Soliton dynamics of an atomic spinor condensate on a Ring Lattice

1.5 MB·English

by  Indubala I Satija

SolitondynamicsofanatomicspinorcondensateonaRingLattice
Indubala I. Satija1, Carlos L. Pando L.2 and Eite Tiesinga3
1SchoolofPhysics,AstronomyandComputationalSciences,GeorgeMasonUniversity,Fairfax,VA22030,USA
2IFUAP, Universidad Autonoma de Puebla, Apdo, Postal J-48, Puebla 72570, Mexico and
3JointQuantumInstitute,NationalInstituteofStandardsandTechnologyandtheUniversityofMaryland,
100 Bureau Drive, Stop 8423, Gaithersburg, MD 20899-8423, USA
(Dated:January25,2013)
Westudythedynamicsofmacroscopically-coherentmatterwavesofanultra-coldatomicspin-1orspinor
condensate on a ring lattice of six sites and demonstrate a novel type of spatio-temporal internal Josephson
effect. Usingadiscretesolitarymodeofuncoupledspincomponentsasaninitialcondition,thetimeevolution
3
ofthismany-bodysystemisfoundtobecharacterizedbytwodominantfrequenciesleadingtoquasiperiodic
1
dynamicsatvarioussites.Thedynamicsofspatially-averagedandspin-averageddegreesoffreedom,however,
0
isperiodicenablinganuniqueidentificationofthetwofrequencies. Byincreasingthespin-dependentatom-
2
atominteractionstrengthweobservearesonancestate,wheretheratioofthetwofrequenciesisacharacteristic

n integermultipleandthespin-and-spatialdegreesoffreedomoscillatein“unison”.Crucially,thisresonantstate
a isfoundtosignaltheonsettochaoticdynamicscharacterizedbyabroadbandspectrum. Inaferromagnetic
J spinor condensate with attractive spin-dependent interactions, the resonance is accompanied by a transition

4 fromoscillatory-torotational-typedynamicsasthetimeevolutionoftherelativephaseofthematterwaveof
2 theindividualspinprojectionschangesfromboundedtounbounded.

] PACSnumbers:03.75.Ss,03.75.Mn,42.50.Lc,73.43.Nq
s
a
g
- I. INTRODUCTION bodysystems.[17–19]Therehasalsobeentheoreticalstudies
t
n ofsolitonsinahomogeneousspin-1condensate[20,21].Fur-
a thermore,theoreticalandexperimentalstudiesofdouble-well
u Spinor condensates are atomic Bose-Einstein condensates bosonicJosephsonjunctionshaveunveilednovelphenomena
q (BEC) with an internal spin degree of freedom that com-
suchasbrokensymmetrymacroscopicquantumself-trapping
. bine magnetism with condensation. Examples are optically-
t [22]andπ-modes[23–25]aswellassymmetryrestoredswap-
a trapped spinor condensates of atomic rubidium-87[1–4],
pingmodes[22]. Inaddition,numericalinvestigationofBEC
m
sodium[5,6],orchromium[7]witheitherspinorangularmo-
systems, spatially separated into a ring lattice have demon-
- mentum f = 1, 2, or 3 condensates with a three-, five-, or
d stratedchaoticdynamics, deterministicdynamicswithsensi-
seven-componentvectororderparameter. Incontrasttomix-
n tivedependenceoninitialconditions.[26–28]
tures of two or more atomic states[8] or mixtures of several
o In this paper, we explore the quantum coherent time evo-
c atomic species, spin-changing collisions in spinor conden- lution of a spin-1 BEC that is spatially separated into six
[ sates permit coherent dynamics among the hyperfine states.
