Interacting Bosons beyond the Gross-Pitaevskii Mean-Field

0.25 MB·English

by  M. Heimsoth

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0 InteractingBosonsbeyondtheGross-PitaevskiiMean-Field
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M.HeimsothandM.Bonitz

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Christian-AlbrechtsUniversit¨atzuKiel,Institutfu¨rTheoretischeundAstrophysik,24098Kiel,Germany
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J

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Abstract

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l SystemsconsistingofcoldinteractingbosonsshowinterestingcollectivephenomenasuchasBose-Einsteincondensationor
l
a superfluidityandarecurrentlystudiedincondensedmatterandatomicphysics.Ofparticularinterestarenonidealbosons
h whichexhibitstrongcorrelationandspatiallocalizationeffects.Hereweanalysetheground-stateofatwo-dimensionalBose
- systemwithaHartree-FocktypeapproximationthatwasfirstintroducedbyRomanovskyetal.[Phys.Rev.Lett.93,230405
s
e (2004)].Weapplythismethodtoaonedimensionalsystemofchargedbosonsandanalyzethebehaviouratstrongcoupling.
m
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t Keywords: bosesystems,Hartree-Fock
a
m
-
d 1. Introduction tionstrength.
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Inthemany-bodytheory,Hartree-Fockisastandard
o
c With the first experimental realisation of a Bose- method to analyse systems of few to many fermions.
[ Einsteincondensatein1995[1],thetheoryofultracold Therelatedequationscanbederivedinseveraldiffer-

bosonicsystemsbecameafieldofexceptionalinterest. ent ways. One is obtained by expanding the reduced
1
v Inthefirstexperiments,theatomswereweaklyinter- two-particle density matrix in terms of one-particle
2 acting,andforsuchsystemstheGross-Pitaevskiimean densitymatricesinordertoobtainaneffectivesingle-
5 fieldapproximation(GP)isanadequateapproachfor particlehamiltonian.Theresultingsolutionstothisap-
8 theanalysisofthecondensate[2].However,recentex- proach are many-particle states consisting of one sin-
1
perimentssucceededinstronglyincreasingtheinterac- gleanti-symmetrisedproduct-state.Anothercommon
.
1 tion strengths of the investigated atomic systems [3]. wayproceedsintheoppositedirectionbytheassump-
0 With this tuning of the coupling, it was possible, to tion, that the many-particle state is given by a single
9
0 produce a phase transition from the superfluid to the anti-symmetrised product state (Slater determinant).
: Mott insulator phase [4]. Also, for charged bosons in By applying the Ritzian principle to this ansatz, one
v
traps crystallization and inhomogeneous distribution obtainstheHartree-Fockequations.
i
X ofthesuperfluiddensityhasbeenpredicted[5].Fora In contrast to Fermi systems, for bosonic particles
r theoretical description of these effects an approxima- the two approaches lead to different systems of equa-
a tion beyond GP is required [6,7,8]. In this paper we tions.EventheansatzassumingasingleSlaterperma-
analyseanapproximationmethodforbosonicsystems nent can be implemented in various ways differing in
which is based on a Hartree-Fock type factorization thechoiceofsupplementaryconditions[6,7].
