Variational determination of approximate bright matter-wave soliton solutions in anisotropic traps

1.9 MB·English

by  T. P. Billam

Variationaldetermination ofapproximatebright matter-wavesolitonsolutions inanisotropictraps
T. P. Billam, S. A. Wrathmall, and S. A. Gardiner
Department of Physics, Durham University, Durham DH1 3LE, United Kingdom
(Dated:January10,2012)
We consider the ground state of an attractively-interacting atomic Bose-Einstein condensate in a prolate,
cylindricallysymmetricharmonic trap. Ifatruequasi-one-dimensional limitisrealized, thenfor sufficiently
weakaxialtrappingthisgroundstatetakestheformofabrightsolitonsolutionofthenonlinear Schro¨dinger
equation. UsinganalyticvariationalandhighlyaccuratenumericalsolutionsoftheGross-Pitaevskiiequation
2 we systematically and quantitatively assess how soliton-like this ground state is, over a wide range of trap
1
andinteractionstrengths. Ouranalysisrevealsthattheregimeinwhichthegroundstateishighlysoliton-like
0
is significantly restricted, and occurs only for experimentally challenging trap anisotropies. This result, and
2
our broader identification of regimes in which the ground state is well-approximated by our simple analytic

n variational solution, are relevant to a range of potential experiments involving attractively-interacting Bose-
a Einsteincondensates.
J
9 PACSnumbers: 03.75.Lm67.85.Bc

]
s I. INTRODUCTION its own right, would be highly advantageous in experiments
a seeking to probe quantum effects beyond the mean-field de-
g
scription[27–29],andpossiblytoexploittheeffectsofmacro-
- Bright solitons are self-focusing, non-dispersive, particle-
t scopic quantum superposition to enhance metrological pre-
n like solitary waves occurring in integrable systems [1, 2].
cision [31, 32]. Similar concerns regardingadverse residual
a They behave in a particle-like manner, emerging from mu-
u 3Deffectsininterferometricprotocolspromptedarecentper-
tualcollisionsintactexceptforshiftsintheirpositionandrel-
q turbative study of residual 3D effects in highly anisotropic,
ative phase. Bright soliton solutions of the one-dimensional
. repulsively-interactingBECs[33].
t
a nonlinearSchro¨dingerequation(NLSE)canbedescribedan-
m alytically using the inverse scattering technique [3, 4] and The potential instability to collapse of attractively-
are well-known in the context of focusing nonlinearities in interacting BECs [34–44] is the key obstacle to realizing
-
d optical fibers [4, 5]. Bright solitary matter-waves in an at- soliton-likebehaviorinaBEC.Previousstudiesofbrightsoli-
n tractivelyinteractingatomicBose-Einsteincondensate(BEC) tarywavedynamics,usingvariationalandnumericalsolutions
o representan intriguingalternativephysicalrealization [6–8]. of partially-quasi-1D GPEs [12, 13, 21, 45] [reductions of
c
In a mean-fielddescriptionan atomic BEC obeysthe Gross- the GPE to a 1D equation which retain some 3D character,
[
Pitaevskii equation (GPE) [9], a three-dimensional NLSE. in contrast to the full quasi-1D limit] and the 3D GPE [16–

2 While in general non-integrable, in a homogeneous, quasi- 18, 34], have shown the collapse instability to be associated
v one-dimensional(quasi-1D)limittheGPEreducestotheone- withnon-soliton-likebehavior. However,previousstudiesof
1
dimensionalNLSE,thussupportingbrightsolitons[10–15]. brightsolitarywavegroundstateshavefocusedonidentifying
4
thecriticalparametersatwhichcollapseoccurs. Approaches
8 Outside the quasi-1D limit the GPE continues to support
6 brightsolitarymatter-waves. Theseexhibitmanysoliton-like usedinthesestudiesincludepartially-quasi-1Dmethods[12],
. variationalmethods[46]usingGaussian[10,34,47]andsoli-
1 characteristicsandhavebeenthesubjectofmuchexperimen-
ton (sech) [34, 48] ansatzes, perturbative methods [49], and
1 tal [6–8] and theoretical [16–29] investigation. Both bright
numericalsolutionstothe3DGPE[34,35,43,44,48]. Inthe
1 solitonsandbrightsolitarywavesareexcellentcandidatesfor
1 lattercase,thecollapsethresholdparametershavebeenexten-
useinatominterferometry[30],astheircoherence,spatiallo-
: sivelymappedoutforarangeoftrapgeometries[43,44].
v calization and soliton-like dynamicsoffer a metrologicalad-
i vantagein,e.g.,thestudyofatom-surfaceinteractions[7,20]. Inthispaperweuseanalyticvariationalandhighlyaccurate
X
Towards this end, proposals to phase-coherently split bright numerical solutions of the stationary GPE to systematically
r
solitonsandbrightsolitarywavesusingascatteringpotential andquantitativelyassesshowsoliton-likethegroundstateof
a
[27–29]andaninternalstateinterferenceprotocol[18],andto anattractively-interactingBECinaprolate,cylindricallysym-
formsolitonmolecules[26]havebeenexploredinthelitera- metricharmonictrapis,overawideregimeoftrapandinter-
ture. However, while the dynamics and collisions of bright actionstrengths. Beginningwithpreviously-consideredvari-
solitary waves have been explored in detail and have been ational ansatzes based on Gaussian [10, 34, 47] and soliton
shown to be soliton-like in three-dimensional (3D) parame- [34,48]profiles,weobtainnew,analyticvariationalsolutions
ter regimes [16–18], less attention has been directed at the fortheGPEgroundstate. Comparingthesoliton-ansatzvari-
question of exactly how soliton-like the ground state of the ationalsolutiontohighlyaccuratenumericalsolutionsofthe
systemis. Inparticular,theexperimentalfeasibilityofreach- stationaryGPE,whichwecalculateoveranextensiveparam-
ingthequasi-1Dlimitofanattractively-interactingBEC,and eter space, gives a quantitative measure of how soliton-like
hence obtaining a highly soliton-like ground state, remains the groundstate is. In the regimewhere the axialand radial
an area lacking a thoroughquantitativeexploration. Obtain- trap strengths dominate over the interactions, we show that
ing such a ground state, in addition to being interesting in theGaussianansatzvariationalsolutiongivesanexcellentap-
2
proximation to the true ground state for all anisotropies; in II. SYSTEMOVERVIEW
thisregimethegroundstateisnotsoliton-like. Intheregime
inwhichtheinteractionsdominateovertheaxial,butnotthe
We consideraBECof N atomsofmassmand(attractive)
radial, trap strength we demonstrate that the soliton-ansatz
s-wave scattering length a < 0, held within a cylindrically
s
variationalsolutiondoesapproximatethetrue,highlysoliton-
symmetric,prolate(theradialfrequencyω isgreaterthanthe
like ground state. However, we show that the goodness of r
axial frequency ω ) harmonic trap. The ground state is de-
x
theapproximationandtheextentofthisregime,whereitex-
scribedbythestationaryGross-Pitaevskiiequation
istsatall,ishighlyrestrictedbythecollapseinstability;even
atlargeanisotropiesitoccupiesanarrowwindowadjacentto
~2 4πNa ~2
theregimewhereinteractionsbegintodominateoveralltrap 2+V(r) | s| ψ(r)2 λ ψ(r)=0, (1)
strengths,leadingtonon-quasi-1D,non-soliton-likesolutions −2m∇ − m | | −
" #
and,ultimately,collapse.
