Table Of ContentQuantum corrections to the thermodynamic potential of interacting Bosons in a trap
Subhasis Sinha
The Institute of Mathematical Sciences, Madras 600 113, India.
(February 1, 2008)
8 We calculate the quantum corrections of the thermodynamic quantities of a system of confined
9
Bosons at finite temperature. Systematically quantum corrections are written in a series of ¯h, which
9
isconvergentwhenkT ismuchlargerthanthespacingbetweenenergylevelsofthesystem. Weapply
1
this method to calculate analytically the thermodynamic potential of a weakly interacting Bose-gas
n confined in 3-d harmonic oscillator potential. For large number of particles, quantum corrections
a become small, and contributeto thefinite size corrections to scaling.
J
5 PACS numbers:03.75.Fi,32.80.Pj
1
There has been renewed interest in the Bose-Einstein is defined as
] condensation(BEC)afteritsexperimentaldemonstration
ch byAndersonetal[1]. Ithasnowbecomepossibletomea- Z0(β)= e−βEn, (2)
e sure the relevant thermodynamic quantities of a system Xn
m of weaklyinteracting Bosons in a magnetic trap [2,3]. In
where β is, for our purposes, a parameter with the di-
- these experiments, the temperature dependence of the
mension of inverse energy. We first briefly review the
t condensatefractionandthereleaseenergyaremeasured.
a method as applied to the finite fermion system [11]. To
t Theoretically,the thermodynamic properties of homoge-
s extract thermodynamic quantites of the finite fermion
. neous Bose gas was studied by several authors starting
t system from the canonical partition function, a weight
a from London [4]. The thermodynamic properties of the functionisdefinedasw = πβT . Byusingthisweight
m inhomogeneousBosegaswasstudiedbyLeggetetal[5,6]. F sin(πβT)
function one can write the grandpotential of the system
After the recent experimental developments, there is a
-
d new impetus to understand the thermodynamic prop- in the following way,
n erties of interacting inhomogeneous Bose gas. Within
o kTq(T,z)=µN E+TS
the localdensity approximation(LDA),Stringarietal[7] −
[c have calculated the thermodynamic quantities numeri- =L−1[Z0(β)wF]. (3)
callyandshownscalingpropertiesofstronglyinteracting µ β2
1 bosons in a trapand since then severalcalculations have
v In integral representation,
been reported by various authors [8,9]. In these papers
2
6 thethermodynamicpropertiesofinhomogeneousgashas 1 ǫ+i∞ Z (β)w
1 beencalculatedwithintheLDA,neglectingthequantum kTq(T,z)= eβµ 0 Fdβ (4)
1 corrections which may be important. 2πiZǫ−i∞ β2
0 In this paper we develop a systematic semiclassical ¯h-
8 expansionforthermodynamicquantitiesofaninteracting where the contour is closed from left, enclosing all the
9 negative poles. Summing all residues we obtain the
bose gas which takes into account the quantum correc-
/ known result [12],
t tions. We follow a method analogous to the extended
a
Thomas-Fermi method(ETF) which has been success-
m q(T,z)= (1+ze−En/kT), (5)
fully applied in the case of finite fermion systems, like
- nuclei, atoms, clusters etc. It is well known that the Xn
d
n semiclassical expansion of the smooth part of the quan- where z = eβµ is the fugacity. We may extend this
o tum density of states can be expanded in powers of h¯.
methodtoBosonsalso,by takingasuitableweightfunc-
c The leading term in this expansionis the Thomas-Fermi
: tion,
v leveldensity. Systematich¯ correctionstothe leadingap-
i proximationmay then be obtained using the well known cos(πβT)πβT
X
Wigner-Kirkwood(WK)expansion[10]. In the caseof fi- wB = − . (6)
sin(πβT)
r nitefermionsystemsthisiswhatisusuallyreferredtoas
a
the extended Thomas-Fermi approximation [11]. The grand potential of an ensemble of bosons can be
Wefirstdevelopasystematic¯hexpansionusingtheW- written in terms of the following integral,
K method [10]. Consider the single particle Hamiltonian
H for a system of particles, which satisfies the equation, 1 η+i∞ Z (β)w
kTq(T,z)= eβµ 0 Bdβ. (7)
Hψn =Enψn (1) 2πiZη−i∞ β2
whereE areeigenvaluesandψ arecorrespondingwave The line of integration is choosen in such a way that all
n n
functions. Thecanonicalpartitionfunctionofthesystem the positive poles(excluding zero) are to the right side
1
of the line. Since the chemical potential µ < E for equation(7),weobtainthegrandpotentialofthenormal
n
bosons, we close the contour from right, and summing state.
