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Ultracold atoms confined in an optical lattice plus parabolic potential: a closed-form approach

Ana Maria Rey∗, Guido Pupillo†, Charles W. Clark, and Carl J. WilliamsNational Institute of Standards and Technology, Gaithersburg, MD 20899

(Dated: September 18, 2018)

We discuss interacting and non-interacting one dimensional atomic systems trapped in an optical lattice plusa parabolic potential. We show that, in the tight-binding approximation, the non-interacting problem is exactlysolvable in terms of Mathieu functions. We use the analytic solutions to study the collective oscillations ofideal bosonic and fermionic ensembles induced by small displacements of the parabolic potential. We treat theinteracting boson problem by numerical diagonalization ofthe Bose-Hubbard Hamiltonian. From analysis ofthe dependence upon lattice depth of the low-energy excitation spectrum of the interacting system, we considerthe problems of ”fermionization” of a Bose gas, and the superfluid-Mott insulator transition. The spectrum ofthe noninteracting system turns out to provide a useful guide to understanding the collective oscillations of theinteracting system, throughout a large and experimentallyrelevant parameter regime.

I. INTRODUCTION

In recent experiments [1, 2, 3, 4, 5], quasi-one dimensionalsystems have been realized by tight confinement of gases intwo dimensions. Due to the enhanced importance of quan-tum correlations as dimensionality is reduced, such systemsexhibit physical phenomena not present in higher dimensions,such as ”fermionization” of a Bose-Einstein gas.

The degree of correlation is measured by the ratio of theinteraction energy to the kinetic energy,γ. For γ ≪ 1 thesystem is weakly interacting, and at zero temperature mostatoms are Bose-condensed. In this limit quantum correlationsare negligible and the dynamics is governed by the mean-fieldGross-Pitaevski equation. Forγ ≫ 1, the system is highlycorrelated and becomes fermionized, in that repulsive inter-actions mimic the effects of the Pauli exclusion principle.Inthis ”Tonks-Girardeau” regime, [6, 7], the low energy exci-tation spectrum of a bosonic gas resembles that of a gas ofnon-interacting fermions.

To date, one dimensional systems have been obtained byloading a Bose-Einstein condensate into a two-dimensionaloptical lattice which is deep enough to restrict the dynamicsto one dimension. This procedure creates an array of indepen-dent 1D tubes. In most experiments, a weak quadratic poten-tial is superimposed upon the lattice, in order to confine theatoms during the loading proccess. The combined presenceof the periodic and quadratic potentials substantially modi-fies the dynamics of the trapped atoms compared to the caseswhen only one of the two potentials is present, as shown bothexperimentally and theoretically [8, 9, 10, 11, 12, 13].

In this paper we study both ideal and interacting bosonicsystems in such potentials. In contrast to previous stud-ies of ideal systems, which used numerical[10] or approxi-mate solutions [8, 11, 12, 13], here we show that the single-particle problem is exactly solvable in terms of Mathieufunctions[14, 15, 16]. We use analytic solutions to fully char-acterize the energy spectrum and eigenfunctions in the vari-ous regimes of the trapping potentials and to provide analytic

∗Electronic address: ana.rey@nist.gov†Electronic address: guido.pupillo@nist.gov

expressions for the oscillations of the center of mass of bothideal bosons and fermions that are induced by small displace-ments of the parabolic trap. These expressions for the dipolarmotion may be tested in experiments.

We further analyze the low-energy spectrum of interact-ing bosons by means of exact diagonalizations of the Bose-Hubbard Hamiltonian, and identify the conditions requiredfor fermionization to occur. Moreover, we show that spe-cific changes in the spectrum of the fermionized system canbe used to describe the characteristics of a Mott insulator statewith unit filling at the trap center.

Center-of-mass oscillations are also studied in the weaklyinteracting and fermionized regimes by comparing exactnumerical solutions for the interacting system to solutions forideal bosons and fermions, respectively. Because fermioniza-tion occurs over a large range of trap parameters, knowledgeof the properties of the single-particle solutions turns out toprovide useful insights in the understanding of the complexmany-body dynamics. In addition, we numerically analyzethe distribution of frequencies pertaining the modes excitedduring the collective dynamics, for different values ofγ.This helps us gain qualitative insight in the dynamics even inthe intermediate regime where interaction and kinetic ener-gies are comparable and no mapping to ideal gases is possible.

The presentation of the results is organized as follows:In Sec.II we discuss the analytic solutions that describe thesingle-particle physics. These show the existence of two typesof behavior, according to whether the energy is dominatedby site hopping or parabolic contributions. Different asymp-totic expansions of the eigenfunctions and eigenvalues applyto these two regimes, and can be effectively combined to de-scribe the full spectrum.

In Sec. III, we then apply the analytic solutions and asymp-totic expansions to the description of the collective dynamicsof non-interacting ensembles of bosons and fermions subjectto a sudden displacement of the parabolic trap. In contrastto the well-known case of the displaced harmonic oscillator,a non-trivial modulation of the center of mass motion occursdue to the presence of the lattice potential. We derive explicitexpressions for the initial decay of the amplitude of the oscil-lation, to which we refer as effective damping.

In Sec.IV A, the low-energy spectrum of interacting bosons

2

is studied as a function of the lattice depth. In particular,wefollow the evolution of the spectrum from ideal bosonic toideal fermionic as the lattice deepens, by means of numeri-cal diagonalization of the Hamiltonian for a moderate numberof atoms and wells. We specify the necessary conditions forthe formation of a Mott state at the center of the trap, and usethe Fermi-Bose mapping to link its appearance and the reduc-tion of number fluctuations to the population of high-energylocalized states at the Fermi level.

Section IV B is dedicated to the study of the collectivedipole dynamics of an interacting bosonic gas by numericalcalculations of the exact quantal dynamics. The dynamics isstudied for two different scenarios that are experimentally re-alizable. The first corresponds to experiments in which thetrapping potentials are kept fixed and the interatomic scatter-ing length is varied (e.g. by use of a Feshbach resonance),Sec.IV B1. Here, parameters have been chosen such that noMott insulator at the center of the trap is formed in the largeγ limit. The second scenario corresponds to experiments inwhich the lattice depth is increased while the frequency of theparabolic potential is kept fixed, Sec.IV B2. In this case a unitfilled Mott insulator is formed at the trap center and the inhi-bition of the transport properties of the system is observedasthe lattice deepens.

II. SINGLE PARTICLE PROBLEM

A. Tight binding solution

The dynamics of a single atom in a one dimensional op-tical lattice plus a parabolic potential is described by theSchrodinger equation

i~∂Φ

∂t= − ~

2

2m

∂2Φ

∂x2+ Vo sin

2(πax)Φ +

mω2T

2x2Φ, (1)

whereΦ(x) is the atomic wave function,ωT is the trappingfrequency of the external quadratic potential,Vo is the opticallattice depth which is determined by the intensity of the laserbeams,a = λ/2 is the lattice spacing,λ is the wavelength ofthe lasers andm is the atomic mass.

If the atom is loaded into the lowest vibrational state ofeach lattice well and the dynamics induced by external per-turbations does not generate interband transitions, the wavefunctionΦ(x) can be expanded in terms of first-band Wannierfunctions only [17, 18]

Φ(x, t) =∑

j

zj(t)w0(x− ja), (2)

wherew0(x− ja) is the first-band Wannier function centeredat lattice sitej, t is time, andzj(t) are complex amplitudes.For a deep enough lattice, tunneling to second nearest neigh-bors can be ignored. This approximation known as the tight

binding approximation yields the following equations of mo-tion for the amplitudeszj(t):

i~∂zj∂t

= −J (zj+1 + zj−1) + Ωj2zj , (3)

with

Ω =1

2ma2ω2

T , (4)

J = −∫

dxw∗0(x)How0(x− a)dx, (5)

Ho = − ~2

2m

∂2

∂x2+ Vo sin

2(πax)

(6)

where J is the tunneling matrix element between nearestneighboring lattice sites. In Eq.(3) the overall energy shiftεo given by

εo =

∫dxw∗

0(x)How0(x)dx, (7)

has been set to zero.

B. Stationary solutions

The stationary solutions of Eq.(3) are of the formz(n)j (t) =

f(n)j e−iEnt/~, with En and f

(n)j the nth eigenenergy and

eigenstate, respectively. Substitution into Eq.(3) yields

Enf(n)j = −J

(f(n)j+1 + f

(n)j−1

)+Ωj2f

(n)j (8)

Equation (8) is formally equivalent to the recursion relationsatisfied by the Fourier coefficients of the periodic Mathieufunctions with periodπ. Therefore the eigenvalue problemcan be exactly solved by identifying the amplitudesf

(n)j and

eigenenergiesEn with the Fourier coefficients and character-istic values of such functions , respectively [14]. In termsofMathieu parameters the symmetric and antisymmetric solu-tions are

f(n=2r)j =

1

π

∫ 2π

0

ce2r (x,−q) cos(2jx)dx, (9)

f(n=2r+1)j =

1

π

∫ 2π

0

se2r (x,−q) sin(2jx)dx, (10)

En=2r =Ω

4a2r (q) En=2r+1 =

Ω

4b2r (q) , (11)

with r = 0, 1, 2... and ce2r(x, q) and se2r(x, q) the evenand odd periodπ solutions of the Mathieu equation withparameterq = 4J

Ω and characteristic parametera2r(q) and

b2r(q) respectively:d2ce2rdx2 + (a2r(q) − 2q cos(2x))ce2r = 0

and d2se2rdx2 + (b2r(q)− 2q cos(2x))se2r = 0.