wherethetrappingpotentialV(r)=m[ω2x2/2+ω2(y2+z2)/2],
x r
Ourresultshavesubstantialpracticalvalueforexperiments
λisarealeigenvalue,andtheGross-Pitaevskiiwavefunction
usingattractively-interactingBECs;primarilytheydefinethe ψ(r) isnormalizedto one. Thisequationis generatedby the
challengingexperimentalregimerequiredto realize a highly
classicalfield Hamiltonian(throughthefunctionalderivative
soliton-like ground state, which would be extremely useful δH[ψ]/δψ =λψ)
∗
to observe quantum effects beyond the mean-field descrip-
tion such as macroscopic superposition of solitons [27–29].
~2
We note that bright solitary wave experiments to date have H[ψ]= dr ψ(r)2+V(r)ψ(r)2
not reached this regime [6–8]. Secondarily, our quantita- 2m|∇ | | |
Z "
tive analysis of a wide parameter space provides a picture 2πNa ~2
of the ground state in a wide range of possible attractively- | s| ψ(r)4 . (2)
− m | |
interacting BEC experiments. In particular, it indicates the #
regimes in which a full numerical solution of the 3D GPE
Thisfunctionaloftheclassicalfieldψdescribesthetotalen-
iswell-approximatedbyoneofouranalyticvariationalsolu-
ergyperparticle,andthegroundstatesolutionminimizesthe
tions,whicharesignificantlyeasierandlesstime-consuming
valueofthisfunctional.
todetermine.
Whendealingwithvariationalansatzesforthegroundstate
The remainder of the paper is structured as follows: Af- solution, we proceed by analytically minimizing an energy
ter introducing the most general classical field Hamiltonian functional in the same form as Eq. (2) for a given ansatz.
andstationaryGPEinSectionII,webeginbydiscussingthe Incontrast,highlyaccuratenumericalgroundstatesaremore
quasi-1DlimitinSectionIII.InSectionIIIAwedefinethedi- convenientlyobtainedbysolvingastationaryGPEofthesame
mensionlesstrapfrequencyγ;inthequasi-1Dlimitthisisthe formasEq.(1).
onlyfreeparameter,andallourresultsareexpressedinterms
of thisquantity. Similarly, ourvariationalansatzes aremoti-
vatedbythelimitingbehaviorsofthesolutioninthequasi-1D
case;inthiscasewedefinethemasGaussianandsolitonpro- III. QUASI-1DLIMIT
files, parametrized by their axial lengths. In Sections IIIB
and IIIC we find, analytically, the energy-minimizing axial A. Reductionto1Dandrescaling
lengthsforeachansatzasafunctionofγ. Comparisonofthe
resulting ansatz solutions to highly accurate numericalsolu-
For sufficiently tight radial confinement (ω ω ), such
tionsofthestationaryquasi-1DGPEallowsustodetermine, r ≫ x
thattheatom-atominteractionsarenonethelessessentially3D
in the quasi-1D limit, the regimes of low γ in which highly
[a (~/mω )1/2]itisconventional[10–15]toassumeare-
soliton-likegroundstates canberealized(SectionIIID).We s ≪ r
ductiontoaquasi-1DstationaryGPE
thenconsiderthe3DGPEinSectionIV.Thesystemthenhas
a second free parameter in addition to γ; we choose this to
~2 ∂2 mω2x2
beκ, the(dimensionless)trapanisotropy,whichisdefinedin + x g Nψ(x)2 λ ψ(x)=0. (3)
SectionIVA.InSectionsIVBtoIVEwedefine3DGaussian −2m∂x2 2 − 1D | | −
" #
andsolitonansatzes,adaptedfromtheirquasi-1Danalogsand
each parametrized by an axial and a radial length, and find Typicallyψ(r)istakentobefactorizedintoψ(x)andtheradial
theenergy-minimizinglengthsforeachansatz. Ingeneralthis harmonic ground state (mω /π~)1/2exp( mω [y2 + z2]/2~),
r r
requires only a very simple numerical procedure, and in the such that g = 2~ω a . Alternative fac−torizationsare also
1D r s
| |
limit of a waveguide-like trap can be expressed analytically possible, which lead to an effective 1D equation retaining
(Section IVF). In Section IVG we comparethe ansatz solu- more3DcharacterthanEq.(3)[12,13,21,45];similarfactor-
tionstohighlyaccuratenumericalsolutionsofthestationary izationshave also been introducedfor axially rotatingBECs
3DGPEand,inSectionIVHassessthepotentialforrealizing [50] and for quasi-2D BECs in oblate traps [51]. In the ab-
trulysoliton-likegroundstates. Finally,SectionVcomprises sence of the axial harmonic confining potential (ω 0),
x
→
theconclusions. thereexistexactbrightsolitonsolutionstoEq.(3)ofthegen-
3
eralform1 where the variational parameter, ℓ , quantifies the axial
G
1 [x vt+C] length. In the trap-dominatedlimit (γ ), the true solu-
2b1x/2sech −2bx !eiv(x−vt)m/~eimg21DN2t/8~3eimv2t/2~eiD, Etioqn.(t7e)nydisetlodsa(GusaiunsgsiiadnenwtiittihesℓGfr=om1.ASpup→besntid∞tiuxtiAng)Eq.(8)into
(4)
whereb ~2/mg Nisalengthscalecharacterizingthesoli-
x 1D γ 1 2
ton’sspat≡ialextent,visthesolitonvelocity,C isanarbitrary H1D(ℓG)= 4 ℓG2 + ℓ2 − (2πγ)1/2ℓ , (9)
displacement,andDisanarbitraryphase.  G G
keyThleinsgetffhescctaivlees:1tDheGaxroiasls-hPairtmaeovnsikciileenqgutahtiaoxn co(~n/taminωsx)t1w/2o, wℓ h.eSreetHti1nDgi∂sHnow/∂eℓxpr=es0serdevaesaalsfuthnacttitohneovfartihaetiaoxniaallelnenerggthy
≡ G 1D G
and the soliton length bx. A mathematicallyconvenientway describedbyEq.(9)isminimizedwhenℓ isapositive,real
G
toexpressthesinglefreeparameterofEq.(3)isasthesquare
solutiontothequarticequation
oftheratioofthesetwolengthscales;
ℓ
b 2 ~ω ℓ4 + G 1=0. (10)
γ x x . (5) G (2πγ)1/2 −
≡ a ≡ 4mω2a 2N2
x! r| s| The positive, real solution to this quartic is (see solution in
This parametrization is achieved by working in “soliton
AppendixB)
units”; lengths are expressed in units of b and energies are
x
expressedinunitsofmg21DN2/~2. Thissystemcanbecodified χ(γ) 1/2 2 3/2 1/2
as~=m =g1DN =1,andyieldsthedimensionlessquasi-1D ℓG = 24/3(πγ)1/6 χ(γ) −1 −1 , (11)
GPE (cid:2) (cid:3)  !  
