Stability of the homogeneous Bose-Einstein condensate at large gas parameter

0.27 MB·English

by  Abdulla Rakhimov

Stability of the homogeneous Bose-Einstein condensate at large gas parameter
Abdulla Rakhimova,b,∗ Chul Koo Kima,† Sang-Hoon Kimc,‡ and Jae Hyung Yeea§
a Institute of Physics and Applied Physics, Yonsei University, Seoul 120-749, R.O. Korea
bInstitute of Nuclear Physics, Tashkent 702132, Uzbekistan
cDivision of Liberal Arts and Sciences, Mokpo National Maritime University, Mokpo 530-729, R.O. Korea
The properties of the uniform Bose gas is studied within the optimized variational perturbation
theory (Gaussian approximation) in a self-consistent way. It is shown that the atomic BEC with
8 a repulsive interaction becomes unstable when the gas parameter γ = ρa3 exceeds a critical value
0 γ 0.01. The quantum corrections beyond the Bogoliubov-Popov approximation to the energy
crit
0 densi≈ty,chemical potential and pressure in powers of √γ expansions are presented.
2

n PACSnumbers: 03.75.Hh,05.30.Jp. 05.30.Ch
a Keywords: Bosecondensate, criticaldensity,fieldtheoretical methods,nonperturbative approach
J

2
1 I. INTRODUCTION naryand,hence,theatomicBECisdynamicallyunstable
against molecular formation.

] It is well known that many of the basic properties of
h Thelongwaitafterit’sprediction(morethan70years)
the condensate of dilute Bose gases in existing experi-
c for realization of the Bose-Einstein condensate (BEC) is
e possibly related to the meta-stability of the initial Bose ments can be described reasonably well using the mean
m fieldapproximation(MFA)whichreducestheproblemto
gas. In fact, we require first, an atomic system would
theclassicalGross-Pitaevskiiequation(GPE)[7]. How-
- stay gaseous and meta-stable at very low temperature
t ever, fluctuations of the quantum field around the mean
a all the way to the BEC transition, and secondly, devel-
t opment of cooling and trapping techniques to reach the field provide corrections which become increasingly im-
s
portant as higher condensate densities (say large γ ) are
. requiredregimesoftemperatureanddensity[1]. Clearly,
at withouta propercoolingtechnique, anyordinaryatomic achieved. It is therefore important to understand the ef-
m fects ofquantumfieldfluctuations especiallyatlargegas
gas would undergo into a liquid or a solid state at low
parameters.
- temperatures,soameta-stalestatecouldbecreatedonly
d In the present paper, we study the properties of a ho-
with low pressure and weak interaction between atoms.
n mogeneous atomic Bose gas using optimized Gaussian
o Even once created, the condensate still remains as a
approximation [8]. It has been proven that the corre-
c fragile and subtle object [2]. The enemies of BEC such
sponding Gaussian effective potential contains one loop,
[ as crystallization, disassociation and three-body recom-
sum of all daisy and superdaisy graphs of perturbation
bination may easily destroy it within very short time.
2 theory [9] and leading order in 1/N expansion.
v When the sign of interaction (or equivalently of the s-
The first application of the Gaussian variational ap-
7 wave scattering length, a ) is suddenly changed into a
proachtoauniformBECwasdonebyBijlsmaandStoof
8 negative value, the BEC collapses and then undergoes
ten years ago [10]. However, it was pointed out in ex-
3 anexplosioninwhichasubstantialfractionofthe atoms
1 were blown off (Bosenova) [3, 4]. cellent review by Andersen that [11] evena modified (by
. introducing many body T-matrix) Gaussian approxima-
1 Due to the ”badcollisions”,evenanatomic BEC with
tion of Ref. [10] does not satisfy the Hugenholtz-Pines
0 a repulsive interaction has a limited life time. Recently,
8 Cornish et. al. [5] carried out an ingenious experiment (H-P)theoremespeciallyatverylowtemperatures. This
0 isparticularlycausedbyalongstandingproblemencoun-
withspinpolarizedatomic85Rb. Intheexperiment,they
: tered in the most of field theoretical approximations: it
v showedthatonecouldcontrolthestrengthofinteratomic
is impossible to satisfy the H-P theorem, namely mak-
i
X interactionfortheBECbyemployingtheFeshbachreso- ing the theory gaplessat the same time and maintaining
nancemethod. Averylargevalueofthescatteringlength
r thenumberofparticleswiththesamevalueofthe chem-
a (a 4500˚A)hasbeenachievedinthisexperiment,which
≈ ical potential. In other words, the chemical potential
correspondstothegasparameterofthecondensatetobe
defined by the H-P theorem does not coincide with the
about γ 0.01. This phenomenon has been recently
max ≈ chemical potential found from the minimization of the
studiedbyYin[6]inRandomPhaseApproximation. The
thermodynamicpotentialwithrespecttothe condensate
author has shown that when γ exceeds a certain criti-
density. Note that, even the T-matrix approximation
calvaluethemolecularexcitationenergybecomesimagi-
cannotresolvethis problemcompletely since in this case
onegets“mismatchofapproximations”whichmakesthe
approachas non self-consistent.