all the residues at the positive poles inside the contour,
we obtain the well known result [12], q(z,T)= 1 d3p d3r[log(1 ze−β(2pM2 +V(~r)))
−(2π¯h)3 Z Z −
q(z,T)=−Xn log(1−ze−En/kT). (8) −24¯hM2 ∇2V(~r)∂∂µ22log(1−ze−β(2pM2 +V(~r)))] (16)
From the above thermodynamic potential we can calcu-
After doing the p integration we get the following form
late all thermodynamic quantities. of the density of the grand potential upto order h¯2.
To do a systematic ¯h expansion of Bosons, we start
from high temperature limit. Neglecting zero point en- 1 ¯h2 ∂2
ergyand the occupationnumber ofgroundstate, we can F(~r)= λ3[g5/2(z˜)− 24M∇2V(~r)∂µ2g5/2(z˜)] (17)
do W-K expansion of canonical partition function. For
thesemiclassicalexpansionofthepartitionfunction,itis where the g (x) and λ are defined as,
n
most convenient to take plane wave basis for calculation
of the canonical partition function. 2π¯h ∞ xi
λ= ; g (x)= . (18)
Z (β)= 1 d3p d3re−ip~.~r/h¯e−βHeip~.~r/h¯ (9) √2πMkT n Xi=1 in
0 h3 Z Z
and the effective fugacity is z˜=ze−βV(r).
To evaluate the partition function, we write, To check the correctness of the above formalism, we
first apply it to a system of bosons in 3-d isotropic
u(~r,~p,β)=e−βHei~r.p~/h¯ harmonic oscillator(h.o) confinement. Following our
=e−βHclei~r.p~/h¯w(~r,~p,β), (10) method, grand potential of the normal state is given by,
where Hcl is the classical hamiltonian Hcl = 2pM2 +V(~r) q(z,T)=(k¯hTω)3[g4(z)− 18(β¯hω)2g2(z)] (19)
and u obeys the Bloch equation,
From this grand potential we can derive the number of
∂u
+Hu=0 (11) particles and the energy of the normal state.
∂β
∂q
with the boundary condition Ne =z( )T
∂z
lim u=eip~.~r/h¯. (12) =(kT)3[g (z) 1(β¯hω)2g (z)] (20)
β→∞ ¯hω 3 − 8 1
∂q
Substituting u in the above equation, we obtain the fol- U = ( )
z
− ∂β
lowing equation for w.
kT 1
=kT( )3[3g (z) (β¯hω)2g (z)] (21)
∂w β 1 ¯hω 4 − 8 2
= i¯h[ (p~. V)w (p~. V)]+
∂β − M ∇ − M ∇
Abovequantitiesmayalsobederivedbyusingdensityof
¯h2
[β2( V)2w β( 2V)w+ 2w 2β( V. w)] (13) state of the system. Exact canonical partition function
2M ∇ − ∇ ∇ − ∇ ∇ of the 3-d h.o is given by,
The above equation can be solved order by order in h¯,
1
by expanding w as a power series in ¯h. Z(β)= . (22)
[2sinh(βh¯ω)]3
2
w =1+h¯w +h¯2w +... (14)
1 2
Density of states can be calculated by taking inverse
Uptothe orderh¯2,canonicalpartitionmaythenbe writ- laplacetransformofthecanonicalpartitionfunctionwith
ten as [11], respect to β. Also the same results may be obtained by
Eular-Maclaurin summation metod [9]. From the above
Zwk(β)= (2π1¯h)3 Z d3pe−β2pM2 Z d3re−βV(~r) cmaalcyublaeteioxnpawnedhedaviensaeesnertiehsaot,ftahdeirmmeondsyionnalmesiscpqauraanmtiettieesr
β¯hω. This series will converge when β¯hω 1. However
β2¯h2 ≤
[1 2V(~r)], (15) at low temperatures, β¯hω 1, and Wigner-Kirkwood
× − 24M ▽ ≈
expansion breaks down. At these temperature, only few
low energy states are occupied, so energy levels can not
where V(r) is the effective single particle potential. In-
be taken as continuous.