3

Solutions of the single particle problem are entirelydetermined by the parameterq = 4J

Ω , which is proportionalto the ratio of the nearest-neighbor hopping energyJ to theenergy costΩ for moving a particle from the central site toits nearest-neighbor. The eigenvalues and eigenstates of thesystem are in general complicated functions ofq. However,asymptotic expansions exist in literature which can helpunveil the underlying physics for different values ofq. Mostof the asymptotic expansions have been available for almost50 years due to the work of Meixner and Schafke [15]. Inthe remainder of this section we introduce such asymptoticexpansions and use them to describe the physics of the systemin the tunneling dominated regime whereq > 1, which wecall highq regime, and theq < 1, or low q regime.

1. Highq regime (4J & Ω)

Most experiments have been developed in the parameterregime whereq ≫ 1. For example for87Rb atoms trappedin a lattice withλ = 810 nm, a value ofq ≥ 10 is obtained fora lattice depth of 2ER if ωT < 2π × 538 Hz and for a latticeof 50 ER if ωT < 2π × 7.5 Hz. HereER is the photon re-coil energyER = h2/(2mλ2), corresponding to a frequencyER/h = 3.47 kHz. Throughout this paper, in our exampleswe use atoms with the mass of87Rb.

In the high q limit the periodic plus harmonic potentialpossesses two different classes of eigenstates dependingon their energy. In fact, as we will show, eigenmodescan be classified as low or high-energy depending on thequantum numbern being smaller or larger than2‖

√q/2‖,

respectively, where‖x‖ denotes the closest integer tox.Physically, this classification depends on which one of thetwo energy scales in the system, the tunneling or the trappingenergy, is dominant. To the energy classification correspondsa classification based on localization of the modes in thepotentials. In particular, the low-energy (LE) modes areextended around the trap center and high-energy (HE) modes

are localized on the sides of the potential. The existence oflocalized and extended states has been tested experimentally[8, 9] and studied theoretically [10, 11, 12, 13]. Here weshow how the asymptotic expansions of the Mathieu solutionscan be used to characterize them quantitatively.

Low-energy modes (n ≪ √q) in the highq regime

In the LE regime the average hopping energyJ is largerthanΩ. In this regime the eigenmodes have been shown tobe approximately harmonic oscillator eigenstates [10, 11,12,13]. Asymptotic expansions valid to describe LE eigenmodesin the highq limit have been studied in detail in Ref. [16],where the eigenstates are described in terms of generalizedHermite polynomials. Here we simply outline the basic resultsand refer the interested reader to [16] and references thereinfor further details.

The asymptotic expansions for the eigenmodes can be writ-ten as

f(n=2r)j ≈ An exp

(−ξ2

(1

2+

(3 + 2n)

16√q

)+

ξ4

48√q

)

·r∑

k=0

h(r)k ξ2k

(1 +

(3k − k2 + 10kr)

24√q

)(12)

f(n=2r+1)j ≈ An exp

(−ξ2

(1

2+

(3 + 2n)

16√q

)+

ξ4

48√q

)

·r∑

k=0

h(r)k ξ2k+1

(1 +

(7k − k2 + 10kr)

24√q

)(13)

with ξ = j 4

√4q , h

(r)k = (−1)r+k22k2r!

(2k)!(r−k)! , h(r)k =

(−1)r+k22k+1(2r+1)!(2k+1)!(r−k)! andAn a normalization constant. Notice

that the coefficientsh(r)k and h(r)

k are related to the Hermite

polynomialHn(x) by the relationsH2r(x) =∑r

k=0 h(r)k x2k

andH2r+1(x) =∑r

k=0 h(r)k x2k+1.

The eigenenergies are approximately given by

Elown ≈ Ω

4

−2q + 4

√q(n+

1

2)− (2n+ 1)2 + 1

8− ((2n+ 1)3 + 3(2n+ 1))

27√q

+O

(1

q

)(14)

If one neglects corrections of order1/√q and higher in Eqs.

(12) and (13) and keeps only the first two terms in Eq. (14),the expressions for the eigenmodes and eigenenergies reduceto

En = −Ωq/2 + Ω√q(n+

1

2) (15)

f(n)j ≈

√ √2

2nn! 4√

qπ2exp

(−ξ2

2

)Hn (ξ) . (16)

The above expressions correspond to the eigenvalues and

eigenenergies (shifted by−Ωq/2) of a harmonic oscillatorwith an effective trapping frequencyω∗ and an effective massm∗. The effective frequency and mass are given by

~ω∗ = Ω√q = ~ωT

√m

m∗, (17)

m∗ =~2

2Ja2, (18)

The harmonic oscillator character of the lowest energy modesin the combined lattice plus harmonic potential is consistent

4

with the fact that near the bottom of the Bloch band the disper-sion relation has the usual free particle form withm replacedby m∗. It is important to emphasize that the expressions forthe effective mass and frequency Eqs. (17) and (18) are onlyvalid in the tight-binding approximation.

Higher order terms introduce corrections to these harmonicoscillator expressions, that become more and more importantas the quantum numbern increases. These corrections comefrom the discrete character of the tight-binding equation.Theycan be calculated by replacingf (n)

j with f (n)(ξ) and takingthe continuous limit of the hopping term proportional toJ inEq.(8). This procedure yields:

(−1

2

∂2

∂ξ2− 1

12aho2∂4

∂ξ4+ · · ·+ 1

2ξ2)f (n) = Enf

(n)(19)

whereξ = j 4

√4q ≡ j

aho, aho =

√~

m∗ω∗= (q/4)1/4 is

a characteristic length of the system in lattice units, whichcan be understood as an effective harmonic oscillator length,and En = (En + 2J)/(~ω∗). To zeroth order in1/

√q,

(aho → ∞), the differential equation (19) reduces to theharmonic oscillator Schrodinger equation. Higher ordercorrections given in Eqs.(12), (13) and (14), can be calculatedby treating the higher order derivatives in Eq.(19) as aperturbation.

High-energy modes (n ≫ √q) in the highq regime

High energy modes are close to position eigenstates sincefor these states the kinetic energy required for an atom to hopfrom one site to the next becomes insufficient to overcomethe potential energy cost [10, 11, 12, 13]. By using asymp-totic expansions of the characteristic Mathieu values and func-tions, we obtain the following expressions for the spectrumand eigenmodes

Ehighn=2r ≈ Ehigh

n=2r−1 ≈ Ω

4

((2r)2 +

q2

2((2r)2 − 1)+

q4(7 + 5 (2r)2

)

32 ((2r)2 − 1)3((2r)2 − 4)

+ . . .

)(20)

f(high)j

n=2r≈ f

(high)j

n=2r−1≈ An

δj,r −

q

4

(δj,r−1

2r − 1− δj,1+r

1 + 2r

)+

q2

32

(δj,r−2

(2r − 2)(2r − 1)− 2(1 + 4r2)δj,r

(2r − 1)2(1 + 2r)2+

δj,2+r

(1 + 2r)(2 + 2r)

)±j → −j

(21)

with An normalization constants. We note that the asymp-totic expansions Eqs.(20) and (21) for the HE-modes in thehigh q regime are identical to the asymptotic expansions forthe modes in the lowq regime reported below. While the lat-ter are well known in literature, to our knowledge they havenever been used to describe the highq regime. We actuallyfound that as long asn ≫ √

q these asymptotic expansions re-produce reasonably well the exact results, even for very largeq. An example of this is given in Figs. 1 and 2, which arediscussed at the end of this section.

The high energy eigenstates are almost two-fold degeneratewith energy spacing mostly determined byΩ. In Ref. [12]the authors show that the localization of these modes can beunderstood by means of a simple semiclassical analysis. Byutilizing a WKB approximation, the localization of the modescan be linked to the appearance of new turning points inaddition to the classical harmonic oscillator ones for energiesgreater than2J . While classical harmonic oscillator turningpoints are reached at zero quasimomentum, the new turningpoints appear when the quasimomentum reaches the end ofthe Brillouin zone and can therefore be associated with Braggscattering induced by the lattice.

