Bose-Hubbard model with occupation dependent parameters
Omjyoti Dutta, Andre Eckardt, Philipp Hauke, Boris Malomed, Maciej Lewenstein
aa r X i v : . [ c ond - m a t . qu a n t - g a s ] D ec Bose-Hubbard model with occupation dependentparameters
O. Dutta , A. Eckardt , P. Hauke , B. Malomed , M.Lewenstein , , ICFO-Institut de Ci`encies Fot`oniques, Mediterranean Technology Park, E-08860Castelldefels (Barcelona), Spain Department of Physical Electronics, School of Electrical Engineering, Faculty ofEngineering, Tel Aviv University, Tel Aviv 69978, Israel ICREA-Instituci`o Catalana de Recerca i Estudis Avan¸cats, Lluis Companys 23,E-08010 Barcelona, Spain Kavli Institute for Theoretical Physics, Kohn Hall, University Of California, SantaBarbara, California 93106-4030E-mail: [email protected]
Abstract.
We study the ground-state properties of ultracold bosons in an opticallattice in the regime of strong interactions. The system is described by a non-standard Bose-Hubbard model with both occupation-dependent tunneling and on-siteinteraction. We find that for sufficiently strong coupling the system features a phase-transition from a Mott insulator with one particle per site to a superfluid of spatiallyextended particle pairs living on top of the Mott background – instead of the usualtransition to a superfluid of single particles/holes. Increasing the interaction further, asuperfluid of particle pairs localized on a single site (rather than being extended) on topof the Mott background appears. This happens at the same interaction strength wherethe Mott-insulator phase with 2 particles per site is destroyed completely by particle-hole fluctuations for arbitrarily small tunneling. In another regime, characterized byweak interaction, but high occupation numbers, we observe a dynamical instability inthe superfluid excitation spectrum. The new ground state is a superfluid, forming a2D slab, localized along one spatial direction that is spontaneously chosen. ose-Hubbard model with occupation dependent parameters
1. Introduction
Systems of ultracold atoms in optical lattices provide a unique playground for controlledrealizations of many-body physics [1, 2]. For sufficiently deep lattices, the kineticsis exhausted by tunneling processes, and an initially weak interparticle interactioneventually becomes important with respect to the kinetics, when the lattice is rampedup. A consequence of this competition is the quantum phase transition from a superfluidof delocalized bosons to a Mott insulator where the particles are localized at minimaof the lattice by a repulsive contact interaction [3]. This effect has been observed inseminal experiments with ultracold rubidium atoms in a cubic lattice [4]. It is describedquantitatively by means of the simple Bose-Hubbard model [3, 5], whose parameters arethe interaction energy U for each pair of particles occupying the same lattice site, andthe matrix element J for tunneling between neighboring sites. Intriguing Hubbard-typephysics can also be observed if the above scenario is extended to fermions, mixturesof several particle species, exotic lattice geometries, or long-ranged dipolar interaction[1, 2, 6].In this paper, we consider a different type of extension of the bosonic Hubbardmodel, becoming relevant when the interaction between the particles is enhanced, e.g,by means of a Feshbach resonance. As long as the interaction is weak compared to thelattice potential, a system of ultracold atoms can be described, to a good approximation,in terms of the lowest-band single-particle Bloch or Wannier states, the latter beinglocalized at the minima of the lattice [7]. Under these conditions, the Hubbardinteraction U and tunneling parameter J are given by respective matrix elementswith respect to the single-particle Wannier states. This approximation correspondsto degenerate perturbation theory up to first order with respect to the interaction,with only the intraband coupling induced by the interaction taken into account.However, if the interaction is stronger, higher-order corrections start playing a role.One may still describe the system in terms of lattice-site occupation numbers n j , butthe occupied Wannier-like orbitals will have admixtures from higher bands, dependingon the occupation. The most significant effect of the repulsive interaction will be abroadening of the Wannier-like orbitals with increasing occupation, effectively enhancing J and decreasing U . In terms of the Hubbard description, we take this into account byreplacing J and U by functions J ˆ n i , ˆ n j and U ˆ n i of the number operators ˆ n i . Quantitativeconsequences of this kind of modification to the plain bosonic Hubbard model have beenstudied by several authors at a theoretical level [8, 9, 10]. Considering an interaction-induced modification of the Wannier functions, also additional Mott-insulator phaseshave been predicted [11, 12]. In Ref. [13], the effect of the interaction-induced couplingto the first excited band on the Mott transition was considered. Re-entrant behaviourin the superfluid-Mott transition has also been predicted due to the interaction-inducedmodification of Hubbard parameters [13, 14]. The effect of interaction on the tunnelingdynamics in one-dimensional double-well and triple-well potentials have been studiedin Refs. [15, 16] where the authors found enhanced correlated pair tunneling near the ose-Hubbard model with occupation dependent parameters U and J [22].In the present work, we show that new quantum phases can arise in Hubbardmodels with number-dependent parameters. After writing down the effective single-bandHamiltonian including the effect of the site occupation, we find that for strong