One of the possible solutions of the above mentioned
∗Electronicaddress: [email protected]
problemhas been proposedrecently by Yukalov [12]. He
†Electronicaddress: [email protected]
‡Electronicaddress: [email protected] has shown that Hartree-Fock approximation (HFA) can
§Electronicaddress: [email protected] be made both conserving and gapless by taking into ac-
2
count of two normalization conditions instead of one. and ψ˜ and v are orthogonaleach other
Hence, two chemical potentials each for the condensed
fraction (µ ) and the uncondensed fractions (µ ) should d~rψ˜(r)v(r)=0. (6)
0 1
Z
be introduced to describe the BEC self-consistently.
In the present paper, we reformulate the field theoret- The condensate order parameter v defines the density of
ical Gaussian approximation following the Yukalov pre- condensedparticleswhileψ˜definesthedensityofuncon-
scription, and apply this self-consistent approach to in- densed particles:
vestigate the properties of a uniform BEC.
ρ =v2, ρ =<ψ˜∗(r)ψ˜(r)>. (7)
0 1
The paper is organized as follows: In Sect. II, we
extend the field theoretical approach by implementing Having performed the Bogoliubov shift, one may in-
Yukalov’sideas. InSect. III,wecalculatethefreeenergy troduce the grandcanonicalthermodynamicpotentialof
in a Gaussian approximation,and also show it’s relation the system Ω as
to the one loop and Bogoliubov-Popov approximations
Ω(µ,v)= (µ,v) . (8)
(BPA). In the next two sections, we present the proce- F |v=hψi
dure of minimization of the free energy. The numerical In a stable equilibrium, Ω attains the minimum:
results and their discussion are presented in section VI.
dΩ(µ,v) d2Ω(µ,v)
Sect. VII summarizes the paper. =0, >0. (9)
dv d2v
Apart from the H-P theorem, the chemical potential
II. QUANTUM FIELD FORMULATION WITH should satisfy the normalization condition
YUKALOV PRESCRIPTION
N = d~rψ∗(r)ψ(r) , (10)
hZ i
A grand canonical ensemble of Bose particles with a
where N is the total number of particles. However, as
short range s - wave interaction is governed by the Eu-
it was pointed out in the above, the chemical potential
clidian action [11], [24]
correspondingto the minimum of Ω may not correspond
β ∇2 tothe chemicalpotentialµdeterminedfromthe normal-
S[ψ,ψ∗]= dτ d~r ψ∗(τ,r)[∂τ µ]ψ(τ,r) ization condition.
Z Z { − 2m −
0 To overcomethis difficulties, Yukalov [12] proposedto
g
+ [ψ∗(τ,r)ψ(τ,r)]2 ,(1)
2 } Introduce one more normalization condition N =
0
•
ρ V. So that for the uniform system
where ψ∗(τ,r) is a complex field operator that creates a 0
boson at the position ~r, µ the chemical potential, g the N +N =N, N = d~rψ˜∗(r)ψ˜(r) , (11)
0 1 1
coupling constant given by 4πa/m, m the atomic mass hZ i
and β = 1/T the inverse of temperature T. The free which simply states that the total number of par-
energy of the system can be determined as
ticles should be equal to the sum of the number of
condensed and uncondensed particles.
(µ)= TlnZ, (2)
F −
Introduce two chemical potentials µ and µ , for
0 1
where Z is the functional integral, •
the condensed and the uncondensed fractions re-
spectively as well as a Lagrange multiplier Λ to
Z = ψ ψ∗exp S[ψ,ψ∗] , (3) satisfy the Eq. (5). The total system chemical po-
Z D D {− }
tential, µ= (∂Ω/∂N), is given by
performed all over Bose fields ψ and ψ∗ periodic in −
µ N +µ N
τ [0,β]. When the temperature in a Bose system falls µ= 0 0 1 1. (12)
∈ N
below the condensation temperature T , breaking of the
c
U(1) gauge symmetry may be taken into account by the These prescriptions lead to the following action,
Bogoluibov shift of the field operator,
β ∇2
ψ(τ,r)=v(τ,r)+ψ˜(τ,r), (4) S[ψ,ψ∗]= dτ d~r ψ∗(τ,r)[∂τ ]ψ(τ,r)
Z Z { − 2m
0
where v(τ,r) is the condensate order parameter. In the µ ψ˜∗(τ,r)ψ˜(τ,r) µ v2 Λψ˜(τ,r)
1 0
− − −
uniform system v(τ,r) is a real constant, v(τ,r) = v, g
Λ∗ψ˜∗(τ,r)+ [ψ∗(τ,r)ψ(τ,r)]2 , (13)
ψ˜(τ,r) is the field operator of the uncondensed parti- − 2 }
cles satisfying the same Bose commutation relation as
which should be used in Eq. (3). Further, µ can be
ψ(τ,r). The conservation of particle numbers requires 0
determined from the minimum condition Eq. (9), while
thatψ˜(τ,r)hasnon-zeromomentumcomponentsothat
µ itself by the requirement of H-P theorem
1
ψ˜ =0, (5) µ =Σ Σ , (14)
1 11 12
h i −
3
where Σ and Σ are the normal and the anomalous parts:
11 12
self-energies. As to the condensed fraction N , it could
0
S = S +S +S .