Intermediate states (n ∼ √q) in the highq regime

In order to reproduce accurately the energy spectrum in theintermediate regime one may expect that many terms in theasymptotic expansions have to be kept. To estimate the en-ergy range where the spectrum changes character from lowto high, we solve for the smallest quantum numbernc whoseenergy calculated by using the low energy asymptotic expan-sion is higher than the one evaluated by using the high energyexpansion. That is

Elownc−1 ≥ Ehigh

nc, (22)

wherenc is required to be even.Solution of Eq. (22) gives

nc ≈ 2‖√q/2‖, (23)

where‖x‖ denotes the closest integer tox.By comparison with the numerically obtained eigenvalues,

we actually find that using Eq. (14) forn < nc − 1 andEq. (20) forn ≥ nc − 1 is enough to reproduce the entirespectrum quite accurately. Moreover, since atnc the energy isapproximately given byEnc

≈ 2J , our analytical findings forthe transition between harmonic oscillator-like and localized

5

eigenstates are in agreement with the approximate solutionsfound in [11, 12] by using a WKB analysis.

FIG. 1: Upper panel (a): Spectrum of a particle in combinedquadratic and periodic potentials as a function of the quantum num-ber n. The quadratic trap frequency and the depth of the periodicpotential areωT = 2π × 60 Hz andVo = 7.4ER, respectively.Points are numerically obtained values, while crosses are asymptoticexpansions of the Mathieu characteristic parameters. The arrow in-dicates the critical valuenc ≈ 2‖

√q/2‖. Lower panel (b): Energy

differenceDEn between the numerically obtained eigenvalues andthe asymptotic expansions.

In Figs. 1 and 2 we compare the above asymptotic approx-imations to exact numerical results for the eigenenergies andeigenfunctions. The parameters used for the plots are thoseof a system with a lattice depth of7.4ER and quadratic trapfrequencyωT = 2π × 60Hz. These values correspond toJ = 0.0357ER, Ω = 0.0009ER and q = 157. Figure 1,upper panel, shows the lowest 35 eigenenergies as a functionof the quantum numbern. The valuenc, which is equal to16 in this case, is indicated by an arrow. The crosses rep-resent the asymptotic solutions, Eq. (14) and (20), and thedots the numerically obtained eigenvalues. On the scale ofthe graph there is essentially no appreciable difference be-tween the two solutions for the entire spectrum. The differ-ence between the the numerically obtained energies and theasymptotic expansions is plotted in the lower panel of Fig. 1.In the upper panel of Fig.2, asymptotic expressions for theLE eigenvectorsn = 2(boxes) andn = 5(crosses) are com-pared to the numerically obtained eigenmodes (triangles anddots respectively). The modes clearly exhibit an harmonicoscillator character, and the agreement between the asymp-totic and numerical solutions is very good. In the lower panel,then = 34 andn = 42 eigenstates belonging to the region

-20 -10 0 10 20j

-0.4

-0.2

0

0.2

0.4

f jHnL

asHn=34L

exHn=34L

asHn=42L

exHn=42L

-20 -10 0 10 20j

-0.4

-0.2

0

0.2

0.4

f jHnL

asHn=2L

exHn=2L

asHn=5L

exHn=5L

FIG. 2: Eigenmodes of a particle in a combined quadratic and peri-odic potentials as a function of the lattice sitej. The quadratic trapfrequency and the depth of the periodic potential areωT = 2π × 60Hz andVo = 7.4ER, respectively. Triangles and dots are numeri-cally obtained values (”ex” in the legend), while boxes and crossesare asymptotic expansions of the Fourier coefficients of theperiodicMathieu functions (”as” in the legend). In the upper panel, trianglesand boxes refer to then = 2 mode, while dots and crosses refer ton = 5. In the lower panel, triangles and boxes refer to then = 34mode, while dots and crosses refer ton = 42.

n > nc are depicted. These states are localized far from thetrap center. While the overall shape of the modes is well repro-duced by the asymptotic solutions, for the chosen values ofnsmall differences between the asymptotic (boxes and crossesrespectively) and numerical solutions (triangles and dotsre-spectively) can be observed. As expected, the convergenceof the asymptotic expansion to the exact solution is better forn = 42 than forn = 34, as the former has a larger value ofnthan the latter.

2. Lowq regime (4J < Ω)

This parameter regime is relevant for deep lattices. When4J < Ω the kinetic energy required for an atom to hop fromone site to the next one is insufficient to overcome the trappingenergy even at the trap center and all the modes are localized.This is consistent with the previous analysis, because when4J . Ω, nc is less than one. The asymptotic expressions thatdescribe this regime are [14]

6

FIG. 3: Upper panel (a): Spectrum of a particle in combinedquadratic and periodic potentials as a function of the quantum num-bern. The quadratic trap frequency and the depth of the periodic po-tential areωT = 2π×60 Hz andVo = 50.0ER, respectively. Pointsare numerically obtained values, while crosses are asymptotic ex-pansions of the Mathieu characteristic parameters in the low q limit.Lower panel (b): Energy differenceDEn between the numericallyobtained eigenvalues and the asymptotic expansions.

ELowqn=0 ≈ Ω

4

(−q2

1

2+ q4

7

128+ . . .

)

ELowqn=1 ≈ Ω

4

(4− q2

1

12+ q4

5

13824+ . . .

)

ELowqn=2 ≈ Ω

4

(4 + q2

5

12− q4

763

13824+ . . .

)

ELowqn=3 ≈ Ω

4

(16 + q2

1

30− q4

317

864000+ . . .

)

ELowqn=4 ≈ Ω

4

(16 + q2

1

30+ q4

433

864000+ . . .

)

ELowqn>5 ≈ Ehigh

n>5 (24)

and

fj(Low)n=0 ≈ A0

δj,0√8+

q√8δj,1 +

q2

32

δj,2 − δj,0√2

+j → −j

fj(Low)n=1 ≈ A1

δj,1 +

q

12δj,2 + q2

(δj,3384

− δj,1288

)

−j → −j

fj(Low)n=2 ≈ A2

δj,1 +

q

12

(δj,2 −

3

2δj,0

)+

q2(δj,3384

− 19δj,1288

)+j → −j

fj(Low)n≥3

= f(high)j

n≥3(25)

with An normalization constants. In Fig. 3 the above expan-sions for the energies are compared with the numerically cal-culated spectrum. Here the lattice is50ER deep and the exter-nal trap frequency isωT = 2π × 60Hz. These lead to valuesof J,Ω andq given byJ = 2.9 × 10−5ER, Ω = 0.0009ER

andq = 0.13. The asymptotic and numerical solutions per-fectly agree on the scale of the graph. The energy differencebetween the numerically obtained eigenvalues and the asymp-totic expansions is plotted in Fig. 3, lower panel.

III. CENTER OF MASS EVOLUTION OF A DISPLACEDSYSTEM

In recent experiments, the transport properties of one di-mensional Bose-Einstein condensates loaded in an optical lat-tice have been studied after a sudden displacement of thequadratic trap [19]. A strong dissipative dynamics was ob-served even for very small displacements and shallow depthsof the optical lattice. This should be contrasted with previ-ous experiments performed with weakly interacting 3D gaseswhere very small damping of the center of mass motion wasobserved for small trap displacements [20, 21]. Recent the-oretical studies have demonstrated that the strongly dampedoscillations observed in one dimensional systems reflect theimportance of quantum fluctuations as the dimensionality isreduced [22, 23, 24, 25].

In this section we study the dipolar motion of ideal bosonicand fermionic gases trapped in the combined lattice and har-monic potentials. We start by writing an expression for theevolution of the center of mass of an ideal gas with generalquantum statistics, and then we use this expression to studythe dipole oscillations for bosonic and fermionic systems.Thesimplicity of the noninteracting treatment allows us to deriveanalytic equations for the dipole dynamics for both statistics.

Later on, in section IV, we show how the knowledge of thebosonic and fermionic ideal gas dynamics can be useful indescribing the dynamics of the interacting bosonic system fora large range of parameters of the trapping potentials.

7

A. Ideal gas dynamics

Consider an ideal gas ofN atoms loaded in the ground stateof an optical lattice plus a quadratic potential initially dis-placed from the trap center byδ lattice sites. The initial stateof the gas is described by a mixed ensemble state with meanoccupation numbers determined by the appropriate quantumstatistics

zj(t = 0) =1

N

∑

n

nnf(n)j−δ, (26)

nn =1

eEn−µkBT ± 1

, (27)

whereT is the initial temperature of the system,kB is theBoltzmann constant,µ is the chemical potential which fixesthe total number of particles toN ,

∑nn = N , and the posi-

tive and negative signs in Eq.(27) are for fermions or bosons,respectively.

The time evolution of the center of mass of the gas is dic-tated by the ensemble average

〈x(t)〉 =1

N

∑

n

nn〈xn(t)〉 (28)

with

〈xn(t)〉 = a∑

k,l

c(n)l c

(n)k e−i(Ek−El)t/~

∑

j

jf(k)j f

(l)∗j

c(n)k =

∑

j

f(n)j−δf

(k)j , (29)

where the quantitiesf (n)j and the energiesE(n) correspond to

eigenvalues and eigenenergies of the undisplaced system. Thecoefficientsc(n)k are given by the projection of then exciteddisplaced eigenstate onto thek excited undisplaced one.