weakly-coupled sites arranged to form a ring geometry, see
Inatypicalprocessfortwof =1atoms,oneinspincompo-
1 Fig. 1. For a large number of atoms such a system is well
nentm = −1andoneinm = +1,canreversiblyscatterinto
v represented by a three-component wavefunction or order pa-
1 twoatomswithspincomponentm = 0,whichconservesthe rameter, Φ(cid:126)((cid:126)x,t), that satisfies a nonlinear Gross-Pitaevskii
5 global magnetization of the condensate. This coherent spin
equation[9,10]. Here,werestrictourcalculationstothecase
8 mixing leads, nevertheless, to oscillations of the spin popu-
withnoexternalmagneticfield.Moreimportantly,weassume
5 lations,andisananalogueofJosephsonoscillationsinultra-
. that all sites and spin components have at all times the same
1 coldatoms[9–13].Hence,justascollisionalinteractionsallow
identical localized spatial-mode function, φ((cid:126)x). That is, for
0 for a single-component condensate to be spatially coherent,
3 spin-changingcollisions,drivenbyinternalinteractions,allow themth-componentofΦ(cid:126)((cid:126)x,t)wehave
1 coherence among internal degrees of freedom. A positive or
: L
v negativesignofthestrengthofthespin-changinginteraction (cid:88)
Φ ((cid:126)x,t)= ψm(t)φ((cid:126)x−(cid:126)x ), (1)
i determineswhetherthesystemsbehaveanti-ferromagneticor m n n
X
ferromagnetic,respectively[10,14]. Spinorphysicshasalso n=1
r
beenstudiedinopticallatticeswithexactlytwoatomsperlat-
a whereListhenumberofsites,(cid:126)x isthecenterofsiten,and
n
ticesite[15]. thedimensionlessψm(t)arecomplextime-dependentampli-
n
In view of the nonlinear nature due to interparticle inter- tudes. Theoverlapbetweenspatial-modefunctionsatdiffer-
actions and high degree of control in the experiments, BEC entsitesleadstotunnelingbetweenneighboringsites.
systems are ideal systems for visualizing a wide variety of More advanced numerical modeling uses time varying
nonlinear phenomena. This includes solitary waves, the lo- mode functions [29, 30] or directly solves for the three-
calizednonlineartravelingwavesthatretaintheirshape,size dimensional Gross-Pitaevskii equation[18]. As a first study
andspeed duringpropagation [16]. Experimentalrealization inthesolitondynamicsofaspinorcondensatewebelievethat
ofsolitonsinahomogeneoussinglecomponentBECsystems our simplified approach is justified. It shows the generaliza-
isahallmarkofthequantumcoherenceassociatedwithmany tionofthespinorJosephsoneffectaswellastheroutetochaos
2
to pendulum-type physics [10, 12]. Our study suggests that
0.8
|Φ(x)|2 the origin of this “regular” global dynamics has its roots in
the strong correlations among different lattice sites and spin
0.7
n=3 n=2 projectionsandisnotrootedinthermalaveragingovermany
degrees of freedom. We note that ω is solely determined by
0.6
thespin-changinginteraction,whereasΩshowsaratherweak
n=4 n=1 ρn dependence. We find that Ω is mainly controlled by the size
0.5 ofthering.
Byincreasingtheabsolutevalueofthespin-dependentin-
n=5 n=6
0.4 teraction strength, the two frequencies can be mode locked.
This resonance condition is found to describe the onset to
0.3 chaotic dynamics. In other words, the spinor condensate ex-
1 2 3 4 5 6
lattice index, n hibits a transition from quasi-periodic dynamics to chaotic
dynamics where at the onset to transition, all local and
FIG.1:(coloronline)Panela)showsaschematicofaringlatticeof globaldegreesoffreedomoscillateinunison. Consequently,
sixsites(reddashedcirclewithlabeledmarkers)containingaf =
the initial soliton profile reappears periodically, providing a
1spinorcondensate. Thesolidblacklinerepresentsthesuperfluid
unique demonstration of a novel-type of spatio-temporal in-
populationalongtheringasapolargraph.Panelb)showstheinitial
ternal Josephson effect. Furthermore, in contrast to the anti-
fractionalpopulation, ρ , asafunctionofsiteindexnusedinour
n
ferromagneticcase,intheferromagneticcondensate,theres-
dynamical simulations. The distribution corresponds to an excited
(period-6) solution of the DNLSE. For each site the population of onance is accompanied by a transition from bounded to un-
thethreemagneticsublevelsofthespinoristhesame. bounded dynamics. In the anti-ferromagnetic case, the dy-
namics remains bounded and oscillatory. Nevertheless, be-
yondtheresonanceadditionalfrequenciesappear.
initsclearestform.Itshouldbenotedthattheringlatticewith For sake of simplicity, we restrict our simulations to zero
lessthanthreesitesisanintegrablesystem. BEConalattice magnetization. This is enforced by choosing initial condi-
withthreeormorethanthreesitesexhibitsamazingdegreeof tions where the m = +1 and m = −1 components have
complexityincludingcoexistenceofregularandirregularbe- sameinitialwavefunction. Infact,wechooseaninitialstate
haviorthatisknowntoaccompanychaoticdynamics. Inour whereallthreecomponentshavethesameinitialwavefunc-
detailed numerical exploration of ring lattices with different tion. An alternate initial state with zero magnetization and
numbersofsites,thekeycharacteristicsofthetimeevolution containsmostlym=±1atomsleadtosimilarresults. Itwill
as discussed in this paper are independent of the number of notbedescribedhere.