1 ∂2 + γ2x2 ψ(x)2 λ ψ(x)=0, (6) wherewehave,fornotationalconvenience,definedχtohave
−2∂x2 2 −| | − γ-dependencesuchthat
" #
in which γ can be interpreted as a dimensionless trap fre-
quency [15]. The corresponding classical field Hamiltonian 1024π2γ2 1/2 1/3
χ(γ)= 1+ 1+
is 27
 ! 
H1D[ψ]= dx 12 ∂∂xψ(x)2+ γ22x2|ψ(x)|2− 12|ψ(x)|4 .  + 1 1+ 1024π2γ2 1/2 1/3. (12)
Z " (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12) #(7)  − 27 ! 
[EqT.h(e6c)]hoaincdetohfeγcfloasrst(cid:12)(cid:12)ihcealsifineglld(cid:12)(cid:12)efHreaempilatroanmiaente[rEinq.th(7e)1]DcaGnPbEe  
mostdirectlypicturedaschoosingtoholdinteractionstrength C. Variationalsolution:solitonansatz
constant while varying the axial trap strength, parametrized
by γ. Experimentally, however, any of ω , ω , a , and N Secondly,weconsiderasolitonansatz
x r s
may be varied in order to vary γ. In the case γ = 0 the ex-
1 x
actgroundstatesolutionisasingle,stationarybrightsoliton: ψ(x)= sech , (13)
ψ(x) = sech(x/2)/2. Inthefollowingsubsectionswedevelop 2ℓS1/2 2ℓS!
analytic variational solutions ψ(x) for general γ. Compar-
wherethevariationalparameter,ℓ ,againquantifiestheaxial
ing these solutionsto highlyaccurate numericalsolutionsof S
length. In the axially un-trappedlimit (γ 0), the true so-
thequasi-1DGPE thengivesapictureofthebehaviorofthe
→
lution tends to a classical brightsoliton, as described by the
groundstatewithγ. Furthermore,thesequasi-1Dvariational
aboveansatzwithℓ = 1. Thevariationalenergyperparticle
solutionsmotivatethelater3Dvariationalsolutionsandyield S
isgivenby(usingidentitiesfromAppendixA)
severalmathematicalexpressionswhichreappearinthemore
complex3Dcalculations.
π2γ2 1 1
H (ℓ )= ℓ2+ , (14)
1D S 6 S 4π2γ2ℓ2 − 2π2γ2ℓ
 S S
B. Variationalsolution:Gaussianansatz whichisminimizedwhen 
WefirstconsidertheGaussianvariationalansatz ℓ4+ ℓS 1 =0. (15)
1/4 S 4π2γ2 − 4π2γ2
γ
ψ(x)= πℓ2 e−γx2/2ℓG2, (8) Again,thisquarticcanbesolvedanalytically(seesolutionin
 G
  AppendixB)togivethepositive,realminimizingvalueofℓS;
1Equation(4)describessolutionsofunitnorm. Moregeneralsolitonsolu- χ(γ) 1/2 2 3/2 1/2
ℓ = 1 1 , (16)
tions(B/2b1x/2)sech(B[x−vt+C]/2bx)eiv(x−vt)m/~eiB2mg21DN2t/8~3eimv2t/2~eiD S 21(cid:2)1/6(π(cid:3)γ)2/3  χ(γ)! −  − 
ehraavlesnoolirtmonsBs(iamnudlteaffneecotuivsely.massη=B/4),asarisewhenconsideringsev- withχdefinedasinEq.(12).  
4
1.0 ground state ψ0(x), and the corresponding ground state en-
ℓ 0.8 (a) SGoaliutossnaannssaattzz,,ℓℓGS(cid:4)• mergetyricE1GDa,uussse-Hs earmpsieteudfuonspceticotnrasl;mtheisthiosdainsimapblaifisiesdovfesrsyimon-
xialngth, 00..46 oisfethxeplpasineueddoisnpmecotrraeldmeteatihloidnuthseednfeoxrt3sDectciaolnc.ulSaetivoenrsa,lwquhaicnh-
Ae
l 0.2 tities are compared in Fig. 1(b–d): the variational minimum
0.0 energies H1D for each ansatz and the numericalgroundstate
0.01 energyE areshowninFig.1(b);therelativeerrorbetween
1D
nalH1D 0.00 (b) 0.4 eHa1cDhaanndsaEtz1Din,dFeigfi.n1e(dc)a;sa∆nd=th(eHm1Da−ximE1uDm)/d|Eiff1Der|,einscsehboewtwnefeonr
riatioergy,--00..0021 H′1D00..23 tahnedmthoestnaupmperoripcrailatgeraonusnadtzstwataevewfuavnecftuionnct(itohna,tewxipthrelsoswedesats∆a)
Vaen-0.03 0.1 percentage of the maximum value of the numerically exact
groundstate, ∆ψ = max(ψ ψ )/max(ψ ) [Fig. 1(d)].
-0.04 0.0 | Ansatz − 0| 0
All the shown computed quantities are insensitive to a dou-
nal -1 (c) blingofthenumericalbasissizefrom500to1000states.
ariatio∆g()10 -3 lenBtoatphptrhoexiGmaautisosniatnoatnhde seoxlaictotnsoalnustaiotznessopvreorvaidlearagneerxacnegle-
vo -5 ofγ. In the regimeswhere the relativeerrorin the energy∆
iny,l becomes significantly lower than 10 9 in particular, the dif-
rorerg -7 ferencebetweentheansatzsolutions−andnumericalsolutions
rn
Ee -9 becomes generally indistinguishable from numerical round-
3 off error. For the Gaussian ansatz the convergence to this
x.%) (d) regime is noticeably slower than for the soliton ansatz [Fig.
a(
mψ 2 1(c)]. This effect is a consequenceof the parametrizationin
d∆
zeψ, terms of γ and the corresponding“soliton units”: increasing
alin 1 γ leadsnotonlyto tohighertrapstrength,butalsoto higher
mri peakdensities ψ(x)2,andhenceastrongernonlineareffect.
