Semiclassical theory of the quasi two-dimensional trapped Bose gas

0.18 MB·English

by  Markus Holzmann

Semiclassical theory of the quasi two-dimensional trapped Bose gas
Markus Holzmann1, Maguelonne Chevallier2, Werner Krauth2
1LPTMC, Universit´e Pierre et Marie Curie, 4 Place Jussieu, 75005 Paris,
France; and LPMMC, CNRS-UJF, BP 166, 38042 Grenoble, France and
2CNRS-Laboratoire de Physique Statistique, Ecole Normale Sup´erieure,
24 rue Lhomond, 75231 Paris Cedex 05, France
(Dated: February 2, 2008)
8 Wediscuss thequasi two-dimensional trapped Bose gas where the thermaloccupation of excited
0 states in the tightly confined direction is small but remains finite in the thermodynamic limit.
0 We show that the semiclassical theory describes very accurately the density profile obtained by
2 QuantumMonteCarlocalculationsinthenormalphaseabovetheKosterlitz–Thoulesstemperature
n TKT,butdiffersstronglyfromthepredictionsofstrictlytwo-dimensionalmean-fieldtheory,evenat
a relatively high temperature. Wediscuss therelevance of our findings for analyzing ultra-cold-atom
J experimentsin quasitwo-dimensional traps.

7
PACSnumbers: 05.30.Jp,03.75.Hh
1

]
h For many years, the physics of two-dimensional quan- fine the quasi two-dimensional regime where the ideal-
c tumgaseshasbeenundercloseexperimentalandtheoret- gas critical temperature is always lower than that of the
e
icalscrutiny. Thequestforquantumphasetransitionsin strictlytwo-dimensionalidealgas. Includinginteractions
m
two-dimensional atomic Bose gases has started with ex- on the level of mean-field, we obtain the density pro-
-
t periments on spin-polarized atomic hydrogen adsorbed files in the semiclassical approximation and solve the
a on liquid 4He surface reaching the quantum degenerate self-consistent mean-field equations directly. Remark-
t
s regime [1] andobservingthe onsetofquasi-condensation able agreement of the semiclassical density profiles with
.
t [2]. The Kosterlitz–Thouless transition [3] was observed the results of Quantum Monte Carlo calculations is ob-
a
m recentlyintrappedatomicgasesofultra-cold87Rbatoms tainedinthehigh-temperaturenormalphasedowntothe
in an optical lattice potential with a tightly confined z- Kosterlitz–Thoulesstemperature. The profilesshouldbe
-
d direction [4]. veryconvenientforcalibratingthetemperatureinexper-
n In contrast to experiments with liquid 4He films [5], iments of quasi two-dimensional Bose gases. Compari-
o
where the Kosterlitz–Thouless transition is realized di- sonofexperimentaldensityprofileswithQuantumMonte
c
[ rectly, the typical extension in the z-direction in the ex- Carlo data has already removedthe originaldiscrepancy
periments on two-dimensional gases is much larger than ofthe Kosterlitz–Thoulesstemperature betweencalcula-
1
v the three-dimensional scattering length. For this rea- tion and experiment [8].
8 son,theeffectivetwo-dimensionalinteractionstrengthre-
We consider an anisotropic trap with oscillator fre-
5 mainssensitivetothedensitydistributioninz[6]. Never-
7 theless,thegasiskinematicallytwo-dimensionalbecause quencies ω ≡ ωx = ωy ≪ ωz. At temperature T ∼ ~ωz,
2 the motion is semiclassical in the coordinates x, y and
of the strong out-of-plane confinement.