Once theEn andf (n)j are known, the center of mass evo-

lution can be calculated. In the following we discuss the zerotemperature dynamics for the ideal bosonic and fermionic sys-tems.

1. Bosonic system

At zero temperature the bosons are Bose condensed andnn = Nδn0, whereδn0 is the Kronecker delta function. Thecenter of mass motion is then given by

〈x(t)〉 = a∑

k,l

c

(0)l c

(0)k e−i(Ek−El)t/~

∑

j

jf(k)j f

(l)∗j

c(0)k =

∑

j

f(0)j−δf

(k)j . (30)

If the initial displacement of the atomic cloud is small,2δ ≪ nc, and the lattice is not very deep (q ≫ 1), local-ized eigenstates are initially not populated. Then, only low-energy states are relevant for the dynamics and the latter canbe modeled by utilizing the asymptotic expansions derived inSec. II B. To simplify the calculations, we use the harmonicoscillator approximation for the eigenmodes (Eq.(16)), and in-clude up to the quadratic corrections inn in the eigenenergies,which corresponds to keep the first three terms of Eq.(14)).Even though this treatment is not exact, we found that it prop-erly accounts for the period and amplitudes of the center ofmass oscillations for small trap displacement. After some al-gebra it is possible to show that the time evolution of the centerof mass is given by

〈x〉 = aδe−

(δ2

a2ho

sin2(Ωt8~ )

)

cos

(ω∗ot−

δ2

2a2hosin

(Ωt

4~

))

(31)

with ~ω∗o = ~ω∗ − Ω/4.

In Fig. 4 we plot the average center of mass position inunits of the lattice spacinga as a function of time for an idealbosonic system of atoms with87Rb mass. The solid line is ob-tained by numerically solving the tight binding Schrodingerequation, Eq.(3), while the dotted line is the analytical so-lution Eq.(31). For the plot we usedVo = 7.4ER, ωT =2π × 60Hz and δ = 3. The time is shown in units ofTo = 2π/ω∗, a characteristic time scale. The two solutionsexhibit very good agreement for the times shown.

The modulation of the dipole oscillations predicted byEq.(31) can be observed in the plot. At early times,t ≪ ~/Ω,the amplitude decreases exponentially asexp(−Γot

2), with

Γo =(

Ωδ8~aho

)2, and the frequency is shifted fromω∗ by

Ω4~ (1 − δ2

2a2ho

). The initial decay does not correspond to real

damping in a dissipative sense, as in a closed system the en-ergy is conserved. The decay is just an initial modulation andafter some time revivals must be observed. Because in thelargeq limit ω∗

0 ≪ Ω/~, the revival time is approximatelygiven by4h/Ω.

It is a general result that the dipole oscillations of a harmon-ically confined gas in absence of the lattice are undamped.The undamped behavior holds independently of the temper-ature, quantum statistics and interaction effects (generalizedKohn theorem [26]). Equation (31) shows how this result doesnot apply when the optical lattice is present even for an idealBose gas.

Recent experimental developments have opened the possi-bility to create a non-interacting gas for any given strength ofthe trapping potentials [27, 28]. The techniques utilize Fes-hbach resonances for tuning the atomic scattering length tozero. These developments should allow for the experimentalobservation of the modulation of the dipole oscillations ofanideal gas predicted in this section.

8

0 10 20 30 40tTo

-3

-2

-1

0

1

2

3

xa

exact analytic

FIG. 4: Center of mass motion in lattice units as a function oftimefor an ideal bosonic gas. In the plot,Vo = 7.4ER,ωT = 2π×60Hzandδ = 3. The time has been rescaled byTo = 2π/ω∗. The solidand dotted lines are the numerical and analytical solution Eq. (31),respectively.

0 10 20 30 40tTo

-3

-2

-1

0

1

2

3

xa

analytic

exact

FIG. 5: Center of mass motion in lattice units as a function oftimefor N = 15 fermions. HereωT = 2π×20Hz andVo = 7.4ER andδ = 3. The time is in units ofTo = 2π/ω∗. The solid and dottedlines are the numerical and analytical solution Eq. (34), respectively.

2. Fermionic system

At zero temperature the Pauli exclusion principle forcesfermions to occupy the lowestN eigenmodes, and thereforenn = 1 for 0 ≤ n ≤ N − 1 and zero elsewhere. The oc-cupation of the firstN displaced modes makes the conditionof occupying only low-energy eigenstates of the undisplacedpotentials more restrictive than in the bosonic case. Neverthe-less, if the initial displacement, lattice depth and atom numberare chosen such that only LE eigenstates are initially popu-lated,

∑∞n=nc−1 |c

(N−1)n |2 ≪ 1, it is possible to derive simple

analytic expressions for the dipole dynamics. This is the fo-cus of the remainder of this section. Population of localizedstates considerably complicates the system’s dynamics andanumerical analysis is therefore required. This is postponed toSec.IV B 2.

When only LE undisplaced eigenstates are occupied, asexplained for the bosonic system, to a good approximationthe eigenmodes can be assumed to be the harmonic oscilla-tor eigenstates and only corrections quadratic in the quantumnumbern are relevant in Eq.(14). After some algebra, theabove approximations yield the following expression for thetime evolution of the center of mass:

〈x(t)〉 =1

N

N−1∑

n=0

〈xn(t)〉 (32)

〈xn(t)〉 ≡ aδRe

(exp

iω∗

ot−(δ2(1− χ(t))

2a2ho

)xn(t)

)(33)

xn(t) ≡n−1∑

k=0

((δ

aho

)2k(χ(t)− 1)2kχ(t)n−1−k((n+ 1)χ(t) + k − n)

(2k)!!(k + 1)!

k∏

s=1

(n+ 1− s)

)+

(δ

aho

)2n(1− χ(t))2n

(2n)!!(34)

whereχ(t) = exp(−iΩt/(4~)) and~ωo∗ = ~ω∗ − Ω/4.

The parameterχ(t) takes into account the quadratic cor-rections to the harmonic oscillator energies. The correctionsare proportional toΩ/4, and due to the presence of the lat-

tice. In the limitχ(t) → 1, xn(t) → 1 ∀n and therefore theamplitude of the dipole oscillations remains constant in time,〈x〉 = dδ cos(ω∗t), as predicted by Kohn theorem. The cor-

9

FIG. 6: Center of mass motion in lattice units as a function oftime for some initially occupied modes. The modes are labeled by the quantumnumbern. As in Fig. 5, the parameters areωT = 2π × 20Hz, Vo = 7.4ER, N = 15, δ = 3 and the time is in units ofTo = 2π/ω∗. Thesolid line corresponds to the numerical solution and the dotted line to the solution given by Eq. (34). When the center of mass evolution of allthe different modes is added, fromn = 0 to n = N − 1 one recovers the total center of mass evolution shown in Fig.5.

rections quadratic inn cause the modulation of center of massoscillations.

The modulation is caused not only by the over-all envelope generated by the exponential termexp

(−δ2(1− χ(t))/(2a2ho)

), which was also present in

the bosonic case, but mainly from the interference createdby the different evolution of theN average positions〈xn〉 inthe sum Eq.(32). The latter induces a fast initial decay of theamplitude of the dipole oscillations.

In Fig.5 we plot the center of mass motion of the fermionicgas composed ofN = 15 atoms with the mass of87Rb. Thesolid and dotted lines correspond to the numerical and analytic

solutions, respectively. Here the depth of the optical lattice is7.4ER, andωT = 2π × 20Hz. The amplitude of oscillationshows a rapid decay in time. The analytic solution capturesthe overall qualitative behavior of the numerical curve. Nev-ertheless, only at short times the agreement is quantitativelygood. Population of eigenstates which are not fully harmonicin character is responsible for the disagreement at later times.This effect is particularly relevant for the evolution of the dis-placed states with larger quantum number, as explicitly shownin Fig. 6 where the time evolution of some displaced modes isplotted. Again, the solid line is the numerical solution andthedotted line is the analytic one. For the lowest energy modes,

10

n = 0 andn = 3, the agreement between the two curves isalmost perfect. For the higher energy modesn = 6 andn = 9the analytic solution is underdamped and overestimates thecollapse time.

Interestingly, the dynamics of the displaced excited modesexhibits an initial growth of the amplitude. This behavioris a pure quantum mechanical phenomenon due to theconstructive interference between the different phases oftheundisplaced eigenmodes during the evolution. We explicitlychecked for energy conservation during the time evolution.The amplitude increase is captured by the analytic solutionand it allowed us to show that the growth happens onlywhen the ratio between the initial displacementδ and ahois less than one. While such a behavior is not observable inthe evolution of a fermionic cloud, as the observable is thecenter of mass position summed over all initially populatedmodes〈x(t)〉, the experimental observation of growth for anindividual mode may be possible if an ideal bosonic gas isinitially loaded in a particular excited state, and then suddenlydisplaced.

