Quantum Dynamics of Solitons in Strongly Interacting Systems on Optical Lattices

Chester P. Rubbo,1 Indubala I. Satija,2, 3 William P. Reinhardt,3, 4

Radha Balakrishnan,5 Ana Maria Rey,1 and Salvatore R. Manmana1

1JILA, (NIST and University of Colorado), and Department of Physics,

University of Colorado, Boulder, Colorado 80309-0440, USA.2Department of Physics, George Mason University, Fairfax, VA 22030, USA.

3Joint Quantum Institute, National Institute of Standards and Technology and University of Maryland, Gaithersburg, MD 20899, USA.

4Department of Chemistry, University of Washington, Seattle, WA 98195-1700, USA.

5The Institute of Mathematical Sciences, Chennai 600113, India

(Dated: February 16, 2012)

Mean-field dynamics of strongly interacting bosons has been shown to support two species of solitons: one ofGross-Pitaevski (GP)-type where the condensate fraction remains dark and a novel non-GP-type characterizedby brightening of the condensate fraction. Here we study the effects of quantum fluctuations on these solitonsusing the adaptive time-dependent density matrix renormalization group method, which takes into account theeffect of strong correlations. We use local observables as the density, condensate density and correlation func-tions as well as the entanglement entropy to characterize the stability of the initial states. We find both species ofsolitons to be stable under quantum evolution for a finite duration, their tolerance to quantum fluctuations beingenhanced as the width of the soliton increases. We describe possible experimental realizations in atomic BoseEinstein Condensates, polarized degenerate Fermi gases, and in systems of polar molecules on optical lattices.

PACS numbers: 03.75.Lm, 03.75.-b, 67.85.De

I. INTRODUCTION

Solitary waves and solitons (i.e., solitary waves whoseshape remains unchanged at collisions) are encountered insystems as diverse as classical water waves [1], magneticmaterials [1–17], fiber-optic communication [1, 18, 19], andBose-Einstein condensates (BEC) [1, 20–22]. Rooted in thenonlinearity of the system which balances dispersive effects,solitons are fascinating non-linear waves that encode collec-tive behavior in the system. Intrinsically nonlinear in naturedue to inter-particle interactions, and due to the high degreeof control in the experiments, the BEC systems are a naturalfertile ground for exploring solitons. In dilute atomic gaseousBECs which are simply described in terms of the propertiesof the non-linear Schrodinger equation (NLSE), or Gross-Pitaevski equation (GPE), bright (density elevation) solitonsexist for attractive interparticle interactions and dark (densitynotch) solitons in the repulsive case. These solutions are char-acterized not only by persistent density profiles, but also bycharacteristic modulations of the quantum phase across theirprofiles, which differs for the bright (attractive condensate)[23] and the dark (repulsive condensate) cases [24–26]. How-ever, we want to emphasize that, on general grounds, othersystems with intrinsic nonlinearities should be able to realizesolitons if the conditions are chosen appropriately.

In this paper, we follow two goals: First, we want to de-scribe the realization of solitons in lattice systems in a broaderframework which includes BEC, quantum degenerate Fermigases, hard-core bosons, and polar molecules on optical lat-tices, as well as certain condensed matter systems. The com-mon aspect of these various systems is that the soliton dy-namics can be described in terms of a simple S = 1/2 spinchain which, in turn, can be the effective model for a varietyof situations, as the ones mentioned above. This description isfooted on an extension of the standard GPE treatment of soli-

tons, and leads us to the second scope of our paper which is todescribe new effects which go beyond mean-field dynamics.

Investigations of effects beyond GPE dynamics have beena subject of various studies in the past decade. For short rangerepulsive systems in the Tonks-Girardeau regime, the cubicnon-linearity of GPE was replaced by a quintic repulsive non-linearity and the resulting modified GPE was shown to sup-port dark solitary waves of GP-type [27]. This 1D systemwas further investigated in the presence of dipolar interactions[28] and was shown to support bright solitons whose stabilityand mobility depended on the dipolar interaction strength. In2D systems, bright solitons were found to be stable given asufficient dipole-dipole strength [64]. Further studies of thestability and dynamics of solitons have been extended to twocomponent BECs [65] and multilayered BECs [66]. Existenceof dark and bright solitary waves was also shown numericallyin systems describing multicomponent BECs [29]. In addi-tion to the continuum systems, solitary waves have been ex-tensively studied in systems described by a discrete non-linearSchrodinger equation (DNLSE) [30, 67], BECs in deep opti-cal lattices and also to optical beams in wave guides.

