Mean field effects on the scattered atoms in condensate collisions

1.1 MB·English

by  P. Deuar

Meanfield effects onthescattered atomsincondensate collisions
P. Deuar,1 P. Zin´,2 J. Chweden´czuk,3 and M. Trippenbach3
1InstituteofPhysics, PolishAcademyofSciences, Al. Lotników32/46, 02-668 Warsaw, Poland
2TheAndrzej Sołtan InstituteforNuclear Studies, Hoz˙a 69, PL-00-681 Warsaw, Poland
3Institute of Theoretical Physics, Physics Department,
1 University of Warsaw, Hoz˙a 69, PL-00-681 Warsaw, Poland
1 (Dated:January31,2011)
0
WeconsiderthecollisionoftwoBoseEinsteincondensatesatsupersonicvelocitiesandfocusonthehaloof
2
scatteredatoms. Thishaloisthemostimportantfeatureforexperimentsandisalsoanexcellenttestingground

n forvarioustheoreticalapproaches. InparticularwefindthatthetypicalreducedBogoliubovdescription,com-
a monlyused,isoftennotaccurateintheregionofparameterswhereexperimentsareperformed. Surprisingly,
J
besidesthehalopaircreationterms,oneshouldtakeintoaccounttheevolvingmeanfieldoftheremainingcon-

8 densateandon-condensatepaircreation. Wepresentexampleswherethedifferenceisclearlyseen,andwhere
2 thereduceddescriptionstillholds.

]
s
a I. INTRODUCTION ofinteractionstrengthforwhichtheRBMisquantitatively
g accurate.
-
t When two Bose-Einstein condensates(BECs) collide at
n
sufficiently high velocity, pairs of atoms are scattered out
a
u ofthecondensates. Aftermanyscatteringevents,adistinct II. BOGOLIUBOVAPPROACHTOTHESCATTERING
q haloofatompairsisformedinmomentumspace. Thiswas
OFATOMSINABECCOLLISION
. observedin manyexperiments[1–17] andanalyzedin nu-
t
a meroustheoreticalworks[1,7,11,18–34]. Theformation
m A. Scatteringincondensatecollisions
ofahalostartsspontaneouslyandisanalogoustothegener-
- ationofphotonpairsinparametricdownconversion. Such
d
photonpairswereusedtoobserveBellinequalityviolation Initially the gas is trapped by a harmonic potential. In
n
[35],andcanbeappliedforquantumcryptography[36]or order to start the (half-) collision, a superposition of two
o
c quantumteleportation[37].Inanalogy,atomsformedinthe counter-propagating, mutually coherent, atomic clouds of
[ collisionoftwoBECshaveapotentialapplicationforpreci- equal density is prepared by shining a Bragg pulse. The

sionmeasurements[38],interferometry[2,39–42],ortests trap is simultaneouslyturned off. The two fractionsmove
1
ofquantummechanics[43]. apartwithrelativespeed2v alongthez axis,wherev
v rec rec
3 Thesimplestmodelthatcapturestheformationofahalo istheatomicrecoilvelocity.Iftherelativevelocityislarger
3 in the condensate collision is the “reduced Bogoliubov” thanthespeedofsoundatthecondensatecenter,thecolli-
5 model (RBM). In this formulation, the two condensates sionissupersonic,superfluiditybreaksdownandahaloof
5 counter-propagateat a constantrelative velocity and with- scattered atoms is formed. The physicalpropertiesof this
.
1 outchangeofshape. Thewave-packetsenterintheBogoli- haloarethemainsubjectofourstudies.
0 ubov equation for the field of scattered atoms as a classi- Inourapproachwemodeltheprocessofthehaloforma-
1 cal source [17–22, 25, 26, 28, 31, 32]. The RBM can be tionusingatimedependentBogoliubovmethod,wherethe
1
usedtocalculatevariousobservables,suchasthedensityof fieldoperatorisdefinedas
:
v scatteredatomsortheirsecond-ordercorrelationfunctions,
Xi whichcanbedirectlycomparedwiththeexperimentaldata. Ψ(x,t)=φ(x,t)+δ(x,t). (1)
The predictions of the RBM are often in good agreement
r
a withexperiment[31],buttherearecaseswheremorecom- Here,φ(x,t) isbthecondensatewavebfunctiongovernedby
plete models have shown significant departures from the
theGPequation
RBM[1,29,32].