sites. Here,wewillpresentourresultsforaringlatticeofsix In section II we review the mean-field equations that de-
sitesonly. scribe the evolution of the spinor condensate on a lattice as
well as the underlying single-mode approximation approxi-
Our initial state is a discrete solitary wave that is an ex-
mationoneachlatticesite. Thelatterhasbeenshowntopro-
cited eigenstate of the discrete nonlinear Schro¨dinger equa-
tion (DNLSE) satisfied by the ψm(t) in the absence of spin- vide a reasonable description for the continuum system. For
n
our ring lattice, the SMA suggests collective coordinates to
changing interactions. The spin-changing interaction then
describe the global dynamics. The initial state is described
induces oscillatory dynamics, an internal Josephson effect
in Sec. III. In section IV, we show numerical simulations
wheredifferentspincomponentsevolvewithdifferentampli-
forweakpositive,anti-ferromagneticspin-dependentintegra-
tudes and phases. In reality the spin-changing Hamiltonian
tionsandshowthatthedynamics,althoughcomplex,isdom-
can not be turned off and this state needs to be engineered
inatedbytwofrequencies. InsectionV,wediscussthereso-
by a combination of resonant electro-magnetic radiation that
controllably creates superpositions of m-states [6] and off- nanceconditionthatoccurswhenthespin-dependentinterac-
tionstrengthisincreased. Atresonanceweobserveanonset
resonant spatially-dependent light forces that induce solitons
tochaos. SectionVIbrieflyillustratestheferromagneticcase.
[17,18].
WeconcludeinSec.VII.
Inthisarticleweshowthatforweakspin-changinginterac-
tions the dynamics on the ring lattice is dominated by two
frequencies. Interestingly, one of these frequencies, which
wedenoteasω, describestheperiodicbehaviorofspatially- II. MEAN-FIELDEQUATIONSFORASPINOR
averagedspinpopulations. Thesecondfrequency,denotedby CONDENSATEANDCOLLECTIVECOORDINATES
Ω, describes the dynamics of the site-dependent population
summedoverthethreespincomponents. Inotherwords,the Anatomicspinorcondensatewithlargeatomnumberisde-
dynamicsofcollectiveoraveragedcoordinatesispendulum- scribed within mean-field theory by a complex vector order
like. Aswediscussbelow,thisissomewhatsurprisingasthe parameterorcondensatewavefunctionΨ((cid:126)x,t),whoseevolu-
spatial profiles of the three spin components evolve differ- tionisgovernedbyamulti-componentGross-Pitaevskiiequa-
ently and the dynamics on the ring should deviate from that tion [9, 10]. We will assume that the order parameter can
for a “simple” trapped spinor condensate. The single-mode be approximated by Eq. 1, where a single time-independent
approximation for the latter system has been shown to lead mode function, φ((cid:126)x), determines the spatial dependence in
3
eachwell. WedenotethisbytheL-sitesingle-modeapprox- which are independent of the spin-independent interaction
imation (L-SMA) in analogy to the single-mode approxima- with strength c . For small Z and γ the dynamics are har-
0
tion(SMA)foraspinorcondensateinadipoletrap. monic. Ingeneral,however,thesecoupledequationsdescribe
The interactions between two spin-1 atoms have a nonlinear pendulum whose length depends upon the mo-
a spin-independent and a spin-dependent or (spin- mentum.
changing) contribution. Within a mean-field the- ForaL > 2ringoflatticesitesthecollectivevariablesZ
ory and the L-SMA their strength is given by andγ satisfy
2cqf=or4thπe(cid:126)2sgpqin/-(i2nµde)p(cid:82)endd(cid:126)xe|φnt((cid:126)xa)n|d4/s(cid:82)pind-(cid:126)xd|eφp(e(cid:126)xn)d|2enwticthonqtr=ibu0tiaonnd, Z˙ = c(cid:126)2 (cid:88)(σn2 −Zn2)sinγn (8)
respectively. Here µ is the reduced mass for two atoms n
and (cid:126) is the reduced Planck constant. The lengths gq are γ˙ = −2c2 (cid:88)Z (1+cosγ )+ JW (9)
g =(a +2a )/3andg =(a −a )/3,wherea anda are (cid:126) n n (cid:126)
0 0 2 2 2 0 0 2 n
scatteringlengthsfors-wavecollisionsoftwof = 1bosons
with total angular momentum F = 0 and 2, respectively. whereW =Re[(cid:80)n(2ψn0+1/ψn0−ψn++11/ψn+1−ψn−+11/ψn−1)].