1. in the momenta ~kx, ~ky, whereas the quantization in
In this letter, we consider the quasi two-dimensional
0 the z-direction is best described through the energy lev-
8 regime of the trapped Bose gas, where the temperature els ν~ω (ν = 0,1,...) of the corresponding harmonic
z
0 T is of the order of the level spacing in z. This corre-
oscillator. Semiclassically, the number dN of particles
: sponds tothe experimentalsituationwithsmall,butnot
v per phase space element dk dk dxdy in the energy level
x y
completely negligible, thermal occupation of a few ex-
i ν is given by [10]
X
cited states of the tightly confining potential[4, 7]. The
r quasi two-dimensional regime crosses over to the three-
a
dimensional and the strictly two-dimensionalBose gases
as the potential in the z-direction is varied. 1 dkxdkydxdy
dN = , (1)
Itwasnoticedinexperiments[7,8]andindirectQuan- (2π)2exp β(~22mk2 +v(r)+ν~ωz −µ) −1
tum MonteCarlocalculations[9] thatthe density profile
(cid:2) (cid:3)
of the gas deviates stronglyfrom two-dimensionalmean-
field theory, even at relatively high temperature. We
where β = 1/T, k2 = k2 + k2, and where v(r) is an
point out that these deviations can be accounted for by x y
arbitrary two-dimensional potential energy (with r2 =
a quasi two-dimensional mean-field theory which incor-
x2+y2).
porates correctionsdue to the tightly confined direction.
We first discuss Bose–Einstein condensation of the Equation (1) can be integrated over all momenta and
ideal gas in strongly anisotropic harmonic traps. We de- summed over all oscillator levels to obtain the two-
2
dimensional particle density
∞ dk2 1 1
n(r)=
4π exp β(~2k2 +v(r)+ν~ω µ) 1
Xν Z0 2m z − − 0.8
∞ 1
1 (cid:0) (cid:1)
= ln 1 exp[β(µ v(r) ν~ω )] , (2)
−λ2 { − − − z } 0.6
where λ = νX=20π~2β/m is the thermal wavelength. The tBEC 0.4 tBEC N=111000462
1012
potential v(r) can itself contain the interaction with the
densityn(r)p,sothateq.(2)isingeneralaself-consistency 0.2 0
0 1
equation. Theintegralofn(r)overspaceyieldstheequa- δ
tion of state, that is, the total number of particles as a 0
0 1 2 3 4 5
function of temperature and chemical potential.
ω/T2d
z BEC
Let us first consider the ideal gas, where the potential
energy v(r) = mω2r2/2 is due only to the trapping po-
tential, so that the rhs of eq. (2) is independent of the FIG. 1: Bose–Einstein condensation temperature tBEC of
the ideal quasi two-dimensional gas in a harmonic trap with
density n(r). We get ωz/ω ∝N1/2(expansionsfromeq.(8)). TheinsetshowstBEC
for scaling ω /ω =Nδ as a function of δ for different N. At
∞ ∞ z
N = π d(r2)ln 1 eβ(µ−ν~ωz−mω2r2/2) δ = 0, the critical temperature TB3dEC = [N/ζ(3)]1/3 of the
−λ2 − ideal gas in a three-dimensional isotropic trap is recovered.
Xν=0Z0 h i
∞
T2
= F ( µβ+νβ~ω ), (3)
~2ω2 2 − z as a relation between the temperature t, the chemical
ν=0
X potential µ˜, and the oscillator strength ω˜ ,
z
where we have defined
t=f(t,µ˜,ω˜ ), (6)
z
∞ e−nx
Fs(x)= ns . with
n=1
X ∞ −1/2
6
The saturationnumber N (T)is the maximumnum- f(t,µ˜,ω˜ )= F ( µ˜+νω˜ /t) . (7)
ber of excited particles (reasacthed at µ = 0) at a given z π2 ν=0 2 − z !
X
temperature. We have
Equation (6) is solved numerically by iterating t =
n+1
∞ f(t ) from an arbitrary starting temperature t to the
T2 n 0
Nsqa2td = ~2ω2 F2(νβ~ωz). (4) fixedpoint. ThecriticaltemperaturetBEC =TBqE2dC/TB2dEC
Xν=0 (as a function of ω˜z) of the quasi two-dimensional ideal
Bose gas is the solution for µ˜ = 0 (see fig. 1). The re-
The above relation between the saturation number and
ductionwithrespecttothe strictlytwo-dimensionalcase
the temperature defines the dependence of the Bose–
is notable for systems of experimental interest. For ex-
Einstein condensation temperature on the particle num-
ample, we find t = 0.78 for the experimental value
BEC
berN. Asmentionedbefore,thestrictlytwo-dimensional
ω˜ =0.55 [4, 7] consideredin the QuantumMonte Carlo
limit is characterized by the limit β~ω (the level z
z → ∞ calculations [9].
