Localization of Bose-Einstein condensates in optical lattices

Cent. Eur. J. Phys.DOI: 10.2478/s11534-011-0035-2

Central European Journal of Physics

Localization of Bose-Einstein condensates in opticallattices

Research Article

Roberto Franzosi1,2∗, Salvatore M. Giampaolo3,4, Fabrizio Illuminati3,4, Roberto Livi5, Gian-Luca Oppo6,Antonio Politi7

1 C.N.I.S.M. UdR di Firenze, Dipartimento di Fisica, Università degli Studi di Firenze,Via Sansone 1, I-50019 Sesto Fiorentino, Italy

2 I.P.S.I.A. C. Cennini,Via dei Mille 12/a, I-53034 Colle di Val d’Elsa (SI), Italy

3 Dipartimento di Matematica e Informatica, Università degli Studi di Salerno,Via Ponte don Melillo, I-84084 Fisciano (SA), Italy

4 INFN Sez. di Napoli, Gruppo collegato di Salerno,Via Ponte don Melillo, I-84084 Fisciano (SA), Italy

5 Dipartimento di Fisica, Università di Firenze and INFN, Sez. di Firenze,Via G. Sansone 1 Sesto Fiorentino, I-50019, Italy

6 Department of Physics, University of Strathclyde,107 Rottenrow, Glasgow G4 ONG, Scotland, U.K.

7 Istituto dei Sistemi Complessi, Consiglio Nazionale delle Ricerche,Via Madonna del Piano 10 Sesto Fiorentino, I-50019, Italy

Received 24 November 2010; accepted 27 March 2011

Abstract: The dynamics of repulsive bosons condensed in an optical lattice is effectively described bythe Bose-Hubbard model. The classical limit of this model, reproduces the dynamics of Bose-Einsteincondensates, in a periodic potential, and in the superfluid regime. Such dynamics is governed by a dis-crete nonlinear Schrödinger equation. Several papers, addressing the study of the discrete nonlinearSchrödinger dynamics, have predicted the spontaneous generation of (classical) breathers in coupled con-densates. In the present contribute, we shall focus on localized solutions (quantum breathers) of the fullBose-Hubbard model. We will show that solutions exponentially localized in space and periodic in timeexist also in absence of randomness. Thus, this kind of states, reproduce a novel quantum localizationphenomenon due to the interplay between bounded energy spectrum and non-linearity.

PACS (2008): 03.75.Lm, 05.30.Jp, 03.65.Sq, 31.15.Pf

Keywords: breathers • Bose-Einstein condensates • localized states • optical lattices© Versita Sp. z o.o.

∗E-mail: [email protected]

Localization of Bose-Einstein condensates in optical lattices

1. Introduction

Localization of atoms in optical lattices is among themacroscopic quantum-effects that have been theoreticallystudied and experimentally observed in Bose-Einsteincondensates (BECs). Localized states of BECs along anoptical lattice are the consequence of the nonlinear char-acter of the classical equations of motion of these sys-tems. These kinds of solutions have been predicted onthe bases of the analogy between the Gross-Pitaevskiiequation, that describe the superfluid (classical) dynam-ics of BECs, and the nonlinear Schrödinger equation usedin nonlinear optics [1–8]. In continuous-space modelsthe localized solution have the form of gap solitons andmatter-wave solitons [5–7]. While in nonlinear classicallattice-Hamiltonians, solutions have the form of discretebreathers. The latter are, in fact, periodic in time, andexponentially localized in space, solutions distinctive ofnonlinear discrete systems. The superfluid regime for asystem of BECs in an optical lattice is described by a dis-crete nonlinear Schrödinger equation (DNSE). In recentpapers [23, 24], by numerically integrating the DNSE, thespontaneous generation of breathers in BECs in opticallattices were predicted. In particular, in [23] a techniqueto obtain localization of BECs by means of boundary dis-sipations, that is by removing atoms at the optical latticeends, was presented.The present paper focuses on exact (numeric) localized so-lutions of the full (quantum) Bose-Hubbard model (BHM).We show that the solutions are energy-eigenstates andare exponentially localized in space and periodic in time.Furthermore, we proceed by a dynamic variational methodin order to determine an analytic approximation to themaximally excited state. We compare the results de-rived with those obtained by exact diagonalization. Inorder to derive such analytic approximation, we follow theroute adopted in previous papers [25, 26], by introducinga macroscopic trial state of the form eiS/~|x〉, where S isa time-dependent phase and |x〉 is an SU(M) coherentstate. The requirement that the trial state satisfies theSchrödinger equation, on average, means that S is an ef-fective action. By a variational on S, one can derive anapproximation to the maximally excited energy eigenstateof the system. By comparing the atomic density distri-bution of the approximate state with the one of the exactsolution we show that, in a wide range of the Hamiltonianparameters, the approximated state (named semiclassicalsolution) provides an excellent approximation to the exactmaximally excited state.Quantum localized states in a BHM of repulsive atoms,can be realized by virtue of the combination of two fea-tures of this Hamiltonian: it has a non-linear depen-