As described above, the evolution of LE modes can behandled analytically. On the other hand, when high-energyeigenmodes are populated the dynamics is much more com-plicated. Nevertheless, there is another simple limiting casethat can be actually solved. This corresponds to the casewhen the displacement is large enough or the lattice deepenough that the displaced cloud has non-vanishing projec-tion amplitudes only onto high-energy undisplaced modeswhich can be roughly approximated by position eigenstates,f2r,2r−1j ≈ (δj,r±δ−j,r)/

√2. Then, one finds that〈xn〉 ≈ aδ

for all n and thus〈x〉 ≈ aδ. That is, when only high-energyeigenmodes are populated the dynamics is completely over-damped and the cloud tends to remain frozen at the initialdisplaced position. We show later on in Sec. IV B 2 wherewe treat interacting atoms, that in the so-calledMott insulatorregime most populated modes are actually localized, and thiskind of overdamped behavior characterizes the dipole dynam-ics.

IV. MANY-BODY SYSTEM

A. Spectrum of the BH-Hamiltonian in presence of an externalquadratic potential

The Bose-Hubbard (BH) Hamiltonian describes the sys-tem’s dynamics when the lattice is loaded such that only thelowest vibrational level of each lattice site is occupied [29]

HBH =∑

j

[Ωj2nj − J(a†j aj+1 + a†j+1aj) +

U

2nj(nj − 1)

].

(35)Here aj(a

†j) is the bosonic annihilation(creation) operator of

a particle at sitej and nj = a†j aj. Ω andJ are defined asin Eqs. (4) and (5), whileU is an on-site interaction energygiven byU = 4πas~

2

m

∫dx|w(x0)|4, with as the s-wave scat-

tering length andw0(x) the first-band Wannier state centered

at the origin. The quantityU is the energy cost for having twoatoms at the same lattice site. The tunneling rate decreasesforsinusoidal lattices with the axial lattice depthVo as

J = A

(Vo

ER

)B

exp

(−C

√Vo

ER

)ER, (36)

where the numerically obtained constants areA = 1.397,B = 1.051 andC = 2.121. The interaction energy increaseswith Vo as

U = βER

(Vo

ER

)1/4

, (37)

where β is a dimensionless constant proportional toas.In current experiments, the one-dimensional lattice is ob-tained by tightly confining in two directions atoms loadedin a three-dimensional lattice. In this caseβ =4√2π(as/λ)(V⊥/ER)

1/2, whereV⊥ is the depth of the latticein the transverse directions [30, 31]. The parameterγ = U/Jtherefore increases as a function of the axial lattice depthas

γ(Vo) =β

A

(Vo

ER

)1/4−B

exp

(C

√Vo

ER

). (38)

In the absence of the external quadratic potential, thebosonic spectrum is fully characterized by the ratio betweenthe interaction and kinetic energiesγ and the filling factorN/M , whereM is the number of lattice sites [32]. Forγ ≪ 1and anyN/M ratio the system is weakly interacting and su-perfluid. Forγ ≫ 1 andM ≥ N the system fermionizesto minimize the inter-particle repulsion. In this regime thebosonic energy spectrum mimics the fermionic one, the corre-spondence being exact in the limit of infinitely strong interac-tions. In particular, in a lattice model the onset of fermioniza-tion is characterized by suppression of multiple particle occu-pancy of single sites. This implies that fermionization occursfor eigenstates whose energy is lower than the interaction en-ergyU . If M > N there areM !/(N !(M −N)!) fermionizedeigenmodes and the dynamics at energies much lower thanUcan be accounted for by using these states only. On the otherhand, if the lattice is commensurately filled,N = M , thereis only one fermionized eigenstate, and it corresponds to theground state. All excited states have at least one multiply oc-cupied site, and therefore excitations are not fermionized. Theground state corresponds to the Mott state with a single par-ticle per site and reduced number fluctuations. The transitionfrom the superfluid to the Mott state is a quantum phase tran-sition, and the critical point for one-dimensional unit filledlattices isγc ≃ 4.65 [32, 33].

In the presence of the quadratic trap the spectrum is deter-mined by an interplay ofU , J , Ω andN . In trapped systemsthe notion of lattice commensurability becomes meaninglessbecause the size of the wave-function is explicitly determinedby these parameters. As a consequence, for any value ofNthe ground state can be made to be a Mott insulator with oneatom per site at the trap center by an appropriate choice ofU , J andΩ [34], and the lowest energy modes can always be

11

0 2 4 6 8 10 12 14 16V

o / E

R

0

0.01

0.02

0.03

0.04

0.05(E

-Eo)

/ ER

γ10 30 1501 100

FIG. 7: Energy spectra as a function of the depth of the axial opticallattice Vo. The continuous, dotted and dashed lines correspond toN = 5 interacting bosons, non-interacting bosons and fermions,re-spectively. The dashed-dotted line corresponds to~ω∗. The horizon-tal axis on the top of the figure isγ = U/J , and is only meaningfulfor interacting bosons. For each energy spectrum, the correspondingground-state energyE0 has been subtracted.

made to be fermionized in the largeU limit. The purpose ofthis section is to characterize fermionization and localizationof the many-body wave-function when both the quadratic andperiodic potentials are present, by relating the occurrenceof the different regimes to changes in the spectrum at lowenergies.

We performed exact diagonalizations of theBH-Hamiltonian forN = 5 particles andM = 19 sites inpresence of a quadratic trap of frequencyωT = 2π× 150 Hz.For the calculations, we chose87Rb atoms with scatteringlengthas = 5.31 nm, a lattice constanta = 405 nm, andthereforeΩ ≃ 0.0046ER. We fixed the transverse latticeconfinement toV⊥ ≃ 25.5 Er and varied the depth of theoptical latticeVo in the parallel direction from 2ER to about17 ER. For these lattice depths, the energiesU andJ bothvary so that their ratioγ increases from3.3 to about150.Due to the changes inJ , the ratioq = 4J/Ω characterizingthe single-particle solutions decreases from 130 to 4 with in-creasingVo. The effective harmonic-oscillator energy spacing~ω∗ decreases approximately from 0.0525ER to 0.0092ER.We used these parameters because they are experimentallyfeasible and fulfill the conditionU − Ω((N − 1)/2)2 > 0for the entire range of the trapping potentials. Later on wediscuss that, for deep enough lattices, fulfillment of the lastinequality ensures the existence of an energy range in whicheigenmodes are fermionized.

Figure 7 shows the lowest eigenergies of theBH-Hamiltonian as a function ofVo. The continuous line is theexact solution forN = 5 interacting bosons. The dottedand dashed lines are the exact spectra for 5 non-interactingbosons and fermions, respectively. They have the same massand are trapped in the same potentials as the interacting

bosons. Their spectra are shown for comparison purposes.For each spectrum the energyE0 of the ground state has beensubtracted.

In the absence of the optical lattice,Vo = 0, the energydifference∆E0 = E1 − E0, or energy spacing, between thefirst excited and ground state equals the harmonic oscillatorlevel spacing~ωT , independent of statistics and interactionstrength. Figure 7 show that this is no longer the case in thepresence of the lattice.

The dependence of∆E0 on statistics is evident in theplot, as the level spacing is different for ideal bosons andfermions. In particular, the energy spacing for bosons is onlyshifted from~ω∗ =

√4ΩJ (dashed-dotted line) by an amount

which is almost constant for all lattice depths, while∆E0 forfermions clearly deviates from~ω∗, especially for deep lat-tices. The behavior of∆E0 in the two cases can be understoodby using the asymptotic solutions of the single-particle prob-lem. For ideal bosons∆E0 is equal to the energy differencebetween the first-excited and ground single-particle eigenen-ergies. The ratioq used for the plots is such that the criticalvaluenc of Eq. (23) is always larger than 2, and therefore theground and first-excited eigenenergies are well described byEq. (14). The calculation of the energy difference using thisequation yields∆E0 ≈ ~ω∗ − Ω/4. For fermions,∆E0 isequal to the difference between the energies of then = N andn = N−1 single-particle excited states. ForVo < 9.6ER, thecritical valuenc is larger thanN , and therefore the energiesof then = N andn = N − 1 single-particle excited statesare also well described by Eq. (14). Then,∆E0 for fermionsis smaller than for bosons because lattice corrections are moreimportant for higher quantum numbers, and have all negativesign. On the other hand, forVo > 9.6ER, nc is smaller thanN − 1 and the energies of then = N andn = N − 1 single-particle excited states are described by Eq. (20). The transitionof the single-particle eigenmodes at the Fermi level from LEto HE aroundV0 = 9.6ER is signaled by the minimum of∆E0 for fermions. In general, an estimate for the value ofJat which the minimum takes place is

J ∼ Ω((N − 1)/2)2/2. (39)

This value is obtained by equating the Fermi energyEN−1,which is of orderΩ((N−1)/2)2 from Eq. (20), toEnc

, whichis approximately2J . The transition of the single-particleeigenmode at the Fermi level from LE to HE is also con-nected to the formation of a region of particle localizationat the trap center in the many-body density profile. As ex-plained in [13], whenEN−1 is equal toEnc

the on-site den-sity in the central site of the trap approaches 1 with reducedfluctuations. ForV0 > 9.6ER, Fig. 7 shows that∆E0 ap-proaches an asymptotic valueΩN , value that can be derivedfrom Eq. (20). When the asymptotic valueΩN is reached,most single-particle states below the Fermi level are localized,and this yields a many-body density profile withN unit-filledlattice sites at the trap center.