be found by solving the normalization Eq. (11) where clas free int
the uncondensed fraction N1 is given by S = Vβ( v2µ + gv4),
clas 0
− 2
1 β ∇2
∂Ω S = dτ d~r[iǫ ψ ∂ ψ +ψ ( +X )ψ
N1 =−„∂µ1«. (15) free 2Z0 Z ab a τ b 1 −2m 1 1
∇2
+ψ ( +X )ψ ]. (19)
2 2 2
−2m
S = S(2) +S(3) +S(4).
int int int int
III. GAUSSIAN, ONE - LOOP AND 1 β
BOGOLIUBOV-POPOV APPROXIMATIONS S(2) = dτ d~r[ψ2(3gv2 Π )+ψ2(gv2 Π )],
int 2Z Z 1 − 11 2 − 22
0
g β
Inthepresentsection,weshowhowthisschemecanbe Si(n3)t = √2Z dτZ d~rvψ1(ψ12+ψ22),
realized in practice. Substituting Eq. (4) into Eq. (13), 0
one may rewrite the action in powers of v and ψ˜, S(4) = g βdτ d~r(ψ4+2ψ2ψ2+ψ4). (20)
int 8Z Z 1 1 2 2
0
S = S(0)+S(1)+S(2)+S(3)+S(4). Here, ǫ (a,b = 1,2) is the antisymmetric tensor in
ab
β gv4 two dimensions with ǫ12 =1 and following notations are
S(0) = dτ d~r v2µ0+ , introduced,
Z Z {− 2 }
0
β Π11 =Σ11+Σ12, Π22 =Σ11 Σ12,
S(1) = dτ d~r[gv3 Λ∗ Λ][ψ˜∗+ψ˜], − (21)
Z0 Z − − X1 =Π11−µ1, X2 =Π22−µ1.
β ∇2
S(2) = dτ d~r ψ˜∗[∂ µ ]ψ˜ In accordance with Refs. [11, 16], Πab are the compo-
τ 1
Z Z { − 2m − nents of the 2 2 self-energy matrix.
0 ×
gv2 The free part of the action, Sfree in Eq. (20) gives
+ [ψ˜∗ψ˜∗+4ψ˜∗ψ˜+ψ˜ψ˜] , rise to a propagator, which can be used in perturbative
2 }
framework. In a momentum space,
β
S(3) = g dτ d~rv ψ˜∗ψ˜∗ψ˜+ψ˜∗ψ˜ψ˜ ,
∞
Z0 Z { } ψ˜ (τ,r)= 1 ψ˜ (ω ,~k)exp iω τ +i~k~r , (22)
a a n n
S(4) = g βdτ d~rψ˜∗ψ˜∗ψ˜ψ˜. (16) √βV n=X−∞Xk { }
2Z Z
0 where = V d~k/(2π)3, and ω = 2πnT is the Mat-
n
Z
Xk
In the following, S(1) will be omitted since it can be set subara frequency. The propagator is given by
to zero by an appropriate choice of Λ in order to satisfy
1 ε +X ω
Eq. (5). G(ω ,k)= k 2 n ,
Now in accordance with the variational perturbation n ωn2 +Ek2 (cid:18) −ωn εk+X1 (cid:19)
(23)
theory, we add and subtract the following term:
S(Σ) =Z0βdτZ d~r»Σ11ψ˜∗ψ˜+ 21Σ12(ψ˜∗ψ˜∗+ψ˜ψ˜)–, (17) wanitdhεtkh=e d~ki2s/p2emrs.ion relation, Ek2 = (εk +X1)(εk +X2)
With this Green’s function using Eqs. (2) and (3),
and neglecting terms S(3) and S(4), one may get the
assuming Σ and Σ as real constants. Further, we int int
write the qu11antum fl1u2ctuating field ψ˜ in terms of two thermodynamicpotentialintheoneloopapproximation:
real fields gv4
Ω(1L)(µ ,µ ,v)=V µ v2+
0 1 „− 0 2 «
1
ψ˜= 1 (ψ +iψ ), ψ˜∗ = 1 (ψ iψ ). (18) +2 Ek+T ln[1−e−βEk] (24)
√2 1 2 √2 1− 2 Xk Xk
1
+ B(3gv2 Π )+A(gv2 Π ) ,
2 − 11 − 22
ˆ ˜
After some algebraic manipulations [14, 15], one can with Π =3gv2 and Π =gv2 ( A and B will be given
11 22
splittheactioninto“classical”,“free”,and“interaction” below), so that the last term in square bracket can be
4