For stable condensed gases we require that c0 > 0. We Thespin-independentinteractionstrengthc0 doesnotexplic-
note that c > 0 for anti-ferromagnetic Na and c < 0 for itly appear in these equations. As already mentioned in the
2 2
ferromagnetic87Rb. introduction and further discussed in next section, this sug-
Thedynamicsoftheψm(t)aregovernedbytheDNLSE geststhatthespatially-averagedZ andγ willexhibitperiodic
n
oscillations, with frequency ω, that are solely determined by
i(cid:126)ψ˙−1 =−J(ψ−1 +ψ−1 ) (2) thespin-changinginteraction. Finally,thespin-averagedpop-
n n−1 n+1
+(c +c )(|ψ−1|2+|ψ0|2)ψ−1 ulationsateachsitesatisfy
0 2 n n n
+(c0−c2)|ψn1|2ψn−1+c2(ψn0)2(ψn1)∗ σ˙ =−J Im(cid:32) (cid:88)1 (ψm)∗[ψm +ψm ](cid:33) , (10)
i(cid:126)ψ˙0 =−J(ψ0 +ψ0 )+c |ψ0|2ψ0 (3) n (cid:126) n n+1 n−1
n n−1 n+1 0 n n m=−1
+(c0+c2)(|ψn1|2+|ψn−1|2)ψn0 +2c2ψn1ψn−1(ψn0)∗ which does not explicitly depend on c2. Hence, we expect
i(cid:126)ψ˙n1 =−J(ψn1−1+ψn1+1)+(c0+c2)(|ψn1|2+|ψn0|2)ψn1 that the σn oscillate periodically, characterized by frequency
Ωanditsharmonics.
+(c −c )|ψ−1|2ψ1 +c (ψ0)2(ψ−1)∗, (4)
0 2 n n 2 n n
where J is the positive site-to-site tunneling energy and we
III. INITIALDISCRETESOLITON
use periodic boundary conditions. In the absence of the last
termontherighthandsideofEqs.2-4,thesetofequationsde-
scribesathree-speciescondensate.Spin-changingtermsmake For a ring lattice of six sites, we use the initial condition
aspinorcondensateuniqueastheyinducepopulationoscilla- showninFig.1. Commonlyreferredtoasadiscretesoliton,
tions between m levels. Equations 2-4 conserve total atom itisastationarysolutionoftheDNLSEintheabsenceofthe
nanudm(cid:80)bernmanmd|mψnmag(nte)t|i2zaartieocno.nsInervoethde.r words, (cid:80)nm|ψnm(t)|2 sJp.inA-clohnagngwinitghipnhtearsaectφiomnn=[260,3o1r]π(if.oer. weviethnco2r=od0d)saitnedicn0de=x
We will monitor the local population of each spin state as n, respectively, the initial state populations are the same for
wellasglobalorcollectivecoordinates,suchasspatially-and thethreecomponents.
spin-averaged population and phases. It is therefore conve- This stationary solution, corresponding to a solution
nient to define ψnm(t) = (cid:112)ρmn(t)exp[iφmn(t)] with popula- ψnm(t) = ψnm(t = 0)exp(−iµt/(cid:126)) where µ is the chemi-
tions ρm(t) and phases φm(t). Following Refs. [9, 10] for a cal potential, is obtained by numerically solving the result-
n n
single-modespinorcondensatenaturallocalcanonicalcoordi- ing nonlinear map, as Eqs. 2-4 reduce to a set of L two-
natesare dimensionalcubicmaps. FollowingRef.[28],wefindthatfor
L=6thesolutionsare6-folddegenerateandforarangeofµ
Z =ρ0−(ρ+1+ρ−1) and γ =φ−1+φ1−2φ0 . (5) thelocalizedsolitonmodeinFig.1istheonlystablesolution.
