Quantum bright solitons in a quasi-one-dimensional optical lattice

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by  Luca Barbiero

Quantum bright solitons in a quasi-one-dimensional optical lattice
Luca Barbiero1 and Luca Salasnich1,2
1Dipartimento di Fisica e Astronomia “Galileo Galilei” and CNISM,
Universita` di Padova, Via Marzolo 8, 35131 Padova, Italy
2Istituto Nazionale di Ottica (INO) del Consiglio Nazionale delle Ricerche (CNR),
Sezione di Sesto Fiorentino, Via Nello Carrara, 1 - 50019 Sesto Fiorentino, Italy
(Dated: June5, 2014)
We study a quasi-one-dimensional attractive Bose gas confined in an optical lattice with a
4
super-imposed harmonic potential by analyzing the one-dimensional Bose-Hubbard Hamiltonian
1
of the system. Starting from the three-dimensional many-body quantum Hamiltonian we derive
0
strong inequalities involving the transverse degrees of freedom under which the one-dimensional
2
Bose-Hubbard Hamiltonian can be safely used. In order to have a reliable description of the

n one-dimensional ground-state, that we call quantum bright soliton, we use the Density-Matrix-
u Renormalization-Group (DMRG) technique. By comparing DMRG results with mean-field (MF)
J ones we find that beyond-mean-field effects become relevant by increasing the attraction between

4 bosonsorbydecreasingthefrequencyoftheharmonicconfinement. Inparticularwefindthat,con-
trary to the MF predictions based on the discrete nonlinear Schr¨odinger equation, average density

] profiles of quantum bright solitons are not shape invariant. We also use the time-evolving-block-
s
decimation (TEBD) method to investigate dynamical properties of bright solitons when the fre-
a
g quencyof theharmonic potential is suddenlyincreased. This quantumquenchinducesa breathing
- mode whose period crucially dependson thefinal strength of the super-imposed harmonic confine-
nt ment.
a
PACSnumbers: 03.70.+k,05.70.Fh,03.65.Yz
u
q
.
t I. INTRODUCTION ous brightsolitons havebeen observedin various experi-
a
m ments[13–16]involvingattractivebosonsof7Liand85Rb
vapors. Instead, discrete (gap) bright solitons in quasi-
- Ultracold bosonic gases in reduced dimensionality are
d 1D optical lattices have been observed [17] only with re-
an ideal platform for probing many-body phenomena
n pulsive bosons made of 87Rb atoms.
where quantum fluctuations play a fundamental role [1,
o
2]. In particular, the use of optical lattices has allowed In this paper we first derive an effective 1D Bose-
c
[ the experimental realization [3] of the well-known Bose- Hubbard Hamiltonian which takes into account the
HubbardHamiltonian[4]withdiluteandultracoldalkali- transversewidth of the 3D atomic cloud. In this way we
3
metal atoms. This achievement has been of tremendous determine a strong inequality under which the effective
v
impact on several communities [5] since it is one of the 1D Bose-Hubbard Hamiltonian reduces to the familiar
2
2 first experimental realization of a model presenting a one and the collapse of discrete bright solitons is fully
2 pure quantum phase transition, namely the metal-Mott avoided. We then work in this strictly 1D regime ana-
8 insulator transition. At the same time new experimen- lyzing the 1D Bose-Hubbard Hamiltonian by using the
1. tal techniques, like in-situ imaging [6], are now available Density-Matrix-Renormalization-Group (DMRG) tech-
0 to detect many-body correlations and density profiles. nique [18]. We evaluate density profiles and quantum
4 Furthermore,thesetechniquesofferthepossibilitytoob- fluctuations finding that, for a fixed number of atoms,
1 serve intriguing many-body effects in regimes which are there are regimeswhere the MF results (obtained with a
v: far from equilibrium. In this contest the relaxation dy- discrete nonlinear Schr¨odinger equation) strongly differ
i namicsregimes[7]andlight-cone-likeeffects [8]inaone- from the DMRG ones. Finally, we impose a quantum
X
dimensional (1D) Bose gas loaded on a optical lattice quench to the discrete bright solitons by suddenly in-
r have been recently observed. creasingthe frequency ofthe harmonic potential. By us-
a
ing the time-evolving-block-decimation(TEBD) method
The 1D Bose-Hubbard Hamiltonian, which accurately
[19] we find that this quantum quench induces a breath-
describesdiluteandultracoldatomsinastrictly1Dopti-
ingoscillationinthe bosoniccloud. Also inthis dynami-
callattice, isusuallyanalyzedinthe caseofrepulsivein-
cal case we find that the MF predictions are not reliable
teraction strength which corresponds to a positive inter-
when the on-site attractive energy is large.