In recent studies [47, 48], solitary waves in a system of hardcore bosons (HCB) have been studied using mean field equa-tions obtained from mapping the HCB system to a S = 1/2spin chain and in 2D. The continuum limit of the lattice pop-ulated with HCB is described by a generalized GPE, whichwe will refer to as ”HGPE” [See Eq. (6)]. In contrast to GPE,HGPE contains both the normal and condensate density. Thissystem describing strongly repulsive BEC was shown to sup-port both dark and bright solitary waves, the existence of bothspecies being rooted in the particle-hole duality/symmetry inHCB systems. Unlike other studies, HGPE solitary wavesare obtained as an analytic solution which was shown to pro-vide an almost exact solution of the equations of motion [47].These two species of solitons can be referred to as the GP-

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type and the non-GP-type as the former corresponds to a darkcondensate fraction that dies beyond sound velocity while thelater is associated with brightening of the condensate and per-sists all the way up to sound velocity and transforms into asoliton train for supersonic velocities. Recent numerical stud-ies investigating collision properties of these nonlinear modessuggest that these solitary waves are in fact solitons [31].

An important question that we investigate here is whetherthese mean field solitons survive quantum fluctuations. In pre-vious work, the quantum dynamics of GP dark solitons in thesuperfluid regime of the Bose-Hubbard Hamiltonian (BHH)has been studied numerically by Mishmash et al. [32, 33].The main findings are that for weak interactions the dark soli-ton is stable on a time-scale of the order of ∼ 20− 40 units ofthe hopping and afterwards decays due to two-particle scatter-ing processes. The larger the on-site interaction, the strongerthe scattering and the faster the decay of the solitons. In ad-dition, these studies treated collisions between the solitarywaves which confirm the soliton nature of the states on thetime scales treated. These studies focus on the limit of smallinteractions. Here, we treat the strong coupling case and studythe fate of the soliton solutions obtained in the HGPE frame-work. We do this by generalizing the BHH of Refs. [32, 33]to include on-site and nearest neighbor density-density inter-actions. As discussed in Ref. [47], in the continuum limitthis gives rise to the two distinct types of solitons mentionedabove, which, as we shall see, are found to be stable in boththe mean field approximation to the lattice dynamics of thesystem, as well as in the full quantum dynamics on the lattice.

More specifically, we describe the exact quantum evolu-tion of an initial mean field soliton solution on 1D lattice sys-tems. The soliton and the Hamiltonian driving the dynamicsare thereby formulated in terms of a S = 1/2 spin language.It is so possible to envisage a realization of the described soli-ton solutions in both, experiments with ultracold bosonic andspin polarized fermionic atoms, as well as in experiments withpolar molecules [34–37] on optical lattices which can be usedto emulate spin-1/2 systems [38, 39]. We combine an analyticsolution of the HGPE which provides a continuum approx-imation to the lattice problem, a numerical treatment of themean-field equations on the lattice, and a full quantum treat-ment of the dynamics by applying the time-dependent DMRG[40–44]. Both, mean-field and numerical results indicate thatfor a certain range of parameters the solutions found are in-deed stable solitons on the time scale treated. The non-GP-type soliton is found to be somewhat less tolerant of quantumeffects compared to the GP-type. We characterize the stabil-ity of the solitons by considering the entanglement in the sys-tem: since in our set-up the initial soliton solutions are productstates on the lattice, the entanglement entropy [45] should re-main zero for a stable soliton solution and hence is a measurefor the stability of the soliton in the course of the time evolu-tion. In addition, we consider correlation functions which, onsimilar grounds, can be used to characterize its stability.

The paper is organized as follows. In section II, we intro-duce the effective spin model and its derivation from HCB andspinless fermions on a lattice, the dynamical equation (HGPE)that describes the continuum approximation to the mean-field

equations of the lattice system, and we summarize the analyticsolution of the HGPE. In Sec. III we describe the mean-fieldansatz and some details of the DMRG approach to the dynam-ics. In Sec. IV, we analyze the stability of the soliton solutionsby comparing the mean-field results on a lattice to the DMRGresults. As a measure for the quality of the soliton solution, weuse in Sec. IV B the von Neumann or entanglement entropy aswell as correlation functions which also should remain zero inthe course of the time evolution if the mean field state were tosurvive quantum fluctuations. In Sec. V we propose possibleexperimental realizations of the HGPE solitons. In Sec. VI,we summarize.

II. HAMILTONIAN AND EQUATIONS OF MOTION

In this paper, we treat the dynamics of initial soliton statesdriven by the spin Hamiltonian

HS = −�

j

�J Sj · Sj+1 − g S

zj S

zj+1

�− g

�

j

Szj (1)

on a one-dimensional lattice, i.e., we are treating the dynam-ics of a XXZ-chain with a global external magnetic field ofmagnitude g. One way to obtain this effective Hamiltonian isas the limiting case of the extended Bose Hubbard model,

H =−�

j

�J

2

�b†jbj+1 + h.c.

�+ V njnj+1

�+

�

j

Unj

�nj − 1

�

− (µ− 2J)nj .