spacing in z is much larger than the temperature). In
We can expand eq. (7) for small and for large ω˜ and
z
this limit, only the first term in eq. (3) contributes. Us-
find
ing F (0)=π2/6, we find
2
1/3
ζ(2) ω˜1/3 1ζ(2)ω˜ for ω˜ 1
T2 π2 √6N~ω t ζ(3) z − 6ζ(3) z z ≪ , (8)
Ns2adt(T)= ~2ω2 6 ⇔TB2dEC(N)= π . (5) BEC ∼1h− 2iζ(21)3/2 exp(−ω˜z) for ω˜z ≫1
In the quasi two-dimensional case, with finite β~ω , the where we have used ζ(s) F (0) (note that ζ(2)=π2/6
z s
≡
occupation of the oscillator levels ν = 1,2,... increases and ζ(3) 1.202). The expansions are indicated in
≃
the saturation number and therefore lowers the critical fig. 1. They give the critical temperature to better than
temperature. ItisconvenienttoexpressinunitsofT2d 1.2% for all values of ω˜ (the low-ω˜ expansion is used
BEC z z
both the temperature t = T/T2d and the oscillator for ω˜ < 1.8 and the high-ω˜ expansion for ω˜ > 1.8).
BEC z
strength ω˜ = ~ω /T2d , and to write µ˜ = βµ. Us- The first term in the small-ω˜ expansion of eq. (8) cor-
z z BEC z
ingeq.(5),wemayrewritethe equationofstate,eq.(3), responds to the three-dimensional gas. Indeed, t
BEC
∼
3
[ζ(2)/ζ(3)]1/3ω˜1/3 is equivalent to us to determine the equation of state explicitly:
z
∞ ∞ ∂(r2)
ζ(2) 1/3 N =π d(r2) n(r)=π dv n(v)
T (~ω )1/3(T2d )2/3 ∂v
BEC ∼ ζ(3) z BEC Z0 Z0 (cid:20) (cid:21)
(cid:20) (cid:21) 2π ∞ ∂n
~3ωzω2/ζ(3) 1/3N1/3, (9) = mω2 dv 1−2g∂v n(v)
∼ Z0 (cid:20) (cid:21)
∞
the well-known co(cid:2)ndensation te(cid:3)mperature for the three- =~T2ω22 F2(−µ˜+νβ~ωz)+2mπ~g2 n(0)λ2 2!, (12)
dimensionalBose–Einsteingasinananisotropictrap[11]. ν=0
X (cid:2) (cid:3)
Equation(9) also follows directly fromeq. (4) by replac-
where the central density,
ingthesumovertheoscillatorlevelsbyanintegral. Like-
wise, the first term in the large-ω˜ expansion of eq. (8) ∞
z
represents the strictly two-dimensional gas. n(0)λ2 = ln 1 exp(µ˜ νω˜z/t) , (13)
− { − − }
The inset of fig. 1 further analyzes the expansions of Xν=0
eq. (8). Indeed, we can choose a scaling ωz/ω Nδ dif- is directly expressed in terms of µ˜, independent of the
∼
ferentfromthequasitwo-dimensionalcaseδ =1/2. Any interaction, due to the subtraction performed in our ef-
choice of δ < 1/2 corresponds to ω˜z 0 for N fectivepotential,eq.(10). (Notethatthefirstintegralon
→ → ∞
so that asymptotically the three-dimensional regime is thesecondlineofeq.(12)hasalreadyappearedineq.(3)
reached. Analogously, δ > 1/2 corresponds to ω˜
z and that the second integral is a total derivative.) The
→ ∞
for N , driving the transition into the strictly two-
equationofstatecanbewrittenintermsofthetempera-
→ ∞
dimensional regime. The rescaled transition tempera- ture t. This yields the following generalization of eq. (7)
tures are plotted for ω /ω = Nδ where the case δ = 0
z to the mean-field gas:
corresponds to the three-dimensional isotropic trap [12].