sertingthis expressionforcanonicalpartitionfunctionin
2
In the systems which show macroscopic occupation of z˜ =z˜exp[β(ǫ V(r) 2uρ (r)
eff 1 c
− −
ground state at finite temperature, chemical potential +2u(ρ (0) ρ (r)))] (28)
nc nc
of the system tends to zero. As z 1, the higher or- −
der terms in h¯ expansion show infra→red divergence. To where,z˜=eβ(µ−ǫ0), whichis alwaysless than1. Assum-
regulate such divergences fugacity of the system can be ing weak coupling, we expand the functions in terms of
replaced by ze−∆E/kT, where ∆E is the natural energy 2u(ρnc(0) ρnc(r)). The perturbative expansion in this
−
gap of finite size systems [13]. This is equivalent to in- method turns out to be well behaved [8]. Excluding the
troducing a infrared momentum cutoff 1/L, where L non-condensatepart,theeffectivefugacitycanbewritten
is the system size. ≈ as,
Having checked the formalism, we now apply above z˜ =z˜eβ(V(r)−ǫ1) for r <r
method to real physical situation that of a non ideal eff 0
Bosegas. InrecentexperimentEnsheretal[2]measured =z˜eβ(ǫ1−V(r)) for r >r0, (29)
the thermodynamic quantities of 40000 Rb87 atoms, for
where r is the turning point of Thomas-Fermi density
which s-wave scattering length is 100a , where a is 0
Bohr radius. ≈ 0 0 ρc. Fromthe above expressionwe can see there is no ex-
pansionparameterforcondensatepart. We approximate
WithinHartree-Focktheoryofbosons,non-condensate
the localchemicalpotentialwithin the condensate by its
part satisfies the following Schro¨dinger’s equation [5],
mean value. To calculate the free energy of the system
¯h2 1 withinmeanfieldtheory,wesubtractselfenergyfromthe
[ 2+ Mω2r2+2uρ(r)]ψ (r)=e ψ (r), (23) total energy of the system. The self energy contribution
i i i
−2M∇ 2
is given by,
and the condensate part satisfies Gross-Pitaevskii equa-
tion, Es =uZ ρ2nc(r)d3r. (30)
¯h2 1 In the following calculation we replace the coupling by
[ 2+ Mω2r2+2uρ (r)+uρ (r)]φ(r)
−2M∇ 2 nc c a dimensionless parameter η = ( 2aN1/6)2/5. Upto
π l
=ǫ φ(r), (24) q
0 order η5/2 and h¯2 the thermodynamic potential is given
by,
where ρ (r) and ρ (r) are the densities of non-
nc c
condensate and condensate part, and ρ = ρc+ρnc. The q(z,T)/N =t3g˜(4,z˜,1/2,αηβ˜)+
interaction strength is given by, u = 4πh¯2a. Neglecting
the kinetic energy of the condensate, thMe lowest energy √2πt3q(αηβ˜)[g7/2(z˜)+ 32αηβ˜ g5/2(z˜e−2/5αηβ˜)]
eigenvalue may be approximated by,
1
+2η5/2t7/2[ζ(3/2)g (z˜) F(3/2,3/2,3/2,˜z)]
1 a 3 − 2
ǫ ¯hω(15N )2/5+2uρ (0). (25)
0 ≈ 2 0l nc 1 t[g˜(2,z˜,1/2,αηβ˜)+
−8N2/3
Further within the same approximationcondensate den-
sity can be written as, 2 (αηβ˜)[g (z˜) 2αηβ˜g (z˜e−2/5αηβ˜)]
√πq 3/2 − 3 1/2
1 +2η5/2t1/2(ζ(3/2)g (z˜) F(1/2, 1/2,5/2,˜z))], (31)
ρ (r)= (ǫ V(r)), (26) 1
c 1 − −
u −
where the new functions and parameters are defined by,
where ǫ1 = 12¯hω(15N0al)2/5. The above approximation ∞ xi
isvalidforlargenumberofcondensateatomsN andfor
0 g˜(s,x,t,y)= I(t,iy) (32)
strong repulsive interaction. This approximation breaks is
Xi=1
down within a small region near the critical tempera- ∞
1
ture,wherethenumberofcondensateatomsbecomevery I(t,y)= dxe−x(x+y)(t−1) (33)
√π Z
small. For the non-condensate part we may use the for- 0
∞ ∞
malism developed earlier. Using equation (17), the local x(i+j)
F(α,β,γ,x)= (34)
thermodynamic potential of the non-condensate atoms iαjβ(i+j)γ
can be written as, Xi=1Xj=1
1 π
F(r)= 1 [g (z˜ ) ¯h2 2V g (z˜ )], (27) α= 2(15r2N0/N)2/5 (35)
λ3 5/2 eff − 24M∇ eff 1/2 eff t=1/β˜=kT/(h¯ωN1/3) (36)
where z˜eff =eβ(µ−Veff(r)) and Veff(r)=V(r)+2uρ(r). For small value of αη, the function g˜(s,z˜,1/2,αηβ˜) can
Forperturbativeexpansionwewritetheeffectivefugacity
be written as g (z˜). Condensate fraction can be calcu-
s
in the following form.