dropped. Note that, hereafter we perform explicit sum- IV. THE GAP EQUATIONS AND THE
mation by Matsubara frequencies (see e.g. [15]). As to THERMODYNAMIC POTENTIAL AT T =0
the BPA, it can be obtained by introducing an auxiliary
expansion parameter η1L as it was shown by Kleinert Inthissection,thevariationalparametersΠ11 andΠ22
[17]. will be determined using the principle of minimal sensi-
Loop expansion of Ω may be organized by using the tivity [8]. From Eqs. (26) and (27), the gap equations
propagator G(ω ,~k) with constraints X = 2gv2 and may be found
n 1
X = 0 as illustrated in Ref. [18]. To take into account
2
∂Ω(X ,X ,v,µ ,µ )
higher order quantum fluctuations, one has to calculate 1 2 0 1
S(3) and S(4) . Although these quantities can not be ∂X1
h inti h inti = 1 A′[gv2 µ X ]+B′[3gv2 µ X ] (30)
evaluatedexactly,theymaybeestimatedintheGaussian 2{ 1 − 1− 2 1 − 1− 1
approximation [25], where for the homogeneous system: +g[A′(3A+B)+B′(3B+A)]/2V =0,
1 1 }
S(3) =0, and
h inti
hψa2i=Gaa(r−r′)|r→r′ ≡Gaa(0), ∂Ω(X1,X2,v,µ0,µ1)
∂X
hψ12ψ22i=hψ12ihψ22i, hψa4i=3G2aa(0), (25) = 1 A′[gv22 µ X ]+B′[3gv2 µ X ] (31)
∞ 2{ 2 − 1− 2 2 − 1− 1
1
G (0)= G (ω ,~k)=V−1B, +g[A′(3A+B)+B′(3B+A)]/2V =0.
11 Vβ 11 n 2 2 }
n=X−∞Xk
G (0)=V−1A. where
22
∂A 1 1 ∂B
Finally,combiningEqs. (24)and(25),wegetthefollow- A′ = = B′,
ing expressions for the thermodynamic potential: 1 ≡ ∂X1 4Xk Ek ∂X2 ≡ 2
∂A 1 (ε +X )2
Ω(X ,X ,v,µ ,µ )=V( µ v2+ gv4) A′2≡ ∂X2 =−4Xk k Ek3 1 , (32)
1 2 0 1 − 0 2 ∂B 1 (ε +X )2
+12Xk Ek+TXk ln[1−e−βEk] (26) B1′ ≡ ∂X1 =−4Xk k Ek3 2 .
+1 B(3gv2 Π )+A(gv2 Π ) Above, we have two equations (30) and (31) with re-
2 − 11 − 22 spect to three unknown quantities X ,X ,µ . An ad-
+gρˆ 3(A2+B2)+2AB , ˜ ditional equation is supplied from{the1rel2atio1n}between
8N
ˆ ˜ the chemical potential and the self-energies given by the
where H-P theorem. So, from Eqs. (14) and (21), one can
immediately conclude X = 0, and, hence, in the long
2
ε +X 1 1
A k 1 + , wavelength limit (k 0), the quasiparticle energy Ek
≡Xk Ek »2 exp(βEk)−1– behaves as ck (with c→= X1/2m ) thus being gapless,
(27) asexpected. Withthisconpstraint,thegapequationsmay
ε +X 1 1 be simplified as
B k 2 + .
≡Xk Ek »2 exp(βEk)−1– 2X +µ 5gv2 11 gI (X )=0, (33)
1 1− − 8 V 1,1 1
ThefreeenergyinEq. (26)is supposedtohaveallthe
information about the system. Particularly taking it’s I (X )[3gv2 µ X ] I (X )[gv2 µ ]
0,1 1 1 1 −2,−1 1 1
− − − −
derivative with respectto µ , one gets the expressionfor (34)
1 gI (X )
the uncondensed fraction N1: + 1,81V 1 [5I0,1(X1)+I−2,−1(X1)]=0,
and Eq. (28) as
∂Ω
N =
1 −„∂µ1« 1
1 N = I (X )
= 2g{A[(3+AB+−B)(A3g′v+2−(3BΠ1+1)AB)′B−′](g,v2−Π22)A′ (28) +1[I0,18(X11,)1−21I−2,−1(X1)][8Vgv2−gI1,1(X1)−8Vµ1].