enoughinteraction (characterized by the s -wave scattering length a s ), there is a transition froma Mott state with one particle localized at each lattice site to a superfluid of pairs extended over neighboring sites, rather than to a superfluid of single atoms. Thisfeature is novel, considering the fact that the extended pairs emerge in the single-species repulsive bosonic system without the presence of any long-range interaction.For even higher interaction strengths, the n = 1 Mott state becomes unstable towardsa superfluid of pairs that are localized on single sites. Moreover, the n = 2 Mott statebecomes unstable towards pair fluctuations already for very low tunneling amplitudes.Finally, we consider the regime where interaction effects are important not because oflarge scattering lengths, but rather because of large site-occupations numbers. In thislimit, starting from the Bogoliubov approach to the homogeneous system, we find aphonon instability at a critical filling fraction. Above that fraction, the new groundstate is a Bose condensate with the particle density being localized along one spatialdirection that is chosen spontaneously.This paper is arranged in the following way: In section 2, we introduce theoccupation-dependent Bose-Hubbard model. In section 3, we start discussing theproperties of this model. Namely, we study the instability of the Mott-insulator phasewith respect to simple particle and hole excitations, leading to the usual single-particlesuperfluidity. In section 4, we then investigate the instability of the Mott phasewith respect to the excitation of bond-centered pairs of particles being extended overneighboring lattice sites. We show that this mechanism will eventually become relevantwhen the s -wave scattering length is increased, and that one finds a phase transitionto a superfluid of extended pairs. In section 5, proceeding to even stronger interaction,the instability of the Mott phase towards a superfluid of site-centered pairs is discussed.In this regime, moreover, the Mott insulator at a filling of two particles per site candisappear completely. Finally, in section 6, we focus on the limit where interaction-induced orbital effects play an important role because of large filling. We find that, withthe increasing superfluid density, the condensate may become dynamically unstable. ose-Hubbard model with occupation dependent parameters
2. The Bose-Hubbard model
The Hamiltonian in the presence of a periodic potential with lattice constant a , givenby V per ( ~r ) = V [sin ( πx/a ) + sin ( πy/a ) + sin ( πz/a )], reads H = Z d r ˆ ψ † ( ~r ) (cid:20) − ~ m ∇ + V per ( ~r ) + g | ˆ ψ ( ~r ) | (cid:21) ˆ ψ ( ~r ) , (1)with bosonic field operators ˆ ψ , mass m , and interaction strength g = 4 π ~ a s /m , where a s is the s -wave scattering length. To derive a Hubbard-type description, the fieldoperators ˆ ψ ( ~r ) are expanded in terms of Wannier-like orbitals φ i ( ~r, ˆ n i ) = φ ( ~r − ~R i , ˆ n i )localized at the lattice minima ~R i , namely ˆ ψ ( ~r ) = P i ˆ b i φ ( ~r − ~R i ; ˆ n i ) with bosonicannihilation and number operators ˆ b i and ˆ n i = ˆ b † i ˆ b i . Note that the “wave function” φ i depends on the number operator ˆ n i in order to take into account interaction-inducedoccupation-dependent broadening. Keeping only on-site interaction, we arrive at theeffective single-band Hamiltonian H = − X ij J ˆ n i , ˆ n j ˆ b † i b j + 12 X i U ˆ n i ˆ n i (ˆ n i − − X µ ˆ n i , (2)where J ˆ n i , ˆ n j = − Z d rφ ( ~r − ~R i ; ˆ n i ) h − ~ m ∇ + V per ( ~r ) i φ ( ~r − ~R j ; ˆ n j + 1) ,U ˆ n i = g Z d rφ ( ~r − R i ; ˆ n i ) φ ( ~r − ~R i ; ˆ n i − , (3)and we have introduced the chemical potential µ to control the particle number. Wewould like to mention that in the presence of an optical lattice for high interactions thepseudo-potential form of contact interaction can still be used, when a modified scatteringlength which is different from the bare scattering length is applied [23, 24, 25, 26].In order to estimate the occupation number dependence in a mean-field way,we make a Gaussian ansatz for the Wannier-like wave functions, φ ( ~r − ~R i ; n i ) =exp( − ( ~r − ~R ) /d ( n i )), where the width d ( n i ) is a variational parameter depending onthe particle number n i , and minimize the Gross-Pitaevskii energy functional. The ideato use the width of the Wannier function as a variational parameter has also been usedin Refs. [27, 28, 29]. Taking into account the full lattice potential (i.e., not employinga quadratic approximation for the lattice minima), for a given n i this leads to (cid:20) d ( n i ) d (cid:21) exp (cid:20) − π d ( n i ) a (cid:21) = d ( n i ) d + √ π (cid:20) V E R (cid:21) / a s a ( n i − . (4)We have introduced d /a = h V E R i − / /π for the width of φ in the limit V ≫ E R , where E R = π ~ / ma denotes the recoil energy. Note that Eq. (4) has a solution only aslong as p V /E R ≫ d ( n i ) /d . Using the variational result, the tunneling parameterbetween two adjacent sites can be approximated by J n i ,n j E R ≈ (cid:18) π − (cid:19) V E R exp (cid:20) − a d ( n i + 1) + d ( n j )) (cid:21) . (5) ose-Hubbard model with occupation dependent parameters U n i E R = √ π (cid:18) V E R (cid:19) / (cid:20) d ( d ( n i ) + d ( n i − (cid:21) / a s a . (6)The single-particle tunneling term arising from the non-on-site contributions ofthe quartic interaction term in Eq. (1) is exponentially smaller than J ( n i , n j ) byapproximately a factor of exp( − π p V /E R / a s /a . Similarly the pair tunneling termis smaller than J ( n i , n j ) by approximately a factor of exp( − π p V /E R / a s /a . Sincewe are in the limit of V /E R ≫
1, these terms are neglected in Eq. (2).