dence from the site occupation number operators and ithas a bounded energy spectrum. The last condition, infact, guarantees the existence of an upper-bounded en-ergy spectrum with spatially bounded states. Whereasthe first property makes it possible to have a percepti-ble separation in the energy spectrum between the higherlevels. A further important aspect is the lack of invari-ance for spatial system translation. The open bound-ary conditions make the central lattice site, where thestates localize, special. Thus, finite-size systems are re-quired in order to observe a finite energy separation be-tween the maximum Hamiltonian-eigenstate and the oth-ers. When translational symmetry is preserved, as wouldbe the case for periodic boundary conditions, no local-ized states can be realized. In fact, the symmetry makesall of the Hamiltonian-eigenstates, that differ for cyclicpermutation of the site indices, degenerate. Thereby, amaximum energy eigenstate will be a generic superposi-tion of all these states, since there is not reason to favorany of them over others. It is characterized by a flat dis-tribution of the atomic density along the lattice. We willshow in the following sections that, for finite lattice sizeand a wide range of values of the non-linear parameter,the highest-energy levels of the repulsive-BHM are char-acterized by a level (with maximum energy) separated bythe quasi-degenerate multiplet of the close lower levels.2. Bose-Einstein condensates in anoptical latticeThe second-quantized Hamiltonian

H = ∫ drψ+(r) [− ~22m∇2 + Vext(r)] ψ(r)+ 4π~2as2m

∫drψ+(r)ψ+(r)ψ(r)ψ(r) (1)

describes the quantum dynamics of an ultracold dilute gasof bosonic atoms. Here Vext is the external trapping po-tential and the boson-field operators ψ(r) (ψ+(r)) annihi-late (create) atoms at r = (rx , ry, rz) in a given internalstate. The nonlinear self-interaction term depends on thes-wave scattering length as and on the atomic mass m.We consider repulsive atoms, i.e. as > 0. In the case ofa one-dimensional optical lattices, the external trappingpotential reads

Vext(r) = ~2ω2 sin2 kLrx4Er +mΩ2 r2y + r2

z2 ,

where kL is the laser mode which traps the atoms andEr = ~2k2

L2m

Roberto Franzosi, Salvatore M. Giampaolo, Fabrizio Illuminati, Roberto Livi,Gian-Luca Oppo, Antonio Politi

is the recoil energy. In the x-direction one has an opticallattice structure of length L, while the trapping frequencyΩ is assumed to be strong enough to confine the atomiccondensate in the y and z directions by harmonic wells.Thus, the physics is effectively one-dimensional.In the presence of a periodic external potential, we can de-rive from (1) an effective quantum (Bose-Hubbard) Hamil-tonian that describes, within the second quantizationformalism, the boson dynamics along the optical lat-tice. In order to derive this effective dynamics we fol-low Ref. [14]. Since the one-dimensional potential Vextis periodic, the “single atom” eigenstates of Hamiltonianin Eq. (1) are Bloch functions [14] Φ(n)q (r), labelled by thequasi-momentum q and the band index n. Already forshallow optical potentials, that is for values of ~2ω24Er of theorder of a few (∼ 5) recoil energies Er , the energy sepa-ration between the two lowest bands is much larger thantheir spread (see [13]). In Ref. [13] it is also shown that,for moderate optical potential depths, a good approxima-tion for the lowest energy gap is given by ~ω, where ω isthe longitudinal oscillation frequency of a particle trappedin the harmonic approximation of the potential Vext closeto a minimum. As done in [14], let us assume that onlythe fundamental energy band is involved into the dynam-ics and we describe the system using Wannier functions.Kohn, in [15], has shown that, by properly choosing thebranch points of the Bloch’s functions, for each band thereexists one and only one Wannier function that is real anddecays exponentially with the distance. The lowest-bandWannier functions uj (r) defined by Kohn in Ref. [15], arelocalized at the minima (xj , 0, 0) of the optical potential,where

