix B.

(c) Varying the dipolar trapping along the tube (layer)dimensions gives rise to trap squeezing, which can be easilykept adiabatic for weakly interacting particles [60].

(d) At this stage, the optical lattice can be adiabaticallyramped up along the tubes (layers), finally realizing a 1d/2d

Bose-Hubbard Hamiltonian at a given particle number and ina given longitudinal trapping potential.

V. IMAGING TECHNIQUES

The two previous sections have discussed how to achievefull control on the chemical potential of the Bose-Hubbardmodel realized in optical lattice experiments via the con-trol on the trapping potential at a fixed particle number.As discussed in Sec. II, the overall chemical potential ofthe trapped system corresponds in particular to the localchemical potential at the trap center: Hence, getting accessto the average central density in the trap as a function ofthe chemical potential gives the possibility of extractingexperimentally the density curve of the bulk Bose-Hubbardmodel in the grand-canonical ensemble, and therefore itscompressibility.

As it is seen in the next sections, measuring an averagecentral density which mimics closely the one of the bulksystem requires imaging the atomic cloud over a typicallength scale of 10–20 lattice sites in each spatial directionfor typical atom numbers and trapping frequencies used incurrent experiments, corresponding to a spatial resolution of∼5–10 µm. To achieve this resolution, a possible technique is

(b)

2 11

2

F ′=3

F ′=2

F=2

F=1

2

2

W

n

n S1/2

1/2 P

(a)

FIG. 15. (Color online) Sketch of the proposed imaging protocolfor alkali atoms. A repumping (ω2) and a cooling (ω1) beam aretightly focused on the center of the trap, and they are crossed so as tooverlap in a region of width ≈ W (minimal beam waist) in all spatialdirections. This setup realizes a spatially resolved optical cyclingtransition.

provided by spatially resolved fluorescence imaging for atoms(see Fig. 15). In the case of alkali atoms initially prepared in anF = 1 hyperfine state of the n2S1/2 level, a repumping beam,resonant with the transition to an F ′ = 2 state of the n2P3/2

level, optically pumps the atoms into an F = 2 state [61]; asecond beam resonant with the F = 2 → F ′ = 3 transition(cooling beam) is then used to image the atoms. If the twobeams are tightly focused to a beam waist of ∼5–10 µm andare perpendicular to each other such that they meet at the focalpoint, they selectively image atoms in a region of space of thedesired size. Typical optical lattice experiments have n > 1particles per lattice site in the trap center, so imaging a centralregion with 103–203 lattice sites involves imaging at least asmany atoms, which is possible over a time of <∼100 ms (see,for example, Ref. [62]). Before imaging, the optical latticecould be rapidly increased to the maximum height in order tofreeze the atomic cloud profile during the successive imagingtime.

The proposed imaging technique has the advantage of beinggenerally compatible with common atom-trapping setupsbased on magneto-optical or dipolar traps and requires only anintermediate imaging resolution. On the other hand, ultrahighresolution optics have given access to few-site/few-atom oreven single-site/single-atom imaging, in a variety of veryrecent experiments [25–29]. This level of resolution far ex-ceeds the one required by our present proposal. An alternativetechnique for the indirect extraction of the local density inthe trap consists in the high-resolution in situ imaging ofthe atomic cloud density integrated along the line of sight,from which the full three-dimensional atomic distribution isreconstructed via inverse Abel transformation [63,64]. Giventhe diversity of techniques mentioned previously, we areconfident that the extraction of the local-density propertiesin selected regions of the trap will become an experimentalroutine in the near future.

After the in situ imaging stage, turning off all trappingpotentials and imaging the expanded cloud gives access tothe total number of atoms. This piece of information isfundamental to postselect the measurements with a given totalatom number N , which corresponds to the desired value of thechemical potential to be realized in the experiment. Strictly

023601-13


speaking, once the calibration curve relating µ to N is known,the final measurement of the particle number has only the roleof assigning the measurement of the central density to theproper place in the grand-canonical phase diagram of the bulksystem. Hence, one can regard the fluctuations in the totalparticle number N as a source of random sampling of theµ axis. In this perspective, all measurements give usefulinformation, provided that particle-number fluctuations are notbringing the chemical potential µ too far from the region ofinterest in the phase diagram.

VI. APPLICATIONS: PHASE DIAGRAM OF THE 1dBOSE-HUBBARD MODEL

As a first application of trap squeezing, we investigatethe ground-state phase diagram of the one-dimensional Bose-Hubbard model. We consider the situation of N = 100 bosonstrapped in the central tube of a strongly anisotropic opticallattice [21] and addressed with spatially resolved imagingdescribed in the previous section. We present quantum MonteCarlo results for the central properties in the trap, as well as forthe global time-of-flight properties. The chemical potential µ

is adjusted so that the ground state of the grand-Hamiltonian,Eq. (4), has the desired number of bosons.

A. Central compressibility

Investigating the phase diagram of the Hubbard model viatrap squeezing implies that the SF-MI transitions that one canprobe are accessed via a variation of the chemical potential,namely that they are of the commensurate-incommensuratekind. Such transitions have the advantage of being very sharp,as they are characterized by the appearance (or disappearance)of an SF component in the system, induced by doping particlesor holes into an MI state at integer filling. The SF-MI transi-tion can be completely characterized by the compressibilityκ = ∂n/∂µ, which, similarly to the SF density, is identically

zero in the MI phase and finite in the SF phase, for all spatialdimensions. In particular, the compressibility has quite a sharpvariation at the transition in low-dimensional systems. Indeed,its critical scaling with system size L obeys the law Ld−z [18]where z = 2 is the dynamical critical exponent; hence, ford = 1, it jumps from a value diverging linearly with systemsize to zero at the transition point (while in d = 2 the jumptakes place from a value diverging logarithmically with thesystem size). This very sharp feature makes it possible todetect the transition unambiguously even on a small systemsize, which is effectively the case here. In the following, wedeliberately limit our attention to the n > 1 region of the phasediagram, as this is a typical situation in the experiments.