atomic s-wave scattering length [9]. Indeed, a negative
s-wavescatteringlengthimpliesanattractiveinteraction
strength which may bring to the collapse [10, 11] due to
the shrink of the transversewidth of a realistic quasi-1D
II. THE MODEL
bosoniccloud. Moreoverincertainregimesofinteraction
the quasi-1D mean-field (MF) theory predicts the exis-
tenceofmeta-stableconfigurations[12]whichareusually Weconsideradiluteandultracoldgasofbosonicatoms
called discrete bright solitons. We remark that continu- confined in the plane (x,y) by the transverse harmonic
2
potential where n =b+b is the on-site number operator,
i i i
m
U(x,y)= ω2 x2+y2 . (1) 1 ∂2
2 ⊥ ǫ = w∗(z) +V(z) w (z) dz (7)
(cid:0) (cid:1) i Z i (cid:20)−2∂z2 (cid:21) i
In addition, we suppose that the axial potential is the
combination of periodic and harmonic potentials, i.e. is the on-site axial energy, which can be written as
V(z)=V0cos2(2k0z)+ 12ωz2z2 . (2) ǫtoi =thVeSp+erVioTdi(c2ip−otLen−ti1a)l2awndithVTVSthtehestorne-nsgittehe(nhearrgmyodnuiec
constant)ofthesuper-imposedharmonicpotential,while
This potential models the quasi-1D optical lattice pro-
J and U are the familiar adimensional hopping (tunnel-
ducedin experimentswith Bose-Einsteincondensatesby
ing) energy and adimensional on-site energy which are
using counter-propagating laser beams [20]. We choose
λ = ω /ω 1 which implies a weak axial har- experimentally tunable via V0 and as [9]. J and U are
z ⊥ ≪ given by
monic confinement. The characteristic harmonic length
of transverse confinement is given by a⊥ = ~/(mω⊥) 1 ∂2
and, forsimplicity, we choosea⊥ and~ω⊥ asplength unit J = −Z wi∗+1(z)(cid:20)−2∂z2 +V(z)(cid:21)wi(z) dz , (8)
and energy unit respectively. In the rest of the paper we
use scaled variables. U = g w (z)4 dz . (9)
i
We assume that the system is well described by the Z | |
quantum-field-theory Hamiltonian (in scaled units)
Remember that now V(z) is in units of ~ω and z in
⊥
1 units of a . Even if J and U actually depend on the
H = d3r ψ+(r) 2+U(x,y)+V(z) ⊥
Z h− 2∇ siteindexi,thechoiceofconsideringlowVT allowsusto
keep them constant.
+πg ψ+(r)ψ(r) ψ(r), (3)
i
where ψ(r) is the bosonic field operator and g =2a /a
s ⊥ B. Dimensional reduction
with a the s-wave scattering length of the inter-atomic
s
potential [21].
OurEq. (6)takesintoaccountdeviationswithrespect
to the strictly 1D case due to the transverse width σ of
i
A. Discretization the bosonic field. We call the Hamiltonian (6) quasi-1D
because it depends explicitly on a transverse width σ
i
which is not equal to the characteristic length a of the
We perform a discretization of the 3D Hamiltonian ⊥
transverse harmonic confinement.
