Extension of the Thomas-Fermi approximation for trapped Bose-Einstein condensates with an arbitrary number of atoms

0.15 MB·English

by  A. Muñoz Mateo

Phys.Rev. A74,065602(2006)
Extension of theThomas-Fermi approximation for trappedBose-Einstein condensateswith an
arbitrary number of atoms
A. Mun˜oz Mateo∗ and V. Delgado†
DepartamentodeF´ısicaFundamental II,UniversidaddeLaLaguna, LaLaguna, Tenerife,CanaryIslands,Spain
By incorporating the zero-point energy contribution we derive simple and accurate extensions of the
7 usual Thomas-Fermi (TF) expressions for the ground-state properties of trapped Bose-Einsteinconden-
0 satesthatremainvalidforanarbitrarynumberofatomsinthemean-fieldregime.Specifically,weobtain
0 approximate analytical expressions for the ground-state properties of spherical, cigar-shaped, and disk-
2
shaped condensates that reduce to the correct analytical formulas in both the TF and the perturbative

n regimes,andremainvalidandaccurateinbetweenthesetwolimitingcases.Mean-fieldquasi-1Dand-2D
a condensatesappearassimpleparticularcasesofourformulation. Thevalidityofourresultsiscorrobo-
J ratedbyanindependentnumericalcomputationbasedonthe3DGross-Pitaevskiiequation.

9
2 PACSnumbers:03.75.Hh,05.30.Jp,32.80.Pj

]
r
e The experimental realization of Bose-Einstein conden- ground state, ψ(r) = (πa2)−3/4exp( r2/2a2), and the
r − r
h sates(BECs)ofdiluteatomicgasesconfinedinopticaland chemicalpotentialsatisfies
ot magnetictraps[1,2,3]hasstimulatedgreatactivityinthe
. characterization of these quantum systems. Of particular (3/2)~ω +gn¯ = µ (3)
t
a interest are the ground-state properties of trapped BECs
m wheren¯ = N/(√2πa )3 isthemeanatomdensity. Away
with repulsive interatomic interactions [4]. These proper- r
- tiesderivefromthecondensatewavefunctionψ(r)which, fromthesetwolimitingcases,inprinciple,onehastosolve
d theGPEnumerically. Veryfewtheoreticalworkshavead-
inthezero-temperaturelimit,satisfiesthestationaryGross-
n
dressed the question of looking for approximate analyti-
o Pitaevskiiequation(GPE)[5]
calsolutionsvalidinbetweenthetwoanalyticallysolvable
c
[ regimes. Themostrelevantproposalsarebasedonavaria-
~2
1 2+V(r)+gN ψ 2 ψ = µψ, (1) tionaltrialwavefunction[6],oronthesemiclassicallimit
−2m∇ | | oftheWignerphase-spacedistributionfunctionofthecon-
v (cid:18) (cid:19)
5 where N is the number of atoms, g = 4π~2a/m is densate[7]. However,thepracticalusefulnessoftheseap-
9 proaches turns out to be somewhat limited in comparison
the interaction strength, a is the s-wave scattering length,
6 withthesimpleTFapproximation.
1 V(r) = 1m(ω2r2 +ω2z2) is the harmonic potential of
2 ⊥ ⊥ z In this work we address the above question from a dif-
0 theconfiningtrap,andµisthechemicalpotential.
ferent point of view. We start from the usual TF approxi-
7 OnlyintwolimitingcasescanEq. (1)besolvedanalyti-
0 mationandmodify itconvenientlyto account,in a simple
cally: intheThomas-Fermi(TF)andperturbativeregimes.