−2V } 128Vm
(35)
where A′ = ∂A/∂µ and B′ = ∂B/∂µ . Note that the Here, the following dimensionless integral
1 1
same expression for the uncondensed fraction could be
εimj−i
obtained in an alternative way as I (X )= k (36)
i,j 1 Ej
Xk k
N 1
ρ = 1 =<ψ˜∗ψ˜>= ψ˜ ψ˜∗exp S[ψ,ψ∗] ψ˜∗ψ˜
1 V ZZ D D {− } isintroduced. Theirexplicitexpressionsandtherelations
(29) betweenthemevaluatedindimensionalregularizationare
5
presented in the Appendix of Ref. [11]. In particular, where n = v2/ρ = N /N and X′ = (dX¯ /dn ). Note
0 0 1 1 0
that the same equation could be obtained from the orig-
I (X )= V(2mX1)3/2 =2B = 4A , inal equation (26) as:
1,1 1 3π2 |X2=0 − |X2=0
(37) dΩ(X1,X2,v,µ0,µ1) = ∂Ω
dI1,1(X1) = I0,1(X1). dn0 ∂n0 (47)
dX1 − m + ∂Ω ∂µ1 + ∂Ω ∂X1 + ∂Ω ∂X2 =0,
„∂µ «∂n „∂X « ∂n „∂X « ∂n
1 0 1 0 2 0
Note that Eqs. (34) and (35) include I (X ),
−2,−1 1
which is infrared divergent, I (X ) 1/ǫ + where the last two terms may be omitted due to the gap
−2,−1 1
lnκ2/mX (with ǫ 0). Below we show th∼at this in- Eqs. (30) and (31), and the factor in the second term is
1
→
tegral will be canceled exactly. In fact, eliminating X1 related to N1 by (15).
from Eq. (33) as Clearly, the optimal value of v2, i.e. v¯2 defined by Eq.
(46), should correspond to the normalization condition
X = µ1 + 5gv2 + 11gI1,1(X1), (38) in Eq. (11) (constraint):
1 − 2 2 16V
I (X¯ )
and substituting it into Eq. (34), one observes that the v¯2+ρ1(X¯1)=v¯2+ 1,18V 1 =ρ, (48)
latter is factorized:
which may be considered as a nonlinear equation with
[I0,1(X1)−2I−2,−1(X1)][gI1,1(X1)−8Vgv2+8Vµ1]=0. (39) respect to the c-number v¯2 with a fixed ρ and X¯1(v¯2).
Strictly speaking, v¯2 must be determined from Eq.
Finally, from the last two equations, we find the formal
(46) as a function of µ , and after substituting it into
0
solutions of the gap equations
Eq. (48), the latter should be solved with respect to µ .
0
3g However, this would be a rather complicated way, since
X =2gv2+ I (X ), (40)
1 4V 1,1 1 Eq. (46) is a highly nonlinear equation. On the other
hand, one may assume that v¯2 is known as a solution of
g
µ1 =gv2− 8V I1,1(X1). (41) Eq. (48) and µ0 could be extracted from Eq. (46).
Following this strategy, we obtain
We denote these optimum values of X and µ by X¯
1 1 1
avn2.dNµ¯o1w, ,recsopmecptaivreinlyg,Ewqhsi.ch(3a5r)eaenxdpl(ic3i9t)ly, odneepecnadneenatsiolny µ¯0 =gρn¯0+ X1′I41V,1(ρX¯1)»1+ 11g1I60V,1m(X¯1)–, (49)
see that only the first term in Eq. (35) survives
inparticular,neglectingthesecondterminsquarebrack-
1 etsandtakingintoaccountX1L =2gρn ,wehaveµ for
N1 = 8I1,1(X1). (42) the one-loop approximation 1 0 0
Now,insertingtheseformalsolutionsintoEq. (26)and
I (X¯1L)
using the relations between the integrals,Eq. (37), gives µ10L =gρ»n0+ 1,12Vρ1 –. (50)
the following form for Ω
Further simplification, by introducing an auxiliary ex-
gv4 m
Ω(X¯1,v,µ0)=V(−µ0v2+ 2 )+ 2I0,−1(X¯1) pansion parameter η1L as in Ref. [17], gives µ0 for the
(43) BPA
11gI12,1(X¯1), µBP =gρ 1 gI0,1(X1 =2gρ) . (51)
− 128V 0 » − 2Vm –
where As to the total system chemical potential µ, it follows
from Eqs. (11) and (12) as
1 2√2V(mX¯ )5/2
I (X¯ )= √ε ε +X¯ = 1 .