(2)

Here, b(†)j are the annihilation (creation) operators for a bo-son at the lattice site j, nj is the number operator, J/2 is then.n. tunneling strength, and µ is the chemical potential. Anattractive nearest-neighbor interaction V < 0 is introduced tosoften the effect of a strong onsite interaction |U | � 0. TheHCB limit (|U | → ∞) corresponds to the constraint that twobosons cannot occupy the same site. This HCB system canthen be mapped to the model Eq. (1) [51], where the two spinstates correspond to two allowed boson number states |0� and|1�, and setting g = J − V . Note that this is in contrast tothe study of Refs. [32, 33] in which the quantum dynamicswas investigated in the original Bose-Hubbard system and notfor the effective model Eq. (1). This is interesting since theexistence of the proposed soliton solutions for this spin modelhas implications for further systems than the ultracold bosonicatoms usually considered when describing soliton phenomenain cold gases. In particular it should be noted that the XXZmodel in 1D can be obtained using the Jordan-Wigner trans-form from a system of spinless fermions

HSF = −J

2

�

j

�c†j+1cj + h.c.

�+V

�

j

njnj+1 +µ

�

j

nj ,

(3)with c

(†)j the fermionic annihilation (creation) operators on

site j, and nj = c†jcj the density on site j. Therefore, it should

3

be possible to investigate the soliton dynamics in experimentswith bosonic atoms, in spin systems, and in fermionic sys-tems. In Sec. V we discuss possible implementations in ex-periments with cold gases. Note that both models, Eq. (1)and Eq. (3) are fundamental models for describing condensedmatter systems such as quantum magnets and systems of itin-erant electrons. It is therefore conceivable that, in principle,the proposed soliton solutions can be realized in such systemsas well.

For simplicity, we set up our discussion in the frameworkof bosonic systems, without loosing generality. Then, the spinflip operators S± = Sx ± iSy correspond to the annihilationand creation operators of the corresponding bosonic Hamil-tonian, bj → S

+j . Thus the order parameter that describes

a BEC wave function is ψsj = �S+

j �, where the expectationvalue is obtained using spin coherent states [51]. In this mean-field description, the evolution equation for the order parame-ter is obtained by taking the spin-coherent state average of theHeisenberg equation of motion. The spin coherent state |τj�at each site j can be parametrized as:

|τj� = eiφj2

�e−i

φj2 cos

θj

2| ↑�+ e

iφj2 sin

θj

2| ↓�

�. (4)

With this choice, the HCB system is mapped toa system of classical spins [47, 51] via S =�12 sin(θ) cos(φ), 1

2 sin(θ) sin(φ), 12 cos(θ)

�. Note that

the particle density ρj and the condensate density ρsj satisfy

the relation ρsj = ρj ρ

hj , with ρ

hj = 1 − ρj the hole density.

In this representation, ψsj =

�ρsje

iφ. This mean fieldtreatment is contrasted to the standard GPE derived fromthe Bose-Hubbard model by taking the expectation value ofthe Heisenberg equation of motion with Glauber coherentstates [46]. We cast the equations of motion in terms of thecanonical variables φj and δj ≡ cos(θj) = (1 − 2ρj) andobtain

δj =J

2

�

i=±1

�(1− δ

2j )(1− δ

2j+i) sin(φj+i − φj)

φj =J

2

δj�(1− δ

2j )

�

i=±1

�1− δ

2j+i cos(φj+i − φj)

−V

2

�

i=±1

δj+i − (J − V )δ0 . (5)

A. Solitary Waves in the Continuum Approximation

In the continuum approximation, the equations for the or-der parameter are derived from a Taylor series in the latticespacing a [54],

i�ψs = − �22m

(1−2ρ)∇2ψs−Veψ

s∇2ρ+Ueρψ

s−µψs (6)

where Ja2 = �2

m , Ue = 2(J − V ) and Ve = V a2.

These equations have been shown to support solitary waves

FIG. 1. (Color online) Bright (top) and dark (bottom) soliton solu-tion in the continuum [Eqs. (7) and (9)]. Left panels show the den-sity, right panels the condensate density as a function of position andspeed. Note that the condensate density of the bright soliton shows a’brightening’, around the notch, i.e., it grows above the backgroundvalue, whereas the dark soliton does not show this effect.

[47] riding upon a background density ρ0: ρ(z) = ρ0 + f(z),with z = x− vt. We obtain for the soliton solution

f(z, ρ0)± =

2γ2ρ0ρ

h0

±�

(ρh0 − ρ0)2 + 4γ2ρ0ρh0 cosh z

Γ − (ρh0 − ρ0),

(7)where γ =

√1− v2, and v being the speed of the solitary

wave in units of cs =�2ρs0(1− V/J), which is the speed of

sound of the Bose gas system determined from its Bogoliubovspectrum [51]. Γ is the width of the soliton,

Γ−1 = γ

�2(1− V

J )ρ0ρh0

14 (ρ

h0 − ρ0)2 +

VJ ρ0ρ

h0

. (8)

The characteristic phase jump associated with the solitarywaves is

∆φ± = (�1− 2c2s) cos

−1 v(1− 2ρ0)