/ manner, for the zero-point energy contribution. This en-
t When N is sufficiently large that µ ~ω , ~ω , one
a ≫ ⊥ z ables us to derive simple and accurate extensions of the
m enters the TF regime. In this case the kinetic energy can
TF expressions that remain valid for an arbitrary number
beneglectedincomparisonwiththeinteractionenergyand
- ofatomsinthemean-fieldregime. Specifically,weobtain
d the GPE reduces to a simple algebraic equation. Useful
general analytical expressions for the ground-state prop-
n analytical expressions can then be obtained for the con-
o erties of spherical, cigar-shaped,anddisk-shapedconden-
densate ground-state properties [4]. In the simple case
c satesthatreduceto thecorrectanalyticalformulasinboth
of a spherical trap characterized by an oscillator length
: theTFandthe perturbativeregimes,andremainvalidand
v a = ~/mω,Eq. (1)leadsintheTFlimitto
i r accurateinbetweenthesetwolimitingcases.
X
We begin by considering a BEC in a spherical trap. In
p
ar 1mω2r2+gN ψ(r) 2 = µ, 0 r R (2) principle, we start from the TF relation of Eq. (2). How-
2 | | ≤ ≤ ever, since we intend to apply this equation to arbitrarily
small condensates, we introduce a lower cutoff radius r ,
where the condensate radius R = 2µ/~ωar is deter- defined through 1mω2r2 = 3~ω, in order to be consis0-
mined from the condition |ψ(r)|2 ≥p0, and the chemical tent with the fact2that th0e cont2ribution from the harmonic
potential µ = 1 (15Na/a )2/5~ω follows from the nor- oscillatorenergycannotbesmallerthanthezero-pointen-
2 r
malizationofψ(r). ergy. As for the small volume V a3 correspondingto
0 ∼ r
In the opposite limit, when N is small enough that the r r ,wedonotaspiretogetapreciseknowledgeofthe
0
≤
interaction energy can be treated as a weak perturbation, wavefunction therein. Instead, we contentourselveswith
oneentersthe(idealgas)perturbativeregime. Inthiscase, an effective condensate density n¯ in that region. As we
0
tothelowestorder,ψ(r)isgivenbytheharmonicoscillator shall see, this is all that is needed to obtain very accurate
2
approximateformulas for mostof the condensateground- 5 20
(a) (b)
stateproperties. Thuswestartfromtheansatz R
4
µ 15
1 2
mω2r2+gN ψ(r) 2 = µ, r0 < r R (4a) 3 µ
2 | | ≤ µ 1,5 3
3 TF 0 1 10 R
2~ω+g 6/πn¯0 = µ, 0 ≤ r ≤ r0 (4b) 2 εpot ε 2 RTF
with ψ(r) = 0 fopr r > R. A renormalization constant int 5 1
1
κ−1 6/π hasbeenintroducedin Eq. (4b)to guaran- ε 0
tee th≡e cporrect perturbative limit. In this limit µ → 32~ω 00 5 10 1k5in 20 00 0 55 10 5
and R r0 = √3ar. Under these circumstances, only χ 1xχ10 2x10
Eq. (4b→) contributes significantly to the chemical poten- 0 0
tial, and in this case n¯ = N/V . This corresponds to a
0 0 FIG.1:(Coloronline)Theoreticalpredictionfortheground-state
uniformsphericalcondensate,definedinthefinite volume
propertiesofsphericalcondensates(solidlines).Theopencircles
V . In order for this uniform density to producethe same
0 aretheexactnumericalresults.Forcomparisonpurposeswehave
chemicalpotentialasthegroundstateoftheharmonicos- alsoincludedtheTFprediction(dashedlines).