3. Insulator to single-particle superfluid transition
Having written down a suitable model Hamiltonian describing the regime of stronginteraction, we now study the transition from the Mott insulator having on average n particles per site to a superfluid of single particles/holes.For this purpose, we use a product ansatz Q i | Φ i i for the many-body state, withthe variational coherent spin-representation state [30, 31], | Φ i i = cos θ | n i i + sin θ sin ψ | n + 1 i i + sin θ cos ψ | n − i i (7)at each site i , with occupation number basis states | n i i i . Here we only take intoaccount states with one additional particle or hole, which in the Mott phase and closeto the transition to the superfluid, where particle fluctuations are small, is sufficient.Accordingly, the variational mean-field energy is given by E ss N = − zH J θ + (cid:20) H U µ cos 2 ψ (cid:21) sin θ, (8)where H J = ( n + n ) J n,n sin 2 ψ/
2+ ( n + 1) J n +1 ,n sin ψ + nJ n,n − cos ψ,H U = n ( n − U n cos θ + n ( n + 1) U n +1 sin θ sin ψ + ( n − n − U n − sin θ cos ψ. (9)Minimizing the energy determines θ and ψ . While θ = 0 corresponds to anincompressible Mott-insulator state with an integer number of particles n per site (foundwithin a finite interval of the chemical potential µ ), the superfluid state is characterizedby θ = 0 with order parameter h b i i ∼ sin 2 θ . In the superfluid phase, the average particlenumber per site is characterized by ψ depending smoothly on the chemical potential. For ose-Hubbard model with occupation dependent parameters ψ ≪ π/
4, the transition to the superfluid occurs mainly via the creation of holes, whilefor ψ near π/ U n +1 n ( n + 1) / − µ , is overcome by the reductionin energy due to tunneling of that particle, which is on the order of z ( n + 1) J n +1 ,n , withcoordination number z = 6 for the cubic lattice. Thus, when E ss minimizes for non-zero θ , the Mott state becomes unstable with respect to single particle and hole excitations.For interaction strength a s /a = 0 .
15 and n = 1 , this happens at the black lines (solidor dotted) in the plane spanned by µ/V and J , /V in Fig. 1. Figure 1.
Mott-insulator-to-superfluid phase transition for a s /a = 0 .
15. Inside theregion marked by the black solid line and the blue dashed line, the system is a Mott-insulator with n = 1 particles per site. Leaving this region by crossing the black solidline, a simple superfluid of single particles (or, equivalently, holes) is formed (SF). Incontrast, crossing the blue dashed line one arrives at a superfluid phase of extended(bond-centered) pairs (ePSF). In technical terms of our variational approaches: outsidethe black solid and dotted line minimizing the energy (8) gives θ = 0, while on ther.h.s. of the blue dashed line θ e = 0 is obtained from minimizing expression (12).
4. Superfluidity of extended (bond-centered) pairs
So far, we have described the usual scenario of the Mott phase becoming instable withrespect to particle and hole delocalization, as it is also found for non-number-dependentHubbard coupling J and U . However, we will now show that—as a consequence ofoccupation dependent hopping and on-site interaction—the Mott insulator with n = 1can become unstable with respect to the creation of pairs of particles already beforethe creation of single particles becomes favorable. Consider a pair excitation with oneadditional particle at site i and another one at the neighboring site j , corresponding to ose-Hubbard model with occupation dependent parameters | P h ij i i ≡ ˆ b † i ˆ b † j |{ n i = 1 }i . Such a bond-centered or extended pair excitation at h ij i can tunnel coherently to a neighboring bond, say h ik i , with k = j being anotherneighbor of i . Generally, bonds are considered neighbors if they share a commonsite. Such a pair tunneling processes occurs in second order with respect to single-particle tunneling via the virtual site-centered pair state | P i i ≡ √ ˆ b † i ˆ b † i |{ n i = 1 }i ,which has larger energy. According to second-order degenerate perturbation theory,the amplitude of the pair tunneling process is given by J eff = 6 J , / (3 U − U ). Onthe same footing, perturbation theory gives the binding energy of − J eff for the bond-centered pair due to number fluctuations within the pair. For a cubic lattice of sites, thebond-centered pair excitations live on an exotic lattice of coordination number z ′ = 10,being a generalization of the two-dimensional checkerboard lattice (see the rightmostdrawing in Fig. 2) to three dimensions. This allows the pair to reduce its energy by10 J eff when delocalizing. In contrast, two additional particles, not forming a pair, canreduce their energy by 2 × × J , when delocalizing on the cubic lattice of sites(coordination number 6). Thus, according to perturbation theory, the formation of abond-centered pair is favorable if − (10+2) J eff > J , . For certain scattering lengths a s ,this condition can be fulfilled, since the Wannier-broadening with increasing scatteringlengths leads to an increase of both J , /J , and U /U . In such a situation, the Mott-insulator state becomes unstable with respect to the creation of bond-centered pairsrather than with respect to the creation of single-particle excitations. This happenswhen the delocalization energy − J eff overcomes the energy 2( U − µ ) − J eff neededto create a pair excitation. It is interesting to note that an equivalent scenario does nothappen for hole excitations, since hole excitations decrease the occupation number andwith that the tunneling amplitudes. Figure 2.