n n n n n n n n
Infact,wehaveusedµ=2.5J. Intriguingly,forthesevalues
(cid:80)
aswellasglobally-averagedcoordinatesZ = Z /Land
(cid:80) n n of the chemical potential the homogeneous solution with the
γ = γ /L. Throughout this article we call γ and γ
n n n samedensityatallsitesisunstable.
spinorphases. Otherusefulpopulationaveragesare
It should also be noted that known localized soliton-type
solutionswithzerophaseforallsitescorrespondtoattractive
L 1
ρm = L1 (cid:88)ρmn and σn = 31 (cid:88) ρmn . (6) sizpeidn-sionlduetpioennsdefonrtrienpteurlascivtieoinnstewraicthtiocn0s,<on0e.nTeoedosbttoacinonlosicdaelr-
n=1 m=−1
solutionswithphasesdifferentfromzero.Ingeneral,itcanbe
Inadipoletrap,orequivalentlyforL = 1,asimplespinor shownthatforalatticewithevennumberofsites,themapping
modelisgivenbytheSMA.ThecanonicalvariablesZ andγ ψm →(−1)n(ψm)∗relatesasolitonsolutionforattractivein-
n n
thensatisfythepairofequations teractionswiththosewithrepulsiveinteractions. Forasingle-
c c componentcondensatediscretesolitonshavebeenstudiedex-
Z˙ = (cid:126)2(1−Z2)sinγ and γ˙ =−2(cid:126)2Z(1+cosγ), (7) tensivelyforringlatticesofvarioussizes[26–28].
4
FIG. 3: (color online) Power spectrum or Fourier transform of
FIG.2:(coloronline)Timedynamicsofananti-ferromagneticspinor
thetimeevolutionofthespatially-averagedspinpopulationρ
m=+1
solitononasix-siteringforseveralspin-and/orspatially-averaged
(red), spatially-averagedspinorphaseγ (green), andspin-averaged
degrees of freedom assuming a small positive spin-changing inter-
population σ at site n = 1 (blue). The parameters and initial
action energy. Time is in units of (cid:126)/J, where J is the tunneling stateareasinn=F1ig.2andthefrequencyisinunitsofJ/(cid:126). Thetop
energybetweenthesites. Calculationsareperformedforc =0.1J
2 andbottompanelshowthepowerspectraonalinearandlogarith-
andc = J. Panela)showsthedynamicsofthespatially-averaged
0 micscale,respectively. Alsoindicatedarethedominantfrequencies
spinor population Z (black line) and the spinor population Z for
n ω and Ω, which have been assigned as due to the spin-dependent
the individual sites n (colored lines). The symmetry of the initial
andspin-independentinteractions,respectively. Inthebottompanel
solitonaroundn = 3impliesthatonlyfourofthesixsiteshavea
higherharmonicsofΩcanbeobserved.
distinct time evolution. Panel b) shows the spinor phases γ (black
line)andγ (coloredlines). Finally,panelc)showsthesite-specific
n
populationσ ,averagedoverthethreespincomponents.
n
Experiments with single-component Bose condensates in
sions around the average γ. Finally, we note that the spinor
double-wellpotentials[24]haveobservedJosephsonoscilla-
tions and quantum self-trapping in the limit J (cid:28) c . The phasesareboundedforoscillatorymotion.
0
opposite limit can also be reached leading to tunneling of
(nearly-)independent atoms. Here, we chose a compromise
Figure 3 shows the power spectrum of three of the time
with c = J. As an aside we note that with our initial state
0 traces shown in Fig. 2. It highlights the existence of two
andc = J wehaveimplicitlyspecifiedthechemicalpoten-
0 dominant frequencies, ω and Ω, as well as weaker higher
tialandthusatomnumberineachsite.
harmonics in Ω indicating non-sinusoidal periodic behavior.