andisexpectedtobevalidforawiderangeofinterac- Thebestknownapproximationforanultracoldnon-
ideal dilute Bose gas is the Gross-Pitaevskii approxi-
PreprintsubmittedtoPhysicaE 13January2009
mationinwhichitisassumed,thatallparticlesremain 2. UnrestrictedbosonicHartree-Fockansatz
in the same one-particle orbital. Thus the GP equa- (UBHF)
tiondescribesonlythecondensatefractionofthesys-
temwithoutanyinteractionwiththeremainiggas.By The UBHF method is derived from the following
definitiontheGPapproximationcannotdescribephe- ansatzforthemany-particlestate
nomena such as condensate depletion and fragmenta-
N
1 X O
tionorMott-insulatorphasetransitionsofcoldatomic |Φ(cid:105)=|12...N(cid:105):= √ |π(s)(cid:105), (1)
gasesonalattice.Theresultsbecomeevenworseifthe N!π∈SN s=1
interactions increase or if systems with small particle whichmeansthateveryparticleremainsinacertainor-
numbers are considered. As we will see later, this ap- bitalandtheunderlyingmany-particlestateisasym-
proximationcannotdescribeanumberoffundamental metrisedproduct-state.Dependingontheimposedad-
properties, such as the localisation of the particles at ditional restrictions, this ansatz contains, as limiting
highcouplingstrengths. cases, well-known approximations. In particular, the
Anothercommonnumericalmethodtodealwithin- GP approximation is obtained by the additional re-
teractingsystemsistheConfigurationInteraction(CI) striction, that all orbitals are identical [8], thus the
inwichnoapproximation,exceptthelimitationofthe many-particlestateisassumedtobetotallyBosecon-
chosenbasisset,ismade,e.g.[9].Withnowadayscom- densed. A more general approximation would be ob-
puters and highly optimised programs CI can be ap- tainedbytheassumption,thattwoorbitalsareeither
pliedtosystemswithupto10particles. equalororthogonaltoeachother.Thiskindofansatz
In this paper we want to analyse an approximation hasbeenintroducedandanalysedextensivlybyCeder-
with substantially less numerical effort than CI, that baumetal.[6,12,13,14].
provideshighqualitynumericalresultsforstronglycor- In contrast, the UBHF approximation is the most
related bosons in traps. This approximation, the Un- generalcaseofthisAnsatzwherenofurtherrestriction
restrictedbosonicHartree-Fockmethod(UBHF),was to the underlying one-particle orbitals |1(cid:105),...,|N(cid:105) is
first introduced by Romanovsky et al. [10,7,11] and imposed. We only require the many-particle state to
is less restricted than the related ansatz proposed by benormalised,sothetotalenergyisgivenby
Cederbaum [6]. While Romanovsky et al. used an ex-
E =(cid:104)Φ|Hˆ|Φ(cid:105). (2)
plicit analytical ansatz for the single-particle orbitals
(discplaced Gaussians) we develop a completely gen- ThusminimisingE leadstothefunctional
eralschemewithoutanysuchrestriction.Thishasthe E(|1(cid:105),...,|N(cid:105),E)=(cid:104)Φ|Hˆ|Φ(cid:105)−E((cid:104)Φ|Φ(cid:105)−1), (3)
advantagethatourmethodisapplicabletoanyinter-
whereE isaLagrangemultiplyerforthenormalisation
actingBosesystem.
of |Φ(cid:105). The needed equations to determine the one-
Furthermore, our goal is to compare the UBHF re-
particle orbitals are obtained by applying the Ritzian
sults with CI results to obtain a quantitative conclu-
principletothisAnsatzviathesearchedorbitals.This
sion about the accuracy of this method. To this end
ansatzhasbeenproposedandimplementedforthefirst
weperformUBHFandCIcalculationsforN =2...6
timebyRomanovskyetal.[11,10],however,theyused
particlesinaharmonictrapusingthesamebasissets
the additional assumption that every orbital is a dis-
for both cases. Our comparisons concentrates on sys-
placedGaussian.Inthepresentpaper,thisrestriction
temswithacouplingparameterintherangefromzero
willbedropped.
tofiveandshowstheexcellentqualityofthisapproxi-
Theexpectationvalueoftheenergyofanormalised
mation.
singlepermanentmanyparticlestatewithinteracting
particlesisgivenby[15]
(cid:104)Φ|Hˆ|Φ(cid:105)= √1 X XN “Y(cid:104)s|π(s)(cid:105)(cid:104)l|hˆ|π(l)(cid:105)
N!