0,−1 1 m k k 1 15m2π2 µ=µ¯ n¯ +µ¯ n¯ , (52)
Xk p 1 1 0 0
(44)
wheren¯ =1 n¯ , andµ¯ andµ¯ aregivenby Eqs. (41)
In particular, neglecting in Eq. (43) the last term gives 1 − 0 1 0
and (49), respectively. Now, substituting (49) into (43),
the one-loop result:
one may obtain the pressure as P = Ω/V
−
Ω(X¯1,v,µ0)|X¯1=2gv2 =Ω(1L)(µ0,µ1,v)|µ1=gv2. (45) P = 1gn2ρ2+ 1 [n X¯′I (X¯ ) 2mI (X¯ )]
presented in the previous section. In the stable equilib- 2 0 4V 0 1 1,1 1 − 0,−1 1
(53)
rium, the grand canonical potential reaches the global
11gI (X¯ )
minimum as a function of v : + 1,1 1 [2n X¯′I (X¯ )+mI (X¯ )].
128mV2 0 1 0,1 1 1,1 1
dΩ(X¯ ,v,µ )
+X1′Id11n,10(X¯10)[1=+V1ρ1(g−Iµ0,01(+Xg1ρ)n]0=)0 (46) bTehTioshbmetaagiyrnoebudenbdeyasstaialywteedleolnnkeenrgboywynrdeefwnorsriimttyiunlgoaftEhthe=eteB(rΩEm+Cµ,µEvN,2)Vm/Vainy.
4 16Vm 0
6
Eq. (43) as µ v2V = µN µ n N (which follows from obtains a well known result of the one-loop approxima-
0 1 1
Eq. (50)) and using Eq. (4−1): tion: X1L = 2gv2, and further, assuming here v2 = ρ
1
gives the self-energy for BPA : XBP =2ρg.
1
Ω(X¯ ,v¯)= µN+ Vρ2gn¯20 + mI (X¯ ) In general, the Eq. (61) can be rewritten in a dimen-
1 − 2 2 0,−1 1 (54) sionless form
ρgn¯ 13g
+ 0I (X¯ ) I2 (X¯ ).
8 1,1 1 − 128V 1,1 1 432Z Z 3/2
N = 3456 , (62)
γ π − „π«
Now, one may immediately obtain
where the following dimensionless quantities were intro-
gv¯4 m gv¯2 13g
= + I (X¯ )+ I (X¯ ) I2 (X¯ ), duced
E 2 2V 0,−1 1 8V 1,1 1 − 128V2 1,1 (155) Z =γX1/2gρ, Nγ = 432πγn0, (63)
and
with γ = a3ρ is the gas parameter. Analysis shows that
gρ2 m
EBP = 2 + 2V I0,−1(X1)|X1=2gρ, (56) Eq. (62)hasnorealpositivesolutionwhenNγ >1. This
is illustrated in Fig. 1 where the solid curve presents
fortheGaussianandBogoliubov-Popovapproximations
RHS, and the dashed straight lines present LHS of Eq.
respectively.
(62) for N = 0.1;0.3;0.7;1.0;1.1 from the bottom to
ItiswellknownthatintheBPA,thenormal(Σ )and γ
11 the top, respectively. It is seen that when N <1, there
the anomalous self-energies (Σ ) are rather simple [17]: γ
12 are two different solutions (denoted as crosses in Fig.1
) which overlap at N = 1 and Z = π/144 = 0.0218,
ΣBP =2gρ, ΣBP =gρ. (57) γ
11 12 and then disappear. This is one of our main results
confirming that there is a critical value of γ, or more
In the Gaussian approximation using Eqs. (21), (40),
exactly critical value of N γ/N which controls the sta-
and (41), one obtains 0
bility of the uniform Bose condensate at T = 0. When
X g N = 432n γ/π exceeds unity, (N > 1), X and hence
Σ = 1 +µ =2gv2+ I (X ), γ 0 γ 1
11 2 1 4V 1,1 1 the self-energy becomes complex, and the BEC will be
(58)
Σ = X1 =gv2+ 3gI (X ), unstable.
12 2 8V 1,1 1 Differentiating Eq. (62) by N and solving with re-
γ
spect to dZ/dN , one obtains:
whichcanbefurthersimplifiedatthestationarypointas γ
Σ¯11 =2gρ Z′ dZ = π3/2 , (64)
Σ¯12 =gρ(1+2n¯1). (59) ≡ dNγ 432(√π−12√Z)
whichis singularatZ =π/144,i.e., atN =1. Thus,at
γ
Clearly, neglecting the uncondensed fraction n¯ in the
1 the critical point, N =1,
γ
lastequation,werecovertheBogoliubov-Popovapprox-
imation, (57). The dimensionless sound velocity defined ∂X
lim 1 = , (65)
as c= limE /k= X¯ /2m is simply related to Σ as Nγ→1 ∂n0 ∞
k 1 12
k→0 q
and hence, at this point the chemical potential of the
c2 = Σ¯12. (60) condensate µ0 in Eq. (49), which is responsible for the
m thermodynamicalstability of the system, has a singular-
ity.