cillatoroverthevolumeoftheentirespaceitisonlyneces-
sary to renormalizethe correspondinginteractionstrength
bymultiplyingby 6/π. Equations(4)alsoyieldthecor-
satisfiesEq. (5)witharesidualerror[8]smallerthan0.7%
rectresultintheTFregime. Thisismainlyaconsequence
p foranyχ [0, ). Figures 1(a) and1(b)show, respec-
ofthedirectrelationexistingbetweenthenumberofparti- 0
∈ ∞
clesandthesizeofatrappedBEC.Forlargecondensates, tively,thepredictedchemicalpotential,µ = 1R2,andcon-
such that µ ~ω, one has R r and, as a result, the densate radius, obtained from Eq. (6) (solid2lines), along
0
≫ ≫
relativecontributionfromEq. (4b)tothenormalizationin- with the exact results obtained from the numerical solu-
tegralthatdeterminesµbecomesnegligible.Sincewehave tion of the 3D GPE (opencircles). For the numericalcal-
renouncedanexplicitexpressionforψ(r)inV ,inthisre- culation we have defined the radius through the condition
0
spect,ourapproachcannotprovidemoreinformationthan ψ(R) 2 = 0.05 ψ(0) 2. With this definition, Eq. (6)re-
theTFapproach. OnlywhenR r0 canwehaveasuffi- |produc|esthenum|erical|Rwitharelativeerrorsmallerthan
≫
cientlypreciseknowledgeofthewavefunctionand,inthis 3% for any χ . Most of the error, however, comes from
0
case,itcoincideswiththeTFwavefunction. theregionwhereχ 1(TFlimit)becauseinthatregion
0
The chemical potential follows from the normalization ≫ 2
R R andit rathersatisfies ψ(R) = 0. Theaccu-
TF
ofψ(r). Afterastraightforwardcalculationoneobtains → | |
racywithrespecttothenumericalµisbetterthan0.5%.
Astraightforwardcalculationyieldsthemean-fieldinter-
1 5 √3 2 3√3 3 a actionenergyperparticle,ǫint ǫint/~ω Eint/N~ω,
R + (κ 1)R κ = N , (5) ≡ ≡
15 2 − − 2 − 5 a
where R R/a , and µ = 1R2(cid:18)is the c(cid:19)hemical proten- ǫint = 8χ1 1085R7+√3(κ−1)R4
tial in uni≡ts of ~ωr. As Eq. (25) shows, the ground-state 0 (cid:20)
properties depend on the sole parameter χ Na/a . 6√3 κ 3 R2+9√3 κ 3 . (7)
0 ≡ r − − 5 − 7
When χ 1 (TF limit) the above equation leads to (cid:18) (cid:19) (cid:18) (cid:19)(cid:21)
0
≫
µ = 1(15χ )2/5, as expected. The χ 1 limit cor- For χ 1, one recovers the TF result, ǫ = (2/7)µ.
2 0 0 ≪ 0 ≫ int
responds to the perturbative regime and, in this case, one 2
In the χ 1 limit, using that R = 3+2 2/πχ
0 0
obtainsµ = 3/2+ 2/πχ ,whichisnothingbuttheper- ≪ −
0 (1/π)(1/9+ 2/3π)χ2+O(χ3)isaperturbativesolution
turbative result (3). For arbitrary χ , in principle one has 0 0 p
p 0 ofEq. (5),oneobtainsǫ = χ ~ω/√2π =gn¯/2,which
tosolvenumericallytheabovequinticpolynomialequation p int 0
againisthecorrectresult. Finally,thekineticandpotential
(which has only one physically meaningfulreal solution).
energies can be readily obtained in terms of the previous
This is a simple task thatcan becarried outwith standard
resultsbyusingtheexactrelations[5]
mathematicalsoftwarepackages.Wehavefound,however,
aratheraccurateapproximatesolution.Itcanbeshownthat
ǫ E /N = µ/2 (7/4)E /N, (8a)
kin kin int
theexpression ≡ −
ǫ E /N = µ/2 (1/4)E /N. (8b)
pot pot int
−1 ≡ −
2 1 1 π/2
R = 3+ (15χ0)25 + 25 + 72χ101/15+10 + p2χ0 ! aInndFiǫgp.o1t,(ao)bwtaeinsheodwfrtohmethEeqosr.et(i6ca)–l(p8r)ed(iscotliiodnlfionreǫs)in,ta,lǫokning,
(6) withtheexactnumericalresults(opencircles).