Thespatially-averageddegreesoffreedompredominantlyos-
cillatewithafrequencyω,whichfromsimulationswithother
IV. QUASIPERIODICDYNAMICSOFAN smallc isfoundtobeproportionaltotheabsolutevalueofc .
2 2
ANTI-FERROMAGNETICSPINOR In contrast, spin-averaged local populations oscillate nearly-
sinusoidalwithfrequencyΩ,whichfromothersimulationsis
Figure2showsthetimeevolutionofthespinorsolitonona foundtoweaklydependonthespin-changinginteractionbut
six-sitering,describedintermsofthespinorcoordinatesZ is inversely proportional to the number of lattice sites. Fi-
n
andZ,spinorphasesγ andγ,aswellaspopulationsσ . We nally, the local dynamics for individual spin projections is
n n
useasmallpositivespin-changinginteractionstrengthc for quasiperiodic with a slow frequency ω and a faster beat fre-
2
ananti-ferromagneticspinor. Weobservethatthetimedepen- quencyatmultiplesofΩ. Thisillustratesthecorrelationsthat
dence of Z and γ are nearly sinusoidal. The site-dependent exist among the spatial and spin components. Detailed stud-
Z ,γ andσ ,however,oscillateatahigherfrequency. They ies with various ring sizes indicate that the spinor dynamics
n n n
do so in a non-sinusoidal manner with sharper minima than ischaracterizedbytwofrequencies,irrespectiveofthesizeof
maximaorviceversa. Thephasesγ onlyshowsmallexcur- thering.
n
5
FIG.5: (coloronline)Powerspectrashowingtheresonancecondi-
tionforincreasingspin-changingstrengthc forthesameparame-
2
tersasinFig.4. ThefrequencyisinunitsofJ/(cid:126). Inallpanelsthe
FIG. 4: (color online) Time series for an anti-ferromagnetic spatially-averagedρ (redcurve)andspin-averagedσ (blue
m=+1 n=1
spinorshowingtheresonanceconditionforincreasingspin-changing curve) are shown. Panels on the left and right show spectra on a
strengthc2.Timeisinunitsof(cid:126)/J andc0 =J.InallpanelsZ(red linearandlogarithmicscale,respectively,whilefromtoptobottom
curve)andσn=1 (bluecurve)areshown. Panelsa),b),andc)show c2 =0.5J,0.65J and1.0J.
resultsforc =0.5J,0.65J,and1.0J,respectively.Theresonance
2
condition occurs for c = 0.65J, where the spatially-averaged Z
2
andspin-averagedσ oscillateatthesamerate. Forc > 0.65J
n=1 2
thetracesarechaotic.
V. RESONANCECONDITIONFORTHE
ANTI-FERROMAGNETICCONDENSATE
Figures1and2showedthatforsmallc /c twofrequencies
2 0
ω andΩareverydistinct. Forincreasingpositivec atfixed
2
c , as shown by time traces in Fig. 4 and power spectra in
0
Fig. 5, both frequencies increase although at a different rate.
Foracriticalspin-changingstrengthc whenω =Ωtheanti-
2
ferromagnetic spinor reaches a “resonance” state. All three
spin-componentsatallthesitesoftheringthenoscillateasa
singleentity.Forc0 =J thisresonanceoccursatc2 =0.65J. FIG.6:(coloronline)Spin-averagedsolitonprofiles,σn,fortheanti-
Forlargevaluesofc thebehaviorbecomeschaotic,whichis ferromagneticspinorbeforeandatresonanceforfivetimesspaced
2
apparent as broad-band features in the corresponding power bytimeinterval2π/Ω,startingwiththeinitialstate(redcurve). We
spectrumshowninFig.5. use c0 = J and panels a) and b) correspond to c2 = 0.5J and
c = 0.65J, respectively. The soliton reforms close to its initial
Fourier analysis of collective as well as local coordinates 2
stateatresonancewhenc =0.65J.
showsthattheresonancestatecorrespondstomatchingofnot 2
onlythedominantfrequenciesω andΩ,butalsosomeofthe
otherlessprominentfrequencies(notvisibleinthelinearplot)
asillustratedinthelog-linearplotonthemiddlerowofFig.5.
Infact,atresonance,thedominantortheprimarypeakisac-
companiedbysecondarysatellitepeaks,equallyspacedonthe statewherewavefunctionsofallthecomponentsofthespinor
eithersideoftheprimarypeak. condensateatallL-sitesoftheringlatticeoscillateinunison
Figure6furtherillustratesthatatresonanceallsitesoscil- and the dynamics is well characterized one frequency. The
late in phase with same frequency. The spatial profile of the factthatthesolitonprofilereemergesperiodicallyprovidesa
solitonreemergesperiodicallywithoutanysignificantchange uniquedemonstrationofquantumcoherenceandanovel-type
fromtheinitialprofile.Thus,theresonancestateinanordered ofinternalspatio-temporalJosephsoneffect.