For N 1, the solutions are given as [26]
γ
≤
V. SOLUTIONS TO THE GAP EQUATIONS
π
Z = [2c cos(c )+3]
1 1 2
576
In this section, we analysis possible solutions to the
π π
gap equation (40) which can be written as ≈ 64 − 216Nγ +O(Nγ2), (66)
π
X =2gv2+ g(mX1)(3/2). (61) Z2 = 576[−c1cos(c2)+√3c1sin(c2)+3] (67)
1 √2π2
πN √3πN3/2 πN2
γ + γ + γ +O(N5/2), (68)
Beforesolvingthisequation,weemphasizethatinaccor- ≈ 432 1944 1944 γ
dance with the generalprinciple ofthe variationalGaus-
sian approximation, the constraint in Eq. (48) and the where c1 = 9 8Nγ, c2 = arccos [27 36Nγ +
− { −
procedureofminimizationofthefreeenergywithrespect 8N2]/c3 /3. p
to v2 may be imposed only after finding an explicit ex- Iγtisu1}nderstoodthatonlythe secondsolution,Z ,is a
2
pression for X X (v2) as a function of v2, which can physicalone,since for the case of Z =Z the self-energy
1 1 1
≡
bedonebysolvingEq. (61)analytically. Notethatwhen X is irregular at γ 0. Moreover, only Z = Z corre-
1 2
→
thesecondtermontheRHSofEq. (61)isneglected,one spondstotheminimumofthethermodynamicpotential,
7
N =1.1 1.0
Gaussian
1.0 X N =1.0 One Loop
BPA
0.9
0.8
X X N =0.7
0.6 n0
0.8
0.4
X X N =0.3 0.7
0.2
X X N =0.1
0.0 0.6
0.01 0.02 0.03 0.04 0.05 0.002 0.004 0.006 0.008 0.010 0.012
Z
FIG.1: Graphicalsolutionofthegapequation(62). Thesolid FIG. 2: Condensate fraction n = n (γ) as a function of
0 0
curverepresentsRHS,andthedashedstraightlinesrepresent γ =ρa3. Solid, dashed and dot-dashed curves correspond to
LHS of the equation for N = 0.1;0.3;0.7;1.0;1.1 from the the Gaussian, the one-loop and Bogoliubov-Popov approxi-
γ
bottom to thetop, respectively. mations.
(∂2Ω/∂2X1)|z=z2 >0. Thus, we conclude, X¯1 =2gρZ/γ µ1 ≈gρ{1−136√√πγ −132π8γ}, Σ12 ≈gρ{1+136√√πγ +132π8γ}.
with Z =Z . In particular, taking into account the first
2 (72)
term in the expansion of Z in Eq. (68), one obtains
2
X¯ 2gρn =X¯1L as expected.
1 ≈ 0 1
gρ 16√γ 128γ gρ2 64√γ 64γ
c2 1+ + , P 1+ + .
≈ m{ 3√π 3π } ≈ 2 { 5√π 3π }
VI. RESULTS AND DISCUSSIONS (73)
which are in good agreement with BPA [19].
Expansionforsmallγ. Thestartingpointofournu- Critical density and exact solutions. In order
mericalcalculationsistheEq. (61),whichcanberewrit- to discuss exact solutions of the equation, (69), we first
ten as establish the boundary for γ which is related to the crit-
1 n¯ 8Z3/2(n¯0) =0. (69) ical value of Nγ found in the previous section. This may
− 0− 3γ√π beevaluateddirectlybysubstitutingNγ =1,Z =π/144
into Eq. (69), which immediately gives γ =5π/1296
Before analyzing this nonlinear equation we note that cr
≈
0.012120. It is interesting to observe that when γ ap-
themajorityofexperimentswithultracoldtrappedgases
proaches this critical value, the condensed fraction re-
deal with weakly interacting atoms, so that γ is very
small, i.e. γ 10−9...10−4. Thus, to obtain a low- mains still large, lim n¯0(γ) = π/432γcr = 3/5 = 0.6
∼ γ→γcr
density expansionof physical quantities, one may search but the condensate as a whole become unstable.
for the solutions of Eq. (69) in power series of √γ to get Fig. 2 presents the condensate fraction n¯ (γ) in the
0
Gaussian (solid line), the one-loop (dotted line) and
n¯1 = 1 n¯0 Bogoliubov-Popov approximations. It is seen that due
−
8 γ 1/2 64γ to the quantum fluctuations the condensed fraction de-
= + +O(γ3/2). (70)
3 π 3 π creasesfaster(withincreasingγ)intheGaussianapprox-
(cid:16) (cid:17)
imation than in BPA.