The dependence of the first excitation energy on interac-tions can also be seen in Fig. 7. In fact, by comparing∆E0

12

for the interacting bosons to the value of∆E0 for ideal bosonsand fermions, three different regimes can be considered:1 .γ . 10, 10 . γ . 30 andγ > 30. These regimes cor-respond to the intermediate, fermionized-non-localized andfermionized-Mott regimes, respectively. The weakly inter-acting regimeγ . 1 is only reached forVo ≪ 2ER forour choice of atoms and trapping potentials. For such lat-tice depths the tight-binding approximation is not valid, andtherefore we do not show the spectra for this regime in Fig. 7.In the following we discuss the main features of the differentregimes focusing on the connection to the ideal bosonic andfermionic systems.• Forγ ≤ 1 the interacting bosonic system is in theweakly

interactingregime. In this regime the first excitation energyis almost the same as the ideal bosonic one. Most atomsare Bose-condensed, interaction-induced correlations can betreated as a small perturbation, and the spectrum can be shownto be well reproduced by utilizing Bogoliubov theory [35].• For1 < γ < 10, the system is in theintermediateregime,

where∆E0 for interacting bosons deviates from the idealbosonic energy spacing and approaches the ideal fermionicone. Indeed, Fig.7 shows that forV0 . 4ER, ∆E0 for theinteracting bosons lies closer to the ideal bosonic energy spac-ing, while forV0 > 4 it lies closer to the ideal fermionic one.In the presence of the optical latticeγ increases exponentiallywith Vo, and therefore the intermediate regime occurs for arelatively small range of accessible trapping potentials,herefor 1 . Vo/ER . 5.5.• Forγ ≥ 10 the interacting spectrum approaches the ideal

Fermi spectrum and the system is in thefermionizedregime.The numerical solutions show that the energy difference be-tween the energy spectra of interacting bosons and fermionsis of the order ofJ2/U and slightly increases for larger fre-quencies of the quadratic trap.

In general, fermionization in the presence of the externalquadratic potential occurs forN < M when the two followinginequalities are satisfied

γ = U/J ≫ 1, U > Ω((N − 1)/2)2. (40)

While the first inequality is the same as for homogeneous lat-tices and relates to the building of particle correlations,thesecond inequality is specific to the trapped case and relatestosuppression of double particle occupancy of single sites. If theinteraction energyU is larger than the largest trapping energy,which corresponds to trapping an atom at position(N − 1)/2,it is energetically favorable to have at most one atom per well.The average on-site occupation is therefore less or equal toone.

The second inequality in Eq. (40) poses some limitationson the choice of possibleΩ andU for a given number oftrapped particles. For our choice of the trapping potentials,this inequality is satisfied for anyVo. Indeed, this is notan unrealistic assumption. In recent experiments with87Rbatoms, an array of fermionized gases has been created withat most 18 atoms per tube [4]. For suchN , the conditionU > Ω((N − 1)/2)2 can be fulfilled for many differentchoices of experimentally feasible trapping potentials.

-6 -4 -2 0 2 4 6j

0

0.2

0.4

0.6

0.8

1

nj

FIG. 8: Density profilesnj ≡ 〈nj〉 as a function of the lattice sitejfor different lattice depths. The dash-dotted, dashed and dotted linescorrespond to interacting bosons, while the boxes, dots andtrian-gles correspond to ideal fermions forVo/ER = 7, 12 and15 latticedepths, respectively.

-6 -4 -2 0 2 4 6j

0

0.1

0.2

0.3

0.4

0.5

Dn

j

FIG. 9: Number fluctuations∆nj ≡√

〈n2

j 〉 − 〈nj〉2 as a function

of the lattice sitej for different lattice depths. Conventions and pa-rameters are the same as in Fig.8

• For γ > 30 the system enters the fermionized-Mottregime. In fact, Fig. 7 shows that forγ > 30 the energy spac-ing for the interacting bosons (and also for ideal fermions)begins to increase until it reaches an asymptotic value atγ ≈ 150.

The approaching of the asymptotic value signals the forma-tion of a localized many-body state for the interacting bosons,the so-called Mott insulator state. The relevant relation be-tweenN , Ω andJ for the formation of an extended core ofunit-filled sites at the trap center with fluctuations mainlyatthe outermost occupied sites is

ΩN ≥ J. (41)

This is explained asΩN is the energy cost for moving a par-ticle from position|(N − 1)/2| to position|(N + 1)/2| at theborders of the occupied lattice and it is the lowest excitationenergy deep in the Mott state.

Figure 7 shows that forγ ≈ 150 the energies of the firstfour excited states become degenerate. This degeneracy

13

occurs because deep in the Mott regime the energy requiredto shift all the atoms of one lattice site to the right or leftis the same as the energy required for moving an atomfrom site±(N − 1)/2 to site±(N + 1)/2. For γ ≈ 150,J is approximatelyΩ, and therefore the tunneling energyis barely sufficient to overcome the potential energy costΩ for moving a particle from the central site of the trapto one of its neighbors. In the single-particle picture,whenJ ≈ Ω all the single-particle states belowEN−1 are localized.

In order to better understand the formation of the Mott in-sulator, in Figs. 8 and 9 the on-site particle numbernj = 〈nj〉and fluctuations∆nj =

√〈n2

j 〉 − 〈nj〉2 are plotted as a func-

tion of the lattice sitej, for different lattice depthsVo. Thelines and symbols are the results for interacting bosons andideal fermions, respectively. All theVo-values are such thatthe interacting bosonic system is fermionized. This is mir-rored by the overall good agreement between lines and sym-bols for all the curves. The plots show that forVo = 7ER(γ =15, dashed-dotted line), the largest average particle occupa-tion is n0 ≈ 0.7, and number fluctuations are of the orderof 0.5 in the central 9 sites, while forVo = 12ER(γ = 54,dashed line),n0 approaches 1 in the central 3 sites, and fluc-tuations at the trap center drop to a value∆nj ≈ 0.1. Thesharp drop in particle fluctuations clearly signals the localiza-tion at the trap center, and is in agreement withJ < ΩN forγ = 54. ForVo = 15ER(γ = 104, dotted line), the Mott stateis formed, as the mean particle number in the five central sitesis one, with nearly no fluctuations. Fluctuations are largeratlattice sites far from the center, and due to the tunneling ofparticles to unoccupied sites.Because of the small number of atoms that we use in the cal-culations,Ω((N − 1)/2)2/2 andΩN are of the same order ofmagnitude. It is therefore not possible to clearly distinguishthe value ofJ for which the on-site density at the trap centerapproaches one from the value ofJ for which the Mott stateis fully formed at the trap center, with reduced number fluc-tuations in the centralN sites. Preliminary results obtainedwith a Quantum Monte-Carlo code, Worm Algorithm [41],confirm the existence of these two distinct parameter regionswhen more atoms are considered, and therefore the usefulnessof both the energy scalesΩ((N − 1)/2)2/2 andΩN for inter-acting bosons. These results will be published elsewhere.

B. Many-body dynamics

In this section we study the temporal dipole dynamics of aninteracting bosonic system composed of 5 atoms trapped ina combined quadratic plus periodic confinement. The role ofinteractions on the effective damping of the dipole oscillationsis studied by means of exact diagonalization of the Hamilto-nian. As discussed when dealing with the ideal gas dynamics,such damping is effective because it is due to dephasing anddoes not have a dissipative character.

Assuming the system is initially atT = 0, the evolution ofthe center of mass is given by:

〈x(t)〉 =∑

l,k

Alkeiωlkt (42)

Alk ≡ a∑

j

j〈φl|nj |φk〉C∗l Ck, (43)

with ~ωlk ≡ El − Ek and whereEl and|φl〉 are eigenvaluesand eigenmodes of theBH-Hamiltonian. The coefficientsCl

are the projections of the initial displaced ground state|ϕ(0)〉onto the eigenfunctions|φ〉l of the undisplaced Hamilto-nian,Cl = 〈φl|ϕ(0)〉.

For the exact evolution, the ground-state|ϕ(0)〉 is calcu-lated by shifting the center of the quadratic trap byδ latticesites. The number of wellsM is 19 for all simulations, whichfixes the size of the Hilbert space to 33649. In the time prop-agation we only keep those eigenstates whose coefficientsCl

are such that|Cl|2 > 10−3. The typical number of statesthat fulfil this requirement is about 100. The accuracy of thetruncation of the Hilbert space during the time propagationhas been checked by increasing the number of retained states,finding no appreciable changes in the dynamics.