1− 2ρs0v2. (9)

This solution has some remarkable properties. One directconsequence of the particle-hole symmetry underlying theequations of motion is the presence of two species of solitarywaves, shown in Fig. 1. The existence of f(z, ρ0) superposedon the background particle density ρ0 implies the existenceof a counterpart f(x, ρh0 ), superposed upon a correspondinghole density ρ

h0 . In fact it is easy to see that f±(z, ρ0) =

±f∓(z, ρh0 ). For ρ0 < 1/2, the ± corresponds to bright and

dark solitons, respectively. The bright solitons have the un-usual property of persisting at speeds up to the speed of sound,in sharp contrast to the dark species that resembles the darksoliton of the GPE whose amplitude goes to zero at soundvelocity. In the special case with background density equal to1/2, the two species of solitons become mirror images of eachother, as f+(z, ρ0 = 1/2) = −f

−(z, ρ0 = 1/2). In this case,the condensate density in fact describes the GPE-soliton [49].

4

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FIG. 2. (Color online) Comparison of the soliton profiles at timest = 20/J obtained in the continuum [black solid line, Eq. (6)] andusing the equations of motion approach Eq. (5) on a lattice of L =40 sites. The left panels show the results for V/J = 0.75 (narrowsoliton), the right panels the case V/J = 0.95 (broad soliton). Thetop panels show the bright soliton, the bottom panels the dark solitonsolution of Eqs. (7) and (9). Both, the particle density ρ (∗) and thecondensate density ρs (+) are shown.

It should be noted that for ρ0 > 1/2, the dark and brightsolitons switch their roles. In other words, for ρ0 < 1/2, itis the dark soliton that behaves like a GPE soliton while thebright soliton is the new type of soliton that persists all theway up to sound velocity. In contrast, for ρ0 > 1/2, the brightsoliton is GPE-type while the dark one is the persistent soliton.In view of the particle-hole duality, we will present our resultsfor ρ0 < 1/2 in which the bright solitons have the persistentcharacter noted above.

In the following sections, we will investigate for the ex-istence and the lifetime of these solutions on lattice systemsusing mean-field equations and the DMRG. We will comple-ment this analysis by investigating the stability of further ini-tial states. In particular, we show that an initial Gaussian den-sity distribution for a stationary soliton shows a similar stabil-ity if a phase jump is realized, but becomes unstable withouta phase jump. This is of importance for experimental realiza-tions indicating that imperfections in the creation of the initialstate may not have a strong influence on the soliton dynamics.

III. METHODS: MEAN FIELD ANSATZ AND DMRG

A. Mean field treatment

In this section, we compare the mean-field treatment of thesoliton dynamics on a lattice [governed by Eqs. (5)], to thedynamics in the continuum [Eq. (6)]. More specifically, weapply the equations of motion (5) to an initial state given byEqs. (7) and (9) on a finite lattice, compare to the continuumsolution and compute the difference in the local observables

(density ρ and condensate density ρs, respectively)

δρ(s) =

��

i

(�ρ(s)i �continuum − �ρ(s)i �MF)2

��

i

�ρ(s)i �2continuum. (10)

In Fig. 2 we show the HGPE solitons for different values ofV/J at time t = 20/J . We compare the continuum solu-tion (black line) to the solution obtained on the lattice (dashedlines). As can be seen, for V/J = 0.95, the lattice approxi-mation and the continuum solution show excellent agreement,up to small deviations at the boundaries. For V/J = 0.75,however, significant deviations occur. We further analyze thisbehavior in Fig. 3. The top panel shows the difference of thelattice solution to the continuum solution at t = 20/J for asystem of L = 40 sites as a function of V/J . As can be seen,the difference is significant for all values of V/J <∼ 0.8. Onlyat such large values of V/J the agreement is within a few per-cent.

This discrepancy between the lattice and the continuum so-lution is to be expected: the continuum model is an approx-imation to the lattice model and its validity will break downwhen the size of features (e.g., the width of the soliton) of theanalytic continuum solutions becomes comparable to the lat-tice spacings. This breakdown of validity can be understood interms of the emission of Bogoliubov quasi-particles [55] forsolitons which are too narrow: analogous to the excitationsin a dilute bose gas, the Bogoliubov dispersion spectrum [51]shows that a narrow perturbation excites high energy modes.We further analyze this in Figs. 3 (c) and (d). Quasi-particlesare emitted in the course of the time evolution, and due to

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0.00.51.01.52.0

V�J

∆�Ρ MF�Ρ 0� �a�

��������������������������������������

0.0 0.2 0.4 0.6 0.8 1.00123456

V�J

∆�Ρs MF�Ρs0� �b�

FIG. 3. (Color online) Difference of (a) the local density ρ and (b)the condensate density ρs between the continuum evolution and themean-field evolution on a lattice of L = 40 sites at times t = 20/Jas a function of V/J . (c) and (d): lattice mean-field evolution of thelocal density for a broad soliton (c) with V/J = 0.65 and a narrowsoliton (d) with V/J = 0.45.