π∈SN l=1 s(cid:54)=l
1X Y ”
+ (cid:104)s|π(s)(cid:105)w . (4)
2 kl,π(k)π(l)
k(cid:54)=ls(cid:54)=k,l
2
The norm of such a productstate is proportional to 25
UBHF
the permanent of the Gramian matrix of the orbitals CI
GP
|1(cid:105),...,|N(cid:105): 20
N =4
N
(cid:107)Φ(cid:107)2 = √1 X Y(cid:104)s|π(s)(cid:105). (5) ω 15
N! h¯
π∈SNs=1 Ein 10 N =3
It is advantageous to stay in the abstract representa-
tion of the orbitals to perform the variational deriva-
5 N =2
tive. By applying the rules given in Appendix A one
obtainsthefollowingequation,bydifferentiating(3)
0
( 0 1 2 3 4 5
X P hˆ|π(n)(cid:105)+XP (cid:104)l|hˆ|π(l)(cid:105)|π(n)(cid:105) λ
n nl
π∈SN l(cid:54)=n
) Fig. 1. Comparison of the energies obtained with UBHF,
+XP Jˆ |π(n)(cid:105)+1 X P w |π(n)(cid:105) GPandCIfordifferentcouplingparametersλandfor2,3
nl lπ(l) 2 nlk kl,π(k)π(l) and4particles.
l(cid:54)=n k,l(cid:54)=n
k(cid:54)=l
X
=E Pn|π(n)(cid:105), foralln, (6) xisgiveninoscillatorunitsx =p(cid:126)/mω,andλisthe
π∈SN 0
ratiooftheCoulombandconfinementenergy
whereweintroducedthesubscriptedentitywithavari-
ationalnumberofsubscripts q2
λ= . (11)
Pi1...in =P(π)i1...in = Y (cid:104)s|π(s)(cid:105), (7) 4π(cid:15)x0(cid:126)ω
s(cid:54)=i1...in Thusλ=0correspondstoanidealsystem.Theshield-
andthedoublysubscriptedHartreeoperator
ing parameter κ is needed in 1D to make the two-
Z
Jˆ = dxdyϕ∗(x)w(x,y)ϕ (x)|y(cid:105)(cid:104)y|. (8) particle integrals convergent. For the limit κ −→ 0
ij i j
the system becomes fermionised [16]. For all calcula-
Bymultiplyingequation(6)fromtherightwith(cid:104)n|, tionsinthispaper,wechooseκ=0.1.Asbasisforthe
oneobtainsaclosedexpressionfortheLagrangemul- single-particle orbitals we chose the eigenfunctions of
tiplyer theidealsystemwithatotalnumberofbasisstatesof
E = (cid:104)Φ|Hˆ|Φ(cid:105), (9) nb = 15. We will compare the results with the exact
(cid:104)Φ|Φ(cid:105) solutionobtainedwithCI[9].TheCIcalculationsare
i.eE isthetotalenergyoftheSystem.Thekeyequation doneinexactlythesamebasis,thusthedifferencesin
(6)canbesolvedwithamultidimensionalminimisation theresultsariseexclusivelyfromtheUBHFansatz.
routinewithoutmakinganyassumptionontheexplicit
analyticalformoftheorbitals.
3.1. Totalenergy
3. Numericalresults InFig.1theenergiesobtainedbybothmethodsare
compared.Interestingly,thedifferencebetweentheex-
We applied the UBHF approximation to few inter- act method and the UBHF approximation becomes a
actingbosonsinaone-dimensionalharmonictrapde- constant shift for high interaction values, λ(cid:38)2. This
scribedbythehamiltonian shift still depends on the considered particle numbers
N. As one can see in 1, the shift grows with N. Note
Hˆ =X12“x2i−∂∂x22”+Xp(x −λx )2+κ2, (10) thatforGPtheenergydivergesrapidlyfromtheexact
i i i<j i j resultalreadyforλ>0.5.ThusFig.1isaconvincing
whichhasbeenmadedimensionlessbyusingstandard evidenceforthegoodqualityoftheUBHFapproxima-
oscillatorlengthandenergyscales:thespatialvariable tion.