3
NextweconsideraBECconfinedinacigar-shapedmag- 5 2
netic trap with oscillator lengths a⊥ = ~/mω⊥ and χ1 = 1.0
az = ~/mωz and an aspect ratio λ p= ωz/ω⊥ ≪ 4 µ 2 1
2. We shall restrict ourselves to the mean-field regime,
which rpequires Nλa2/a2 1 [9, 10, 11]. As be- 3 µ µ
fore, we startfrom the⊥usual≫TF expression,whichwe as- TF 10 1 0√ λ
sume to be valid up to a minimum radial distance r0 = 2 ε z
⊥ pot ε
√2a⊥, determined from the condition that the contri- int -1
1
bution from the radial harmonic oscillator energy should
ε
not be smaller than ~ω . This defines an outer region kin
⊥
V (r ,z): r2/R2+z2/Z2 1 r > r0 , 0 -2
w+hich≡isn{oth⊥ingbutt⊥heusualTFellTipFso≤idal∧dens⊥ityclo⊥u}d, 0 5 1χ0 15 20 0 0a,3 n 0,6
1 1
truncated at r = r0. Note that unlike what happens
⊥ ⊥
with the condensate radius R = 2µ/~ω a , which FIG.2:(Coloronline)Theoreticalpredictionfortheground-state
⊥ ⊥
remains the same, now the axial condensate half-length properties of arbitrary cigar-shaped condensates with λ ≪ 2
Z = 2(µ/~ω 1)a /√λ cpoincides with the TF (solid lines). The open circles are exact numerical results ob-
value Z only in⊥ −the limzit µ/~ω 1. For large tainedwithλ=0.2. ThedashedlineistheTFprediction.
TF ⊥
condenspates, when µ ~ω (TF re≫gime), this is the
⊥
≫
only region that contributes significantly. On the con-
trary, in the perturbative regime, as µ ~ω most of µ = 1+1(3χ )2/3andZ = λ−1/2(3χ )1/3,inagreement
→ ⊥ 2 1 1
the contribution comes from the inner cylinder V withpreviousresults[10,11]. Ingeneral,forarbitraryχ ,
− 1
(r ,z): r r0 z Z . In this case, if a ≡ anapproximatesolutionsatisfyingEq. (10)witharesidual
{ ⊥ ⊥ ≤ ⊥ ∧ | | ≤ } ≪
a , the transverse dynamics becomes frozen in the radial errorlessthan0.75%foranyχ [0, )isgiven by
⊥ 1
∈ ∞
groundstateoftheharmonictrapandthecondensatewave
functioncanbefactorizedasψ(r ,z) = ϕ(r )φ(z),with −1
⊥ ⊥ 1 1 1 4
tϕo(ar⊥m)ea=n-fi(πelad2⊥q)u−a1s/i2-1eDxpc(o−ndr⊥e2n/s2aate2⊥.)S.uTbshtiistuctoinrgretshpeonndins √λZ = (15χ1)54 + 13 + 57χ1+345 + (3χ1)43!
Eq. (1)andintegratingouttheradialdynamics,onefinds (11)
Themean-fieldinteractionenergyǫ ǫ /~ω is
1 int int ⊥
~ω + mω2z2+g N φ(z)2 = µ, (9) ≡
⊥ 2 z 1D | |
1 1
where g = g/2πa2 [12], and we have used that µ ǫint = (√λZ)5+ (√λZ)7 . (12)
~ω 1D1~ω to neg⊥lect the axial kinetic energy. No∼te 15χ1 (cid:18) 7 (cid:19)
⊥ ≫ 2 z Forχ 1,Eq. (12)reducestoǫ = (2/7)µ, whilefor
that g can be conveniently rewritten as gn¯ with n¯ = 1 int
1/π(r10D)2,indicatingthatonecanaccountfor2thecontr2ibu- χ1 1≫, itleadstoǫint = (2/5)(µ ~ω⊥), whichagain
⊥ are≪thecorrectanalyticallimits. Asfo−rthecondensateden-
tionfromtheradialgroundstatebyusingauniformmean
2
densityperunitareanormalizedtounityinV−. Guidedby sity per unit length, n1(z) ≡ N 2πr⊥dr⊥|ψ(r⊥,z)| ,
thesesimpleideas,wethenproposethefollowingansatz: afterastraightforwardcalculationonefinds
R
(√λZ)2 z2 (√λZ)4 z2 2
n (z) = 1 + 1
12mω⊥2r⊥2 + 12mωz2z2+gN |ψ(r⊥,z)|2 =µ, r∈ V+ 1 4a (cid:18) − Z2(cid:19) 16a (cid:18) − Z2(1(cid:19)3)
1 ThefirsttermisthecontributionfromV andthusitisthe
~ω + mω2z2+gNn¯ φ(z)2 =µ, r V −
⊥ 2 z 2| | ∈ − onlyonethatcontributessignificantlyintheχ1 1limit.