We focus our attention on the behavior of the centralcompressibility,

κC

= ∂nC

∂µ, (24)

corresponding to the response of the density in the centralregion C to a variation of the chemical potential, drivenexclusively by a change in the trapping potential (whichleaves the parameters J and U unchanged). Upon choosing asufficiently narrow central region C satisfying the condition ofEq. (7), the central density is supposed to reproduce closely thebehavior of the compressibility in the bulk system. Figure 16shows indeed that this is the case. In fact, we observe thatthe central compressibility jumps from from a finite valueto zero for critical values of the chemical potential whichreproduce closely the shape of the MI lobes of the bulk-phasediagram of the 1d Bose-Hubbard model. (The MI boundariesfor the bulk system have been determined in Ref. [40] viadensity-matrix renormalization group (DMRG) for the n = 1and n = 2 lobes, while we have determined the n = 3 lobevia quantum Monte Carlo.) Remarkably, upon doubling thecentral region C from ten to twenty sites (namely halvingthe resolution of the imaging in the experiment), the main

J/U

µ/U

0.05 0.1 0.15 0.2 0.25

0.5

1

1.5

2

2.5

3

0

0.05

0.1

0.15

0.2n = 3

n = 2

n = 1

κCU , 10 sites J/U

µ/U

0.05 0.1 0.15 0.2 0.25

0.5

1

1.5

2

2.5

3

0

0.05

0.1

0.15

0.2n = 3

n = 2

n = 1

κCU , 20 sites

FIG. 16. (Color online) Central compressibility κC

= ∂nC/∂µ (in units of U−1) of N = 100 bosons under variable trapping frequency.

The central density nC

is obtained by imaging over ten sites (left panel) and twenty sites (right panel). The white curves report the boundarylines between SF and MI phases as determined in Ref. [40] (for the n = 1 and n = 2 lobe) via density-matrix renormalization group andindependently in this work (for the n = 3 lobe) via quantum Monte Carlo.

023601-14


features of the reconstructed phase diagram are only slightlyaltered, at least for what concerns the lobes n = 1, 2. Thisshows the robustness of the proposed approach under realisticexperimental conditions. An interesting phenomenon to beobserved in Fig. 16 is the blue shift (toward higher chemicalpotentials) of the n = 2 and n = 3 Mott lobes predicted bythe vanishing of κ

Cwith respect to what is observed in a bulk

system. This shift can be easily understood as a result of thefinite width of the imaging region and as an effect of thefiniteness of the central region exhibiting bulk physics, and, aswe discuss later, it can be corrected for systematically.

1. Blue shift of the lower lobe boundaries

The lower lobe boundaries are associated with an SF-MItransition induced by an increase of the chemical potentialabove a lower critical value µ(l)

c (J/U ). A chemical potentialµ > µ(l)

c forces the atom density to an integer value, for whicha Mott gap opens if J/U < (J/U )c, corresponding to the lobetip. When this phenomenon occurs in the center of the trap, theMI region, which appears at first, is extremely narrow in space[see Fig. 17(a) for a sketch]. Indeed, it extends over a regionof width �r such that the chemical potential modulation dueto the trap over this region is lower than the MI hole gap �h,namely Vt (�r/2)2 ≈ �h. The central compressibility detectsthe appearance of an incompressible MI phase in the trapcenter only when �r reaches the size |C| of the imaging region,namely for a critical bulk gap �C

h ≈ Vt,C(|C|/2)2. Here, Vt,C isthe trapping potential corresponding to the effective chemicalpotential µ(Vt,C,N,J,U ) at which the hole gap takes the value�C

h . This chemical potential is simply given by µ(l)c + �C

h .This implies that, for a strong confinement Vt , as that requiredto obtain n = 2 or n = 3 MI lobes in the system, one observes

(a)

(b)

FIG. 17. (Color online) Mechanisms of blue shift of the Mottlobes obtained from the central compressibility. (a) At the lowercritical chemical potential µ(l)

c , a Mott region only forms on thecentral site, and one needs an increase of the chemical potentialby a term �C

h to establish a Mott phase throughout the imaging (pink)region. (b) At the upper critical potential µ(u)

c of the bulk system,the trapped system has not yet developed an SF core because of theadditional zero-point energy εzp(n) due to trapping in the harmonicoscillator potential. Only when the chemical potential reaches theblue-shifted value µ(u)

c + εzp(n) does the first particle of the SF coreappear (red).

the onset of a MI phase for a blue-shifted critical chemicalpotential: (

µ(l)c

)′ = µ(l)c + �C

h . (25)

Remarkably, this shift depends exclusively on experimentallyknown parameters Vt and |C|, so that one can correct a posteri-ori for it when determining the phase diagram experimentally.

2. Blue shift of the upper lobe boundaries

The upper lobe boundaries are associated with an MI-SFtransition induced by an increase of the chemical potential.Reaching an upper critical value µ(u)

c , the chemical potentialcloses the Mott gap of the MI with n particles per site,and it therefore adds the first particles on top of the MI.In a bulk system, these particles are essentially added in ak = 0 state, and they undergo condensation. In a trap, thecorresponding phenomenon would be the appearance, inducedby trap squeezing, of the first particle over the central Mottplateau with n particles per site [see Fig. 17(b)]. At variancewith the bulk case, these particles are not fully delocalized, butthey still feel the overall trapping potential—hence they have aresidual zero-point energy, due to the confinement in the trap.To estimate this shift in the zero-point energy, we can proceedin the spirit of the extended fermionization approach [55] (seeSec. IV and Appendix A) and approximate the extra particleadded in the trap center as moving on an inert background atfixed density n, whose only effect is to increase the hoppingamplitude of the particle from J to J (n + 1). Hence, thezero-point energy can be estimated from the solution of aharmonic oscillator of effective mass m∗ = h2/[2J (n + 1)]and frequency hω = √

4(n + 1)VtJ . This estimate is valid aslong as the width of the central MI plateau is larger than thewidth of the ground state of the harmonic oscillator, namelyσ = [J (n + 1)/Vt ]1/4. Hence, the shift in zero-point energydue to confinement is εzp(n) = hω/2 = √

(n + 1)VtJ . Thisleads to a shift in the critical chemical potential needed to“dope” a particle onto the central MI plateau with respectto that needed in a bulk system; as a consequence, theupper critical chemical potential µ(u)

c measured by the centralcompressibility is blue-shifted as(

µ(u)c

)′ = µ(u)c + εzp(n). (26)

Similarly to that observed for the lower lobe boundary, onecan correct a posteriori for this shift using the experimentalparameters and hence reconstruct the bulk value of µ(u)

c .