We are interested in the dynamics both when a Mott insu-lator state is not and is present at the trap center. These twocases are discussed in Secs.IV B 1 and IV B 2, respectively. Inparticular, in Sec. IV B 1 the interaction strengthU is varied,while the ratioq, specifying the ideal gas dynamics, is largeand constant. For the chosen values ofq andN , J > ΩNand for increasingU the system fermionizes without forminga Mott insulator at the trap center. In Sec. IV B 2, the dynam-ics of systems that do exhibit a Mott insulator in the largeUlimit is analyzed. In this case,J andU are simultaneouslyvaried by increasing the axial optical lattice depth.

1. Non-localized dynamics

In the absence of the optical lattice, the equations of mo-tion for the center of mass are decoupled from those of therelative coordinates. As only the latter are affected by inter-actions, all modes excited during the collective oscillationshave the harmonic oscillator energy spacing~ωT , and there-fore 〈x(t)〉 = 〈x(0)〉 cos(ωT t).

When the lattice is present, the equations of motion forthe center of mass and relative coordinates are coupled, andthus the many-body dynamics is interaction dependent. Inthis section we fixωT = 2π × 100Hz andVo = 7ER, andstudy the role of interactions in the many-atom dynamics byvaryingγ from 0 to 200, for constantq = 77. This can beexperimentally realized by tuning the scattering length ofthesystem by means of Feshbach resonances. In the following,we analyze the weakly interacting, intermediate and stronglyinteracting regimes separately.

Weakly interacting regime:γ ≤ 1In order to study the role of interactions in the weakly in-

teracting regime, in Fig. 10 the effective damping constantof

14

dipole oscillationsΓ is shown as a function ofγ. The dampingconstant was calculated by fitting the first 10 center of massoscillations to the ansatz〈x(t)〉 = aδ exp (−Γt2) cos(ωt),whereΓ andω are fitting parameters. This ansatz is cho-sen in analogy to the non-interacting model. The effectivedampingΓ is in units ofΓo = δ2Ω2/(8~aho)

2, which is thedamping constant predicted by Eq. 31. The solid and dottedlines correspond toΓ as calculated by means of exact diag-onalizations and by numerically evolving the followingDis-crete Non-Linear Schrodinger Equation(DNLSE) for the am-plitudeszj

i~∂zj∂t

= −J(zj+1 + zj−1) + Ωj2zj + U |zj |2zj , (44)

respectively. Eq.(44) has been obtained by replacing the fieldoperatoraj with the c-numberzj(t) in the Heisenberg equa-tion of motion foraj . Such replacement is justified forγ ≪ 1as the many-body state is almost a product over identicalsingle-particle wave-functions. The amplitudeszj satisfythe normalization condition

∑j |zj|2 = N . The initial state

used in the evolution of the DNLSE was found by numericallysolving for the ground state of Eq. (44), displaced byδ = 2lattice sites.

0 0.2 0.4 0.6 0.8 1Γ

0

0.2

0.4

0.6

0.8

1

GG

0

DNLSE

Exact

FIG. 10: Effective damping constant of the dipole oscillations as afunction ofγ for a system in the weakly interacting regime. Here,q ≈ 77, ωT = 100 Hz andΓo = (δΩ/8~aho)

2.

In Fig. 10, the continuous and dotted lines overlap forγ ≤ 0.05, and show a decrease in the damping constant withincreasing interaction strength in the whole rangeγ ≤ 0.2.For values ofγ > 0.05 the mean field and exact solutions startto disagree. While the exact solution has a minimum aroundγ ≈ 0.2, and then grows for largerγ values, the mean fieldcurve decreases monotonically to zero.

The fact that the mean-field solution decreases to zero forincreasing interactions is explained by noting that when inter-action effects become important the density profile acquiresthe form of an inverted parabola, orThomas-Fermiprofile,|zTF

j (t)|2 ≈ Ω((j − 〈x(0)〉/a)2 −R2

TF

)/U , 〈x(0)〉 = aδ.

Substitution of the Thomas-Fermi profile in Eq. (44) leads tothe exact cancellation of the quadratic potential, and thus, inthe frame co-moving with the atomic cloud, the atoms feel aneffective linear potential. The spectrum of a linear plus peri-odic potential is known to be equally spaced [36], and there-fore no damping due to dephasing is expected.

It is important to note that the mean-field undamped oscil-lation occurs only in a parameter regime far from dynamicalinstabilities. In fact, as shown in previous theoretical andexperimental studies [37, 38, 39], when the initial displace-ment is large enough to populate states above half of thelattice band-width, mean-field dynamical instabilities inducea chaotic dipole dynamics. In the framework of this paper,this critical displacement corresponds toδ ≈ nc/2. Forδ > nc/2, the initial ground-state has a significant overlapwith localized eigenmodes of the undisplaced system, whichare therefore populated during the dipolar dynamics, causingdamping. The importance of the population of these modesis enhanced by the non-linear term, which causes an abruptsuppression of the center of mass oscillations at the criticalpoint in the mean-field solution.

While the mean-field anaysis accounts for the decrease inthe damping constant, Fig. 10 shows that it is not accurate forγ & 0.2. This is due to the fact that the mean-field analysiscompletely neglects interaction-induced correlations. Thesecorrelations are responsible for the quantum depletion of thecondensate, which causes some atoms to be excited to higher-energy single-particle eigenmodes which are more affectedbythe lattice. To show this effect in a quantitative manner, inFig.11 we plot the probability distribution of the frequenciesωlk, Eq. (42), for some values ofγ. In the histograms, theheight of a bar-chart centered at a given frequencyω is theoccurrence probability ofωlk. The probability is proportionalto the normalized sum over all the weight factors|Alk| whosefrequency lies betweenωlk − 0.0025 andωlk + 0.0025. Thefrequencies are in units of the effective harmonic oscillatorfrequencyω∗ of Eq. (17). This approach is similar to the oneused in Ref. [42], where the strength function is used to studythe collective dynamics induced by mono- and dipolar excita-tions.

The histogram for the caseγ = 0 shows a frequency distri-bution with most frequencies in the interval0.85ω∗ ≤ ωlk ≤0.96ω∗. In particular, two large peaks are observed in therange0.9ω∗ ≤ ωlk ≤ 0.96ω∗. This is to be compared tothe case where the lattice is not present, and a single peak atω∗ is expected. The observed frequency spread is due to themodification of the harmonic oscillator spectrum introducedby the lattice and is responsible for the observed damping inthe ideal bosonic gas, as explained in detail in Sec. III A 1.

Figure 11 also shows that forγ = 0.3 the system has anarrower frequency distribution. In this case approximately100% of the frequencies lie in the interval0.9ω∗ ≤ ωlk ≤0.96ω∗. The frequency narrowing fromγ = 0 to γ = 0.3 isconsistent with the decrease in the damping constant shownin Fig. 10. For larger values ofγ, γ = 1 and2, some modeswith frequency smaller than0.7ω∗ and larger than1.1ω∗ startto contribute to the collective dynamics. Population of suchmodes is related to the depletion of the condensate and sig-nal the increased importance of quantum fluctuations in thesystem.

Finally, we note that for our choice ofΩ, N and J ,Ω((N − 1)/2)2/J is approximately0.2, which is the valueof γ = U/J at which the mean-field and exact solutions

15

FIG. 11: Probability distribution of frequencies for different values ofγ, with q ≈ 77 andωT = 100 Hz. The frequenciesωlk are in units ofω∗.

begin to disagree. This suggests that the fulfillment of thesecond inequality in Eq. (40),U > Ω((N − 1)/2)2, is re-lated to the failure of the mean-field approach, even forγ < 1.

Intermediate Regime:1 < γ < 10

In Fig. 12 the numerically obtained damping constant isshown forγ > 1. In this parameter regime we find that thefunctionaδ exp(−Γt2) cos(ωt) does not provide a good fit tothe center of mass evolution. Instead, we find that a better fit-ting ansatz is given byaδ exp(−Γt) cos(ωt) . In the plot, thedamping constant is normalized toΓ1, which is the dampingrate atγ = 1.

We observe that for2 ≤ γ < 5 the damping is almostconstant. This is consistent with the fact that by inspectionthe spectrum of excited frequencies has a similar shape and

width in the entire transition region. An example of such afrequency distribution is given in Fig. 11 forγ = 4. Thedominant peak is aroundωlk ≈ 0.9ω∗, while multiple peaksare noticeable between0.7ω∗ ≤ ωlk ≤ 0.92ω∗. The overallenvelope of the distribution has a long tail, as opposed to thecaseγ = 1 where all the weight is roughly concentrated injust two frequencies.

The increased importance of the tails of the distributionfor γ > 1 qualitatively accounts for the transition from anexponential decay quadratic in time towards an exponentialdecay which is linear in time. In fact, forγ < 1 the proba-bility distribution of frequencies may be fitted by a Gaussian,while for γ > 1 a better fit is provided by a Lorentzian-likeprofile. The Fourier transforms of such distributions giveprecisely the observed functional form of the decay of the

16

1 10 100Γ

2

10

1.

3.

5.

7.