Inordertodescribeinteractioneffectsinthequasitwo- 6 ∞ ω˜ mg −1/2
dimensionalBosegas,wenowaddasemiclassicalcontact t= π2" F2(−µ˜+ν tz)+2π~2[n(0)λ2]2#! . (14)
term to the potential energy of eq. (2): νX=0
An iteration procedure t = f(t ) again obtains the
n+1 n
v(r)=mω2r2/2+2g[n(r) n(0)]. (10) temperature t = T/T2d as a function of the chemical
− BEC
potential µ˜ for given parameters ω˜ and g. (The central
z
densityiscomputedusingeq.(13)duringeachiteration.)
We have subtracted the central density, so that the po-
The mean-field density profile is obtained in two steps
tential still vanishes at the origin. As discussed earlier
[9], the effective interaction g = 4πa~2/m dz [ρ(z)]2 is byfirstcalculatingthedensityprofileasafunctionofthe
scaled effective potential v˜=βv,
proportional to the three-dimensional s-wave scattering
R
lengthaandtotheintegralofthesquareddensitydistri- ∞
bution in z, described by the normalized diagonal den- n(v˜)λ2 = ln 1 exp(µ˜ v˜ νω˜ /t) ,
z
− { − − − }
sity matrix ρ(z). In the temperature range of interest, ν=0
X
this density distribution is well described by the single-
andthen by invertingeq. (10) to obtainr(n) (thus n(r))
particle harmonic-oscillator density matrix in z, leading
for the given v˜ and nλ2:
to
2T mgnλ2
g =a 8πωz~3 tanh[ω˜z/(2t)]. (11) r(nλ2,v˜)=smω2 v˜− π~2 . (15)
m (cid:18) (cid:19)
r
p
In fig. 2, we show the remarkable agreement of the
The effective interaction thus decreases with tempera-
quasi two-dimensional mean-field profile with the one
ture from its zero-temperature value g˜= a 8πω ~3/m.
z obtained by Quantum Monte Carlo simulations as in
To keep the interaction strength fixed, we must keep g
p Ref. [9], for N =10000bosons for parameters ω˜z =0.55
(orequivalentlyg˜)constantinthequasitwo-dimensional t = T/T2d = 1, and mg˜/~2 = 0.13. The simula-
BEC
thermodynamic limit which requires a fixed value of the
tions take into account the full three-dimensional geom-
scaled scattering length a√ωz aω1/2N1/4. etry, and particles interact via the three-dimensional s-
∝
The mean-field density n(r), on the lhs of eq. (2), de- wave scattering length a (see Ref. [13] for a more de-
pends on the variable r only via the potential v(r). In tailed description of finite temperature simulations of
the space integral over the particle density, we can thus trapped Bose gases). Comparison with the ideal quasi
change the integration variable from r to v. This allows two-dimensional gas is also very favorable.
4
1
2 7 0.8
0.1
6
1.5 πk0.01 q2dtKT0.6
5
0.001
2λ 2λ
n n 4 0.4
density 1 0.0001 1 k 8 density 3 0 1 ~ω2z 3 4
ideal q2d
0.5 mQeaMn Cfi eNld= q120d5 2 mifd qea2mld f q (22gdd)
Boltzmann 1 QMC N=105
mf q2d (~g)
0 0
0 0.5 1 1.5 2 0 0.5 1 1.5 2
position rβ1/2 position rβ1/2
FIG. 2: Two-dimensional density profile n(r =px2+y2)λ2 FIG. 3: Two-dimensional density profile n(r = px2+y2)λ2
at temperature T = TB2dEC in a trap with ωz = 0.55TB2dEC, in at temperature T = 0.8TB2dEC in a trap with ωz =0.55TB2dEC,
the ideal Bose gas and for mg˜/~2 =0.13 according to mean- in the ideal Bose gas and for mg˜/~2 = 0.13 according to
field theory, compared to Quantum Monte Carlo simulations strictly two-dimensional and quasi two-dimensional mean-
at N = 10000, and for ideal distinguishable particles (first field theory (using g and g˜), compared to Quantum Monte
term in eq. (16)). The inset shows the cycle weights π for Carlo simulations with N = 10000. The inset shows the
k
the strictly two-dimensional and the quasi two-dimensional Kosterlitz–Thouless temperature within modified mean-field
Bosegas,andfortheQuantumMonteCarlosimulations(from theoryasafunctionofω˜ (thestrictlytwo-dimensionallimit
z
above). is t2KdT =0745 (from eq. (18)).