3. Blue-shift correction of trap squeezing data

Figure 18 shows the lower and upper critical chemicalpotentials of the n = 2 MI lobe, as extracted from the vanishingof the central compressibility of N = 100 bosons imaged overthe central twenty sites and compared with the numericallyexact results of Ref. [40]. The blue shift of the lower criticalchemical potential is quite sizable, but it is seen to be very wellcorrected for by the prediction of Eq. (25). The blue shift ofthe upper critical chemical potential is instead much weaker,in agreement with the prediction of Eq. (26). This shows thattrap squeezing allows us to reconstruct quite accurately thequantum phase transition lines of the Bose-Hubbard model,

023601-15


0 0.05 0.1 0.15 0.2J/U

1

1.2

1.4

1.6

1.8

2

µ/U

DMRG - Ref. 40

µc

(l) (bare)

µc

(l) (corrected)

µc

(u) (bare)

µc

(u) (corrected)

FIG. 18. (Color online) The n = 2 Mott lobe of the 1d Bose-Hubbard model as determined by trap squeezing of N = 100 bosons,imaged over the central twenty sites. The open circles correspond tothe bare values of the lower and upper critical chemical potentialsextracted from the vanishing of the central compressibility for |C| =20 (Fig. 16); the closed circles correspond to the values corrected forblue shift, according to Eqs. (25) and (26). The correction leads to avery good agreement with the DMRG results of Ref. [40].

even in the case of a coarse imaging of a central region of∼20 sites in the trap center.

Trap squeezing supplemented with imaging over |C| sitescan reliably detect MI behavior only as long as the MI plateauoccurring in the trap center can reach the same width as that ofthe central C region: This condition guarantees that (µ(l)

c )′ <

(µ(u)c )′, namely that upon increasing the chemical potential,

the (blue-shifted) lower chemical potential is reached before anew SF core forms. The MI region can no longer be reliablyobserved if Vt reaches the value Vt , satisfying the condition(µ(l)

c )′ = (µ(u)c )′, namely

Vt (|C|/2)2 −√

(n + 1)VtJ = µ(u)c − µ(l)

c = �ph, (27)

where �ph is the particle-hole gap of the MI phase. The abovecondition gives

Vt = �ph

R2−

√(n + 1)J�ph

R3+ (n + 1)J

2R4+ O

(1

R5

)(28)

where R = |C|/2 � 1. The merging of the blue-shifted lowerMott lobe boundary with the upper one is seen to occur, forexample, in Fig. 18 for J/U >∼ 0.12. To be able to observethe tip of the Mott lobe via trap squeezing, it would benecessary to increase the total particle number so that Vt can bedecreased below Vt , maintaining the same chemical potentialin accordance with Eq. (13).

So far, we have focused our discussion on the spectral prop-erties associated with the compressibility. In the following, wealso discuss the evolution of the coherence properties undertrap squeezing as an alternative tool to investigate the phasediagram of the system.

B. Time-of-flight observables and condensate compressibility

At variance with diagonal observables such as density andcompressibility, which can be measured locally, the phase-coherence properties, encoded in the momentum distribution,

are generally measured at the global level. We define themomentum distribution as

n(k) = 1

N

∑ij

eik(ri−rj )〈a†i aj 〉, (29)

where the normalization to the boson number N is chosenso as to account for the actual spatial extent of the cloud[65]. The momentum distribution is probed via time-of-flightmeasurements by collecting the interferogram of the matterwave emitted from all sites of the optical lattice. Given thestrong inhomogeneity of the trapped system, one can suspectthat the momentum distribution of a trapped gas is generallynot a faithful portrait of the corresponding distribution of ahomogeneous system with the same global chemical potential.However, some remarks are in order. For any given toleranceε, one can select a central region C that satisfies Eq. (7) and acomplementary region C. Hence, the momentum distributioncan be decomposed into three contributions, two of whichinvolve the central region C,

n(k) = nC(k) + nC,C(k) + nC(k), (30)

where

nC(C)(k) = 1

N

∑i,j∈C(C)

eik(ri−rj )〈a†i aj 〉 (31)

and

nC,C(k) = 1

N

⎛⎝ ∑

i∈C,j∈C

+∑

i∈C,j∈C

⎞⎠ eik(ri−rj )〈a†

i aj 〉. (32)

We argue in the following that the behavior of the global mo-mentum distribution is strongly dependent on the contributionnC(k) from the central region. Indeed, the central region of thetrap is the only simply connected region, which implies thatthe pairs of sites (ij ) associated with this region are theones at the shortest mutual distance |ri − rj |. This lattertopological aspect has strong consequences on the coherenceproperties of the global system. One can envision in generaltwo situations: (1) If the central region C is locally in an SFstate, its coherence properties are going to give the dominantcontribution to the sum of Eq. (29), because nC(k) containsthe largest number of strongly correlated pairs (ij ) at shortdistance; moreover, if the wings are SF too, the central regionwill also contribute with the nC,C(k) term, which generallydominates over nC(k) because the central region is at a higherdensity than its complement C. (2) If the central region isin a local MI phase, extending over a region of linear size∼RC , there might still be a SF halo in the region C, but itwill give a weak contribution to the n(k) because the pairs ofsites (ij ) involved in nC(k) are separated by a distance ∼RC ,and sites separated by the central MI core are very weaklycorrelated, given that particles cannot coherently propagateacross C. An equivalent statement to the latter one is that in ad-dimensional system, the SF halo surrounding an MI regionis effectively quasi-(d − 1)-dimensional, and consequently ithas reduced coherence properties, due to the enhanced roleof quantum fluctuations in lower dimensions. Hence, it isreasonable to believe that the momentum distribution mainlyreflects the coherence properties of the central C region, at least

023601-16


J/U

µ/U

0.05 0.1 0.15 0.2 0.25

0.5

1

1.5

2

2.5

3

0

5

10

15

20

25

30

n = 3

n = 2

n = 1

n(k = 0) J/U

µ/U

0.05 0.1 0.15 0.2 0.25

0.5

1

1.5

2

2.5

3

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

n = 1

n = 2

n = 3

FWHM ∗ a

FIG. 19. (Color online) Evolution of time-of-flight observables for N = 100 bosons under variable trapping frequency. The left paneldepicts the condensate number n(k = 0), and the right panel the full width at half maximum (FWHM) of the momentum distribution n(k), inunits of a−1, where a is the lattice spacing.

in the d = 1 case explicitly treated here. The case of higherdimensions is discussed toward the end of this subsection.