In this paper we compare results of the RBM and the
~2
complete Bogoliubov equation for a range of parameters. i~∂ φ(x,t)= 2+g φ(x,t)2 φ(x,t). (2)
t
In Section II we introduce both models and the numerical (cid:18)−2m∇ | | (cid:19)
methodofsolvingtheequationsofmotion. Twoadditional
simplified formulations, also introduced in Section II, are ItisnormalizedtoN –thenumberofatomsintheconden-
used to investigate the properties of the halo. The depen- sate, which remainsundepletedduringthe dynamics. The
dence of the halo shape on the model is shown in Sec- couplingconstantgrelatestothescatteringlengthathrough
tion IIIA for two characteristic cases, whereas Sec. IIIB g =4π~2a/m,wheremisamassofanatom.Thefieldop-
explainsindetailthedependenceofthehalopositiononthe eratorδ(x,t)describesthenon-condensedatoms(boththe
degreeofsimplificationsmade. Wealsoindicatetherange quantumdepletionandthescatteredhalo)andsatisfies the
b
2
1 A Complete B theinitialstateisthen
φ(x,0)=Aφ (x) eik0z +e−ik0z , (6)
0
0 whereAisthenormalizationc(cid:0)onstantandk0 =(cid:1) mvrec/~is
thewave-vectorassociatedwiththerecoilvelocity.Theini-
-1 tialstateofthenon-condensedpartistakentobeavacuum
0
k
/
ky1 C RBM D δ(x,0)|0i=0. (7)
Fromthisinitialstatetbhesystemevolvesaccordingtoequa-
0 tions(2)and(3).
-1
B. Dimensionlessparameters
-2 -1 0 1 2-2 -1 0 1 2
kz/k0
We point out that in our approximate description (Bo-
FIG. 1. Cross-sections through the density of non-condensed
patloanmesaifntemrothmeeenntudmofstphaececonllnics(iokn),=forhδbth†e(kb)oδbs(eke)nihianntcheedkcxas=eo0f gnoolni-ucboonvdmenestehdodfi)e,ltdheδr(exi)saisuindievnetriscaallsicnalainllgssyusctehmthsahtathve-
α=27.6,β =31.6.ThelabelsA,B,C,Dcorrespondtothefour ing the same value of gN (or aN, where a is a scatter-
b
models introduced inthe text. Color scalevaries between plots. inglength). Hencethedynamicsisdescribedbythelength
Theblackcircleshows |k| = k0 forreference. Thered(white) scaleaN ratherthanbythescatteringlengthaorthenum-
contourisat0.1(10−5)ofthepeakcondensatedensity. berofparticlesN separately.Otherimportantlengthscales
are a = ~/mω – the harmonic oscillator length, and
ho
1/k0. Alterpnatively, since there is no trap for t > 0, we
followingequation
canusethewidthoftheinitialcondensateσinsteadofa .
ho
~2 IfweapplytheThomas-Fermiapproximation[45],agood
i~∂tδ(x,t)= 2+2g φ(x,t)2 δ(x,t) choice of characteristic width σ is the Thomas Fermi ra-
(cid:18)−2m∇ | | (cid:19)
diusR = (15Ng/4πmω2)1/5. Inconclusion: thereare
b +gφ2(x,t)δ†(x,t). b (3) threereTlFevantlengthscales, henceoursystemcanbechar-
acterized by two dimensionless parameters. We choose:
Fromthisequation,wecantrbacebackaneffectiveBogoli-
β =k σandα=aN/σ(see[25,26,28,31]).
0
ubovHamiltonian:
The first parameterβ is independentof interactionsand
~2 has a kinetic character. It can be viewed as the number
Heff = d3xδ†(x) 2 δ(x) (4a) of fringes created by the Bragg pulses on the initial con-
Z (cid:18)−2m∇ (cid:19)
densate, or as a ratio of the dispersion time scale, mσ2/~
b b b
+2g d3x φ(x)2δ†(x)δ(x) (4b) to the collision timescale, (mσ)/(~k0), see for instance
Z | | [25,26,28,31]. Inrealisticsituationsβ 1.
+ g d3xφ2(x)δ†b(x)δ†b(x)+ h.c. (4c) The second parameter α is proportio≫nal to the ratio of
2Z theinteractionenergyperparticlegN/σ3 tothekineticen-
b b ergyperparticleintheinitialcondensate~2/(2mσ2). Itis
Line (4a) stands for kinetic energy, while line (4b) re-
relatedtoα(past) =mgN/(σ~2π3/2)whichhasbeenused
sults from the interaction of the condensate mean-field
previously[25,26,28,31],butdiffersbyanumericalfactor:
with the scattered atoms. Finally (4c) describes the cre-
α = (√π/4)α(past). Theratiobetweenαandβ quantifies
ation/annihilationprocessofpairsofnon-condensedatoms.
thestrengthandnatureofscattering.Ithasbeenshownthat
MostsalientfeaturesofBECcollisionsarealreadyfound
Boseenhancementofscatteringoccursforα&β [26].
forthecaseofinitiallysphericallysymmetriccondensates.
