The second critical density and anisotropic generalised condensation
In this letter we discuss the relevance of the 3D Perfect Bose gas (PBG) condensation in extremely elongated vessels for the study of anisotropic condensate coherence and the "quasi-condensate". To this end we analyze the case of exponentially anisotropic (van den Berg) boxes, when there a...
Saved in:
| Published in: | Condensed Matter Physics |
|---|---|
| Date: | 2010 |
| Main Authors: | , |
| Format: | Article |
| Language: | English |
| Published: |
Інститут фізики конденсованих систем НАН України
2010
|
| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/32089 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | The second critical density and anisotropic generalised condensation / M. Beau, V.A. Zagrebnov // Condensed Matter Physics. — 2010. — Т. 13, № 2. — С. 23003: 1-10. — Бібліогр.: 23 назв. — англ. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859847802979352576 |
|---|---|
| author | Beau, M. Zagrebnov, V.A. |
| author_facet | Beau, M. Zagrebnov, V.A. |
| citation_txt | The second critical density and anisotropic generalised condensation / M. Beau, V.A. Zagrebnov // Condensed Matter Physics. — 2010. — Т. 13, № 2. — С. 23003: 1-10. — Бібліогр.: 23 назв. — англ. |
| collection | DSpace DC |
| container_title | Condensed Matter Physics |
| description | In this letter we discuss the relevance of the 3D Perfect Bose gas (PBG) condensation in extremely elongated vessels for the study of anisotropic condensate coherence and the "quasi-condensate". To this end we analyze the case of exponentially anisotropic (van den Berg) boxes, when there are two critical densities ρc<ρm for a generalised Bose-Einstein Condensation (BEC). Here ρc is the standard critical density for the PBG. We consider three examples of anisotropic geometry: slabs, squared beams and "cigars" to demonstrate that the "quasi-condensate" which exists in domain ρc<ρ<ρm is in fact the van den Berg-Lewis-Pulé generalised condensation (vdBLP-GC) of the type III with no macroscopic occupation of any mode. We show that for the slab geometry the second critical density ρm is a threshold between quasi-two-dimensional (quasi-2D) condensate and the three dimensional (3D) regime when there is a coexistence of the "quasi-condensate" with the standard one-mode BEC. On the other hand, in the case of squared beams and "cigars" geometries, critical density ρm separates quasi-1D and 3D regimes. We calculate the value of the difference between ρc, ρm (and between corresponding critical temperatures Tm, Tc) to show that the observed space anisotropy of the condensate coherence can be described by a critical exponent γ(T) related to the anisotropic ODLRO. We compare our calculations with physical results for extremely elongated traps that manifest "quasi-condensate".
|
| first_indexed | 2025-12-07T15:40:37Z |
| format | Article |
| fulltext |
Condensed Matter Physics 2010, Vol. 13, No 2, 23003: 1–10
http://www.icmp.lviv.ua/journal
The second critical density and anisotropic generalised
condensation
M. Beau, V.A. Zagrebnov
Université de la Méditerranée and Centre de Physique Théorique – UMR 6207,
Luminy – Case 907, 13288 Marseille, Cedex 09, France
Received February 25, 2010, in final form March 30, 2010
In this letter we discuss the relevance of the 3D Perfect Bose gas (PBG) condensation in extremely elongated
vessels for the study of anisotropic condensate coherence and the “quasi-condensate”. To this end we analyze
the case of exponentially anisotropic (van den Berg) boxes, when there are two critical densities ρc < ρm
for a generalised Bose-Einstein Condensation (BEC). Here ρc is the standard critical density for the PBG. We
consider three examples of anisotropic geometry: slabs, squared beams and “cigars” to demonstrate that the
“quasi-condensate” which exists in domain ρc < ρ < ρm is in fact the van den Berg-Lewis-Pulé generalised
condensation (vdBLP-GC) of the type III with no macroscopic occupation of any mode. We show that for
the slab geometry the second critical density ρm is a threshold between quasi-two-dimensional (quasi-2D)
condensate and the three dimensional (3D) regime when there is a coexistence of the “quasi-condensate” with
the standard one-mode BEC. On the other hand, in the case of squared beams and “cigars” geometries, critical
density ρm separates quasi-1D and 3D regimes. We calculate the value of the difference between ρc, ρm
(and between corresponding critical temperatures Tm, Tc) to show that the observed space anisotropy of the
condensate coherence can be described by a critical exponent γ(T ) related to the anisotropic ODLRO. We
compare our calculations with physical results for extremely elongated traps that manifest “quasi-condensate”.