Figure 19 shows the height of the k = 0 peak and theFWHM of the n(k) in a system of N = 100 bosons undertrap-squeezing control of the chemical potential for variousoptical lattice depths. By plotting these quantities on the(J/U,µ/U ) plane, one sees that sharp drops in the globalcoherence of the trapped system are observed close to theSF-MI boundary lines of the bulk-phase diagram of the Bose-Hubbard model. As in the case of the global compressibility,for sufficiently high density the sharpest features in thecoherence properties are observed for blue-shifted values ofthe chemical potential [66]. This result seems to confirmour expectation that the global properties of the momentumdistribution are strongly dominated by the behavior of thesystem at the trap center, which best approximates the bulkbehavior.

To make this connection even more quantitative, we canconsider again the partition of the system into a centralC region and a complementary peripheral C region as apartition of the trapped gas into a system and a particlereservoir, respectively (Fig. 1). Trap squeezing increases thechemical potential of the reservoir with respect to that ofthe system, and it hence pumps particles from the formerto the latter. If the system experiences (quasi-)condensation,a (quasi-)macroscopic fraction of the pumped particles willenter the (quasi-)condensate. Alternatively, if the central regionhas a density which approaches an integer value from below,nC < int(nC) + 1/2, condensation should be rather thoughtof in terms of holes [of density int(nC) + 1 − nC]; in thatcase, a (quasi-)macroscopic fraction of the holes removedfrom the center under trap squeezing belongs to the (quasi-)condensate. In both cases, if the trap center is in a local(quasi-)condensate state, one expects that the variations ofthe n(k = 0) peak in the momentum distribution will beproportional to (for true condensation) or will go as a powerlaw of (for quasicondensation) the variation of the density in

the trap center. On the other hand, if the center of the trap isin an MI phase, this prevents the particle (hole) transfer, and itinhibits the variations of the momentum distribution.

From these arguments, one expects the behavior of themomentum distribution to be extremely sensitive to trapsqueezing; this sensitivity can be quantified by the condensatecompressibility

κ0 =:∂

∂µn(k = 0), (33)

which can capture the (quasi-)macroscopic transfer of particlesfrom the C reservoir into a condensed state in the centerC (or vice versa for hole condensation). The condensatecompressibility is positive for particle (quasi-)condensationin the trap center and negative for hole (quasi-)condensation.Consequently, |κ0| is expected to be finite in the presenceof condensation in C (apart from the zeroes correspondingto the passage from particle to hole condensation) and to beessentially vanishing corresponding to MI behavior. Figure 20confirms this expectations, showing that |κ0| is vanishingcorresponding to the bulk MI regions (apart from the usual blueshift of the critical chemical potentials), while it has generallyfinite values corresponding to the SF regions. The behavior inthe SF regions is highly inhomogeneous, due to the frequentpassage from particle to hole condensation. In any case, onecan easily tell apart the accidental vanishing of the condensatecompressibility in an SF regime from the systematic vanishingin the MI regime, for example, by inspection of the FWHM,which is much smaller than 2π/a in the SF case and of theorder of 2π/a in the MI case (a being the lattice spacing).

The arguments provided previously on the dominant role ofthe central region C in the global coherence properties applystraightforwardly to the d = 1 case explicitly considered here.Yet, in d = 2 and 3, one might suspect that the C region can stillsustain important coherence properties even though the centralregion undergoes the formation of a local insulator, based upona simple geometrical aspect: In d = 2 and 3, the C region

023601-17


J/U

µ/U

0.05 0.1 0.15 0.2 0.25

0.5

1

1.5

2

2.5

3

0.5

1

1.5

2

2.5

3

3.5

n = 3

n = 2

n = 1

|κ0|U

FIG. 20. (Color online) Absolute value of the condensate com-pressibility κ0 = ∂n(k = 0)/∂µ (in units of U−1) of N = 100 bosonsunder variable trapping frequency.

represents a dominant fraction of the total volume occupied bythe atomic cloud. Yet the topological aspect of connectivity andeffective dimensionality—which distinguishes C from C inany dimension—seems to play a crucial role here. As discussedin Ref. [35], extensive quantum Monte Carlo simulations onthe 2d and 3d Bose-Hubbard model in a trap show that theglobal coherence of the system is strongly suppressed as soonas a Mott plateau appears in the trap center. A fundamental,quantitative distinction between the case d = 1 on one sideand d = 2, 3 on the other seems to be the fact that the global-coherence properties (namely the k = 0 peak and the FWHM)evolve smoothly in d = 2, 3 when the MI forms in C, whilethey exhibit sharp kinks in the case d = 1. (See also Ref. [30]for the case d = 1.)

VII. APPLICATIONS: BOSE GLASS IN ATHREE-COLOR SUPERLATTICE

A. Signatures of a Bose glass

In the previous section, we have seen that trap squeezingallows us to faithfully reconstruct the phase diagram of theordinary Bose-Hubbard model, either via the measurement ofthe diagonal properties in the central region of the trap or viathe measurement of the evolution of the global off-diagonalproperties (momentum distribution). This latter aspect is dueto the fact that spectral properties and coherence propertiescan both characterize fully the two phases (SF and MI)present in the phase diagram of the Bose-Hubbard model.On the other hand, spectral and coherence properties becomecomplementary pieces of information in the presence ofdisorder in the system, inducing a novel phase in the phasediagram: the Bose glass [18]. This extra phase is in factcharacterized by a finite compressibility coexisting with short-range phase coherence, and it needs the joint measurement oflocal diagonal properties and global momentum distributionto be detected.

In Ref. [30], we have shown that trap squeezing canreveal the Bose-glass phase appearing in the Bose-Hubbardmodel with a one-color quasiperiodic potential, realized by a

(a)

(b)

FIG. 21. (Color online) Trap squeezing in the Bose glass phase.(a) Schematic view of the density profile of a boson gas in a disorderedoptical lattice. (b) In the presence of Bose glass behavior in the trapcenter, upon infinitesimal trap squeezing Vt → Vt + δVt , a particlecan be transferred from the trap wings to a localized state in thecenter. Due to its localized nature, this additional particle doesnot contribute significantly to the coherent fraction, which remainsessentially unchanged.

two-color incommensurate optical superlattice [22,36,67,68].The fingerprint of the Bose glass in a trap is the observationof a finite central compressibility κC , coexisting with a weakcondensate compressibility κ0, and with the observation of abroad peak in the momentum distribution.