Color online: The left hand side shows a square lattice of sites (blue squares)connected by bonds (black lines). The lattice of the bonds of the square lattice, wherebonds sharing a site are connected, is given by the checkerboard lattice shown on theright hand side. If a bound pair of two indistinguishable particles can either occupya site or a bond of the cubic lattice (the latter means that the two particles occupyneighboring sites) and if the pair can move (by single-particle tunneling) from a siteto a neighboring bond and vice versa, then the pairs move on the lattice shown in thecenter plot. Sites and bonds are denoted by blue squares and black bullets, respectively.Extending all the considerations shown in this figure to the case of a three-dimensionalcubic lattice of sites is straightforward. ose-Hubbard model with occupation dependent parameters n = 1 Mott-insulator phase within mean-fieldapproximation, we construct a model for the excited bond-centered pair excitations.When the number of pairs is small compared to the number of sites, the Hamiltonianfor the pairs living on top of a Mott state with one particle per site can be written as H pair = − J eff X h LL ′ i ˆ p † L ˆ p L ′ + 2( U − µ − J eff ) X L ˆ n pL . (10)Here, L = h i, j i labels the bonds of the cubic lattice and h LL ′ i denotes pairs of nearestneighbors of these bonds as they are described by the three-dimensional checkerboardlattice (cf. Fig. 2). Moreover, we have defined the bosonic creation and destructionoperators for bond-centered pair-excitations ˆ p † L and ˆ p L , with number operator ˆ n L =ˆ p † L ˆ p L . As a consequence of the diluteness assumption, we have neglected the interactionbetween pairs, arising if pairs occupy neighboring bonds. Since the transition to a pair-superfluid will happen with the creation of a single pair, this approximation will notinfluence the phase boundary. The energy of a condensate of bond-centered pairs cannow be estimated in a similar fashion as before by making a product ansatz Q L | Φ p i L of coherent states being a superposition of zero and one pair at each bond, | Φ p i L = cos θ e | i L + sin θ e | i L . (11)The order parameter of the pair condensate is defined by h ˆ p L i = sin(2 θ e ). Accordingto this ansatz, the variational mean-field energy per site can be written as E ep N = − z ′ J eff θ e + 2( U − J eff − µ ) sin θ e (12)where z ′ = 10 is the coordination number of the three-dimensional checkerboardlattice. The mean-field approach gives the same phase boundary for the appearanceof a pair condensate with finite order parameter h ˆ p L i as the perturbation theoreticalconsiderations of the previous paragraph. The equivalence of both approaches isgenerally given for an ansatz like (11) which includes only two states per site.In Figure 1 we plot the results of minimizing E ss , E ep with respect to θ, θ e for a s /a = 0 .
15. The stable Mott region with respect to single particle-hole excitationis given by the interior of the black solid and dotted line characterized by θ = 0.On the right hand side of the blue dashed line in Fig. 1 one finds a region wheremin[ E ep ] < min[ E ss ] with θ e = 0. Thus, here the system is characterized by h p L i 6 = 0and h b i i = 0, i.e., the state is a superfluid of extended pairs (ePSF).Condensates of extended pairs have also been proposed in the context of dimermodels of reduced dimensions, describing frustrated magnets like SrCu (BO ) [32]. Byapproximating triplet excitations as hard-core bosons, the authors of Ref. [33] argue thatfor correlated hopping these bosons can condense in pairs. Such pairing processes alsobear resemblance to molecular condensation due to Feshbach resonances in an opticallattice [34].We would like to point out that triple, quadruple or higher order excitations donot play a dominant role. The effective tunneling matrix element of such excitationswill be very small since it appears in third or higher order perturbation theory only. ose-Hubbard model with occupation dependent parameters n = 1 phase, triple and higher excitations cannot lower their energyefficiently by delocalization. We can, thus, exclude a superfluid of triples or higher orderobjects. However, there is another possible and competitive scenario we would like tomention. Instead of exciting a triple or quadruple, one can create a huge cluster of extraparticles, i.e., a big spatial domain with doubly occupied sites. In this case, within eachcluster, the energy of the additional particles (on top of the n = 1 Mott background)is not lowered by delocalization, but rather by the attractive interaction between themas it appears in second order perturbation theory. In the bulk of such a cluster, thisgives a binding energy of − J eff per extra particle. In comparison, in the pair superfluideach particle can lower its energy by J eff because of binding and further by another5 J eff because of delocalization (i.e., Bose condensation). Accordingly, in leading order asuperfluid of bond-centered pairs on top of the n = 1 Mott insulator is equally favorableas a phase separated state with spatial domains hosting either a Mott insulator of filling n = 1 or n = 2. As a consequence, we cannot reliably exclude phase separation bymeans of simple variational arguments.Before moving on, let us briefly discuss another issue: In this article, we are workingin a situation with the chemical potential fixed rather than the particle number. Thisapproach is actually quite suitable for the description of experiments with ultracoldatoms, provided the atoms are trapped by a sufficiently shallow potential. In sucha situation, the local density approximation applies and different regions in the trapcorrespond to different values of the chemical potential. However, if the trap is toosteep for the local density approximation to be valid, it might introduce also newphysics. Consider the following example: The phase separated state described in thepreceding paragraph might not be favored in the homogeneous system. But, becauseit is energetically very close to the pair superfluid, it can be favored already when aslight potential difference is introduced, helping to form n = 2 Mott domains in theregion of slightly lower potential energy. Such a scenario can spoil the local densityapproximation already for a very weak trapping potential.