5

momentum conservation the soliton gets a velocity in the op-posite direction so that it starts to move away from the originalposition. The narrower the soliton, the stronger the emissionof quasi-particles, and – as expected – the lattice approxima-tion becomes more and more unstable as the width of the soli-ton decreases, i.e., with decreasing the value of V/J .

Note that this behavior is reminiscent of the mechanismwhich leads to the ’light-cone’ effect in correlation functionsfollowing a quantum quench [56–60]. In this case, the quenchcreates entangled quasi-particles on each lattice site whichthen move ballistically through the system and lead to a lin-ear signature in the time evolution of correlation functions. Inthis way, the velocity of the quasi-particle excitations can beobtained [58–60]. In a similar way, we propose that the linearsignatures in Fig. 3 can be used to further analyze the proper-ties of the quasi-particles. However, this lies beyond the scopeof the present paper so that we leave this issue open for futureinvestigations.

Due to the necessity of having a width of the soliton largerthan a few lattice spacings, we find that we need to investi-gate systems with L ≥ 30 lattice sites. In the following wetherefore compare the lattice mean-field solution to the fullquantum dynamics obtained by the DMRG for systems withL = 40 and L = 100 lattice sites and V/J ≥ 0.9.

B. Details for the DMRG

We apply the adaptive t-DMRG [43, 44] for systems withup to L = 100 lattice sites with open boundary conditions(OBC). The reason we solely use OBC is that the DMRG per-forms far better than in the case of periodic boundary condi-tions, so that we can treat larger system sizes. However, at thispoint it becomes necessary to discuss the effect of the bound-aries: we choose system sizes and initial widths of the soli-tons so that there is a wide region between the soliton and theboundary which can be considered to be ’empty’. In Fig. 4,we compare the initial state for a system with L = 40 andL = 100 sites. As can be seen, the effect of the boundarieson the soliton is completely negligible. This remains so ontime scales on which perturbations either from the boundariesreach the soliton or from the soliton reach the boundaries. Atthese instants of time, we stop the evolution and consider thisto be the maximal reachable time for the given system size.We find that already for systems as small as L = 40 sites, themaximal reachable time is t > 20/J , so that we conclude thatthe analysis which we present in the following is not affectedby boundary effects.

We work with the S = 1/2 spin system [Eq. (1)] and engi-neer the initial state on the lattice by imprinting a phase anddensity profile by applying an external magnetic field. Morespecifically, for the initial state we compute the ground stateof

H0 = −h

�

j

�Bj · �Sj (11)

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tJ � 0.0

100 sites

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tJ � 0.0

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FIG. 4. (Color online) Local particle density ρ (blue ∗) and conden-sate density ρs (red +) obtained by DMRG (symbols) and by themean-field ansatz (solid line) for a stationary bright soliton (v = 0)at times t = 0 (left panels) and at times t = 20/J (right panels). Theplots show results for lattice sizes of L = 100 sites (top) and L = 40sites (bottom). The parameters are V/J = 0.95 and ρ0 = 0.45.

with h a large multiplicative factor (∼ 100) and

�Bj =

��Sx

j � =�

ρj(1− ρj) cosφj ,

�Syj � =

�ρj(1− ρj) sinφj ,

�Szj � = 0.5− ρj

�. (12)

For the treatment of the dynamics of the system, we apply aKrylov-space variant [72] of the adaptive time dependent den-sity matrix renormalization group method. During the evolu-tion, we keep up to 1000 density-matrix eigenstates for sys-tems with up to L = 100 sites. We apply a time step of∆t = 0.05, resulting in a discarded weight of < 10−9 atthe end of the time evolution. We estimate the error bars atthe end of the time evolution to be smaller than the size of thesymbols.

IV. FULL QUANTUM DYNAMICS

A. Soliton Stability

The initial soliton state is prepared as discussed in Sec-tion III B and is propagated with the XXZ spin-1/2 Hamil-tonian Eq. (1). Snapshots of the resulting time evolution forthe density profile of both, the bright and the dark soliton,with speed v = 0 are shown in Fig. 5. Since our results formoving solitons (v > 0) are similar, we restrict in the follow-ing to the case of static solitons. While the mean-field solu-tion remains essentially unchanged in time, the full quantumevolution shows some deformation of the initial state: in thecourse of the evolution, the total density profile widens as thepeak decreases. The amount of change depends on the param-eters V/J and v, and is different for the bright and the dark

6

soliton. However, as further discussed below, for V/J closeenough to unity the difference between the quantum solutionand the initial state remains below a few percent on a timescale t ∼ 20/J , where the hopping amplitude due to the map-ping from the spin system is J/2. This has to be compared totime scales reachable by experiments on optical lattices. Fortypical lattice depths in which a tight binding description isvalid, the tunneling rate varies from 0.1 − 1 kHz, while thetypical time scale of the experiments is on the order of 1-100milliseconds. We therefore conclude that the density profilesuggests a stable soliton on the experimentally accessible timescale in the full quantum evolution. Now we turn to the con-densate density. Here, at t = 20/J , the deviation from themean field solution is larger. Nevertheless, as shown in Fig. 5,the change remains within a few percent for V/t = 0.95, sothat we conclude that both quantities identify a stable solitonsolution on this time scale.