Figure 21 sketches the response to trap squeezing of aBose glass occurring in the trap center. By definition, theBose-glass phase admits particle excitations at arbitrarily lowenergy, which means that a small increase in the trap frequencyVt and, consequently, in the global chemical potential µ isable to transfer a particle to the center of the trap, resultingin a finite central compressibility κC . It is important toemphasize that this is only true on average over the disorderdistribution: Indeed, averaging over the disorder statistics,one expects always the existence of one or several disorderarrangements in the trap center which admit the transfer ofa particle from the wings to the center at an infinitesimalenergy cost. Explicit disorder averaging is crucial, becausewe cannot expect in general the central region of the systemto be self-averaging, due to its relatively small size. In thefollowing, we consider quasiperiodic superlattices, whichadmit random relative spatial phases between the variousFourier components. In this case, the full disorder statisticsin the trap are sampled by considering different values of therelative phases.

If the shift in chemical potential associated with trapsqueezing is not bringing the system across a Bose-glass-to-SFtransition, the particles that are transferred from the wings tothe center will occupy localized quasiparticle states, meaningthat the density variation that their transfer causes will bewell localized in space. This implies that the new particleappearing in the center remains highly spread in momentumand that it only contributes a small amount to the k = 0 peak inthe momentum distribution. As a consequence, the coherence

023601-18


properties are only weakly altered by the particle transfer, andthe condensate compressibility remains small.

While a finite central compressibility κC is a solid cri-terion (and a prerequisite) for the identification of a Boseglass, it is a priori more difficult to fix a criterion on thecondensate compressibility which would distinguish a Boseglass from a weakly correlated SF. In fact, in both cases,the condensate compressibility is generally finite, given thata global increase of the density leads to a global changeof the momentum distribution, regardless of whether thesystem exhibits condensation. The only rigorous approach todiscriminate a compressible insulator from a condensate is tostudy the scaling of the condensate occupation n(k = 0) underincreasing particle number and decreasing trapping frequency,namely at fixed chemical potential; this will be the subject of afuture publication. Nonetheless, a semiquantitative criterioncan be formulated based on the width of the momentumdistribution. Indeed, the inverse of the FWHM of such apeak gives an estimate of the correlation length ξ in thesystem: In the following, we take ξ ≈ 2(FWHM)−1 (basedon the assumption that the momentum distribution n(k) can beapproximated by a Lorentzian with parameter ξ ). The systemis obviously in an insulating state if ξ is far below the widthof the atomic cloud. We can estimate such a width via theparticipation ratio:

PR = N2∑i〈ni〉2

. (34)

Hence, the system can be safely considered as an insulator ifξ � PR.

B. Three-color incommensurate superlattice

We now focus our attention on the 1d Bose-Hubbard ina two-color quasiperiodic potential, Eq. (17), realized viaa three-color incommensurate superlattice. In Ref. [36], wehave shown that this potential, despite being quasirandom,appears to support a Bose-glass phase over a continuousregion of the phase diagram. This is to be contrasted withthe one-color quasiperiodic potential, realized via two-colorsuperlattices [22], which features Bose-glass behavior as wellas incompressible incommensurate-band-insulator behavior[36,67], tightly alternating upon varying the chemical poten-tial. In other words, while the one-color potential still retainsfeatures of its quasiperiodic nature, the two-color potentialseems to mimic very closely the behavior of a purely randompotential.

In the following, we focus on the regime in which thesystem immersed in a two-color potential exhibits an extendedBose-glass phase, namely for potential strength V2 = U ,which removes completely the MI from the phase diagram.In particular, we consider the case of a strong potential,V2 = 20J , which establishes a Bose glass for all the values ofthe chemical potential we have explored. The trapped versionof the system is investigated in the case of N = 100 bosonsand under averaging over the spatial phases φ, φ′ of theFourier components of the potential, Eq. (17). In particular,we investigate the behavior of the phase-averaged quantities:the central density 〈nC〉φ , the momentum distribution 〈n(k)〉φand its FWHM, and the participation ratio 〈PR〉φ . We plotthese quantities as a function of the phase-averaged chemical

0 0.5 1 1.5 2 2.5 3µ / U

1

1.5

2

2.5

3

<nC

> φ

bulk system (L=300)C = 10 sites 16 sites 20 sites

FIG. 22. (Color online) Phase-averaged central density of asystem of N = 100 one-dimensional bosons trapped in a parabolicpotential plus a three-color superlattice, Eq. (17) (see text for theparameters). The results for the trapped system are compared withthose of a bulk system with L = 300 sites.

potential, which in Sec. III has been shown to satisfy asimple scaling relationship as a function of the experimentalparameters Vt and N . The data on the trapped system arecompared with those of the bulk system, realized by a chain ofL = 300 sites for a fixed value of the potential phases φ, φ′; inthis case, disorder averaging is provided by the extended size ofthe system, which is fully self-averaging. (We have explicitlychecked this aspect by comparison with an even longer chain,L = 500.)

Figure 22 shows the evolution of the central density inthe trapped system for a central region C of varying sizecompared with the bulk behavior. The bulk behavior shows acontinuous variation of the density with the chemical potential,signaling a finite compressibility over the whole range we haveconsidered. We observe that the phase-averaged central densityclosely follows the bulk behavior, and, in the case of N =100 bosons, the dependence of the central density on the sizeof the C region is very weak up to densities of n <∼ 2, signalingthat a faithful measurement of the n(µ) curve can be achievedat different imaging resolutions. To faithfully reproduce thebehavior at higher densities, it is necessary to have a goodresolution of around ten sites, or, alternatively, to increasethe number N of trapped particles and reduce the trappingpotential (thereby keeping the product N2/3V

1/3t constant) in

order to increase the quasi-homogeneous region in the trapcenter.

The insulating nature of the phase stabilized in the trapcenter is revealed by the momentum distribution. In Fig. 23, weshow the evolution of the peak in the momentum distributionunder trap squeezing. The comparison with the data of thebulk system shows that the trapped system reproduces themain features of the bulk behavior, including some subtleones (for instance, the anomaly for µ/U ≈ 2); this furtherconfirms that the momentum distribution is dominated by thebehavior in the trap center, which best approximates the bulkbehavior. In the case of the bulk system, the inspection of theSF density [36] allows us to conclude that the system is inan insulating state for all explored chemical potentials, as the

023601-19


0 0.5 1 1.5 2 2.5 3µ / U

0

1

2

3

4

5bulk system (L=300): Σ

ij <b

i

+b

i> / (6L)

trapped system: Σij<b

i

+b

j>φ / N

FIG. 23. (Color online) The k = 0 peak in the momentumdistribution for a system of N = 100 one-dimensional bosons trappedin a parabolic potential plus a three-color superlattice (same as inFig. 22). The results are compared with those of a bulk system withL = 300 sites.