5. Superfluidity of local (site-centered) pairs
Now, let us consider a regime that can be achieved if the interaction strength a s /a isincreased further. Considering again the n = 1 Mott insulator, for increasing interactiona site-centered pair excitation, described by | P i i , eventually becomes more favorablethan the bond-centered excitations described by | P h ij i i . This happens when the ratio U /U is reduced so much that the potential energy 3 U needed to create a pair ofparticles on the same site equals the potential energy 2 U required to create a pairof particles on neighboring sites. Such a situation is possible as can be derived fromEq. (4). In the limit of large V ≫ E R and a s /a we can write d ( n ) /d ≈ ( gn i ) / resulting in 3 U − U ≈ − . U . If | U − U | becomes comparable to or smallerthan J , , a bond centered pair excitation | P h ij i i can transform to a site-centered pairexcitation | P i i by a single-particle tunneling process described by the matrix element ose-Hubbard model with occupation dependent parameters J pair = √ J . In this regime, the pairs occupy the lattice given by both the sites andthe bonds of the cubic lattice (see Fig. 2, center). By delocalizing on this lattice, a paircan reduce its kinetic energy by 12 J pair . As long as this energy is bigger than the kineticenergy reduction 24 J , which two non-paired particles can achieve by delocalization,the pair is stable towards breaking; this is the case for J , > p / J , . Thus, thebinding mechanism of the pair is based solely on the delocalization of its center of mass.At | U − U | ≈ a s /a ≈ .
21 when V /E R ≈ n = 1 Mottinsulator becomes unstable with respect to pair creation when 12 J pair exceeds 3 U − µ .It is fascinating to observe the emergence of exotic lattice geometries as illustrated inFig. 2 as a consequence of pair creation.If the scattering length is increased further, such that 2 U − U ≫ J , , site-centered pair excitations | P i i will be created rather than bond centered ones | P h ij i i .The site-centered pair excitations can then tunnel from site to site coherently via theoccupation of a virtual bond-centered pair excitation. The corresponding tunnelingmatrix element reads J ′ eff = 6 J , U − U = − J eff . Moreover, the pair has a binding energy of6 J ′ eff (stemming from a small perturbative admixture of the 6 neighboring bond-centeredpair states). Therefore, a site-centered pair is more favorable than two single-particleexcitations if 3 U − J ′ eff < U − J , ). If this condition is fulfilled, the Mott insulatorbecomes rather unstable towards the creation of site-centered pair excitations than tothe creation of single particles. The instability occurs when 12 J ′ eff reaches 3 U − µ .As before, a mean-field calculation leads to the same phase boundary. We plot theboundary of the n = 1 Mott phase for a s /a = 0 . U ≫ J , metastable repulsively bound pairs of ultracoldbosons have been observed in optical lattices [35, 36]. Also, two-species mixtures ofbosons with inter-species attraction trapped in an optical lattice have been shown togive rise to superfluidity of pairs [37]. In the context of dipolar atoms in a two-legladder, when no tunneling is present between the two legs, pair superfluidity arises dueto attraction between the dipolar atoms between the two legs of the ladder [38, 39].Also, using a state-dependent optical lattice potential, it is possible to create correlatedtunneling of on-site pairs, which in turn gives rise to superfluidity of local pairs [40, 41].In our present study, we find that such local pairing can emerge due to the strongoccupation-dependence of tunneling and on-site interaction.After having studied the boundaries of the Mott-insulator phase with one particleper site, let us have a look at the n = 2 Mott state. In the limit of vanishing tunneling,a Mott state with two particles localized at each site is favorable for U < µ < U − U .The upper border of this interval is given by the potential energy difference of havingthree and two particles at a site. This difference can, in fact, become lower thanthe potential energy difference U between two and one particle per site marking thelower border. This is the case if 3 U − U <
0; then the n = 2 Mott-insulatorphase is never stable with respect to the creation of particle-hole pairs, irrespective ose-Hubbard model with occupation dependent parameters Figure 3.
Mott-insulator-to-superfluid phase transition for a s /a = 0 .