along the z axis due to the presence on the periodic po-
This on-site transversewidth σ canbe determinedby
tential. In particular we use the decomposition [5] i
averaging the Hamiltonian (6) over a many-body quan-
ψ(r)= φ (x,y) w (z) (4) tumstate GS andminimizingtheresultingenergyfunc-
i i | i
Xi tion
wherew (z)isthe Wannier functionmaximallylocalized 1 1
i GS H GS = ( +σ2)+ǫ n
at the i-th minimum of the axial periodic potential. In h | | i Xi n(cid:2)2 σi2 i i(cid:3)h ii
this paper we consider the case of an even number L of
1 U
sitTeshzisi =tig(h2ti-b−inLd−ing1)aπn/s(a4tkz0)iswriethliaib=le1i,n2,t.h..e,Lca.se of a + 2σi2 (cid:0)hn2ii−hnii(cid:1) (10)
dperoepbleomptwicealselatt(tfiiceeld[-2th2]eo[r2y3]e.xtTenosifounrtohferthseimmpelaifny-fitehlde − J (cid:0)hb+i bi+1i+hb+i bi−1i(cid:1)o
approach developed in [11])
with respect to σ . Notice that the hopping term is in-
i
1 x2+y2 dependent on σi. In this way, by using the Hellmann-
φ (x,y)GS = exp b GS , (5)
i | i π1/2σ (cid:20)−(cid:18) 2σ2 (cid:19)(cid:21) i| i Feynman theorem, one immediately gets
i i
where|GSiis the many-bodygroundstate,while σi and σ4 =1+Uhn2ii−hnii . (11)
biaccountrespectivelyfortheadimensionalon-sitetrans- i ni
verse width (in units of a ) andfor the bosonic field op- h i
⊥
erator. We insert this ansatz into Eq. (3) and we easily Eqs. (6) and (11) must be solved self-consistently to
obtain the effective 1D Bose-Hubbard Hamiltonian obtain the ground-state of the system. Clearly, if U < 0
the transverse width σ is smaller than one (i.e. σ <
1 1 i i
H = 2(σ2 +σi2)+ǫi ni a⊥ in dimensional units) and the collapse happens when
Xi n(cid:2) i (cid:3) σi goes to zero [11]. At the critical strength Uc of the
− Jb+i (bi+1+bi−1)+ 12σUi2ni(ni−1)o, (6) csiotlelasp(sie=allLp/a2rtaicnldesia=reLac/c2u+mu1l)ataerdouinndthtehetwmoinceimnturmal
3
V =0.01 V =0.001 V =0.0001 V =0.01 V =0.001 V =0.0001
T T T 3 T 3 T 3 T
U=-0.0<n>1i468 a) <n>i468 b) <n>i468 c) MDMFRG U=-0.0∆<n>1i21 a) ∆<n>i21 b) ∆<n>i21 c) MDMFRG
2 2 2
0 0 0 0 0 0
-10z/z 0 10 -10z/z 0 10 -10z/z 0 10 z/z -10 0 10 z/z -10 0 10 z/z -10 0 10
0 0 0 3 0 3 0 3 0
U=-0.058 d) 8 e) 8 f) U=-0.05 d) e) f)
<n>i46 <n>i46 <n>i46 ∆<n>i21 ∆<n>i21 ∆<n>i21
2 2 2
0 0 0 0 0 0
-10z/z 0 10 -10z/z 0 10 -10z/z 0 10 z/z -10 0 10 z/z -10 0 10 z/z -10 0 10
0 0 0 3 0 3 0 3 0
8 8 8 g) h) i)
U=-0.10 g) h) i) U=-0.10
<n>i46 <n>i46 <n>i46 ∆<n>i21 ∆<n>i21 ∆<n>i21
2 2 2
0 0 0 0 0 0
-10z/z 0 10 -10z/z 0 10 -10z/z 0 10 z/z -10 0 10 z/z -10 0 10 z/z -10 0 10
0 0 0 0 0 0
FIG. 1: (Color online). MF (squares) and DMRG (circles) FIG. 2: (Color online). MF (squares) and DMRG (circles)
densityprofileshniiofthebrightsolitonwithJ =0.5,L=80 quantumfluctuations∆ni =phn2ii−hnii2ofthebrightsoli-
and N = 20. In the horizontal axis there is the scaled axial tonwithJ =0.5,L=80,andN =20. Inthehorizontalaxis
coordinatez/z0,withz0=π/(4k0). Theresultsofeachpanel there is the scaled axial coordinate z/z0, with z0 = π/(4k0).
are obtained with different values of harmonic strength VT Theresultsofeachpanelareobtainedwithdifferentvaluesof
(columns) and interaction strength U (rows). harmonic strength V (columns) and interaction strength U
T
(rows).
of the harmonic potential and consequently U 2/N
c
(i.e. U /(~ω ) 2/N in dimensional units). ≃ − problem it is relevant to compare MF predictions with
c ⊥
≃−
We stress that, from Eq. (11), the system is strictly DMRG ones in order to observe in which regimes MF
1D only if the following strong inequality can give accurate and reliable results. In particular, we
use a MF approach based on Glauber coherent state
n2 n
Uh ii−h ii 1 (12)
hnii ≪ |GCSi=|β1i⊗...⊗|βLi (14)
is satisfiedforanyi, suchthatσi =1(i.e. σi =a⊥ indi- where|βiiis,bydefinition,suchthatbi|βii=βi|βji[24].
mensional units). Under the condition (12) the problem By minimizing the energy GCS H GCS with respect
h | | i
of collapse is fully avoided. to βi, one finds that the complex numbers βi satisfy the
1D discrete nonlinear Schr¨odinger equation (DNLSE)
III. NUMERICAL RESULTS µβi =ǫiβi J (βi+1+βi−1)+U βi 2βi , (15)
− | |
where µ is the chemical potential of the system fixed
In the remaining part of the paper we shall work in
by the total number of atoms: N = β 2 =
this strictly 1D regime where the effective Hamiltonian i| i|
of Eq. (6) becomes (neglecting the irrelevant constant ihGCS|ni|GCSi. By solving Eq. (15) wPith Crank-
NPicolson predictor-corrector algorithm with imaginary
transverse energy)
time [31]itispossibletoshowthatinthe attractivecase
(U <0) discrete bright solitons exist [11].