≪
Onthecontrary,thesecondterm,whichisthecontribution
withψ = 0elsewhere. Thenormalizationofψleadsto
from V , gives the dominant contribution in the χ 1
+ 1
≫
limit,ingoodagreementwithpreviousresults[11].
1 1 a Figure 2 shows the theoretical predictions for the
(√λZ)5+ (√λZ)3 = Nλ , (10)
15 3 a ground-state properties of arbitrary cigar-shaped conden-
⊥
sates with λ 2, obtained from Eqs. (8) and (11)–(13)
where Z Z/a and R R/a . The chemical poten- ≪
z ⊥ (solid lines), alongwith exactnumericalresults (opencir-
≡ ≡
tial µ ≡ µ/~ω⊥ is given by µ = 1 + 21(√λZ)2. Now cles).
the relevantparameterdeterminingthe ground-stateprop- Finally, we consider a BEC in a disk-shaped trap with
(e1rt0i)esleiasdχs1to≡µN=λa1/(1a5⊥χ. W)2h/e5nanχd1Z≫=1λ(T−F1/r2e(g1i5mχe),)1E/q5.. bλat≫ive2reagnidmaez, w≫hicah. oIncctuhrisscwahseen, iχnthemNeaan/-λfi2ealdpetu1r-,
2 1 1 2 ≡ z ≪
Whenχ 1(mean-fieldquasi-1Dregime), oneobtains thesystemreducestoaquasi-2Dcondensatesatisfying
1
≪
4
5 2
χ = 1.0
1 1 2
2~ωz + 2mω⊥2r⊥2 +g2DN|ϕ(r⊥)|2 = µ, (14) 4 1,5
µ
1,5 µ
wpmgeκiart−2lhiu1znn¯agit21tiD,olenwn=fghatechrtegoa/rnn¯,√da1nκ2d=π−2p1ar1≡zo/p[2o1as3ez2].t/ihsπeWaifsoeultnlhtoihefweoainrpnmpgrreaomwnpersriaaitanetzte:dgre2enDnsoiatrys- 23 µεpToFt 0 0 ε µ TF 1 1r/ λ√⊥
p int 0,5
1
1 1 ε
mω2z2+ mω2r2 +gN ψ(r ,z) 2 =µ, r V kin
2 z 2 ⊥ ⊥ | ⊥ | ∈ + 0 0
0 5 10 15 20 0 0,1 0,2 0,3
1 1 χ a a n
~ω + mω2r2 +gκ−1Nn¯ ϕ(r )2 =µ, r V 2 z 2
2 z 2 ⊥ ⊥ 2 1| ⊥ | ∈ −
(15) FIG.3:(Coloronline)Theoreticalpredictionfortheground-state
propertiesofarbitrarydisk-shapedcondensateswithλ≫2(solid
with ψ = 0 elsewhere. In the above equations, V
(r ,z): r2/R2 +z2/Z2 1 z > z andV+ ≡ lines).Theopencirclesareexactnumericalresultsobtainedwith
{ ⊥ ⊥ TF ≤ ∧ | | 0} − ≡ λ=20. ThedashedlineistheTFprediction.