To obtain a better measure for the life time of the solitons,we analyze in Fig. 6 for the local observables (density ρ andcondensate density ρ

s, respectively) the discrepancy betweenthe mean field solution and the t-DMRG evolution computedusing Eq. (10) for different parameters. As shown in Fig. 6,it decreases significantly as V/J approaches unity or as thespeed of the soliton v (in units of the speed of sound) in-creases. This is associated to a widening of the initial densityprofile when increasing V/J and a reduction of the peak am-plitude for larger v, so that we conclude from this analysis thatfor a variety of initial conditions the GPE and HGPE solitonscan survive quantum fluctuations on the time scales treated.This is further confirmed in the following by the behavior ofthe entanglement entropy and the correlation functions.

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FIG. 5. (Color online) Local particle density ρ (top) and condensatedensity ρs (bottom) obtained by the DMRG (symbols) and by themean-field (solid line) propagation of the stationary (v = 0) bright(red +) and dark (blue ∗) solitons for times t = 0 and 20/J for asystem with L = 40 sites. The parameters are V/J = 0.95 andρ0 = 0.25.

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FIG. 6. (Color online) Difference between the DMRG results and thelattice mean-field results for the total density (left) and for the con-densate density (right) for a stationary (top) and for moving (bottom)bright solitons (ρ0 = 0.25) on a system with L = 40 sites.

B. Entanglement entropy and nearest neighbor correlations

A quantity that reveals the quantum nature of a state isthe von Neumann or entanglement entropy in the system [45]which is defined as

SvNA = −Tr(ρAlogρA), (13)

with ρA the reduced density matrix of a subsystem A obtainedby tracing out the degrees of freedom of the remaining part ofthe system B. From the Schmidt decomposition

|ψ� =�

i

�λi|φi

A�|φiB� (14)

it follows that

SvN = −

�

i

λi log λi, (15)

with |φiA� and |φi

B� the eigenstates of the reduced density ma-trix of subsystem A or B, respectively, and λi the eigenvaluesof the corresponding eigenstates. This quantity gives a mea-sure for the entanglement between two subsystems. Since theinitial states are product states on the lattice, SvN is exactlyzero at the beginning of the time evolution since only one ofthe weights is finite with λi = 1 while the others are exactlyzero. If SvN (t) remains zero (or very small) in the course ofthe time evolution, we conclude that quantum fluctuations donot strongly influence the nature of the initial product state,and so the value of S

vN (t) gives an additional measure forthe stability of the soliton solutions. Note that there are twovariants of this analysis: in Refs. [32, 33], the entanglemententropy for a subsystem of one single site is measured withrespect to the remainder of the system. However, within theDMRG framework it is easier to consider the time evolution ofthe entanglement entropy for all bipartitions of the system asit is automatically computed in the course of the sweeps. For

7

simplicity, and since it gives a similar measure for the stabil-ity of the soliton, we consider here the latter. In addition, thebehavior of this quantity for ground states of finite spin chainsis well known from conformal field theory [61], and the nu-merical values can be obtained easily from the DMRG. Thisallows us to compare the values of the entanglement entropyduring the time evolution to the ones of the strongly corre-lated ground state of the system which serves as a referencefor how strongly entangled the state has become during thetime evolution. In Fig. 7 we show typical result for the entan-glement entropy in ground states of the spin system Eq. (1)with L = 40 sites, V/J = 0.95 and S

totalz corresponding to

ρ0 = 0.1, 0.25 and 0.45, respectively. As can be seen, thenumerical value in the center of the system increases with ρ0

and reaches SvN, center ≈ 1.35 for ρ0 = 0.45.A second estimate for the strength of the entanglement

growth is to compare to the maximal possible entanglemententropy in a generic spin-1/2 chain with L sites. Considera bipartition of the chain into M and L − M spins withM ≤ L − M . Since the dimension of the Hilbert space ofa chain of M spins is 2M , a maximally entangled state is ob-tained when all λj = 1/2M . This state has hence an entropy

SvN,max = −

�

j

λj log λj = M log 2.

For a system of L = 40 sites and a bipartition M = L/2 wetherefore obtain S

vN,max ≈ 13.86, i.e. it is a factor of ∼ 10larger than the one in the ground state for the same bipartition.