Norerro 0 Forlatercom| pari|sontothe3Dcase,itisusefultodefinea
-12 -8 -4 0 4 8 12 benchmarkvalueoftherelativeerror∆thatindicatesexcellent
log (γ) agreementbetweentheansatzandthenumericallyexactsolu-
10
tion. Such a definition, however,will varyaccordingto pur-
pose.Asourobjectivesinthispaperrelatesignificantlytothe
FIG.1.Comparisonofquasi-1Dvariationalandnumericalsolutions:
(a) Energy-minimizing axial lengths ℓ (Gaussian ansatz, squares) shapeof the groundstate, thisformsthe basis of ourbench-
G
andℓ (solitonansatz,circles)forthequasi-1DGPE.(b)Minimum mark; a maximum deformation of the wavefunction below
S
variationalenergycomparedwiththenumericallycalculatedground 0.1%ofthepeakvalue[asmeasuredby∆ψinFig.1(d)]cor-
state energy E (black line) for each ansatz: for low γ we show respondsverycloselyto∆< 10 5. Becausetherelativeerror
1D −
wHe1Dsh(soowlidHs1′yDm=bolHs)1,Dw/γhi(chholtleonwdsstyom−bo1/ls2)4, washiγch→ten0d;sfotor 1h/ig2haγs ∆thesacthuorasteensatonsaabtzaciskginroapupnldicvaablulee,oafv≈al1u0e−o1fin∆rfeoguimroersdwerhseoref
γ (H is equal to the energy expressed in the “harmonic
uni→ts,”∞~=m1′D=ω =1). (c)Relativeerrorinthevariationalenergy, magnitude below this backgroundvalue thus corresponds to
x an excellent match in shape between the ansatz and the nu-
∆=(H E )/E .(d)Normalizedmaximumdeformationofthe
1D− 1D 1D mericallyexactsolution. Withrespecttothisbenchmark,the
best-fitting ansatz wavefunction ψ with respect to the numeri-
Ansatz Gaussianansatzrepresentsanexcellentfitforlog (γ)>1.15,
calgroundstateψ ,∆ψ=max(ψ ψ )/max(ψ ),expressedas 10
0 | Ansatz− 0| 0 whilethegroundstateishighlysoliton-like(thesolitonansatz
a percentage. For clarity in (a,b) [(c)], every 16th [20th] datum is
markedbyasymbol. representsanexcellentfit)forlog10(γ)<−0.95.
D. Analysisandcomparisonto1Dnumericalsolutions IV. BRIGHTSOLITARYWAVEGROUNDSTATESIN3D
Theenergy-minimizingaxiallengthsℓ andℓ ,definedby A. Rescalingtoeffective1Dsolitonunits
G S
Eq.(11)andEq.(16)respectively,areshownasafunctionof
γ in Fig. 1(a). There is no collapse instability in the quasi- We now consider the cylindrically symmetric 3D Gross-
1DGPE, andsolutionsareobtainedforall(positive,real) γ. Pitaevskiiequation[Eq.(1)].Comparedtothequasi-1Deffec-
Asintendedbythechosenformsoftheansatzes,thelimiting tiveGross-Pitaevskiiequation[Eq.(6)],three-dimensionality
casesareℓ 1asγ andℓ 1asγ 0. Toeval- introducesan additionalrelevantlengthscale, the radialhar-
G S
uate the accu→racy of th→e an∞satzes for→generalγ→, we compare monic length a = (~/mω )1/2. We incorporate this into the
r r
each ansatz with the numericallydeterminedgroundstate of dimensionlesstrapanisotropyκ ω /ω ,whichformsanad-
r x
≡
the quasi-1D GPE. The computation of a numerically exact ditionalfreeparameter. Expressedinthesame“solitonunits”
5
asEq.(6),Eq.(1)becomes From Eq. (22) it follows that we must have ℓ >
G
1/(2πγ)1/2κ to obtainaphysicallyreasonablesolution, i.e., a
1 2+V(r) 2π ψ(r)2 λ ψ(r)=0, (17) real, positive value of kG, consistent with our initial ansatz.
−2∇ − κγ| | − Foragivensuchvalueofk ,Eq.(21)issolved(seesolution
" # G
inAppendixB)by
withcorrespondingenergyfunctional
H3D[ψ]=Z dr"12∇ψ(r)+·∇Vψ(∗r()r|ψ)(r)|2− κπγ|ψ(r)|4 , (18) wiℓthGχ=dheχfi(cid:16)n2γe4k/d3G−(a4π(cid:17)sγii1)n/12/E6kqG2/.3(12).χ(cid:16)γ2kG−4(cid:17)3/2−11/2−1, (23)
#
whereV(r)=γ2[x2+κ2(y2+z2)]/2.
Inthefollowingsubsectionsweobtainvariationalsolutions C. AnalysisofGaussianansatzsolution
for general κ and γ using ansatzes similar to the Gaussian
and soliton ansatzes employed in the previous section, with
Contrary to the quasi-1D limit, minimization of the vari-
anadditionalvariable-widthGaussianradialprofile.Contrary
ational energy in 3D requires simultaneous solution of two
to the case in the quasi-1D limit, a self-consistent energy- equationsfor the radiallength, k 1, and the axial length, ℓ .
minimizing solution for both the axial and radial length pa- G− G
These equations are, respectively, Eq. (22) and [rearranged
rameterscannotbeexpressedentirelyanalytically. However,
fromEq.(21)]
we reduce the numerical work required to the simultaneous
solutionoftwoequations,andintroduceastraightforwardit- (2πγ)1/2 1/2
erative technique to achieve this. We also consider the case k = 1 ℓ4 . (24)
of a waveguide-like trap (ωx = 0) separately, where an en- G " ℓG − G #
(cid:16) (cid:17)
tirelyanalyticvariationalsolutionexists(SectionIVF).Sub-
Theseequationsdictatethatphysicalsolutionsmusthave
sequently,inSectionIVG,weagaincomparetheansatzsolu-
tionstohigh-accuracynumerics.
1
<ℓ <1, (25)
(2πγ)1/2κ G
B. Variationalsolution:Gaussianansatz
andhencethatγ>1/2πκ2mustbesatisfiedinorderforphys-
icalsolutionstoexist.