3
0.9 Coulomb interacting trapped bosons. As another ex-
CI
0.8 UBHF ample, in Fig. 4 we show the density of six bosons
GP
0.7 ϕ (x) fortwovaluesofthecouplingparameter.Again,with
1/2
0.6 increasing λ the overlap of the orbitals decreases and
thesystem,asawholeexpands.
y 0.5
t
si
n 0.4
e
d 3.3. Delocalisation and non-orthogonality of the
0.3
orbitalsforsmallλ
0.2
0.1
Thereareseveralpossibilitiestoanalysetheoverlap
0
3 2 1 0 1 2 3 of orbitals in a quantitative way. One that works for
− − −
xinx allparticlenumbersistoconsidertheGramiandeter-
0
minantofthegivensetoforbitals,
Fig.2.ParticledensityofasystemwithN =2andλ=0.9
for the three different approximation methods. The two X YN
orbitalsobtainedwithUBHFarealsogiven. G(|1(cid:105),...,|N(cid:105))= sign(π) (cid:104)s|π(s)(cid:105)
π∈SN s=1
0.8 CI ˛˛(cid:104)1|1(cid:105) (cid:104)1|2(cid:105) ... (cid:104)1|N(cid:105)˛˛
0.7 UBHF ˛ ˛
GP ˛ ˛
0.6 ϕ1/2(x) ˛˛(cid:104)2|1(cid:105) (cid:104)2|2(cid:105) ... (cid:104)2|N(cid:105)˛˛
=˛ ˛ (12)
0.5 ˛˛ ... ... ... ... ˛˛
˛ ˛
nsity 0.4 ˛˛˛(cid:104)N|1(cid:105) (cid:104)N|2(cid:105) ... (cid:104)N|N(cid:105)˛˛˛
e
d 0.3
Thisentityisalwayspositiveandapproachesthevalue
0.2
1,iftheorbitalsformanorthonormalset(thisismost
0.1 easilyseenforthecaseoftwoorbitals).Geometrically,
0 theGramiandeterminantisthesquareofthevolumeof
3 2 1 0 1 2 3
− − − theparallelepipedspannedbythevectors|1(cid:105),...,|N(cid:105)
xinx
0 [21]. In Fig.5 the λ-dependency of G is shown. Obvi-
Fig.3.SameasFig.2,butforλ=1.5 ously,inthelimitoflargeλtheorbitalsbecomepair-
wiseorthogonal.Withincreasingparticlenumberthis
3.2. Localisationoftheorbitalsforλ(cid:29)1 limitisreachedforlargervaluesofthecouplingparam-
eter.
The UBHF approximation scheme offers the possi- Letusnowconsidertheoppositelimitofsmallcou-
bility to analyse the delocalisation of the interacting pling. As Fig. 5 shows, for all N, the determinant G
particles in a special way. As can be seen in Figs. 2, monotonicallydecreaseswhenλisreduced.SinceGis
3 and 4, the overlap of the orbitals vanishes with in- a determinant it is obvious that it vanishes, if one or
creasingλ.Itiscrucialtonotice,thatthedensityfrom moreorbitalsarecollinear.Inthepresentcase,alsothe
UBHF shows a localisation of the particles which, by oppositeistrue:vanishingofGisanindicationofthe
definition, is missing in the GP model. Finally, the orbitals becoming collinear. This is observed for λ →
othertwocurvesinFigs.2and3whichresembleGaus- 0.Duetothesymmetryofthesystemnotonlytwoor-
siansarethetwoorbitalsobtainedbyUBHF.Itisin- bitalsbutallorbitalsarebecomingcollinear,i.e.they
teresting to see that the localisation of the particles areidentical.ButthisisjustthelimitofanidealBose
emerges already in each of the orbitals. This can be gas where we expect that all particles Bose condense
understoodasaprecursoroftheclassicalstronglycor- inthegroundstate.Itisaremarkablepropertyofthe
related limit of the system where the particles form a present UBHF ansatz which does not impose any re-
fullylocalizedcrystal-likearrangement[17,18]. strictionsontheorbitalsthatityieldsthecorrectBEC
This trend is typical for the present system of limitofidenticalHartreeFockorbitalsforallparticles.