Beforethenumericalsimulationofthecollision,wefindthe
condensate wave-function as a ground state of the Gross-
C. SimulationofdynamicsusingtheSTABmethod
Pitaevskiiequationwithharmonictrapoffrequencyω,
~2 1 TheBogoliubovequation(3) canbe solvednumerically
µφ0(x)= 2+ mω2x2+g φ0(x)2 φ0(x). usingthepositive-Prepresentationmethod[44], wherethe
(cid:18)−2m∇ 2 | | (cid:19)
(5) fieldoperatorδisreplacedwithtwocomplex-numberfields
Next, the trap is turned off and at the same time a set of ψ(x)andψ(x). Thedynamicsofthesystemisgovernedby
b
Braggpulses is applied [14]. In the center-of-massframe, apairofstochasticItoequations,
e
3
~2
i~∂ ψ(x,t)= 2+2g φ(x,t)2 ψ(x,t)+gφ(x,t)2ψ(x,t)∗+ i~gφ(x,t)ξ(x,t), (8a)
t
(cid:18)−2m∇ | | (cid:19)
p
~2 e
i~∂ ψ(x,t)= 2+2g φ(x,t)2 ψ(x,t)+gφ(x,t)2ψ(x,t)∗+ i~gφ(x,t)ξ(x,t). (8b)
t
(cid:18)−2m∇ | | (cid:19)
p
e e e
1 A Complete1 B D. PartiallyreducedBogoliubovmodels
A fortuitous “side-effect” of the method formulated
0 0
above is the possibility to systematically add or remove
partsoftheeffectiveHamiltonian(4)andGPequation(2)to
-1 -1 understandtheirimpactonthedynamicsoftheδˆfield. This
0
k
/ -2 -1 0 1 2 includesanumericalsimulationoftheRBM.Todothis,we
ky1 C RBM D firstwritethecondensatewavefunctionasasumofleft-and
right-movingwavepackets,
0 φ(x,t)=φ (x,t)+φ (x,t), (9)
L R
-1 whereφ (x,0)=Aφ (x)e±ik0z withtheGPequation
L/R 0
-2 -1 0 1 2-2 -1 0 1 2 ~2
kz/k0 i~∂ φ (x,t)= 2 (10a)
FIG.2. AsFigure1,butforthecaseofα=9.2,β=31.6where t L/R (cid:18)−2m∇
bosonicstimulationofscatteringisnegligible. +g φ (x,t)2 (10b)
L/R
| |
+2g φ (x,t)2 (10c)
R/L
| |
Here ξ(x,t) and ξ(x,t) are delta-correlated, independent, +gφR/L(x,t)∗φL/R(x,t) (10d)
realstochasticnoisefieldswithzeromean.Thesecondmo-
mentsareequaltoe (cid:19)φL/R(x,t).
ξ(x,t)ξ(x′,t′) =0 and The terms proportional to the coupling constant g can be
h i
ξ(x,t)ξe(x′,t′) = ξ(x,t)ξ(x′,t′) =δ(x x′)δ(t t′). interpreted as the self-interaction of the wavepacket (10b)
h i h i − − and cross-interaction between differentwavepackets (10c)
e e respectively.Theremainingprocessesareinline(10d).
Numerically,ξandξareapproximatedbyrealgaussianran-
Next, the decomposition (9) is put into the Bogoliubov
domvariablesofvariance1/(∆t∆V)thatareindependent
e Hamiltonian(4)giving
ateachpointatthecomputationallattice(ofvolume∆V),
andateachtimestepoflength∆t. ~2
H = d3xδ†(x) 2 δ(x) (11a)
Any physicalquantity is obtained by substituting δ† eff Z (cid:18)−2m∇ (cid:19)
→
ψ∗andδ ψandchangingfromaquantumaverageofthe b b b
normally-→orderedoperatortoastochasticaverage[46b], +2g d3x φL(x)+φR(x)2δ†(x)δ(x) (11b)
e b Z | |
b b
DYj δb†(xj)Yk δb(xk)E=mlri→m∞DYj,k ψe(xj)∗ψ(xk)Est. +g +d3gxZ φd3(xxφ)L2(+x)φφR((xx))2δb†δ(†x()xδb)†δ(†x()x+)+h.ch..c. ((1111dc))
L R
The braces st denote the statistical average over mr re- 2Z (cid:0) (cid:1)
h·i b b
alizations. Observablesink-spacefollowfromtheFourier The resonant term (11c) governs the process where two
transformation.Forexample,theone-particledensityreads atoms, one from φ (x) and one from φ (x), collide and
R L
elastically scatter into the halo localized aroundthe radius
ρ1(k,k′)=φ(k)∗φ(k′)+Rehψ(k)∗ψ(k′)ist. |k| ≈ k0 asintheusualRBM.Theline(11d)leadstocre-
ationoftwoatomsintheδˆfieldoriginatingfromonewave-
e
The stochastic equations (8) strictly reproduce the full packet,andwecallthishere“on-condensatepairing”. This
quantumdynamicsdescribed by H when the numberof incudesquantumdepletionprocesses.