3. Inside theregion enclosed by the black solid and the blue dashed line, the system is a Mott-insulator with n = 1 particles per site. Crossing the dashed blue line, one enters asuperfluid of local, site-centered pairs (PSF). Leaving the Mott-phase by crossing theblack solid line a superfluid of single particles (or, equivalently, holes) is found. Blackdashed line defined as in Fig. 1. of the tunneling strength; it ceases to exist. The disappearance of the n = 2 Mottinsulator coincides with site-centered pair excitations becoming more favorable thanbond-centered ones in the limit of vanishing tunneling. Note that the Mott-insulatorphases with higher filling, n ≥
3, do not disappear for large interaction a s /a withinthe Gaussian approximation. The reason why these phases do not share the fate of the n = 2 Mott insulator is that the broadening on the Wannier-like site-wave functions φ i in response to adding one particle to that site becomes less pronounced with increasingoccupation: U /U ≥ U /U ≥ U /U ≥ · · · . However, one should have in mind that forstrong interaction, sites occupied by three and more particles suffer strong dissipationdue to three-body collisions [42, 43].One might ask about the nature of the system’s ground state at fixed filling n = 2and for 3 U − U <
0, when there is no n = 2 Mott phase. At vanishing tunneling, theground state is highly degenerate consisting of all Fock-states having occupation n i = 1on half of the sites and occupation n i = 3 on the others. Alternatively, one might saythat on top of a n = 1 Mott insulator, half of the sites are occupied by additionalsite-centered pairs. For small but finite hopping this degeneracy will be lifted. Onecan think of three possible scenarios: (i) The pairs gather in one region in space; thiscorresponds to a phase segregation between the n = 1 and the n = 3 Mott phases.(ii) The pairs delocalize to form a superfluid. (iii) The pairs form a checkerboard-typeinsulator avoiding pairs on neighboring sites. In order to decide this question, we write ose-Hubbard model with occupation dependent parameters H pair = − J ′ eff X h i,j i c † i c j − X i (2 µ − J ′ eff ) n ci + ( J ′ eff − ∆) X h ij i n ci n cj (13)with bosonic pair annihilation and creation operators ˆ c i , ˆ c † i , and where we assume ahard-core constraint (ˆ c † i ) = 0. The nearest-neighbor repulsion present in the last term,with ∆ = 2 J , U + U − U , stems from super-exchange processes between neighboring pairs.This model can be mapped to a spin-1/2 XXZ model with the first term correspondingto the XX coupling and the last one to the Z-coupling. Since ( J ′ eff − ∆) ≤ J ′ eff is alwaystrue, the system will neither form the checkerboard pattern (iii) (corresponding to anantiferromagnetic state for the XXZ-magnet) nor show phase segregation (i) [44]. Thesystem forms a superfluid of site centered pairs (ii).
6. Weakly interacting limit
Finally, we investigate the limit where interaction effects are important not because ofa large scattering length but because of large site occupation, i.e., a s /a ≪
1, but themean number of particles per site n ≫
1. We assume small on-site number fluctuations δn ≪ n , i.e., p U n / ( n J n ) ≪
1. In this limit, we can write the modified HubbardHamiltonian as H = − J n X ij ˆ b † i [1 + α ( δ ˆ n i + δ ˆ n j )] b j + U n X i ˆ n i (ˆ n i − β − β (ˆ n i − − X µ ˆ n i , (14)where β = 35 r π (cid:20) V E R (cid:21) / a s a , (15) α = π / √ (cid:20) V E R (cid:21) / a s a , (16) J n V = (cid:18) π − (cid:19) exp " − π r V E R " − √ π (cid:20) V E R (cid:21) / a s a n , (17)and δ ˆ n i,j = ˆ n i − n . Here, we would like to point out the similarity of Hamiltonian (14)to the Quantum Ablowitz-Ladik (AL) model for q -deformed bosons [45], given by H AL = − X i [ B † i B i +1 + B † i +1 B i + 12 γ ln(1 − QB † i B i )] , (18)where [ B i , B † i ] = exp[ − γN i ], and Q = 1 − exp[ − γ ]. In the limit of γ → γN i ≪
1, Eq. (18) reduces to the occupation-dependent modified Hubbard modelEq. (14) with α = γ and U n = 0. It is found that in one and higher dimension the ALmodel contains localized solutions [46, 47]. To investigate this possibility, we first solveEq. (14) in the superfluid limit, where the order parameter reads h b i i = √ n . To look ose-Hubbard model with occupation dependent parameters b i = P i b k exp( − i~k.~r i ), ǫ k = 4 P i =1 , , sin k i a , and γ k = 4 P i =1 , , cos k i a . Neglecting correlations arising from the three-body interactionterm in Eq. (14), one arrives at the Hamiltonian H mod = − n U n X k J n ǫ k b † k b k (19)+ X k (cid:20) n U n β − β ( n − − αJ n n γ k (cid:21) × (cid:16) b † k b k + b † k b †− k + b k b − k (cid:17) . (20)It can be diagonalized via a Bogoliubov transformation, and the excitation spectrumΩ k of the superfluid is found to be given byΩ k = J n ǫ k + 2 U n n (cid:18) β − β ( n − − α J n U n γ k (cid:19) ǫ k . (21)In a cubic lattice, as k → k /J n U = c | k | a , where c is the phonon velocitygiven by c = s (1 + β − β ( n − − α J n U n γ . (22)In Fig. 4, we plot the phonon velocity c as a function of the filling fraction n for a s /a = 0 .