GG

1

FIG. 12: Effective damping rate of the dipole oscillationsΓ as afunction of γ in the intermediate and strongly correlated regimes.Here,q ≈ 77 andωT = 100 Hz. The damping rate has been rescaledsuch thatΓ(γ = 1) = 1. The large and small dots are for interactingbosons and non-interacting fermions, respectively. The continuousline is a guide for the eye. The dashed line is the best-fit curveΓ =10.13 exp(−16.4γ−1.25) to the exact damping rate for interactingbosons.

dipole oscillations.

Strongly interacting regimeγ > 10

In Fig. 12 for γ > 10, the damping rate is shown torapidly increase and approach a finite asymptotic value whichis depicted in the plot by a dotted line. This asymptoticvalueΓ∞ corresponds to the damping rate calculated for anideal fermionic gas. The fermionic damping rate is con-stant because hereJ andΩ are kept constant whileU in-creases. The tendency to approach the fermionic dampingrate asγ increases is a consequence of fermionization of thebosonic wave-functions forγ ≫ 1, Eq. (40). Numericallywe find that the damping rate approachesΓ∞ exponentially,Γ(γ) = Γ∞ exp(−aγα), with a best-fit exponentα of order−1. The fitting curve is shown in the plot with a dashed line.

The dipole dynamics of the bosonic and fermionic systemsare explicitly compared in Fig. 13, where we plot the first 10oscillations of the center of mass after the sudden displace-ment of the trap. In the figure, the dashed line and the dotsare the bosonic dynamics forγ = 13.6 and 100, the solidline is the fermionic evolution, and the crosses are the analyti-cal approximation to the fermionic evolution Eq. (34), respec-tively. Consistent with Fig. 12, we observe that for increasingγ the decrease of the amplitude of oscillation for the bosonsresembles more and more the one for fermions. In particu-lar, for γ = 100, the curves for the interacting bosons andideal fermions nearly overlap. The distribution of excitedfre-quencies forγ = 100 is shown in Fig. 11. The frequencydistribution is broad and centered aroundω ≈ 0.75ω∗. Alsoin the inset small peaks are shown to appear in the range1.6ω∗ ≤ ω ≤ 3ω∗ (notice the different scale in the inset). Thebroad distribution is due to the population of single-particlestates that are not harmonic in character. For the value ofq

0 2 4 6 8 10tTo

-2

-1

0

1

2

xa

Analy. Γ=13.6 Γ=100 Fermi

FIG. 13: Center of mass position in lattice units as a function of timefor interacting bosons, forγ = 13.6(dashed line) andγ = 100(dots).The solid line is the center of mass position for ideal fermions, whilethe crosses are the analytical approximation to the fermionic solutionEq.(34). Here,q ≈ 77 andωT = 2π × 100 Hz andTo = 2π/ω∗.

used for the calculations no single-particle localized modesare occupied in the ground state before the trap displacement.After the displacement about 90 percent of the atoms occupynon localized single-particle modes. The phase mixing be-tween these modes accounts for most of the observed damp-ing. The remaining 10 percent occupy localized states and thepopulation of these modes is responsible for the shift of thepeak of the distribution to lower frequency. In fact we showbelow, Sec. IV B 2, that a large population of localized stateswith n ≫ nc yields a distribution which is peaked atω ≈ 0.

Finally, we note that the small population of localizedmodes after the displacement also explains why the analyticsolution Eq.(34) reproduces the exact fermionic evolutioninFig. 13 only qualitatively. In fact, Eq.(34) was derived underthe condition

∑n=nc−1 |c

(N−1)n |2 ≪ 1, while herenc = 12

and∑

n=11 |c(N−1)n |2 ≈ 0.1.

2. Localized dynamics

In analogy to recent experiments [19], in this section westudy the dipole dynamics when the depthVo of the opticallattice is varied, while the parabolic confinement is kept con-stant. Then, bothJ andU change as a function of the latticedepth, as explained in Sec. IV. The parabolic confinementωT = 2π × 150Hz has been chosen to be the same as inSec. IV, so that the energy spectrum exactly matches the onediscussed there, whenVo is varied.

In Fig. 14 lines and symbols correspond to the time evolu-tion of the interacting bosons and ideal fermions, respectively.In particular, the dashed-dotted, dashed and small-dottedlinesare for bosons, while boxes, large-dots and triangles are forfermions withVo/ER = 7, 12 and15, respectively. For suchlattice depthsγ = 14, 50 and 100, respectively, and the systemis fermionized, as discussed in Sec. IV A. As expected, the

17

0 1 2 3 4tTo

-2

-1

0

1

2

xa

eHVo=7L FHVo=7L eHVo=12L FHVo=12L eHVo=15L FHVo=15L

FIG. 14: Center of mass position (in lattice sites) as a function of timecalculated for different lattice depths and a fixedωT = 2π × 150Hz. The dash-dotted, dashed and small-dotted lines are the exactsolutions for interacting bosons (e in the legend) and the boxes, large-dots and triangles are the solutions for ideal fermions (F inthe legend) for Vo/ER = 7, 12 and15, respectively.To = 2π/ω∗.

FIG. 15: Probability distribution of frequencies forVo/ER = 7, 12and15 (see text). The frequenciesωlk are given in units ofω∗. Be-causeω∗ ∝

√J , ω∗ decreases with increasing lattice depth.

agreement between the bosonic and fermionic solutions im-proves for largerγ-values, and is almost perfect forγ = 100.

Notably, for all the γ-values, no complete oscillationare observed, as the amplitudes of oscillations are stronglydamped at very early times. The inhibition of the trans-port properties in the experiment here envisioned is a direct

consequence of the large population of single-particle stateswhich are localized in character. For the caseVo = 7ER

(ΩN/J ≈ 0.58, γ = 13.6 , q = 134.2 andnc ≈ 8) beforethe displacement the system is fermionized but non-localized.On the other hand, after the displacement about 20 percentof the atoms occupy localized modes of the undisplaced po-tential. The population of the localized modes withn ≫ nc

can be directly linked to the presence of low-frequency peaks(ωlk ≈ 0) in the distribution of frequencies, Fig. 15. Because80 percent of the atoms occupy non-localized modes the cen-ter of mass position can still relax to zero as shown in Fig.14.For the casesVo = 12ER and15ER the Fermi energy is largerthanEnc

, and even before the displacement most states are lo-calized. After the displacement has taken place, about 60 and90 percent of the atoms occupy localized modes respectively,and the dynamics is overdamped. This is mirrored by the ap-pearance of a large peak atωlk ≈ 0 in the probability distri-bution, Fig.15, and by the fact that the center of mass positiondoes not relax to zero as shown in Fig.14.

V. CONCLUSIONS

We studied the spectrum and dipolar motion of interactingand non-interacting one-dimensional atomic gases trappedinan optical lattice plus a parabolic potential using the tight-binding approximation. We showed that the single-atom tight-binding Schrodinger equation can be exactly solved by map-ping it onto the recurrence relation satisfied by the Fouriercoefficients of some periodic Mathieu functions. We usedasymptotic expansions of such functions to fully characterizethe eigenenergies and eigenmodes of the system. Our ana-lytic approach is complementary to previous numerical andsemiclassical analysis for single-atom systems. The advan-tage is that we could explicitly calculate the corrections to theharmonic oscillator spectrum introduced by the lattice. Theknowledge of these corrections allowed us also to provide an-alytic expressions for the modulations of the center of massmotion induced by the periodic potential when trapped idealbosonic and fermionic gases are suddenly displaced from thetrap center.

By means of numerical diagonalizations of the Bose-Hubbard Hamiltonian we studied the interacting many-bodybosonic problem. First, we characterized the changes in thelow-energy excitation spectrum as a function of lattice depth,by comparing it with the ideal Bose and Fermi spectra. Then,we stated the necessary conditions for fermionization to occurand showed that it takes place for a large range of experimen-tally accessible parameters. We clarified the required condi-tions for the formation of a Mott insulator at the trap centerand linked its appearance to the population of localized statesat the Fermi level of the correspondent ideal fermionic system.We then studied the many-body dipole dynamics and showedthat, in the parameter regime where the system is expectedto be fermionized, the knowledge of the single-particle solu-tions is a powerful tool for the understanding of the stronglycorrelated dynamics. By studying the distribution of the fre-quencies pertaining the many-body modes excited during the

18

dipole dynamics, we explicitly showed the connection be-tween the population of single-particle localized states withthe inhibition of the transport properties of the system. Thesespectral analysis allowed us also to gain some insight into thedynamics in the weakly-interacting regime, where an analysisbeyond mean-field is required, and in the complex interme-diate regime, where no mapping to single-particle solutionispossible.

VI. ACKNOWLEDGEMENT

We thank Trey Porto and Eite Tiesinga for useful discus-sions. We acknowledge financial support from an the Ad-

vanced Research and Development Activity (ARDA) contractand the U.S. National Science Foundation under grant PHY-0100767.

Note: We note that some of the points discussed here havebeen just recently pointed out in Ref.[43] where the authorsdiscuss the dipole oscillations of 1D bosons in the hard-corelimit.

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