eff
samples tends to infinity. For finite sample sizes, one ob- Note thatthe specific formof Equations(10–11) allows
b
tains an estimator for the full quantum dynamics, with an forremovalofchosenterms.Thisway,onecaninspecttheir
uncertaintythatiscalculatedbystandardmethods[47]. roleinthedynamicsofthesystem. Forinstance,wecanne-
4
(a) (b) III. RESULTS
A. Theshapeofthehalo
]
.
u
.
a
[ We first investigate the dynamics of the system in two

y
t characteristiccases,varyingonlyα,whileβ =31.6iskept
i
s constant. It has been previously shown, that for Gaussian
n
e colliding clouds, significant Bose enhancement of scatter-
d
ing occurs when α(past)/β 2 or greater, i.e. α & β.
≈
Figures1and2showthecross-sectionthroughthehaloof
scatteredatomsinthek = 0planeattheendofthecolli-
x
sion,forthecaseofappreciable(α = 27.6)andnegligible
0.8 1 1.2 0.8 1 1.2
k /k (α=9.2)bosonicenhancementofscattering. Densitypro-
y 0
filesofthehalointhesetwocasesareshowninFig.3.
Whenthedynamicsoftheδˆfieldisdescribedbythesim-
FIG.3. Cross-sectionsthroughthehaloofscatteredatomsalong
plified models C and D, the density of scattered atoms is
the ky axis after the end of the collision for α = 27.6 (a) and
α= 9.2(b). ThecurvesarecalculatedusingthemodelsA(red), spherically symmetric. On the other hand, this symmetry
B(green),C(black)andD(blue).Triplelinesindicatethebeststa- is lost within models A and B, where the halo of atoms
tisticalestimateandthe±1σstatisticaluncertaintiesinthemean. is weakened near the condensates. This effect can be at-
tributed primarily to the mean field (11b), with some en-
hancementresulting from (11d). Notice that for larger in-
glectalltermsbutthekineticenergy(11a)andpairproduc-
teractionstrength–andconsequentlylargerα–thesephe-
tion(11c)intheBogoliubovHamiltonian. Thissimplifica-
nomenaaremorepronounced. Moreover,modelsAandB
tionwecallthePairProduction(PP)dynamics.Withinthis
predictsomenon-condensedatomsappearingontopofthe
approximation,thePositivePformulationtakestheform:
BECs(inthesamelocationastheBECs). Thelattereffect
~2 2 resultsfromthe term (11d) only,andit asrelatedto quan-
i~∂tψ(x,t)= ∇ ψ(x,t)+2gφL(x,t)φR(x,t)ψ(x,t)∗ tumdepletion.