6
VI. FERROMAGNETICCONDENSATE
We now briefly describe the dynamics of a ferromag-
netic spinor condensate with negative c . For a small spin-
2
changing interaction, the dynamics is similar to that of an
anti-ferromagneticspinor. Namely,thespatially-averagedbe-
haviorispendulum-likewithdominantfrequencyω,whilethe
local population oscillates with frequency Ω. An example is
shown in Fig. 7. The local density for each spin component
oscillateswithbothfrequencies. Aresonancestatecanagain
be achieved by increasing |c | but, for c = J, now occurs
2 0
whenω/Ω=3.
Incontrasttotheanti-ferromagneticcase,however,theres-
onant transition is accompanied by unbounded dynamics or
phasewindingwherethephaseofthecondensatebecomesun-
boundedasshowninFig.8. Thisbehaviormanifestsitselfas
azero-frequencymodeinapowerspectrum. Inotherwords,
chaotic dynamics with its broad-band spectrum is accompa-
niedbyatransitionfromanoscillatorytorotationalmodefor
thecollectivedegreesoffreedom.
FIG.7: (coloronline)Timeevolutionofspatially-averagedpopula- VII. DISCUSSION
tionZ (red)andspin-averagedpopulationσ (blue)foraferro-
n=1
magnetic spinor condensate with negative c . Panel a), b), and c)
2 In summary, numerical explorations of mean-field equa-
show traces for c = −0.075J, −0.09J, and −0.095J, respec-
2 tions of spinor condensate on a ring lattice with a small
tively,correspondingtocasesbefore,at,andbeyondresonance. We
number of sites reveals strikingly correlated dynamics of the
usec =J andtimeisinunitsof(cid:126)/J.
0 many-bodysystem. Eventhoughthecondensateisseparated
intoL-sites,solitondynamicsatallsitescanbecharacterized
by just two frequencies. With quasiperiodic dynamics at lo-
cal sites for individual spin components, spatially averaged
behavior for each spin component as well as spin-averaged
dynamics at each sites is found to be periodic. The fact that
thetimeseriesdescribinglocaldynamicsischaracterizedby
twofrequenciesandthesetwofrequenciesuntangleinthecol-
lectivedegreesoffreedomisrootedincorrelationsamongdif-
ferent spin degrees of freedom at various sites of the lattice.
However,properunderstandingofthesecorrelationsremains
an open challenge. Our study with ring lattices of various
sizesshowthattwo-frequencycharacterizationsofthespin-1
condensateonaringlatticeisvalidirrespectiveofthenumber
ofsitesonthelattice.
Our study provides a new illustration of a quasiperiodic
route to chaotic dynamics in a many-body system where the
critical point is known to be characterized by a resonant
state. Simple models of dynamical systems such as the one-
dimensional circle map[32] are paradigms of quasiperiodic
route to chaos, where the critical point corresponds to pa-
rameters where the two frequencies are mode-locked. The
quasiperiodic route to chaos is a well-established scenario
in dynamical systems exhibiting a transition from regular to
chaoticdynamics[32].However,thefactthatthecriticalpoint
FIG.8: Phaseportraitsorparametricplotsofthetime-evolutionof describes periodic dynamics is a novelty in many-body sys-
the spinor population Z and phase γ for a ferromagnetic spinor at tems rooted in the coherence associated with spinor conden-
c2 =−0.075J,panela),andc2 =−0.095J,panelb). Thesystem sate. Inthiscase,thecriticalpointisahighly-orderedmany-
isatresonanceinpanelb).Thedynamicsinpanelb)isaccompanied
body state exhibiting a new type of spatial-temporal coher-
byunboundedmotionofthespinorphase.Weusec =J.
0 ence. At the critical point where the two dominant frequen-
cies are in resonance, the ring lattice with L-sites oscillates
7
inunisonwithasinglecharacteristicfrequency. Thisinternal temporalcoherence.
Josephsoneffectwhereanunscathedsolitonprofilereemerges ThisresearchissupportedbyOfficeofNavalResearch,the
periodically provides a novel illustration of both spatial and CONACYT-Me´xicoandtheUSArmyResearchOffice.
[1] T.Kuwamoto, K.Araki, T.Eno, andT.Hirano, Phys.Rev.A here.
69,063604(2004). [17] S. Burger, K. Bongs, S. Dettmer, W. Ertmer, K. Sengstock,
[2] M.-S. Chang, C. D. Hamley, J. A. Sauer, K. M. Fortier, A.Sanpera,G.V.Shlyapnikov,andM.Lewenstein,Phys.Rev.
W. Zhang, L. You, and M. S. Chapman, Phys. Rev. Lett. 92, Lett.83,5198(1999).
140403(2004). [18] J.Denschlag,J.E.Simsarian,D.L.Feder,C.W.Clark,L.A.