Key words: anisotropic generalized Bose-condensation, van den Berg-Lewis-Pule condensation of type III,
effect of exponential anisotropy, the second critical point for the BEC
PACS: 05.30.Jp, 03.75.Hh, 67.40.-w
1. Introduction
One can rigorously show that there is no a conventional Bose-Einstein condensation (BEC) in
the one- (1D) and two-dimensional (2D) boson systems or in the three-dimensional squared beams
(cylinders) and slabs (films). For interacting Bose-gas it results from the Bogoliubov-Hohenberg
theorem [1, 2], based on a non-trivial Bogoliubov inequality, see e.g. [3]. For the Perfect Bose-
gas this result is much easier, since it follows from the explicit analysis of the occupation number
density in one-particle eigenstates. A common point is the Bogoliubov 1/q2-theorem [1, 4, 5], which
implies destruction of the macroscopic occupation of the ground-state by thermal fluctuations.
Renewed interest to eventual possibility of the “condensate” in the quasi-one-, or -two-dimen-
sional (quasi-1D or -2D) boson gases (i.e., in cigar-shaped systems or slabs) is motivated by recent
experimental data indicating the existence of the so-called “quasi-condensate” in anisotropic traps
[6–8] and BKT crossover [9].
The aim of this letter is twofold. First we show that a natural modeling of slabs by highly
anisotropic 3D-cuboid implies in the thermodynamic limit the van den Berg-Lewis-Pulé generali-
zed condensation (vdBLP-GC) [10] of the Perfect Bose-Gas (PBG) for densities larger than the
first, i.e., the standard critical ρc(β) for the inverse temperature β = 1/(kBT ). Notice, that a
special case of this (induced by the geometry) condensation was for the first time pointed out by
Casimir [11], although the theoretical concept and the name are due to Girardeau [12]. So, for
the PBG, the “quasi-condensate” is in fact the vdBLP-GC. Here we generalize these results to
the highly anisotropic 3D-cuboid with anisotropy in one-dimension, which is a model for infinite
squared beams or cylinders, and “cigar” type traps.
c© M. Beau, V.A. Zagrebnov 23003-1
http://www.icmp.lviv.ua/journal
M. Beau, V.A. Zagrebnov
Second, we show that for the slab geometry with exponential growing (for α > 0 and L → ∞)
of two edges, L1 = L2 = LeαL, L3 = L, of the anisotropic boxes: Λ = L1 × L2 × L3 ∈ R
3, there
is a second critical density ρm(β) := ρc(β) + 2α/λ2
β > ρc(β) such that the vdBLP-GC changes its
properties when ρ > ρm(β). This surprising behaviour of the BEC for the PBG was discovered by
van den Berg [13], developed in [14], and then in [15, 16] for the spin-wave condensation.
Notice that the exponential anisotropy is not a very common concept for the experimental
implementations. Therefore, it appeals for a re-examination of the standard vdBLP-GC concept
in Casimir boxes [17] and the corresponding version of the Bogoliubov-Hohenberg theorem [18].
Our original observation concerns the coexistence of two types of the vdBLP-GC for ρ > ρm(β)
(or for corresponding temperatures T < Tm(ρ) for a fixed density, see figure 1) and the analysis
of the coherence length (ODLRO) in this anisotropic geometry. We also extend our observations
to obtain another new result proving the existence of the second critical density in the squared
beam and in the “cigar” type traps for exponentially weak harmonic potential confinement in one
direction. We use these results to calculate the temperature dependence of the vdBLP-GC particle
density for the case of two critical densities, ρm(β)>ρc(β) and to apply the recent scaling approach
[17] to the ODLRO asymptotic in this case.