We now compare this values to the ones reached in thetime evolution of the solitons. In Fig. 8 we display the en-tanglement growth of both the dark and the bright soliton atρ0 = 0.1, 0.25 and 0.45, respectively. We obtain that theentanglement growth is strongest at low fillings (ρ0 = 0.1),and it is larger for the bright soliton than for the dark one.For ρ0 = 0.45, the maximum value for the bright soliton isSvN ≈ 0.4, and for the dark soliton S

vN ≈ 0.25 – both val-ues are significantly smaller than the one in the correspondingground state, and much smaller than the one of the maximallyentangled state. This shows that on the time scale treated,the state is significantly closer to a product state than to astrongly correlated ground state of the same system, or thanto a maximally entangled state. Since the entanglement is notnegligible, quantum fluctuations play an important role for thecharacterization of the state towards the end of the consideredtime evolution, but they are not strong enough to fully destroythe product nature of the initial state.

Note that the entanglement growth for the bright soliton forρ0 = 0.1 is significantly larger than for ρ0 = 0.45. This isconnected to the fact that also for the local observables thecorresponding initial state decays much faster. The entangle-ment entropy can be used as a measure to compare the stabil-ity of the initial states at ρ0 = 0.1 and ρ0 = 0.45: it appearsthat the bright soliton at ρ0 = 0.1 is about half as stable asthe one at ρ0 = 0.45. This is reflected in the numerical val-ues of δρ(s)(t) which also show approximately a factor of twobetween the two cases.

While the entanglement entropy at the center of the systemshows a peak for the bright soliton, it possesses a minimum for

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

0 5 10 15 20 25 30 35 40

SvN

position of bipartition l

L = 40, V/J = 0.95!0 = 0.1

!0 = 0.25!0 = 0.45

FIG. 7. (Color online) Entanglement entropy in ground states of thespin system Eq. (1) for V/J = 0.95, L = 40 sites for values ofSztotal corresponding to the background density ρ0 = 0.1, 0.25, and

0.45, respectively.

the dark soliton. This can be understood by the fact that thedark soliton has fewer particles at the center of the system andso quantum fluctuations are less pronounced. Due to particle-hole symmetry, the dark and bright soliton evolutions for ρ0 ≥1/2 possess the same behavior.

The behavior of the entanglement entropy can be com-pared to the local spin fluctuations and correlations in thesystem. In the mean field approach, at all times the coher-ent spin state enforces that ρs(MF )

i = ρi(1 − ρi). This re-lation can be expressed in terms of spin observables, leadingto �Sx

i �2 + �Syi �2 + �Sz

i �2 = 1/4 on each site, realizing aconstraint on the local spin fluctuations. In the full quantumdynamics, this constraint is broken, so that the initial coherentstate becomes modified, and entanglement is induced in thesystem [62].

The entanglement entropy is related to the long distancecorrelations and has been extensively studied in spin systems[63, 68–71]. It is therefore interesting to consider the growthof correlations in our system in the course of the time evolu-tion. For simplicity, and since they are the most relevant onesfor experiments, we consider nearest neighbor spin correlac-tions �Si · Si+1� − �Si� · �Si+1�. The results shown in Fig. 9show similar behavior to the entropy dynamics.

C. Gaussian initial states

In this section, we test the stability of the HGPE solitonsolutions to modifications of the initial state. Specifically, wecompare the time evolution of the HGPE solitons to that of aGaussian initial state

ρ(x) ∼ e− x2

2σ2 , (16)

which might be easier to implement in experiments [24, 25,73]. We analyze the dynamics for initial states with and with-out a phase shift of π across the center in order to compare

8

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the evolution of an initial state with a similar shape and phaseproperties as the HGPE soliton to one which has only a simi-lar shape. As discussed in Sec. III B, the initial state is createdvia a Gaussian external field.

The obtained results are shown in Fig. 10. As can be seen,the Gaussian state with a phase jump remains stable and ap-pears to be a very good approximation to the HGPE soliton.In contrast, without the phase jump, the initial wave packetquickly disperses. Note that due to the lattice the wave packetcan disperse by creating two peaks moving in opposite di-rections. This is due to the deviation of the cos(k) disper-sion of the lattice from the dispersion ∼ k

2 of a free particleand comes into appearance if the number of particles is highenough.

We conclude, therefore, that once the phase jump is imple-mented, it is not necessary in the experiments to implementinitial states which have exactly the form of the HGPE soli-tons.