xj = jπkL, j = 1, . . . ,M

andM = L× kL

π ,

and result to beuj (r) = W − 12 ∑

qe−iqxjΦ(0)

q (r),whereW is a normalization constant. We express the fieldoperators as (see [14])

ψ(r, t) = M∑j=1 u

∗j (r)aj (t), (2)

where the boson operator aj (a+j ) destroys (creates) anatom at the lattice site j .

By substituting Eq. (2) into Eq. (1) we obtainH =∑

jklm

Ujklma†j a†k alam −

12 ∑jk

Tjka†j ak ,

whereUjklm = 4π~2as2m

∫dru∗j (r)u∗k (r)ul(r)um(r)

is the strength of the on-site interaction andTjk = −2 ∫ dru∗j

[p22m + Vext

]uk

is the hopping amplitude. Already for moderate opticalpotential depth~2ω24E2

r& 5,

the numerical values of the off-diagonal elements of theinteraction matrix Ujklm and the tunneling matrix Tjk be-tween sites (other than nearest neighbors) are negligiblewith respect to Ujjjj ≡ U and Tj j+1 = Tj+1 j ≡ T re-spectively (see Ref. [13]). Thus, one obtains the effectiveone-dimensional M-site Bose-Hubbard HamiltonianH = M∑

j=1[Unj (nj − 1) + ξjnj

]− T2 M−1∑

j=1(a+j aj+1 + h.c.) ,

(3)where the boson operators satisfy the boson commutationrelations [ak , a+j ] = δk,j .

3. Classical dynamics in the SU(M)coherent states pictureOriginally introduced in order to describe thesuperfluid/Mott-insulator transition in lattice sys-tems [12], the Bose-Hubbard model is nowadays used todescribe the dynamics of systems of cold bosonic-atomson an optical lattice [13]. In Ref. [14] it has beenpredicted, and then experimentally verified in [16], thatthe dynamics of an ultracold dilute gas of bosonic atomsin an optical-lattice/periodic-potential is well describedby the Bose-Hubbard model.Hamiltonian in Eq. (3) commutes with N =∑k nk and canbe expressed in terms of the set of operators Ekr = a†k arwhich, in turn, give a dimension-independent realization ofthe SU(M) algebra. Thus, one can choose an irreducibleunitary representation of the SU(M) (Lie-)algebra with


the same dimension of the space spanned by the system’sstates:D(M,N) = (M +N − 1)![N!(M − 1)!] .This fact suggest to describe the system dynamics in termsof SU(M) generalized coherent states. Of course, being

H a quadratic form of operators Ekr , this description isexact only in the superfluid (Λ→ 0) limit where the lineardependence is recovered. Nevertheless, a description interms of coherent states, permites a drastic reduction ofdegrees of freedom and analytic or numeric computationsare allowed. Let us underline that, accordingly to the factthat Hamiltonian (3) conserves the total number of atoms,such coherent states are eigenvectors of N .A detailed derivation of the SU(M) coherent states can befound in Refs. [17–19]. We will just summarize a few factsabout them. They are defined as|x, N〉 = 1√

N! M∑

j=1 xj a†j

N

|0, . . . , 0〉, (4)where M and N are the numbers of lattice sites and atomsrespectively. The complex coherent state variables xj ∈ C,for j = 1, . . . ,M , are constrained on the unit-sphere, i.e.∑M

j=1 |xj |2 = 1 [17–19]. The following identities holdaj |x, N〉 = √Nxj |x, N − 1〉,a†j |x, N〉 = 1√