The mean-field density is an exact sum of Gaussians,
π2 N mω2(r√kβ)2 withtheidealquasitwo-dimensionalprofileandwiththe
n(r)λ2 = 6t2 kπkexp − 2 , (16) strictly two-dimensional mean field are important. To
k=1 (cid:20) (cid:21) illustratethetemperature-dependenceoftheeffectivein-
X
teraction, we also show the mean-field profile computed
whose variances correspond to the density distribution
with the zero-temperature interaction parameter g˜ in-
of the harmonic oscillator at temperature kβ. The pre-
stead of the true effective two-dimensional interaction g
factors in eq. (16) contain the cycle weights π . These
k
(see eq. (11)). We note in this context that the differ-
weights give the probability of a particle to be in a cy-
ence in eq. (11) between g and g˜ was determined under
cle of length k in the path-integral representation of the
the condition that the interaction leaves the density dis-
Bosegas,wherethedensitymatrixmustbesymmetrized
tribution in z unchanged. Whenever this condition is
throughasumofpermutations[16,17]. FortheidealBose
violated,the density distributionin z mustbe computed
gas, the π are easily computed. In the distinguishable-
k
byothermeans,asforexamplebyQuantumMonteCarlo
particle limit, at infinite temperature, only cycles of
methods (see [9]).
length k = 1 contribute (π = 1), whereas the Bose–
1
Einstein condensate is characterized through contribu-
Let us finally discuss the Kosterlitz–Thouless transi-
tionsofcyclesoflengthk N. Thecycleweightsπ are
∝ k tion into the low-temperature phase, which is not con-
shown in the inset of fig. 2 for small k. Only very short
tainedinmean-fieldtheory. Thesemiclassicalquasitwo-
cycles contribute,andthe density profilecanthus be de-
dimensionalgasdoesnotBose-condensebecausethepar-
scribed by a very small number of Gaussians. The exact
ticlenumberineq.(12)divergesatµ˜ =0(itsaturatesat
cycle-weight distribution of the Quantum Monte Carlo
a finite value in the ideal Bose gas). This divergence is
does notrigorouslycorrespondto aprofile asineq.(16),
due to the logarithmic divergence of the central density
although the corrections are negligible in our case.
(seeeq.(13)). However,interactioneffectsbeyondmean-
Figure3considersthetemperaturet=0.8intheinter-
fielddriveaKosterlitz–Thoulessphasetransition[3]from
val between the Kosterlitz–Thouless temperature of the
thehigh-temperaturenormalphasetoasuperfluidbelow
quasitwo-dimensionalinteractinggas(att=t 0.70
KT ≃ TKT.
fortheseparameters[9])andthestrictlytwo-dimensional
Bose–Einstein condensation temperature. Again, the As discussed previously [9, 15], the Kosterlitz–
agreement of the quasi two-dimensional mean field with Thouless transition occurs when the central density
the exact density profile obtained by Quantum Monte n(0)λ2 reaches the critical value of the two-dimensional
Carlois remarkable. At this temperature, the deviations homogeneous gas, which has been determined numeri-
5
cally [18] for g 0: icalcalculationsof the algebraicdecayof the condensate
→
density with system size [9].
380~2
n(0)λ2 n λ2 log . (17)
c
≃ ≃ mg
ACKNOWLEDGMENTS
Wecanintroduce(byhand)theconceptofacriticalden-
sity into mean-field theory by selecting among the solu-
tions t(µ˜) of eq. (14) the one satisfying eq. (17). For We are indebted to Jean Dalibard for many inspiring
the interaction parameters used in Ref [9], mg˜/~2 = discussions.
0.13, we find a mean-field critical temperature tq2d =
KT
Tq2d/T2d = 0.69. This value is in excellent agreement
KT BEC
with the Monte Carlo data. The inset of fig. 3 shows
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610 (1972).
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the exact saturation numbersat finiteN.
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