V. EXPERIMENTAL REALIZATIONS

In this section, we discuss possible realizations of the mod-els introduced in Sec. II. We start with the experimental im-plementation of the extended Bose Hubbard model, Eq. (2),and its fermionic variant. The nearest neighbor interactionterm V can be possibly generated in bosonic or spin polar-ized fermionic systems via long-range electric [74] or mag-netic [75] dipolar interactions as discussed below or with a

short-range interaction between atoms in higher bands of thelattice [76]. The hard-core constraint for bosons requires in-creasing the interactions so that there is a large energy dif-ference between states with a different number of bosons persite. This can be achieved by tuning the scattering length via aFeshbach resonance [77]. Note that in this type of implemen-tation, in which the spin 1/2 degrees of freedom correspond tosites with zero and one atom, the sign of the U and V interac-tion is determined by the scattering length and thus is the samefor both. In our proposal, we need attractive V interaction, soalso the Hubbard U will be attractive. However, note that alsoin this case it is possible to realize the hard-core constraint:even though the states with one or zero atom per site do notbelong to the ground state manifold, when prepared, they aremetastable since there is no way to dump the excess energy, atleast when prepared in the lowest band [78]. This is since thebandwidth in a lattice is finite, which is known to lead also torepulsively bound pairs [79]. In our case, however, it preventsdouble occupancies, which for |U | → ∞ corresponds to theHCB limit. A possible way to proceed then is to prepare theground state in the repulsive side of the Feshbach resonanceand then quickly ramp the magnetic field to the attractive sidein which the evolution takes place. The atoms now are still inthe lowest band and need to be promoted to higher bands us-ing, e.g., similar techniques to the ones discussed in Ref. 80.Note that the requirement of populating higher bands can in-deed lead to an additional relaxation. In the fermionic systemthe decay to the lowest band can be blocked by filling the low-est band. The lifetime of bosons in higher bands on the other

9

FIG. 9. (Color online) Top: contour plot of the time evolution ofthe nearest neighbor spin correlations �Si · Si+1� − �Si� · �Si+1�on the whole lattice for the stationary bright soliton (left) and thedark soliton (right) for ρ0 = 0.25 and V/J = 0.95. Bottom: timeevolution of the entanglement entropy for the same parameters.

hand does require further investigation but at least recent ex-periments in 2D [80] reveal that it can be 10−100 times longerthan the characteristic time scale for intersite tunneling.

In a recent proposal, it is shown that the XXZ spin modelEq. (1) and the spinless fermion model Eq. (3) both can be re-alized in systems of polar molecules on optical lattices, eventhough with a long-range 1/r3 decay of the interactions ratherthan nearest-neighbor interactions only. Two different pathsallow the study of the soliton dynamics in such experiments:First, as discussed in detail in Refs. [38, 39], the spin modelEq. (1) can be directly implemented in the case of unit filling(i.e., one molecule per site of the optical lattice) by select-ing two rotational eigenstates of the molecules which emulatethe two spin degrees of freedom of the S = 1/2 chain. Theparameters of the system can then be tuned via external DCelectrical and microwave fields. The second implementationis by populating the lattice with molecules which are all in thesame rotational eigenstate, emulating a spin polarized system.Since the dipolar interaction decays quickly, we presume thatthe effect of the interactions beyond nearest neighbor on thesoliton dynamics should be very small, so that both realiza-tions can be used to study the soliton dynamics. We leavea detailed study of the effect of the interaction terms beyondnearest-neighbor sites on the dynamics of the solitons to fur-ther studies.

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FIG. 10. (Color online) Comparison of the time evolution of a bright(top) and dark (bottom) HGPE soliton (#) to the evolution of aninitial Gaussian state with (+) and without (∗) phase imprinting.

VI. SUMMARY

We have analyzed the stability and lifetime of HGPE soli-tons on 1D lattice systems driven by a XXZ-Hamiltonianwhich can model the behavior of bosonic atoms, fermionicpolar molecules, spin systems, and spin-polarized itinerantfermions on optical lattices and in condensed matter systems.We compare the dynamics obtained in a mean field approxi-mation to the full quantum evolution obtained using the adap-tive t-DMRG and find that the solitons remain stable underthe full quantum evolution on time scales t ∼ 20/J , whereJ/2 is the unit of the hopping. This is quantified by the en-tanglement entropy which remains smaller than the one inthe ground state of the corresponding spin system and sig-nificantly smaller then the one of a maximally entangled stateon this time scale. Similar to the findings of Refs. [32, 33],for longer times the soliton decays. However, given the timescales reachable by ongoing experiments with optical lattices,this should suffice to identify this effect in the lab. In addition,we find that imperfections in the creation of the initial stateshould be of minor importance, as long as the density profileand the phase jump are similar to the ones of the proposedsoliton solutions. This is exemplified by a Gaussian initialstate, which in the case of a phase jump shows good agree-ment with the soliton solution, while in the absence of thephase jump becomes completely unstable. Due to the tunabil-

10

ity of parameters either via Feshbach resonances for atomicsystems or via electric and microwave fields in the case of po-lar molecules, the possibility of realizing both, bright and darksolitons in strongly interacting systems, adds a new paradigmto the existence of coherent non-linear modes in systems ofultracold quantum gases.

ACKNOWLEDGEMENTS

We acknowledge financial support by ONR grant N00014-09-1-1025A, grant 70NANB7H6138 Am 001 by NIST, by

NSF (PHYS 07-03278, PFC, PIF-0904017, and DMR-0955707), the AFOSR, and the ARO (DARPA-OLE). R.B.thanks the Department of Science and Technology, India, forfinancial support.

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