Clearly,the firsttermcorrespondstothe Bogoliubovap-
Thechemicalpotentialµ=µ n +µ n ispresentedin
0 0 1 1
proximation,whiletheothersmaybeconsideredasquan-
Fig. 3. One may observe that, in the Gaussian approxi-
tum corrections to this approximation. Note also that
mationitvariesslowlywithincreasingγ ,almostcoincid-
the above expansion Eq. (70) is exactly the same as the
ing with that for the BPA. However,when γ approaches
one obtained in the modified Hartree-Fock Bogoliubov
the critical value γ = 0.012 it starts to increase very
crit
(HFB) approximation [13]. Now, using Eqs. (41), (49), fast since in this region, when N 1, X′ in Eq. (49)
(53) , (55), (59), (60) and (70) we obtain the following γ → 1
becomesverylargeandsodoesµ . Bearinginmindthat
0
low density expansions for the energy density, chemical
the chemical potential is the energy needed to add (or
potentials, selfenergies,the soundvelocity andthe pres-
extract) one more particle to (or from) the system, one
sure:
may interpret this effect as a particle number saturation
128√γ 128γ 32√γ 224γ of the condensed particles. In other words, when γ (or
gρ2 1+ + , µ gρ 1+ + .
E ≈ { √π 9π } 0 ≈ { 3√π 3π } more exactly Nγ) reaches the critical value, the num-
(71) ber of condensed atoms N0 cannot be further increased,
8
VII. SUMMARY
12
10 Gaussian In conclusion, we have developed a new Bosonic self-
BPA
consistent variational perturbation theory, which can be
8
made the starting point for systematic expansion proce-
/g 6 dure [22]. We have shown that taking into account two
4 normalizationconditions at the same time solves the old
outstandingproblemofBosesystems makingvariational
2
perturbation theory both conserving and gapless.
0 0.002 0.004 0.006 0.008 0.010 0.012 Studying the properties of a system of uniform Bose
gas at zero temperature with repulsive interaction both
analytically and numerically, we have found that in this
FIG. 3: The chemical potential in the Gaussian (solid line) systemthereisadynamicalparameterN n ρa3which
γ 0
∝
and Bogoliubov-Popov approximations. controls the stability of the Bose condensate. When this
parameter remains smaller than the critical value the
phonon spectrum is purely real and the excitations have
since it will lead to a dynamical instability of the BEC. infinite lifetimes. On the contrary, when N exceeds the
γ
The pressuredefined by eq. (53) is positive inthe region criticalvaluethecondensatebecomesunstable,insimilar
γ <γcrit,butnearthecriticalpointγ γcrit itbecomes fashion to the BEC with an attractive interaction. Note
∼
negative and small, as expected. Note that at this point thatthisphenomenacannotbeobtainedinordinaryper-
the energy density, the self-energies and sound velocity turbative framework.
remainfinite, since correspondingexpressions Eqs. (55)-
It would be quite interesting to study the dependence
(60) do not include X′ explicitly.
1 of critical N on temperature. It was discovered long
Nowweconsiderpossibleoriginoftheinstabilityfound γ
ago by Bethe [23] that the inelastic cross section, which
above. It is well known that [20] the BEC is an effect of
tends to destroy the condensate, varies as 1/k, (called
the exchange coupling, which leads to an effective at-
as 1/velocity law), and hence the bad collisions can be
traction between atoms forcing them to accumulate in a
surprisinglylargenearzerotemperature. Thus,the tem-
single state. However, when the density (or scattering
perature dependence of the critical N seems not to be
length) reaches a critical value this effective attraction γ
trivial. This work is on progress.
makes the condensate collapse.
Anotherpossiblereasonisathreebodyrecombination
ofcondensedatoms. Althoughthereisnoexplicit3body
interactioninourstartingLagrangian,itwasshownthat
Acknowledgments
[21],atT 0,therepulsivetwobodyinteractionleadsto
→
athreebodyrecombinationwiththerateconstantα
rec
a4 and, hence, the three body recombination become∝s A.R. appreciates the Yonsei University for hospitality
very significant for a large scattering length i.e. large during his stay, where the main part of this work was
γ. This seems to be one of the main reasons for the performed. We are indebted to V. Yukalov for several
factthata stablecondensatewith largegasparameteris constructiveremarksandhighlyusefuladvice. Thiswork
inaccessible experimentally. When γ exceeds the critical wassupportedbythe secondphaseoftheBrainKorea21
value,the atomsstartto combineintomoleculesandthe Project. C.K.K. acknowledgesthe support from the Ko-
condensate may undergo phase transition into a solid or rea Science and Engineering Foundation (R01-2006-000-
a liquid state. 10083-0).
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