SF density turns out to be identically vanishing everywhere.This conclusion cannot be easily drawn on the basis of theobservation of the momentum distribution. In fact, at variancewith the MI state, the condensate compressibility κ0 appearsto be finite everywhere, although large excursions of thechemical potential (e.g., from 0.5 U to U ) only slightlychange the condensate population (by about 10% in thiscase). As already mentioned, the most dramatic signature ofthe insulating behavior would be revealed upon scaling thesystem size. Yet, in this case, the insulating behavior is sopronounced that it is already revealed at the level of the singlesystem size we considered. In fact, as shown in Fig. 24, thecorrelation length estimated via the inverse of the FWHM isconsiderably smaller than the system size estimated via theparticipation ratio. (It is more than an order of magnitude

0 0.5 1 1.5 2 2.5 3µ / U

100

101

102

<PR>φ / a

2 (FWHM)-1

/ a

FIG. 24. (Color online) Correlation length of a system of N =100 one-dimensional bosons trapped in a parabolic potential plusa three-color superlattice (same as in Fig. 22), estimated via theinverse FWHM of the momentum distribution. This quantity iscompared with the effective cloud size, given by the phase-averagedparticipation ratio.

smaller for µ/U <∼ 2.) Hence, the observation of the centraldensity and its evolution under trap squeezing, togetherwith the measurement of the momentum distribution, wouldexperimentally allow us to draw unambiguous conclusionsabout the Bose-glass nature of the phase realized in thissystem.

VIII. CENTRAL VERSUS GLOBAL COMPRESSIBILITY

Recent experiments have probed for the first time thecompressibility of a fermionic gas in an optical lattice viatrap squeezing and in situ imaging of the cloud width [31].The same technique is obviously applicable to bosons as well,and it has been proposed as a practical tool for the detectionof the Bose glass [69]. Probing the response of the globalwidth of the atomic cloud to trap squeezing has the obviousadvantage that the imaging resolution required in this case ismuch less than that demanded by the technique we propose.Here, we point out a few drawbacks, particularly associatedwith the detection of the Bose-glass phase. A discussion of thedrawbacks for the case of repulsive fermions can be found inRef. [51].

In Ref. [31], the cloud width is estimated via the cloudradius, namely the second moment of the density distribution〈ni〉 [70] (centered around the origin):

R =√

1

N

∑i

|r i |2〈ni〉. (35)

The response of the atomic cloud to the variation of the trapfrequency is then quantified in terms of the variation in thecloud radius R; given that N/Rd gives an estimate of theeffective average density of the cloud, one can define a globalcloud compressibility κg as the derivative of R−d with respectto the trapping potential,

κg = − 1

Rd+1

∂R

∂Vt

. (36)

This is a measure of the compressibility of an inhomogeneoustrapped system, which collects contributions from all itsvarious parts. In particular, no matter what state the system coreis in, the dilute halo around the core is always in a compressiblestate, so that the global compressibility is necessarily alwaysfinite, although it will be significantly higher or smallerdepending on whether the core is in a locally compressiblestate. Hence, the realization of a specific bulk phase in thesystem core does not result in a qualitatively different behaviorof the global compressibility but only a quantitatively differentone. This aspect makes the discrimination of the phasesrealized in the system core very hard, and even more sothe location of the phase boundaries. In particular, as wediscuss later, the identification of some phases might evenbe completely missed.

Figure 25 shows the global density N/R for 100 bosons de-scribed by the 1d Bose-Hubbard model with U = 20J and fora variable trapping potential, controlling the chemical potentialof the system. An alternative definition of the global averagedensity, which turns out to be much closer to the true averagedensity, is N/PR, based on the participation ratio, Eq. (34),which gives the effective volume occupied by the system. The

023601-20


0 0.5 1 1.5 2 2.5 3µ / U

1

1.5

2

2.5

3 N / 4R N / PR< n

C >, 10 sites

n=1

n=2

n=3

FIG. 25. (Color online) Central density (averaged over ten sites)of a system of N = 100 trapped bosons, described by the one-dimensional Bose-Hubbard model with U = 20J in a parabolic trap.This quantity is compared with the effective global density of theatomic cloud, estimated via the inverse of the cloud radius R and viathe inverse of the participation ratio.

participation ratio is very close to the size of the region overwhich the atomic density is nonzero, while the cloud radius R iscloser to the FWHM of the density profile. Despite the differentdefinitions, Fig. 25 shows that N/R and N/PR exhibit pre-cisely the same behavior, and they essentially differ by a mul-tiplicative factor. Hence, we use them interchangeably in thefollowing.

In Fig. 25, the global densities N/R and N/PR arecompared with the central density 〈nC〉 for a central regionof ten sites. As shown in Ref. [30] for the same parameterset, this latter quantity reproduces faithfully the density of thebulk system. For the chemical potential excursion consideredin this study, we observe three Mott plateaus, which are clearlymanifested in the central density and which also show up inthe global densities via a significant change of slope in thedependence on the chemical potential [71]. In this case, onecould argue that the global compressibility contains essentiallythe same information as the central one. At the same time, itis evident that important pieces of information are missing inthe global compressibility: Under the assumption that a loss inglobal compressibility is always equivalent to the appearanceof an incompressible phase in the system core, it is quite hardto locate the phase boundaries between core-compressible andcore-incompressible phases, as the changes in slope are quitesmooth (despite the commensurate-incommensurate transi-tions in the core being very sharp for what concerns the centralcompressibility, as we have already discussed in Sec. VI).Moreover, the global densities contain very little informationon the central density at which the core-incompressible phasesare realized. As a general remark, one could say that the globaldensity and global compressibility can give information onthe alternation of phases and their nature provided that onehas a prior knowledge of the topology of the phase diagramof the system and of the particular phases that are realizedthere.