lated from the thermodynamic potential,
3
∂(q(z,T)/N) the variation of condensate fraction with reduced tem-
N /N =1 z (37)
0
− ∂z perature t, by choosing N = 10,000 atoms and the di-
mensionlessparameterη =0.18. InFig.2wehaveshown
For T < T , z˜ 1. To regulate some divergent terms
appearingicnthe≈h¯2 corrections,wesetz˜ e−βh¯ω,which the finite size correction to the condensate fraction for
≈ same values of the parameters. From the graph we can
is the natural energy gap from ground state. From the
see that the first order quantum correction is two order
above expression we can see that the condensate frac-
smaller than the leading order contribution.
tion and other thermodynamic quantities are not fully
Inconclusion,wehavedevelopedanextendedThomas-
scale invarient with respect to the dimensionless scaling
Fermi method at finite temperature for a system of
parameters η and t. But the extra terms coming from
Bosons. Systematicquantumcorrectionsmaybewritten
the quantum corrections contain a factor 1/N2/3. These
ina series ofa dimensionless parameterβ¯hω, whichcon-
terms depend on the number of particles in the system,
vergesforthetemperatureslargerthantheaveragespac-
and give finite size corrections to the scaling form of the
ing between energy levels. We have applied our method
thermodynamicquantities. Inthe largeN limit andalso
to a system of weakly interacting Bose gas confined in
in the high temperature phase, quantum corrections are
isotropic harmonic oscillator potential. Thermodynamic
negligible. Butfor smallquantum confinedsystems they
potentialand condensate fractionare calculatedanalyti-
give finite corrections to the free energy.
cally upto η5/2 and h¯2 order. Leading terms in h¯ are the
functions of scaling parameterst and η only, but leading
orderquantumcorrectionsgiveN dependent corrections
to the scaling form. In present situation quantum cor-
rections are negligibily small. But quantum corrections
will be important for small number of particles and for
lowdimensionalsystems. Alsoitsmagnitudedependson
the nature of confinement potential. This method may
be applied to investigate the properties of a system of
confined charged bosons in weak magnetic field.
I would like to thank M.V.N. Murthy and R.K.
Bhaduri for helpful discussions.
FIG.1. Condensate Fraction for 10,000 atoms, η=.18.
[1] M. H.Anderson, J. R. Ensher, M. R. Matthews, C. E.
Wieman, and E. A. Cornell, Science 269, 198 (1995).
[2] J. R. Ensher, D. S. Jin, M.R. Matthews, C.E. Weiman
and E. A.Cornell, Phys.Rev.Lett. 77, 416(1996).
[3] M. O. Mews, M.R. Andrews, N.J. van Druten, D.M.
Kurn, D.S. Durfee, and W. Ketterle, Phys. Rev. Lett.
77, 416(1996).
[4] F. London, Nature 141, 643(1938) ; L. Landau, J.phys.
USSR5, 71(1941).
[5] V. V. Goldman, I. F. Silvera and A. J. Leggett, Phys.
Rev.B 24, 2870(1981).
[6] J. Oliva, Phys.Rev. B 39, 4197(1989).
[7] S.Giorgini, L.P. Pitaevskii, and S.Stringari, Phys. Rev.
Lett. 78, 3987(1996).
[8] H. Shiand W.M. Zheng,Phys. Rev.A 56, 2984(1997).
[9] H.Haugerud,T.HaugsetandF.Ravndal,Phys.Lett.A
225, 18(1997).
FIG. 2. Finite size correction to condensate fraction for [10] E. Wigner, Phys. Rev. 40, 749(1932) ; J. G. Kirkwood,
10,000 atoms, where t is reduced temperature. Phys. Rev.44, 31(1933).
[11] Matthias Brack and Rajat. K. Bhaduri, Semiclassical
Physics, (Addison-Wesleypublishing company,1997).
For real physical systems the typical value of the pa- [12] R. K. Pathria, Statistical Mechanics, (Pergamon Press,
rameter η which determines the two-body interaction New York,1972).
strength varies from 0.1 0.25. In Fig.1 we have shown [13] R.MasutandW.J.Mullin,Am.J.Phys.47,493(1979).
−
4