(r ,z): r R z z ,wherez = a ,R =
⊥ ⊥ 0 0 z TF
{ ≤ ∧ | | ≤ }
2µ/~ω √λa , R = 2(µ/~ω 1/2)√λa , and
z ⊥ z ⊥
−
Z = 2µ/~ω a . More precisely, one expects κ−1
p z z p 2 →
2/π intheperturbativeregime(χ 1),whileκ−1
1inthpeTFregime(χ 1). Thefin2a≪lresultsareno2tve→ry ξ[2µ (r ) 1] [2µ (r )]3/2 1
spensitive to the specifi2c≫functional form of κ−21. We thus n2(r⊥) = 4zπa⊥az− + z 6π⊥aaz − , (20)
proposeoneofthesimplestpossibilities:
2
whereξ (κ 1)and2µ (r ) 1+R (1 r2/R2).
≡ 2− z ⊥ ≡ λ − ⊥
InFig. 3weshowtheground-statepropertiesofarbitrary
κ−1(χ ) 2/π +Θ(χ 0.1)
2 2 ≡ 2− disk-shaped condensates with λ 2, obtained from our
≫
p RTF(χ2 = 0.1) analytical formulas [Eqs. (8) and (18)–(20)] (solid lines),
1 2/π 1 , (16)
× − − R (χ ) alongwithexactnumericalresults(opencircles).
(cid:18) TF 2 (cid:19)
(cid:16) p (cid:17) In conclusion, modifying the usual TF approximation
where Θ(x) is the Heaviside function and R (χ ) =
TF 2 conveniently to account for the zero-point energy contri-
(15χ )1/5a is the TF radius. The normalization of ψ
2 ⊥ bution,we havederivedgeneralanalyticalexpressionsfor
yields
theground-statepropertiesofspherical,cigar-shaped,and
disk-shapedcondensatesthatreducetothecorrectanalyti-
4 2 calformulasinboththeTFandthemean-fieldperturbative
1 5 1 R R 1 Na
Z + (κ 1) = , (17) regimes and remain valid and accurate in between these
15 8 2− λ2 − 6λ − 15 λ2a
z two limiting cases. Mean-fieldquasi-1Dand-2Dconden-
whereZ Z/a ,R R/a ,andZ2 R2/λ = 1. The satesappearassimpleparticularcasesofourformulation.
z ⊥
chemical≡potentialisµ≡ µ/~ω = 1(1−+R2/λ). This work has been supported by MEC (Spain) and
≡ z 2 FEDERfund(EU)(ContractNo. Fis2005-02886).
Forχ 1Eq. (17)leadstotheusualTFresults,while
2
≫
forχ 1(mean-fieldquasi-2Dregime),oneobtainsµ =
2
≪
1/2+(2 2/πχ )1/2 andR = λ1/2(8 2/πχ )1/4. An
2 2
approximatesolutionthatsatisfiesEq. (17)witharesidual
errorlesspthan0.95%foranyχ [0, p)isgiven by ∗ Electronicaddress: [email protected]
2
∈ ∞ † Electronicaddress: [email protected]
−1/8
R R/√λ = (1/15χ )8/5+(κ /8χ )2 (18) [1] M.H.Andersonetal.,Science269,198(1995).
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andthecondensatedensityperunitarean (r ):
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ǫint = +ξ λ λ λ [8] Given P(R) = χ, we define the residual error associated
8χ2 105 6 − 3 − 15 − 105! withtheapproximatesolutionRεas[P(Rε)−χ]/χ.
(19) [9] D.S.Petrovetal.,Phys.Rev.Lett.85,3745(2000).
5
[10] V.Dunjko,V.Lorent,andM.Olshanii,Phys.Rev.Lett.86, [12] M.Olshanii,Phys.Rev.Lett.81,938(1998).
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