WefirstconsideranansatzcomposedofGaussianaxialand
Where solutions exist, they must be found numerically.
radialprofiles.Wephrasethisas
However, a very practical method of numericalsolution fol-
lowsfromtheshapeoftheℓ surfacedefinedbyEq.(23),and
κ1/2γ3/4k G
ψ(r)= Ge−κγkG2(y2+z2)/2e−γx2/2ℓG2. (19) shownin Fig. 2(a), which is a decreasingfunctionof kG for
π3/4ℓ1/2 all (real, positive) γ. The method can be consideredgraphi-
G
cally, in termsof locatingthe intersection(s)of Eq. (22) and
Here, the first variational parameter, ℓ , quantifies the axial
G Eq.(24). Thesecurvesareshown,forvariousκ,inFig. 2(b–
lengthoftheansatzinanalogytothequasi-1Dcase. There-
d), along with the lower bound from inequality(25). Below
ciprocal of the second variational parameter, k 1, quantifies
G− aκ-dependentthresholdvalueofγthecurvesfailtointersect,
the radial length of the ansatz. In the trap-dominated limit
indicating instability of the BEC to collapse. At the thresh-
(γ ) both these lengths approach unity ( ℓ ,k 1).
→ ∞ { G G} → old value [dotted curves in Fig. 2(b–d)] there is exactly one
Substitution of this ansatz into Eq. (18) yields (using identi-
intersection, and above the threshold value [other curves in
tiesfromAppendixA)
Fig.2(b–d)]therearetwointersections. Inthelattercasethe
higher-ℓ intersection, which smoothlydeformsto the limit-
γ 1 2k2 2κ G
H3D(ℓG,kG)= 4 ℓG2 + ℓ2 − (2πγ)1G/2ℓ +2κkG2 + k2 (20) ing case {ℓG,kG} → 1 as γ → ∞, represents the physical,
 G G G minimal-energyvariationalsolution. Thissolutioncanbelo-
Setting the partialderivativeswith respectto bothℓG andkG cvaatleudeku¯si,nsgataisfsyiminpgle1“stka¯irc<aske”,minettohEodq:. 2su3bpsrtoitduuticnegsaattrriiaall
equaltozero,wededucethatℓG mustsolvethequarticequa- value,ℓ¯G,satisfyingℓ≤<Gℓ¯ G1,andsubsequentlysubstitut-
tion G G G ≤
ingthistrialvalueintoEq.22producesaniteratedtrialvalue,
ℓG4 + (2kπG2γℓ)G1/2 −1=0, (21) ik¯tG′er,astaiotinsfoyfintghiks¯Gpr<ock¯eG′ss<coknGv.erTgehsusth,ebetrgiianlnvinalguewsitthok¯thGe=tru1e,
k andℓ .
G G
andthatk mustsolve Thephysicalsolutionstoequations(21)and(22)fordiffer-
G
entanisotropiesκareshownontheℓ surface,andprojected
G
k = (2πγ)1/2κℓG 1/4. (22) into the ℓG–γ plane, in Fig. 2(a). These solutions are also
G (2πγ)1/2κℓG−1! shown as black crosses in the ℓG–kG plane in Figs. 2(b–d),
6
1.0
Axiallength,ℓ (a)
G
0.8
1
−G
k
Anisotropy,κ gth,0.6 0.56
n γ ≈
e
l γ=1.2
al
di0.4
a
R γ=4.8
0.2
(b)
γ=900 κ=1
(Radiallength)2,kG−2 log10(γ) 0.00.0 0.2 0.4 0.6 0.8 1.0
Axiallength,ℓ
G
1.0 1.0
2
0
0.8 0.026 0.8 0.≈0
1−G γ ≈ 1−G γ
k k
h,0.6 γ=0.25 h,0.6 γ=0.01
gt gt
n n
e e
l l
al al
di0.4 di0.4
Ra γ=4.8 Ra γ=4.8
0.2 0.2
(c) (d)
γ=900 κ=16 γ=900 κ=256
0.0 0.0
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
Axiallength,ℓ Axiallength,ℓ
G G
FIG.2.Energy-minimizingvariationalparametersforthe3DGPEusingaGaussianansatz:(a)axiallengthℓ asafunctionoftheradiallength
G
k 1 andtheparameterγ[Eq.23]. Linesshowthesimultaneoussolutionsofequations(22)and(24)fortheaxiallengthℓ andradiallength
G− G
k 1,fordifferentanisotropiesκandvaluesofγ. Projectionsofthesesolutionsontheγ–ℓ planearealsoshown;heretheblacklineindicates
G− G
thequasi-1Dresult[fromfigure1(a)]. (b–d)Illustrationoftheintersectionsofequations(22)[lineswithverticalasymptoteℓ =1/(2πγ)1/2κ
G
shownwithfinedashes]and(24)forvariousκ:thehigher-ℓ intersection,whichcorrespondstoaphysicalsolutionfortheaxiallengthℓ and
G G
radiallengthk 1,canbefoundusinga“staircase”methodstartingfromk = 1. Thenumericalsolutionsobtainedthisway, andshownby
G− G
pointsin(a),areshownbycrossesin(b–d). Thelowestvaluesofγplottedin(b–d)arethelowestforwhichaself-consistentGaussianansatz
solutionisfound.
where they form a line connecting the physical-solution in- in this regime. Importantly,for γ abovethe collapse thresh-
tersectionsofEq.(22) andEq.(24) forthevariousγ shown. old the projected curvesfor each anisotropyagree well with
InFig.2(a)thecollapseinstabilityismanifestasarapidrise theGaussianansatzinthequasi-1DGPE,suggestingthatthe
in k — correspondingto a decrease in radial extent— and Gaussianansatzgivesagoodapproximationtothetruesolu-
G
fall in ℓ — corresponding to a decrease in axial extent — tionhere.