U
H =−J b†i(bi+1+bi−1)+ 2 ni(ni−1)+ ǫi ni On general physical grounds one expects that the MF
Xi Xi Xi resultsobtainedfromtheDNLSEofEq. (15)arefullyre-
(13) liableonlywhenU 0andN withUN takencon-
which is the familiar 1D Bose-Hubbard model [4]. We stant. IndeedtheG→laubercoher→en∞tstate GCS istheex-
call the Hamiltonian (13) strictly-1D because the trans- act ground state of the Bose-Hubbard H|amiltoinian only
verse width σi is equal to the characteristic length a⊥ of ifU =0andN . Noticethattheexactgroundstate
transverse harmonic confinement. oftheBose-Hub→bar∞dHamiltonianwithU =0andafinite
number N of bosons is the atomic coherent state ACS
| i
whichreduces to the Glauber coherentstate GCS only
A. Glauber coherent state and DNLSE for N (see for instance [25]). In pract|ice, onie ex-
→ ∞
pects that MF results are meaningful in the superfluid
As already mentioned, in a 1D configurationquantum regimewherethereisaquasi-condensate,i.e. algebricde-
fluctuations, which are actually neglected in mean-field cay of phase correlations [1, 5, 21]. Nevertheless, in gen-
approaches, can play a relevant role. Thus, in our 1D eral it is quite hard to determine this superfluid regime.
4
Forthisreason,workingwithasmallnumberN ofbosons a) Vfin
it is important to compare the Glauber MF theory with VTin T
a quasi-exact method.
5 b) TEBD
4 MF
3
B. DMRG approach
2
0 10 20
5
c)
DMRG is able to take into account the full quantum
4
behavior of the system and it has already given strong 3
evidences of solitonic waves in spin chains [26] and in 2
0 10 20
bosonic models with nearest neighbors interaction [27].
d)
Acrucialpointinordertohaveaccurateresultsbyusing 3
DMRG is played by the size of the Hilbert space we set 2
in our simulations. Clearly, for system sizes and densi- 1
0 10 20 30 40 50
ties comparable with the experimental ones, we can not
e)
3
investigate the collapse phase where all the bosons ”col-
lapse”inonesite. Indeedit requiresasize ofthe Hilbert 2
space which is not approachable with our method. Any- 10 10 20 tJ 30 40 50
way, as shown in Eq. (11) this phase does not happen
forsufficientlylowdensityandon-siteinteractionU. For FIG. 3: (Color online). a) cartoon of the quench procedure.
this reason and in order to fulfill Eq. (11) we consider In the other panels: MF (dotted lines) and TEBD (dashed
regimeswhicharesufficientlyfarfromthisscenario,more lines) central density hnL/2i vs time t with J = 0.5, L =
precisely we use a number N = 20 of bosons in L = 80 40, and N = 10. Panels b,c,: quench from VTin = 0.01 to
lattice sites and interactions U 0.1. Nevertheless if VTfin = 0.05 for respectively U = −0.1,−0.01. Panels: d,e
we allow a too small number of≥bo−sons per site, namely quench from VTin = 0.001 to VTfin = 0.005 for respectively
if we consider a too small Hilbert space, even if we are U =−0.1,−0.01.
far from the collapse, our results might be not reliable
since the shape of density profile is modified by this cut
In order to check if our interpretation is valid we plot
off and not by physical reasons. To treat this problem
in Fig. 2 the expectation value of quantum fluctuations
we consider a maximum number of bosons in each site
nmax = 8 and we checked that increasing this quantity ∆n = n2 n 2 . (16)
does not significantly affect our results. Moreover we i qh ii−h ii
keep up to 512 DMRG states and 6 fine size sweeps [18] For the Glauber coherent state GCS one has ∆n =
i
to have a truncation error lower than 10−10. √n . We expect that ∆n of th|e DMiRG ground-state
i i
GS canbequitedifferentfromtheMFprediction. More
| i
precisely quantum fluctuations are enhanced by the ki-
C. Comparing DMRG with DNLSE netic term J which tries to maximally spread the shape
ofthecloud. Ontheotherhandthevalueof∆n ismini-
i
In Fig. 1 we compare the density profiles given by mizedbothbythestrongon-siteinteraction(becausethe
DMRG with the ones obtained by using the mean-field system gains energy having many particles in the same
DNLSE for different strengths of the harmonic potential site) and by the strong trapping potential (which con-
and interaction. For weak interactions U the particles fines the bosons in the two central sites of the lattice
are substantially free and the shape of the cloud is given where VT is weaker). This behavior is clear in Fig. 2
only by the harmonic strength VT. Of course when the where,forlargeU andsmallVT,∆ni presentsstrongde-
particlesarestronglyconfinedinthecenterofthesystem, viations from MF behavior, whereas mean-field DNLSE
as in panel a) of Fig. 1, the interaction U begins to play and DMRG are in substantialagreementin the opposite
a role due to the relevant number of bosons lying in the regime.