[3] H.Schmaljohann, M.Erhard, J.Kronja¨ger, M.Kottke, S.van Collins, J.Cubizolles, L.Deng, E.W.Hagley, K.Helmerson,
Staa, L. Cacciapuoti, J. J. Arlt, K. Bongs, and K. Sengstock, W.P.Reinhardt,etal.,Science287,97(2000).
Phys.Rev.Lett.92,040402(2004). [19] R.V.Mishmash,I.Danshita,C.W.Clark,andL.D.Carr,Phys.
[4] J.Guzman,G.-B.Jo,A.N.Wenz,K.W.Murch,C.K.Thomas, Rev.A80,053612(2009).
andD.M.Stamper-Kurn,Phys.Rev.A84,063625(2011). [20] M. Wadati and N. Tsuchida, J. Phys. Soc. Jpn. 75, 014301
[5] A.T.Black, E.Gomez, L.D.Turner, S.Jung, andP.D.Lett, (2006).
Phys.Rev.Lett.99,070403(2007). [21] J. Ieda, T. Miyakawa, and M. Wadati, Laser Physics 16, 678
[6] Y.Liu,S.Jung,S.E.Maxwell,L.D.Turner,E.Tiesinga,and (2006).
P.D.Lett,Phys.Rev.Lett.102,125301(2009). [22] I.I.Satija,R.Balakrishnan,P.Naudus,J.Heward,M.Edwards,
[7] B. Pasquiou, E. Mare´chal, G. Bismut, P. Pedri, L. Vernac, andC.W.Clark,Phys.Rev.A79,033616(2009).
O. Gorceix, and B. Laburthe-Tolra, Phys. Rev. Lett. 106, [23] A.Smerzi,S.Fantoni,S.Giovanazzi,andS.R.Shenoy,Phys.
255303(2011). Rev.Lett.79,4950(1997).
[8] C.J.Myatt,E.A.Burt,R.W.Ghrist,E.A.Cornell,andC.E. [24] M.Albiez,R.Gati,J.Folling,S.Hunsmann,M.Cristiani,and
Wieman,Phys.Rev.Lett.78,586(1997). M.K.Oberthaler,Phys.Rev.Lett.95,010402(2005).
[9] H.Pu,C.K.Law,S.Raghavan,J.H.Eberly,andN.P.Bigelow, [25] K.W.Mahmud,H.Perry,andW.P.Reinhardt,Phys.Rev.A71,
Phys.Rev.A60,14631470(1999). 023615(2005).
[10] W. Zhang, D. L. Zhou, M.-S. Chang, M. S. Chapman, and [26] C. L. Pando L. and E. J. Doedel, Phys. Rev. E 69, 036603
L.You,Phys.Rev.A72,013602(2005). (2004).
[11] L.SantosandT.Pfau,Phys.Rev.Lett.96,190404(2006). [27] P.Buonsante,P.G.Kevrekidis,V.Penna,andA.Vezzani,Phys.
[12] R. Barnett, J. D. Sau, and S. Das Sarma, Phys. Rev. A 82, Rev.E75,016212(2007).
031602(2010). [28] C.L.PandoL.andE.Doedel,PhysicaD238,687(2009).
[13] E.YukawaandM.Ueda,Phys.Rev.A86,063614(2012). [29] T. Anker, M. Albiez, R. Gati, S. Hunsmann, B. Eiermann,
[14] R. Barnett, A. Turner, and E. Demler, Phys. Rev. Lett. 97, A. Trombettoni, and M. K. Oberthaler, Phys. Rev. Lett. 94,
180412(2006). 020403(2005).
[15] A.Widera,F.Gerbier,S.Fo¨lling,T.Gericke,O.Mandel,and [30] O.E.Alon,A.I.Streltsov,andL.S.Cederbaum,Phys.Lett.A
I.Bloch,Phys.Rev.Lett.95,190405(2005). 362,453(2007).
[16] A soliton is a solitary wave with a special collision property, [31] C.PandoL.andE.J.Doedel,Phys.Rev.E71,056201(2005).
viz.,itretainsitsidentityevenaftercollisionwithanothersoli- [32] E. Ott, Chaos in Dynamical Systems (Cambridge University
tary wave. Following a common practice in physics lierature, Press,1993),1sted.
wewillrefertononlinearlocalizedsolitarywavesassolitions