A much harder (yet more general) situation is the one inwhich one does not know the succession of phases realizedin the system, although the set of the possible phases mightstill be known. This is a generic situation in which one would

0.5 1 1.5 2 2.5µ / U

1.5

2

2.5

3

3.5

< N / PR >φ< n

C >φ, 10 sites

n=2-α

n=1+α

n=3-α

FIG. 26. (Color online) Central density (averaged over tensites) of a system of N = 100 bosons trapped in a parabolicpotential plus a two-color incommensurate superlattice (see text).This quantity is averaged over different realizations of the relativespatial phase of the superlattice. Comparison is made with theglobal density, estimated via the phase-averaged inverse participationratio.

like an analog quantum simulator to work. As an example of acomplex phase diagram, we consider the case of the Bose-Hubbard model in an incommensurate one-color potential[36,67,68]. As already mentioned, this system exhibits fourbosonic phases: MI, SF, Bose glass, and incommensurateband insulator. Following Ref. [36], we consider a system ofN = 100 bosons in a strong incommensurate potential V2 =U = 20J with incommensuration parameter α = 830/1076under variable trapping potential. Figure 26 shows the phase-averaged global density of such a system 〈N/PR〉φ , comparedwith the phase-averaged central density 〈nC〉φ for a region often sites. As seen in Ref. [36], this latter quantity reproducesmost of the features of the density of a bulk system forsufficiently low values, n <∼ 2.5. In particular, a distinctivefeature is represented by plateaus at incommensurate densities,related to the potential parameter α and corresponding toincommensurate band-insulating phases. The plateau regionsare separated by compressible regions manifesting eitherBose-glass or SF behavior. (For such strong external potential,the MI is completely washed out, and no plateau appears atinteger densities.)

When inspecting the global density, one can still observea weak decrease of its slope as a function of the chemicalpotential corresponding to the incommensurate plateaus.Nonetheless, without prior knowledge of the bulk-phasediagram, it is very hard to understand the nature of thisless compressible behavior. First of all, it is a priori unclearwhether a local minimum in the compressibility should alwaysbe associated with the appearance of a strictly incompress-ible behavior in the core, and even working under thisassumption, the estimate of the phase boundaries is somewhatarbitrary.

Moreover, it is not possible in principle to decide whetherthe less compressible phases correspond to ordinary MIsor to some other form of incompressible insulating state,

023601-21


because one lacks definite information on the filling at thetrap center. Hence, we can conclude that the global densityand compressibility in general do not allow the reconstructionof the phase diagram of the Bose-Hubbard model in aquasiperiodic potential.

IX. CONCLUSIONS

In this article, we have shown how the control on thetrapping potential for bosons in optical lattices provides a verypractical experimental knob on the total chemical potentialof the system. Indeed, even in the strongly correlated caseof bosons described by the Bose-Hubbard model with orwithout an external superlattice potential, the global chemicalpotential at zero temperature turns out to be related to thetrapping potential (and to the other Hamiltonian parameters)via a simple scaling relation. This relation takes the form of amodified TF scaling relation, corrected by quantum effectsand valid in the case of large enough densities or weakenough interactions, in which lattice commensuration effectsare negligible. Corrections beyond this simple expressionare quantitatively captured at the level of a simple latticemean-field theory (GA), which agrees surprisingly well withquantum Monte Carlo results. The possibility of tuning thechemical potential via the trap gives access to the explo-ration of the phase diagram of Hamiltonian models in thegrand-canonical ensemble, even if cold-atom experiments aregenerally performed at a fixed number of particles.

If supplemented with an imaging technique able to measureselectively the average density nC in a finite region C aroundthe trap center, the a priori knowledge of the global chemicalpotential µ of the system in an optical lattice allows usto reconstruct the density versus µ curve of the modelimplemented in the experiment without the trap. In particular,this gives access to the bulk compressibility κ , estimatedvia the central compressibility κC = ∂nC/∂µ. We show thatcentral compressibility measurements tell apart very clearlythe MI from SF phases, and hence they allow us to fullyreconstruct the phase diagram of the Bose-Hubbard modelat zero temperature in the grand-canonical ensemble. Theaddition of an incommensurate superlattice [22] introducesBose-glass regions in the phase diagram, whose unambiguousdetection can be achieved via the joint measurement of thecentral compressibility and of the global coherence of thesystem, as well as its response to trap squeezing.

The broader perspective of a truly useful quantum simula-tion with cold atoms requires us to implement computationallymore challenging models than lattice bosons, for example, tosimulate lattice fermions. To do so with minimal “classicalinput,” one needs a strategy to know in advance all theparameters of the system without the necessity of an ab initioclassical simulation of the model of interest. While it is stillunclear how to gain a priori knowledge of the temperaturewithout extensive classical simulations [12,13], the case studyof the Bose-Hubbard model presented in this work suggeststhat mean-field variational ansatzes might be successfully usedto reconstruct a priori the chemical potential of stronglycorrelated system in its ground state, even when the samevariational ansatzes do not provide an accurate description ofthe microscopic behavior of the system.

ACKNOWLEDGMENTS

We thank L. Fallani for suggesting the imaging techniquedescribed in Sec. V and M. Rigol for stimulating suggestions.

APPENDIX A: JORDAN-WIGNER FERMIONIZATION ANDEXTENDED FERMIONIZATION

Hardcore-boson operators [bi,b†j ] = 0 (i �= j ), {bi,b

†i } = 1

can be mapped onto spinless fermions fi , f†i via the Jordan-

Wigner transformation [43],

b†i = f

†i

i−1∏k=1

e−iπf†k fk , bi =

i−1∏k=1

eiπf†k fkfi, (A1)

so that the Hamiltonian of infinitely repulsive bosons in anexternal potential Vi becomes that of noninteracting spinlessfermions:

H = −J∑

i

(fif†i+1 + H.c.) +

∑i

Vif†i fi, (A2)

for which we assume open boundary conditions.This approach has been recently extended to the case

of strongly repulsive 1d softcore bosons in a trap [55] forU/J > (U/J )c,n, where (U/J )c,n is the critical value for theSF-to-MI transition at the largest integer density n reproducedlocally in the trap. Under this assumption, the density profilein the trap takes a characteristic “wedding-cake” structure;namely, regions at densities in the interval [m − 1,m] (withinteger m) are separated from regions with densities in theinterval [m,m + 1] by a sizable Mott plateau at integerdensity m. If U/J is sufficiently large, density fluctuationsin the plateau can be neglected, which means that densityfluctuations in the [m − 1,m] region are essentially decoupledfrom those in the [m,m + 1] region and that the lattice sitesin each of these regions only experience density fluctuationsbetween the two extremal values of the average density. Thisfundamental restriction of the local Hilbert space to a set oftwo-level systems implies the possibility of fermionizing eachlayer of the cake individually (extended fermionization); thefermionization has to be supplemented with the informationthat particles in the mth layer have an effectively larger hoppingdue to bosonic enhancement J → mJ and that they feel anoverall potential energy U (m − 1) per site. The extended-fermionization Hamiltonian reads then

H =∑i,m

−mJ (fi,mf†i+1,m + H.c.)

+∑i,m

[U (m − 1) + Vi]f†i,mfi,m, (A3)

where fi,m, f†i,m are the spinless fermions associated with the

mth layer.