− 2m
+ 2i~gφ (x,t)φ (x,t)ξ(x,t), e (12a)
L R
p~2 2
i~∂ ψ(x,t)= ∇ ψ(x,t)+2gφ (x,t)φ (x,t)ψ(x,t)∗ B. Positionofthehalocenter
t L R
− 2m
e + 2i~gφeL(x,t)φR(x,t)ξ(x,t). (12b) Inthemomentumspacewe denotelocationofthe max-
p imumofthehalodensitybyk . Thisparameter,briefly
We can also simplify the GP equatione(11) neglecting all max
discussedin[1],variesbetweenmodelsA-D,ascanbeseen
the nonlinear terms and approximating the kinetic energy
inFigs.1and2andFig.3inmoredetail.Theshiftofthepo-
operator. As a result of this treatment we obtain the stiff
sitionofthehalocanbeexplainedwhentheBECsaremod-
movement(SM)ofthecounter-propagatingwave-packets,
elledwithtwocounter-propagatingplane-waves[1,48],and
i~∂ φ (x,t)= ~2k0 k 2i ∂ φ (x,t) (13a) usingenergyconservationargument. The energyof a par-
t L − 2m (cid:18) 0− ∂z(cid:19) L ticlereleasedfromthecondensatesdependsontheformof
~2k ∂ theGPequationweusetodescribeit. ThefullGPequation,
i~∂tφR(x,t)= 0 k0+2i φR(x,t). (13b) withn-themeandensityofthesystem,gives ~2k02 + 3gn,
− 2m (cid:18) ∂z(cid:19) 2m 2
whiletheSM(13)givesapurelykineticvalue ~2k02. Onthe
The RBM, described in literature, consists of both 2m
other hand the energyneeded to place a particle in a non-
the PP and SM simplifications. Also, using alternative
combinations of the approximations described above, we condensedmodewithmomentum~kisequalto ~2k2+2gn
2m
cangeneratefourdifferentmodels: inthecaseoffullBogoliubovand ~2k2 inthecaseofthePP
2m
equation(12). Hence,usingenergyconservationweobtain
A: GPequation(2)+fullBogoliubov(8) A: ~2k2 = ~2k02 1gn
2m 2m − 2
B: SM(13)recombinedto(9)+fullBogoliubov(8).
B: ~2k2 = ~2k02 2gn
C: SM(13)+PP(12)equation wecallitRBM 2m 2m −
→
D: GPequation(reducedasin(10))+PP(12)equation. C: ~2k2 = ~2k02
2m 2m
Inthefollowingsectionwecomparethedistributionsof
scatteredatomsobtainedusingthefourabovemodels. D: ~2k2 = ~2k02 + 3gn
2m 2m 2
5
Finally,wenotethatforsmallenoughα,thediscrepancy
1.2
betweentheRBMandfullBogoliubovtreatmentisnegligi-
ble,asexpected.Theregionofagreement(α.β)matches
1 approximatelytheregionwherestimulatedscattering(Bose
enhancement)isabsent,althoughinvestigationwithvarious
2
k valuesofβ wouldbenecessarytoconfirmorrefutealink.
max0.8
k2
0
0.6
IV. CONCLUSIONS
0.4
We have demonstrated how differentapproximationsof
thefullBogoliubovequationdramaticallyinfluencethere-
0 0.02 α/β2 0.04 0.06
sultsofnumericalsimulationsofBECcollisions. Although
the simpliest RBM approach predicts a spherical density
FIG.4. Dependenceofthehalopeakpositionkmaxonα/β2,cal- ofscatteredatoms,moreinvolvedmodelsshowsignificant
culatedwithA(redcircles),B(greensquares),C(blacktriangles) discrepanciesfromthisdistribution. Thesedifferencescan
andD(blue×). Errorbarsindicatethe±1σstatisticaluncertain- beunderstoodasaresultoftheactionofmean-fieldterms
tiesinthemean. in the GP and Bogoliubov equations. Also, the position
of the density maximum changes from model to model.
Thiseffect,whichwasinvestigatedwhentheBECsareap-
Thissimplifiedmodelgivesquantitativeagreementwiththe
proximatedbytheplane-waves,ishereinterpretedfornon-
resultspresentedinFigure3.
uniform condensates. The RBM model is quantitatively
Notice thatin allmodels(k/k )2 is a linear functionof
0 correct when the interaction strength, as quantified by α,
the parameter gn 2m . In order to generalize this depen-
~2k02 issufficientlysmall.
dence to the case of non-uniformcondensates, we replace
n with half of the maximaldensity, which in the Thomas-
Fermiapproximationisn = (15/8π)N/R3 . Recall-
max TF
ing the definitionsof α and β, we obtain the shift propor- ACKNOWLEDGMENTS
tionalto α/β2. To verifythis conjecture, in Fig.4 we plot
(k /k )2 asafunctionofα/β2 forallfourmodels. We We acknowledge fruitful discussions with Chris West-
max 0
observethatforgrowingα/β2 (andthusgrowingα,aswe brook, Denis Boiron and Karen Kheruntsyan. M. T. ac-
keepconstantβ = 31.6),thereissomedeviationfromthe knowledgessupportofPolishGovernmentResearchGrants
linearbehaviourformodelsAandD.Whatthesetwomod- for2007-2011,P.D.supportbytheEUcontractPERG06-
elshaveincommonisthefullevolutionofthecondensates, GA-2009-256291and J. C. was supported by Foundation
whichmightmakethepictureofplanewavecollisionswith for Polish Science International TEAM Programme co-
atime-independentdensitydifficulttouphold. financedbytheEUEuropeanRegionalDevelopmentFund.
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