4
1.0 λ=0.5 chargedbosonsintrapswhenthedimensionlessinter-
x) action strengths λ exceeds 0.5. For its resulting ener-
φ( 0.5 giesaresubstaniallyhigherthantheexactonesandit
cannotreproduceimportantqualitativefeaturesofthe
0.0
λ=1.0
systemsuchasthelocalisationofthetotalparticleden-
φx() 0.5 sity.InthelimitofvanishingλtheUBHFapproxima-
tionyieldssingleparticleorbitalswhicharecollinear–
0.0 inotherwords,weautomaticallyrecovertheGPmodel.
Wementionthatthecomputertimerequiredtosolve
x) 1.0
ρ( 0.5 λ=0.5 theequations(6),withinourcurrentimplementation,
λ=1.0 hasanunfavorabledependenceontheparticlenumber
3 2 1 0 1 2 3 N scaling as N ·N!. This arises from performing the
− − −
xinx sumoverallpermutationsin(6).However,itispossi-
0
ble to reduce the calculation of the entities that turn
Fig.4.Connectionoftheobtainedorbitals(uppertwofig-
up in this equation to the calculation of permanents.
ures) and the resulting total particle density of a system
By using the Ryser algorythm, the N-dependency of
withN =6particlesandλ=0.5and1.0.
the complexity becomes of order N ·2N [19]. This is
1 stillarapidlygrowingcalculationtime,butitismuch
0.9 betterthanthefirstversion.Thecalculationtimealso
0.8 dependsonthesizeofthechosenbasissetnb.Forboth
0.7 implementationsitisofordern4.Thisarisesfromthe
b
0.6 calculation of the two-particle-integrals in each step
G 0.5 [20].
0.4 Presentlyweareworkingonfurtheroptimizingthe
N
0.3 2 implementationofthisansatzinordertoextenditto
0.2 3 systems with higher particle numbers and higher di-
4
0.1 5 mensions. Furthermore, the development of anexten-
6
0 siontothetimedependentregimeisinprogress.
0 1 2 3 4 5
λ
Fig. 5. Gramian determinant of the set of orbitals for dif-
ferentparticlenumbersversusλ
Acknowledgements We thank K. Balzer for useful
4. Discussion
discussions.
In this paper we have studied a Hartree-Fock ap-
proximationforinteractingbosonsinatrapwhichhas
proposed by Romanovsky et al. In contrast to their Appendix A. Variational differentiation of
workwehavenotusedanyassumptionontheexplicit abstractHilbertspacevectors
formoftheorbitalsbutobtainedthemselfconsistently.
WehavetestedtheUBHFschemeindetailbycompar- In order to obtain the key equations for the UBHF
ing the resulting energies and densities with the ones approximation,onehastoperformavariationalderiva-
obtained with CI. Due to the identical choice of the tive with respect to all N orbitals. In this appendix
basis set for both approximations, the differences in wederivesomedifferentiationrulesthatareneededto
theresultsariseexclusivelyduetotheapproximation deduce(6).Inthefollowingthewavefunctionofthen-
made in UBHF. The comparison shows that this is a th orbital will be denoted ϕ (x). Let us now consider
n
very accurate model. In contrast, it is confirmed that the variational derivative of the matrix element of an
the GP approximation is not applicable to nonideal arbitraryoperatorOˆ:
5
δ δ Z
(cid:104)i|Oˆ|j(cid:105)= ϕ∗(z)O(x,y)ϕ (y)dxdy
δϕ∗(x) δϕ∗(x) i j
n n
Z
=δ O(x,y)ϕ (y)dy=δ (cid:104)x|Oˆ|j(cid:105) References
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7