G
just abovea κ-dependentthreshold value of γ. There are no
self-consistent solutions for these quantities below this col-
lapse threshold. For increasing anisotropies κ, this collapse D. Variationalsolution:solitonansatz
thresholdoccursatlowervaluesofγ. Forthehighesttwoval-
ues of κ considered the collapse threshold lies in the regime Secondly,weconsiderasolitonansatzcomposedofaaxial
whereℓ isalreadyapproaching0;ouranalysisoftheGaus- sechprofileandaradialGaussianprofile.Wephrasethisas
G
sian ansatz in thequasi-1Dlimitindicatesthatthe3D Gaus-
γ1/2κ1/2k
sian ansatz will be a poorapproximationto the true solution ψ(r)= (2πℓ )1/2Se−κγkS2(y2+z2)/2sech(x/2ℓS). (26)
S
7
1
(a)
Axiallength,ℓ
S
0.8
1
Anisotropy,κ −S
gth,k0.6 γ ≈0.59
n
e
l γ=1.2
al
di0.4
a
R γ=4.8
2 0.2
2,k −S (b)
ngth) γ=900 κ=1
le
log10(γ) (Radial 00.0 0.2Scaled0a.x4iallengt0h.6, √2πγℓ0.8 1.0
S
1 1
2
0
0.8 0.027 0.8 0.≈0
1−S γ ≈ 1−S γ
k k
h,0.6 γ=0.25 h,0.6 γ=0.01
gt gt
n n
e e
l l
al al
di0.4 di0.4
Ra γ=4.8 Ra γ=4.8
0.2 0.2
(c) (d)
γ=900 κ=16 γ=900 κ=256
0 0
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0
Scaledaxiallength, √2πγℓ Scaledaxiallength, √2πγℓ
S S
FIG.3. Energy-minimizingvariationalparametersforthe3DGPEusingasolitonansatz: (a)axiallengthℓ asafunctionoftheradiallength
S
k 1andtheparameterγ[Eq.30].Linesshowthesimultaneoussolutionsofequations(33)and(34)fortheaxiallengthℓ andradiallengthk 1,
S− S S−
fordifferentanisotropiesκandvaluesofγ. Projectionsofthesesolutionsontheγ–ℓ planearealsoshown;heretheblacklineindicatesthe
S
quasi-1Dresult[fromfigure1(a)]. (b–d)Illustrationoftheintersectionsofequations(33)[lineswithverticalasymptoteℓ =(π/3)/(2πγ)1/2κ
S
shownwithfinedashes]and(34)forvariousκ:thehigher-ℓ intersection,whichcorrespondstoaphysicalsolutionfortheaxiallengthℓ and
S S
radiallengthk 1, canbefoundusinga“staircase”methodstartingfromk = 1. Thenumericalsolutionsobtainedthisway, andshownby
S− S
pointsin(a),areshownbycrossesin(b–d). Thelowestvaluesofγplottedin(b–d)arethelowestforwhichaself-consistentsolitonansatz
solutionisfound.
Aswiththe3DGaussianansatz,thefirstvariationalparame- Onceagain,settingpartialderivativeswithrespecttobothℓ
S
ter,ℓ ,quantifiestheaxiallengthoftheansatzandtherecip- andk equaltozeroallowsustodeducethat:
G S
rocal of the second variational parameter, k 1, quantifies its
radiallength.Inthequasi-1DlimitbothlengtG−hsconsequently ℓ4+ kS2ℓS 1 =0, (28)
approach unity ( ℓG,kG 1). Substituting this ansatz into S 4π2γ2 − 4π2γ2
{ } →
Eq.(18)yields(usingidentitiesfromAppendixA)
andthatk mustsolve
S
6κγℓ 1/4
π2γ2 1 k2 k = S . (29)
H (ℓ ,k )= ℓ2+ S S 6κγℓ 1
3D S S 6 S 4π2γ2ℓ2 − 2π2γ2ℓ S− !
 S S
 + 3κkS2 + 3κ . (27) toFobrotaminEaqp.h(y2s9i)cailtlyforlelaoswosnatbhlaetswoleutmiouns,ti.hea.,vaerℓeSal,>po1s/i6tiκvγe
π2γ π2γk2
S valueofkS,consistentwithourinitialansatz.Foragivensuch

8
valueofk ,Eq.(28)issolved(seesolutioninAppendixB)by self-consistentsolutionsexist. For increasinganisotropiesκ,
S
this collapse threshold again occurs at lower values of γ. In
ℓ = χ γkS−4 1/2kS2/3 2 3/2 1 1/2 1 , (30) claopnsteraisntsttoabtihlietycapsreecolfudtheessGoaluutsisoinansiannesxatazc,tlhyotwheevleimr,itthwehceorle-
withSχdhefi2(cid:16)n1e1d/6a((cid:17)πsiγin)2E/3q.(12).χ(cid:16)γkS−4(cid:17) −  −  opbnriloeitpyeexrotpfyeoocbtfsstehthrevecinosgolllahitpiogsnehliaynnsbstaaritbgzihltitot-ysboseelivtaoecnrce-ullyirkareteesgt(rrγoicu→tnsdth0set)a.pteoTsshsiiins-
3D. The solution curvesin Fig. 3(a) illustrate that this effect
isworstforlowtrapanisotropiesκ,butistosomeextentmit-
E. Analysisofsolitonansatzsolution igatedforhigherκ. However,afullcomparisonwithnumeri-
callyexactsolutionsisnecessarytoquantifytheseeffects;we
AsinthecaseoftheGaussianansatz,minimizationofthe undertakesuchacomparisoninSectionIVG.
variational energy in 3D requires the simultaneous solution
ofequationsfortheradiallengthk 1 andtheaxiallengthℓ .
S− S
These equations are, respectively, Eq. (29) and [rearranged F. Variationalsolution:waveguideconfiguration
fromEq.(28)]
In broad experimental terms, the collapse instability sets
1 1/2 a maximumvalue forthe ratio ofinteractionstrengthto trap
k = 1 4π2γ2ℓ4 (31)
S ℓ − S strength(equivalenttoaminimumvalueofγ)whichincreases
" S (cid:16) (cid:17)# (and hencethe minimumvalue of γ decreases) with the trap
Theseequationsdictatethatphysicalsolutionsmusthave anisotropy κ. In the context of atomic BEC experiments
one would typically think of controlling the interaction–trap
1 1
6κγ <ℓS < (2πγ)1/2 (32) sωtrecnognthstraantti;oibnythviasrysiintugaetiiothnetrh|easc|oolrlaNpswehinilsetahboillditiyngplωarceasnda
x
trap-anisotropy-dependent upper limit on the product a N.
andhencethatγ > (π/3)2/2πκ2 mustbesatisfiedinorderfor | s|
However,the minimumvalueofγ doesnotincreasewithout
physical solutions to exist. These equations and constraints
limit in the trap anisotropy κ: In an experiment one can, in
can be further simplified by casting them in terms of ℓ =
S′ principle,removeallaxialtrappingtocreateawaveguide-like
(2πγ)1/2ℓS;thisyieldstwoequations, configuration;inthiscaseωx =0andthetrapanisotropyκ
→
,whiletheparameterγ 0. Inthislimitareparametriza-
k = (2πγ)1/2κℓS′ 1/4 (33) ∞tionisnecessary,andonly→needstobeperformedforthesoli-
S (2πγ)1/2κℓ π/3 tonansatz,whichisclearlymoreappropriateinthiscontext.