01. We find that initially, for increasing n , the phonon velocity increases.But for higher n the phonon velocity starts decreasing due to the attractive effect ofthe occupation dependent tunneling term, until the phonon velocity becomes imaginaryfor a critical n . This results in a dynamical instability of the superfluid when we arewithin the limit a s a n <
1. This instability occurs due to the attractive effect of theoccupation dependent tunneling, which can overcome the decreased repulsive on-siteinteraction depending on the number of particles per site n . To understand the effectof this instability, we first make a transition from the discrete Hubbard model to acontinuous model applicable for ka ≪ φ ( r ), H cont = − Z d rφ ∗ ( r ) ∇ φ ( r ) + U Z V eff ( r − r ′ ) | φ ( r ) | | φ ( r ′ ) | . (23)Here, the distance is expressed with respect to the lattice constant a , and theeffective interaction potential is given by V eff ( r − r ′ ) = F − [1 + β − β ( n − − α J n U n γ k ], where F − stands for the inverse Fourier transformation. Using a Gaussianansatz along one direction, say x , and uniform in the other directions, φ ( r ) =1 /π / d / s exp( − x / d s ), the energy functional for the self-trapped state reads E sol =1 /d s + U n J n √ π (cid:16) β − β ( n − − α J n U (5 + exp( − /d s )) (cid:17) /d s . When n exceeds acritical density, E sol is minimized for a finite d s ≫
1. Thus, the homogeneous superfluidbecomes dynamically unstable towards a state which is localized only in one direction,forming a 2D slab. ose-Hubbard model with occupation dependent parameters Figure 4.
Phonon velocity c as a function of the superfluid occupation number n . We find that after a critical occupation number, the phonon velocity becomesimaginary, denoting a dynamical instability. The fixed parameters are a s /a = 0 . V /E r = 10.
7. Conclusion and Outlook
In this paper we have predicted various effects resulting from interaction-induced bandmixing in systems of ultracold bosonic atoms in optical lattice potentials. We havederived the modified bosonic Hubbard model (2) having occupation-number-dependentparameters. This model comprises an effective interaction-induced broadening of theWannier-like single-particle orbitals, and, thus, captures also the situation when thes-wave scattering length becomes comparable to the lattice spacing, a s /a →
1. Usingthis model, we find that for scattering lengths a s ∼ . a and lattice depths V ∼ E R ,the n = 1 Mott-insulator state can become unstable towards a superfluid which consistsof bond-centered pair excitations. This scenario is novel considering the fact that theextended pairs emerge due to the occupation dependence of both the tunneling strengthand the on-site interaction. For even higher interaction, the nature of the superfluidpair excitations (destroying the insulator) changes. The pairs can now occupy both thebonds on the lattice (i.e., two neighboring sites) or its sites; in that way an exotic latticegeometry as shown in the center plot of Fig. 2 emerges. Increasing the interactionfurther, eventually the pairs live only on the sites of the lattice. In this regime ofhigh interaction strength, the n = 2 Mott state gets completely destroyed by thesite-centered pair fluctuations. We have also looked into the regime where interactioninduced Wannier-broadening arises from large filling n ≫ a s ≪ a . In this limit, we found that the superfluid becomes dynamically unstabledue to the attractive nature of the occupation-dependent tunneling. The system then ose-Hubbard model with occupation dependent parameters J n i ,n j and U n i will be required for a quantitative description of the effectsdescribed here. Finally, it would also be worth studying in detail the role of a trappingpotential, as it is present in experiments.