twocentralsites. MorepreciselyU triestodropquantum
fluctuations induced by J and it explains the small but
significant discrepancies we find. D. Dynamical properties
When the interaction U is sufficiently strong (pan-
els g),h),i) of Fig. 1) the MF results becomes insensi- Another relevant aspect of bright solitons is given by
tivetothe super-imposedharmonicpotentialofstrength its dynamical properties. In particular it is predicted by
V since the shape of the cloud remains practically un- time-dependent DNLSE [12] that discrete bright soliton
T
changed giving rise to self-localized profiles. Instead can give rise to a breathing mode. To study the time
DMRG results do not show this self-localization. In fact evolution of the system we use the time-evolving-block-
quantum fluctuations try, in opposition to U, to maxi- decimation(TEBD)algorithm[19],whichisstillaquasi-
mally delocalize the bosonic cloud. exact method recently used to study the appearance of
5
a dark soliton [28] and its entanglement properties[29] obtained by the MF nonlinear Schr¨odinger equation
[30]. We compare TEBD results with time-dependent are not found with the DMRG results (quantum bright
DNLSE ones, which are immediately obtained from Eq. solitons). In other words, we have found that with a
(15) with the position µ id/dt. We determine the small number N of bosons the average of the quantum
→
ground-state of the Bose system for a chosen value Vin density profile, that is experimentally obtained with
T
of transverse confinement and then we perform the time repeated measures of the atomic cloud, is not shape
evolutionwithalargervalueVfin. Inthiswaywemimic invariant. This remarkable effect can be explained by
T
asuddenchangeinthestrengthV ofthesuper-imposed considering a quantum bright soliton as a MF bright
T
harmonic confinement (see panel a) of Fig. 3). soliton with a center of mass [33, 34] that is randomly
Inpanelsb),c),d),e)ofFig. 3wereportthe densityof distributed due to quantum fluctuations, which are
atomsinthe twocentralsites(where ittakesthe highest suppressed only for large values of N [35]. This is
valuesincetheeffectofV isweaker)asafunctionoftime the same kind of reasoning adopted some years ago to
T
t. Thepanelsshowaperiodicoscillationwheretheperiod explain the distributed vorticity of superfluid liquid 4He
τ of this breathing mode strongly growsby reducing the [36], and, more recently, the Anderson localization of
harmonic strength V . Moreover, τ is slightly enhanced particles in one dimensional system [37] and the filling
T
by a smaller U . Remarkably, as in the static case, of a dark soliton [28]. For the sake of completeness, we
| |
beyond-mean-field effects become relevant for a strong havealsoanalyzedthe breathingmode ofdiscrete bright
U and they are instead less evident for high values of solitons after a sudden quench finding that also in the
| |
V . Indeed, in Fig. 3 the relative difference between dynamics beyond mean-field effects become relevant for
T
TEBD and MF in the period τ is below 1% in panels c) astronginteractionstrengthU andforasmallharmonic
ande),whileitisaround8%inpanelb)andaround37% constant VT [38]. Our paper gives strong evidence on
in panel d). the limits of MF theory in the study of bright solitons,
suggesting that DMRG calculations must be used to
simulate and analyze them.
IV. CONCLUSIONS
The authors acknowledge for partial support Univer-
In this paper we have obtained a strong inequality, sit`adiPadova(grantNo. CPDA118083),CariparoFoun-
Eq. (12), under which the 3D system is reduced to a dation(Eccellenzagrant11/12),andMIUR(PRINgrant
strictly-1D one and the collapse is fully avoided. More- No. 2010LLKJBX). The authors thank L.D. Carr, B.
over, we have compared MF theory with the DMRG Malomed, S. Manmana, A. Parola, V. Penna, and F.
looking for beyond-mean-field effects in the effective Toigo for fruitful discussions,and S. Saha for useful sug-
1D system of bosons in a lattice. From our results we gestions.
conclude that the self-localized discrete bright solitons
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