and  S′ −  parEalmimeitnerastioonf othfeth3eDaxGiaPlEtra[pEqe.lim(1i7n)a]t.esTohneeroefmthaeintiwngoffrreeee
parameterisΓ=γκ =(a /2a N)2,wherea =(~/mω )1/2is
k = (2πγ)1/2 1 ℓ 4 1/2, (34) theradialharmonicoscillrato|rlse|ngthscale. Trhesolitonransatz
S ℓ − S′ maybere-writtenintermsofΓas
" S′ #
(cid:16) (cid:17)
andaninequality, ψ(r)= (2Γπ1ℓ/2k)1S/2e−ΓkS2(y2+z2)/2sech(x/2ℓS). (36)
S
π/3
<ℓ <1, (35)
(2πγ)1/2κ S′ SubstitutingthisintoEq.(18)withωx =0yields(usingiden-
titiesfromAppendixA),
whichareextremelysimilarto thoseencounteredinthecase
oftheGaussianansatz. Thenumericalsolutionoftheseequa- 1 k2 Γk2 Γ
H (ℓ ,k )= S + S + , (37)
tγGioa>nuss(sπfioa/n3r)at2hn/e2saπptκzh2.y,sfioclallowsosluthtieosna,mwehpicrhocceadnuroenalsyuesxeidstfowrhthene fromwhich3DweSdedSuceth2a4tℓtS2he−e1n2eℓrgSy-mi2nimiz2inkgS2variational
Variational-energy-minimizing solutions to the soliton parameterssatisfy
ansatzequationsfordifferentanisotropiesκareshowninFig.
1
3; these are shown superimposed on the ℓS surface and pro- ℓS = k2 (38)
jectedintotheℓS–γplaneinFig.3(a),andalongsideequations S
(28)and(29)andinequality(35)inFig.3(b–d).Thecollapse
and
instabilityisevenmoreevidentinthesolitonansatzthaninthe
Gaussianansatz,sinceitoccursinaregionwithalargerback- 6Γℓ 1/4
groundvalueofℓS. Onceagain,thecollapseismanifestasa kS = 6Γℓ S 1 . (39)
rapidriseink anddropinℓ —correspondingtobothaxial S− !
S S
andradialcontractionofthe solution—immediatelypriorto Contrarytothemoregeneral3Dcase,ananalyticsimultane-
a κ-dependentthresholdvalueof γ. Below the threshold, no oussolutionofEq.(38)andEq.(39)existswhenℓ satisfies
S
9
1.0 (a) 1.0 (a)
1
ℓgth,Sngth,k−S 00..89 xialℓngth, 00..68
ne Ae 0.4
el l
xialladial 00..67 0.2
AR 0.0
al -1 (b) D1.00 (b)
rrorinvariation∆nergy,log()10 ---753 Scaledgroundstateenergy,H′30000....88990505 Anκiκκs=oκκ==t2r==o516p66414y Gauss Soliton
Ee -9
-1 0 1log (Γ)2 3 4 nal -1.0 (c)
FIG. 4. Comparison of 3D variation1a0l and numerical solutions in ariatio∆g()10--32..00
tiaiaolwnlsae,vnegggitvuheidnℓeSbycaonEndfiq.rgau(d4ria1at)li,olneexn(igsωtthxfok=rS−1al0lf:oΓr(a=th)eκEγsnoe>lrigtoy3n-1m/2ai/nn4si.amtzi(z.bi)ngSRoealluxa--- rrorinvnergy,lo ---654...000
Ee
tive error in the minimum variational energy of the soliton ansatz, (d)
∆ = (H3D − E3D)/E3D, where E3D is the numerically determined nal -2.0
groundstateenergy. ariatio∆g()10-3.0
vo
thedepressedcubicequation iny,l -4.0
rg
or
1 rrne
ℓ3 ℓ + =0. (40) Ee -5.0
S− S 6Γ -3 -2 -1 0 1 2 3 4
log (γ)
10
Usingthegeneralsolutionforadepressedcubicequationfrom
AppendixB,onefindsthatthephysicalroot(withreal,posi- FIG.5. Comparisonof3Dvariationalandnumericalsolutions: (a)
tiveℓSsatisfyingthelimitℓS →1asΓ→∞)isgivenby Energy-minimizingaxiallengthsℓG(Gaussianansatz,solidsymbols)
andℓ (solitonansatz,hollowsymbols). (b)Scaledvariationalener-
S
1 1 3 1/2 1/3 gies H3′D = κH3D/γ(κ +1/2) (a similarly scaled ground state en-
ℓS = −12Γ + 33/2Γ 16 −Γ2 earllgyanEis3′oDtro=pieκsEκ3)Dc/oγm(κp+are1d/2w)itthentdhsentoum1eirnictahlleylcimalcitulγate→dg∞roufnodr
 ! 
 + 1 1  3 Γ2 1/2 1/3. (41) svtaartieateinoenraglieenseErg3Dy∆(bl=ac(kHd3Dot−s)E.3(Dc),d/)EN3DofromratlhiezeGdarueslsaitaivne(ecr)raonrdinsothlie-
−12Γ − 33/2Γ 16 − ton(d)ansatzes. Forclarityevery4thdatumismarkedbyasymbol
Consequently,solutionsonlyexistfor Γ>31/2/!4,asshownin in(a–d).
Fig.4(a).
generalizedLaguerrefunctions(radialdirection). Theansatz
withthelowestvariationalenergyisusedbothtooptimizethe
G. Comparisonto3Dnumericalsolutions scalingofthebasisfunctionsandasaninitialestimateforthe
solution. Expanding the stationary 3D GPE in such a basis
Thevariationalenergy-minimizingaxiallengthsℓ andℓ producesasystemofnonlinearequationswhicharesolvedit-
G S
are shown as functions of γ in Fig. 5(a) for the general 3D erativelyusingamodifiedNewtonmethod. Asimilarmethod
case; for the waveguide limit both axial and radial lengths wasusedtosolveasimilarcylindricallysymmetric,stationary
ℓ and k 1 are shown as functions of Γ in Fig. 4(a). As in 3DGPE,withrepulsiveinteractions,inRef. [52].
S S−
thequasi-1Dcase,wequantitativelyevaluatetheaccuracyof Asinthequasi-1Dcase,wecompareseveralquantitiesbe-
theansatzsolutionsforgeneralγ(Γ)bycomparingthevaria- tweentheansatzandnumericalsolutions.Fig.5(b)showsthe
tionalminimumenergyH withthenumericallydetermined scaled energy H = (H /γ)/(1+1/2κ) in the general 3D
3D 3′D 3D
ground state energy E . We calculate E using a pseu- case. Thisscalingissuchthat E —whichisdefinedanal-
3D 3D 3′D
dospectral method in a basis of optimally-scaled harmonic ogouslyto H withrespectto E —tendsto1asγ .