8. Acknowledgments
This work is financially supported by Spanish MICINN (FIS2008-00784 and ConsoliderQOIT), EU Integrated Project AQUTE, ERC Advanced Grant QUAGATUA, the EUSTREP NAMEQUAM, the Caixa Manresa, the Alexander-von-Humboldt foundation,and the German-Israel Foundation (grants 149/2006 and I-1024-2.7/2009). B.A.M.appreciates hospitality of the Institut de Ci`encies Fot`oniques (Barcelona, Spain)
References [1] M. Lewenstein, A. Sanpera, V. Ahufinger, B. Damski, A. Sen, U. Sen, Adv. Phys. , 243 (2007).[2] I. Bloch, J. Dalibard, W. Zwerger, Rev. Mod. Phys. , 885 (2008).[3] M. P. A. Fisher, P. B. Weichman, G. Grinstein, D. S. Fisher, Phys. Rev. B , 546 (1989).[4] M. Greiner, O. Mandel, T. Esslinger, T. W. H¨ansch, I. Bloch, Nature , 39 (2002).[5] D. Jaksch, C. Bruder, J. I. Cirac, C. W. Gardiner, and P. Zoller, Phys. Rev. Lett. , 3108 (1998).[6] T. Lahaye, C. Menotti, L. Santos, M. Lewenstein, T. Pfau, Rep. Prog. Phys. , 126401 (2009).[7] N.W. Ashcroft, N.D. Mermin, Solid State Physics , Saunders College Publishing (1976).[8] P. R. Johnson, E. Tiesinga, J. V. Porto, C. J. Williams, New Jour. Phys. , 093022 (2009)[9] J. Li, Y. Yu, A. M. Dudarev, Q. Niu, New Jour. Phys. , 154 (2006).[10] K. R. A. Hazzard, E. J. Mueller, Phys. Rev. A , 031602. (2010)[11] O. E. Alon, A. I. Streltsov, L. S. Cederbaum, Phys. Rev. Lett. , 030405 (2005).[12] K. Sakmann, A. I. Streltsov, O. E. Alon, L. S. Cederbaum, arXiv:1006.3530.[13] J. Larson, A. Collin, J.-P. Martikainen, Phys. Rev. A , 033603 (2009).[14] A. Cetoli, E. Lundh, Euro. Phys. Lett. , 46001 (2010).[15] S. Z¨ollner, H.-D. Meyer, P. Schmelcher, Phys. Rev. Lett. , 040401 (2008).[16] L. Cao, I. Brouzos, S. Z¨ollner, Peter Schmelcher, Arxiv: 1011.4219.[17] S. Will, T. Best, U. Schneider, L. Hackerm¨uller, D. L¨uhmann, I. Bloch, Nature , 197 (2010).[18] J. Catani, L. De Sarlo, G. Barontini, F. Minardi, M. Inguscio, Phys. Rev. A , 011603(R) (2008).[19] S. Ospelkaus, C. Ospelkaus, O. Wille, M. Succo, P. Ernst, K. Sengstock, K. Bongs, Phys. Rev.Lett. , 180403 (2006).[20] K. G¨unter, T. St¨oferle, H. Moritz, M. K¨ohl, T. Esslinger, Phys. Rev. Lett. , 180402 (2006).[21] S. Will, T. Best, S. Braun, U. Schneider, I. Bloch, arXiv:1011.3807.[22] D.-S. L¨uhmann, K. Bongs, K. Sengstock, D. Pfannkuche, Phys. Rev. Lett. , 050402 (2008).[23] T. Busch, B.-G. Englert, K. Rza¸˙zewski, Martin Wilkens, Found. Phys. , 549 (1998).[24] P.O. Fedichev, M.J. Bijlsma, P. Zoller, Phys. Rev. Lett. , 080401 (2004).[25] X. Cui, Y. Wang, F. Zhou, Phys. Rev. Lett. , 153201 (2010).[26] H. P. B¨uchler, Phys. Rev. Lett. , 090402 (2010).[27] M.L. Chiofalo, M. Polini, M.P. Tosi, Eur. Phys. J. D , 371 (2000).[28] P. Vignolo, Z. Akdeniz, M. P. Tosi, J. Phys. B , 4535 (2003).[29] J.-F. Schaff, Z. Akdeniz, P. Vignolo, Phys. Rev. A , 041604 (2010); ose-Hubbard model with occupation dependent parameters Phys. Rev. A 81, 041604 (2010)[30] E. Altman and A. Auerbach, Phys. Rev. Lett. , 250404 (2002).[31] S. D. Huber, E. Altman, H. P. B¨uchler, and G. Blatter, Phys. Rev. B , 085106 (2007).[32] F. Mila, K.P. Schmidt, Arxiv: 1005.2495.[33] R. Bendjama, B. Kumar, F. Mila, Phys. Rev. Lett. , 110406 (2005).[34] M. W. H. Romans, R. A. Duine, S. Sachdev, H. T. C. Stoof, Phys. Rev. Lett. , 020405 (2004).[35] K. Winkler et al., Nature , 853 (2006).[36] D. Petrosyan, B. Schmidt, J. R. Anglin, M. Fleischhauer, Phys. Rev. A , 033606 (2007).[37] A. Kuklov, N. Prokof’ev, B. Svistunov, Phys. Rev. Lett. , 050402 (2004).[38] A. Arg¨uelles, L. Santos, Phys. Rev. A , 053613 (2007).[39] C. Trefzger, C. Menotti, M. Lewenstein, Phys. Rev. Lett. , 035304 (2009).[40] M. Eckholt and J. J. Garc´ıa-Ripoll, Phys. Rev. A , 063603 (2008)[41] L. Mazza, M. Rizzi, M. Lewenstein, J. I. Cirac, arXiv:1007.2344.[42] N. Syassen, D. M. Bauer, M. Lettner, T. Volz, D. Dietze, J. J. Garc´ıa-Ripoll, J. I. Cirac, G. Rempe,and S. D¨urr, Science , 1329 (2008).[43] A. J. Daley, J. M. Taylor, S. Diehl, M. Baranov, and P. Zoller, Phys. Rev. Lett. 102, 040402 (2009).[44] G. G. Batrouni, R. T. Scalettar, G. T. Zimanyi, and A. P. Kampf, Phys. Rev. Lett. , 2527(1995).[45] M. J. Ablowitz and J. F. Ladik, J. Math. Phys. , 1011 (1976).[46] D. Cai, A. R. Bishop, N. Grønbech-Jensen, Phys. Rev. Lett.72