N + 1∂xj |x, N + 1〉,a†j ak |x, N〉 = xk∂xj |x, N〉,

〈x, N|a†j ak |x, N〉 = Nxkx∗j ,

(5)

and will be used below to derive a semiclassical descrip-tion for the dynamics of the system. The classical equa-tions of motion involve the order parameters xj , which arejust complex numbers. They will be derived by a time de-pendent variational principle (TDVP) [20, 20, 21]. By solv-ing such equations, we will obtain xj (t) for j = 1, . . . ,M ,and from these, the state of the quantum system via Eq. (4).The TDVP procedure [20, 20, 21] is based on a suitablechoice of the quantum trial state for the system. Thus wecan make the ansatz|Φ(t)〉 = exp( iS(t)

~

)|x(t)〉, (6)

where |x(t)〉 is a time-dependent SU(M) coherent state,whereas exp( iS(t)~

) is a phase factor.In the following we derive a closed set of dynamical equa-tions for the variables x(t) and the expression for S(t).

In particular, we will show that S(t) is an effective semi-classical action. In Eq. (6), and hereafter, we dropthe explicit dependence on N in the coherent statesof Eq. (4). The TDVP method amounts to constrain-ing the time evolution of |Φ(t)〉 by using, instead of thefull, just the weaker form of the Schrödinger equation〈Φ|(i~∂t − H)|Φ〉 = 0 (note the explicit dependence ontime t has been dropped). Using Eq. (6) one obtainsthe identity S = ∫

dt(i~〈x|∂t |x〉 − 〈x|H|x〉), which is in-deed an (effective) action. The equations of motion forthe dynamical variables xj can be derived from the actionS = ∫

dt(iΣj xjx∗j − H), where H(x, x∗) := 〈x|H|x〉 is theclassical Hamiltonian. Thus, the set of variables xjj re-sult canonical (complex) variables that satisfy the Poissonbracketsx∗j , x` = iδj`

~N

and describe the condensate through population |xj |2 andphase arg(xj ) of each site. The explicit form of the classicalHamiltonian isH = M∑

j=1[UN(N − 1)|xj |4 + ξjN|xj |2]−

+ T2 N M′∑j=1(x∗j xj+1 + c.c.) , (7)

whereas the corresponding equations of motion areidxjdτ = [Λ(N − 1)|xj |2 + εj

]xj −

12 [xj−1 + xj+1] . (8)We have introduced the parameters Λ = 2U

T , εj = ξj ,and time has been rescaled via τ = T t~ so that τ is adimensionless quantity. Finally, let us also emphasizethat Eq. (8) is a discrete version of the Gross-Pitaevskiiequation [2].

4. Classical and quantum breathersA technique to obtain localization of BECs in optical lat-tices via boundary dissipations was introduced in [23].In [23, 24], we have numerically integrated the set of equa-tions Eqs. (8) with boundary dissipation terms. Fig. 1shows two typical evolutions of the atomic density alongthe optical lattice with losses at the ends of the trappingpotential. In Fig. 1a two static breathers have been ex-cited via progressive losses of atoms at the boundaries,whereas in Fig. 1b the atomic losses have generated asingle localized state.


Figure 1. Time evolution of the atomic density for γ1 = γM = 0.3,εj = 0 for all j, M = 128, (a) Λ(N(0) − 1) = 89.6 and(b) Λ(N(0)− 1) = 44.8, where N(0) is the initial number ofatoms.

The static breathers are stable solutions; exponentiallylocalized in space and periodic in time. They have a sim-ple mathematical form which can be derived as follows.These static solutions satisfy Eqs. (8) with a time de-pendence characterized by only one frequency ω. Let usreformulate Eqs. (8) by introducing the variables of sitepopulation and phase. By substituting the new variablesnj = |xj |2, φj = − i2 ln( xjx∗j

), (9)

into Eqs. (8) we getnj =− [√nj+1nj sin(∆φj )−√njnj−1 sin(∆φj−1)] ,φj =− [Λ(N − 1)nj − µ]

+ 12[√nj+1

njcos(∆φj ) +√nj−1

njcos(∆φj−1)] , (10)

where we have assumed εj = 0, ∆φj := φj+1−φj , and µ isa Lagrange multiplier introduced for taking into accountthe conserved quantity ∑j nj = N . The solution for astatic breather requires nj = 0, which corresponds to theπ-state configuration

φj − φj+1 = ±π. (11)