APPENDIX B: THOMAS-FERMI RESULTSFOR THE ATOM NUMBER DISTRIBUTION IN

LOW-DIMENSIONAL SYSTEMS

We consider a gas of weakly interacting bosons in athree-dimensional trap of frequencies (ωx,ωy,ωz) and in

023601-22


a standing-wave potential along 3 − d spatial dimensions,defining an array of d-dimensional trapping elements (tubesfor d = 1, pancakes for d = 2). In the weakly interacting case,we describe the system within GP theory in continuum spacefor d dimensions and on a lattice for the remaining 3 − d

ones. To calculate the density profile of the atoms, we makeuse of the TF approximation, which amounts to minimizingthe potential-energy part of the GP energy functional.

A. d = 1 (tubes)

In this case, the system is immersed in two standing wavesalong the x and y directions, with wavelength λx and λy

respectively, defining a 2d array of tubes. With dx = (dx,0),dy = (0,dy) being the vectors of the 2d array, where dα =λα/2 (α = x,y), we identify the position of the tubes asR(ix,iy) = ix dx + iy dy . Imagining that the atoms can onlyoccupy the lowest Wannier states of each tube in the x andy directions (wx and wy , respectively), we can discretize thespace in those directions, so that the GP wave function can bewritten as � = �(ix,iy ; z). The potential-energy part of theassociated GP functional reads:

E(pot)GP [�,�∗] =

∑ix ,iy

∫dz

[g(1d)

2|�(ix,iy ; z)|4

+(

εxi2x + εyi

2y + 1

2mω2

zz2 − µ(1d)

)

× |�(ix,iy ; z)|2], (B1)

where εα = mω2αd2

α/2, and

g(1d) = g

∫dxdy|wx(x)|4|wy(y)|4 (B2)

is the effective coupling constant for particles trapped in thelowest Wannier states in the x and y directions, related tothe bare coupling constant 4πh2as/m (as = s-wave scatteringlength). By taking a Gaussian approximation for the Wannierstates [72], one obtains

g(1d) ≈ gkxky

2π

(V0xV0y

ErxEry

)1/4

, (B3)

where kα = 2π/λα , V0α is the lattice depth in the α

direction, and Erα = h2k2α/(2m) is the associated recoil

energy.

The TF approximation gives for the number of atoms ineach tube the expression

N (1d)(ix,iy) =∫

dzµ(1d) − (

εxi2x + εyi

2y + 1

2mω2zz

2)

g(1d), (B4)

where the chemical potential µ(1d) is consistently given by

µ(1d) =[

15g(1d)dxdyN

8π

(mω2

2

)3/2]2/5

, (B5)

with ω = (ωxωyωz)1/3. By integrating in Eq. (B4) on the regionin which the integrand is positive, one finally obtains theexpression [73]

N (1d)(ix,iy) = 4√

2

3g(1d)√

mω2z

(µ(1d) − εxi

2x − εyi

2y

)3/2. (B6)

B. d = 2 (pancakes)

In this case, the system is immersed in a standing wavealong the z direction with wavelength λz, defining a 1d array ofpancakes. In analogy with what done in the previous section,we define a discretized GP wave function � = �(x,y; iz),whose associated potential energy is described by the GPfunctional

E(pot)GP [�,�∗] =

∑iz

∫dxdy

[g(2d)

2|�(x,y,iz)|4

+(

1

2mω2

xx2 + 1

2mω2

xy2 + εzi

2z − µ(2d)

)

× |�(x,y; iz)|2], (B7)

where we have introduced the effective coupling constant

g(2d) = g

∫dz|wz(z)|4 ≈ g

kz√2π

(V0z

Erz

)1/4

(B8)

with evident meaning of the symbols.By integrating the TF expression for the density within each

pancake, we obtain the atom number distribution

N (2d)(iz) = π

mg(1d)ωxωy

(µ(2d) − εzi

2z

)2(B9)

with chemical potential

µ(2d) =[

15g(2d)dzN

8π

(mω2

2

)3/2]2/5

. (B10)

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[65] The standard definition of the lattice momentum distributionwould involve a normalization to the volume Ld of thequantization box, but in the case of a trapped system, thequantization box is arbitrary. As an alternative, here we imaginethat the quantization box is fixed to the number of bosons N ,which is a convenient convention, given that this number is keptfixed in our study of a trapped system. In this case, the condensateoccupation n(k = 0) takes the meaning of the number of particleswith momenta in the box k � π/(aN ), where a is the latticespacing.

[66] Reference [49] has documented that local-density signaturesof the onset of the Mott phase in a trap show up at a higherU/J value than in the bulk for one- and two-dimensionalsystems and that this blue shift is much more pronounced forhigher trapping potentials Vt . A similar shift has been exper-imentally documented in K. Jimenez-Garcıa, R. L. Compton,Y.-J. Lin, W. D. Phillips, J. V. Porto, and I. B. Spielman,e-print arXiv:1003.1541, based on time-of-flight measurementsof bosons trapped in quasi-two-dimensional optical lattices.We find that these observations are fully consistent with ourobservations for the central compressibility (Fig. 16) and forthe time-of-flight observables (Fig. 19) of a one-dimensionalsystem and that they can be easily explained with the fact thatthe onset of an MI phase with a small gap � is hindered by thetrap inhomogeneity: Indeed, the trapping potential discards the

MI behavior beyond a length scale ri − r0 from the trap center,such that Vt (ri − r0)2 ≈ �.

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[70] In the case of Ref. [31], 〈ni〉 is actually the column-integrateddensity of a three-dimensional system, but this aspect is actuallyirrelevant for our discussion.

[71] We avoid extracting the central and global compressibil-ities by differentiating the related densities, because ofthe significant numerical noise associated with the deriva-tive. Nonetheless, the changes in slope in Fig. 25 areevident.

[72] W. Zwerger, J. Opt. B 5, S9 (2003).[73] A similar formula appears in B. Paredes, A. Widera, V. Murg,

O. Mandel, S. Folling, I. Cirac, G. V. Shlyapnikov, T. W. Hansch,and I. Bloch, Nature 429, 277 (2004). That formula matches withours for the case ωx = ωy = ωz if the factor of 5N/[2πN (0,0)]appearing there is changed into its inverse, 2πN (0,0)/(5N ). Webelieve our formula to be correct, as it satisfies the sum rule∑

ix ,iyN (1d)(ix,iy) = N , which is not satisfied in turn by the

formula of Paredes et al.

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