3′D 3D → ∞
oscillator eigenstates; this is formed from a tensor product Figs. 5(c) and (d) show the relative error in the variational
of symmetric Gauss-Hermite functions (axial direction) and minimumenergy∆ = (H E )/E fortheGaussianand
3D 3D 3D
−
10
solitonansatzes, respectively. Thesamequantity∆isshown H. Discussion
forthe waveguidelimit in Fig. 4(b). Allquantitiesshownin
Figs. 5 and 4 are computed using between 2000 and 12000
Aphysicalinterpretationoftheaboveresultsfollowsfrom
basisstates(κ-dependent)andareinsensitivetoadoublingof
consideringtwo conditionsthat must be satisfied in order to
thenumberofbasisstates.
realizeasoliton-likegroundstate;(1)theradialprofileshould
be“frozen”toaGaussian,thusrealizingaquasi-1Dlimit;and
(2)interactionsshould dominateoverthe axialtrapping. On
In the general 3D case, a close inspection of Fig. 5(b–d)
first inspection these conditions seem mutually compatible,
is necessary to revealthe overallrelationbetween the ansatz
andsatisfiablesimplybyincreasingtheradialtrapfrequency
solutions and the numerically obtained ground state. In the
ω with other parametersheld constant. However, condition
high-γ limit Fig. 5(b) shows that both the Gaussian varia- r
(1)canonlybesatisfiedifthemaximumdensityremainslow
tional energies (solid symbols) and the ground state energy
enoughto avoid any deformationof the radial profile due to
E (black dots) approach1 as γ , whereasthe soliton
an3sDatzenergies(hollowsymbols)te→ndt∞ohigherenergies.This thecollapseinstability.Increasingωrleadstoexactlysuchde-
formation,andultimatelytocollapse,asithasthesecondary
correspondstotheactualgroundstatemostcloselymatching
effectofstronglyincreasingthedensity. Thisstrongincrease
the Gaussian ansatz in this limit, as one would expect. In-
indensitywithω isparticulartothecaseofattractiveinterac-
deed,therelativeerrorinvariationalenergy,∆,fortheGaus- r
tions.Increasingω inarepulsively-interactingBEClikewise
sianansatz[Fig.5(c)]continuestodropexponentiallywithγ r
acts to increase the density, but this increase is counteracted
forallanisotropiesκ, makingitpossibletofindregimesofγ
bytheinteractions;theseacttoreducethedensity,andcause
where the Gaussian ansatz gives an excellent approximation
theBECtoexpandaxially.Intheattractively-interactingcase
tothetruegroundstate.
theresponseoftheinteractionsistheopposite: increasingω
r
leads to axial contraction of the BEC. Consequently condi-
In the opposite, low-γ limit, collapse occurs at a κ- tion (1) is far harder to satisfy for an attractively-interacting
dependent value of γ; this corresponds to the points in Fig. BEC than a repulsively-interacting one. Responding to this
5(a–d) where solution curves abruptly cease. Prior to col- problem simply by reducing the interaction strength (either
lapse(athighervaluesofγ)therelationbetweentheGaussian through as or N) leads to violation of condition (2). The
| |
ansatz,thesolitonansatz,andtheactualgroundstateishighly nature of the problem is made particularly clear by consid-
dependent on the trap anisotropy κ [Fig. 5(b)]. In the case ering the waveguide limit: here condition (2) is automati-
ofa sphericallysymmetrictrap, wherethe anisotropyκ = 1, cally satisfied (ωx = 0). This makes it possible to achieve
thesolitonansatzvariationalenergyisneverclosertothetrue a highly soliton-like groundstate by satisfying condition(1)
groundstateenergyE thantheGaussianansatzvariational alone. However,suchagroundstateisachievedbylowering
3D
energy. A regime of soliton-like groundstates consequently the product ωr1/2asN, and thus by progressing towards the
| |
cannot exist at this low anisotropy; as the soliton ansatz is limitofextremediluteness.
intrinsically asymmetric, this is to be expected. For higher Thisphysicalbehaviorofthesystempresentsconsiderable
anisotropies, the soliton ansatz energy is closer to E than challengesforexperimentsaimingtorealizeahighlysoliton-
3D
theGaussianansatzenergyinasmallregimepriortocollapse. likegroundstate. Inessence,themostdesirableconfiguration
Exactlyhowsoliton-likethegroundstateisinthisregimecan istohaveextremelyhighanisotropiesκ,whilekeepingωr as
be quantitativelyassessed using the relative error ∆. This is low as possible. Realizing such a configuration through ex-
shown for the soliton ansatz in [Fig. 5(d)]. For each κ the tremelylow,orzero,axialtrapfrequenciesωxisproblematic:
“background”valueof∆inthelimitγ isdifferent;this such frequencies are hard to set precisely experimentally as
→ ∞
effectisduetothedecreasingsizeoftheaxialpartoftheen- they require a very smooth potential to be generated, poten-
ergywithrespecttotheradialpartforincreasingγ. Intheop- tially over a considerable length. Furthermore, in the case
posite, low-γ, limit ∆ increases sharplyclose to the collapse ωx =0themean-fieldapproximationceasestobevalidforan
pointasthegroundstatewavefunctionrapidlycontracts. The attractively-interactingBEC;thetruewavefunctionshouldbe
maximumextentto which∆ decreasesfromits high-γlimit, translationallyinvariantin thiscase, butthe mean-fieldsolu-
beforethisincreaseduetocollapse-relatedcontractionatlow tion breaks this symmetry [53]. Even for very low but non-
γ,quantifieshowsoliton-likethegroundstatebecomesinthis zeroωx themean-fieldapproximationcanlosevaliditydueto
regime. Evenforthehighestanisotropyshown,κ = 256,the the extremedilutenessof the BEC, and the energygap from
regime of γ over which ∆ dropsbelow its backgroundvalue the ground state to states with excited axial modes can be-
is rather narrow, and the actual drop in ∆ is only one order come low enough to cause significant population of the ex-
ofmagnitude. ComparedtobenchmarkofSectionIIID,this citedstatesatexperimentallyfeasibletemperatures.
indicates that the true groundstate remains considerablyde- It is informativeto consider the parametersused in bright
formedwithrespecttothesolitonansatz. Theminimumerror solitarywaveexperimentstodate[6–8]. Noneoftheseaimed
in the soliton ansatz energydoes, however,improvewith in- torealizehighlysoliton-likegroundstatesinthesenseconsid-
creasing anisotropy κ. Excellent agreementcan be achieved eredhere. However,theynonethelessindicateregimeswhich
in thewaveguidelimit(κ ): Fig.4 showsthatexcellent haveprovedtobeexperimentallyaccessibleandofferaguide
→ ∞
agreement, with respect to the benchmark figure of Section to future possibilities. All have operated outside the regime
IIID,canbeobtainedforΓ>103/2. ofhighlysoliton-likegroundstates; directcomparisonofthe