Thus, the second of Eqs. (10) becomesφj = −(Λnj − µ)− 12

[√nj+1nj

+√nj−1nj

]. (12)

By setting φj = ω = (µ − χ) in the latter equation, weget the algebraic set of equations− 2(χ − Λnj ) + [√nj+1

nj+√nj−1

nj

] = 0, (13)where the solutions nj have to be worked out in a self-consistent way together with χ which is fixed by the con-train∑j nj = N . We have solved numerically Eqs. (13) byNewton’s method, for Λ = 1, 10, 100, N = 7 and M = 9.The atomic density distributions of the classical breathersobtained in this way, have been compared with the anal-ogous quantity derived by the quantum breathers of theBose-Hubbard Hamiltonian.In analogy with the classical case, a static quantumbreather, is characterized by a single frequency, namelyit is an Hamiltonian’s eigenstate. For this reason, thefull quantum spectrum of Bose-Hubbard Hamiltonian hasbeen numerically calculated for several of the Λ values,by a full diagonalization method. We have found thatin a wide range of values of Λ, there exist eigenstateswhich are exponentially localized along the lattice. Forany given value of Λ, the quantum-breather state corre-sponds to the highest energy eigenstate. Furthermore, wehave found that the higher energy levels have the struc-ture shown in the inset of Fig. 2, with the highest levelseparated by a multiplet of levels, nearly degenerate. Itis worth emphasizing that the energy difference betweenthe highest level and the near multiplet, goes to zero veryfast when increasing the relevant physical parameter ΛN .Already for ΛN ' 104 this energy difference is no moreappreciable. Whereas we have been able to observe lo-calized quantum states, for values of ΛN in the interval[1, 103]. The energy gap between the highest level and themultiplet of levels also depends on M . And by increasingthe lattice size this difference rapidly decreases. In otherwords, our results can be observed in systems with sizessmall enough that the translation symmetry breaking pro-duces tangible consequences. When the latter symmetryis restored, quantum states with localization around what-ever lattice site are, of course, equivalent, and the maxi-mum energy eigenstate is a superposition of these states.In this case no localization can be seen.The atomic density distribution of the quantum breathershas been calculated for several values of Λ. We havecompared these distributions to the ones of the classicalbreathers obtained by numerically solving Eqs. (13) with


Figure 2. Energy spectrum for Λ = 10, N = 7, andM = 9. The insetmagnifies the four highest levels.

the same values of Λ. A good agreement between thesequantities is evident in Fig. 3; which summarizes the re-sults obtained for Λ = 1, 10, 100, N = 7 and M = 9.

Figure 3. The figure compares the atomic density distributions forthe maximum energy eigenstates (continuous lines) andthe static breathers obtained by solving Eqs. (13) (dashedlines) for N = 7 and M = 9, and Λ = 1 (circles), Λ = 10(squares), and Λ = 100 (diamonds).

5. Concluding remarksIn the last few years, Bose-Einstein condensates have be-come one of the most versatile test-beds for unexpectedbehavior of quantum physics. A variety of macroscopicquantum effects have been theoretically investigated andexperimentally observed in BECs. Breather states inDNSE have been studied for the last few decades. Sincethe equations of motion that describe the dynamics of

BECs in optical lattices, in the superfluid limit, belong tothe class of DNSEs, one also expects, to observe breathersin such systems.In the present paper we have focused our attention onthe Bose-Hubbard model which describes the quantumdynamics of BECs in optical lattices. We have given aneffective description of its dynamics by describing the sys-tem states in term of SU(M) coherent states. Although thesystem’s equations of motion are reformulated as classicalHamiltonian equations, they describe quantum dynamicsof the system pretty well. Moreover, we have numericallydiagonalized the full Bose-Hubbard Hamiltonian and wehave shown that the breathers can also be observed in thequantum regime.We have shown that they are the highest-energy eigen-states and exists for a wide range values of Hamiltonianparameters. These quantum breathers, as well as the clas-sical ones, are exponentially localized in space and pe-riodic in time. Remarkably, these kinds of states existsin the absence of randomness. Thus they reproduce anovel quantum localization phenomenon, essentially dueto the interplay between bounded energy spectrum andnon-linearity.References

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Localization of Bose-Einstein condensates in optical lattices

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