2. Conventional BEC of the Perfect Bose-Gas
It is known that all kinds of BEC in the PBG are defined by the limiting spectrum of the
one-particle Hamiltonian T
(N=1)
Λ = −~
2∆/(2m), when cuboid Λ ↑ R
3. In this paper we make this
operator self-adjoint by fixing the Dirichlet boundary conditions on ∂Λ, although our results are
valid for all non-attractive boundary conditions. Then the spectrum is the set
εs =
~
2
2m
3∑
j=1
(πsj/Lj)
2
sj∈N
(1)
and {φs,Λ(x) =
∏3
j=1
√
2/Lj sin(πsjxj/Lj)}sj∈N are the eigenfunctions. Here N is the set of the
natural numbers and s = (s1, s2, s3) ∈ N
3 is the multi-index.
In the grand-canonical ensemble (T, V, µ), here V = L1L2L3 is the volume of Λ, the mean
occupation number of the state φs,Λ is Ns(β, µ) = (eβ(εs−µ) − 1)−1, where µ < infs εs,Λ. Then,
for the fixed total particle density ρ the corresponding value of the chemical potential µΛ(β, ρ)
is a unique solution of the equation ρ =
∑
s∈N3 Ns(β, µ)/V =: NΛ(β, µ)/V . Independent of the
way Λ ↑ R
3, one gets the limit ρ(β, µ) = limV →∞ NΛ(β, µ)/V , which is the total particle density
for µ 6 limV →∞ infs εs = 0. Since ρc(β) := supµ60 ρ(β, µ) = ρ(β, µ = 0) < ∞, it is the (first)
critical density for the 3D PBG: ρc(β) = ζ(3/2)/λ3
β. Here ζ(s) is the Riemann ζ-function and
λβ := ~
√
2πβ/m is the de Broglie thermal length.
3. The second critical density for generalised BEC in slabs
For Λ = LeαL ×LeαL ×L one gets ([13, 14]) that for any µ 6 0 the limit of Darboux-Riemann
sums
lim
L→∞
∑
s6=(s1,s2,1)
Ns(β, µ)
VL
=
1
(2π)3
∫
R3
d3k
eβ(~2k2/2m−µ) − 1
. (2)
We denote by µL(β, ρ) := ε(1,1,1) − ∆L(β, ρ), where ∆L(β, ρ) > 0 is a unique solution of the
equation:
ρ =
∑
s=(s1,s2,1)
Ns(β, µ)
VL
+
∑
s6=(s1,s2,1)
Ns(β, µ)
VL
. (3)
23003-2
The second critical density and anisotropic generalised condensation
Figure 1. For the slab geometry, the blue curve ρc(1/(kBT )) is the first critical line for the BEC
transition as a function of T , the red curve ρm(1/(kBT )) = ρc(1/(kBT )) + 2α/λ2
β is the second
critical line. Notice that above the red curve there is a coexistence between “quasi-condensate”
(vdBLP-GC of type III) and the conventional condensate in the ground state (vdBLP-GC of
type I), between two curve there is only “quasi-condensates” phase and below the blue curve
there is no condensate.
Since by (2): limL→∞
∑
s6=(s1,s2,1) Ns(β, µ = 0)/VL = ρc(β), for ρ > ρc(β) the limit L → ∞ of the
first sum in (3) is equal to
lim
L→∞
∑
s=(s1,s2,1)
Ns(β, µ)
VL
= lim
L→∞
1
L
1
(2π)2
∫
R2
d2k
eβ(~2k2/2m+∆L(β,ρ)) − 1
= lim
L→∞
− 1
λ2
βL
ln [β∆L(β, ρ)] = ρ − ρc(β). (4)
This implies the asymptotics:
∆L(β, ρ) =
1
β
e−λ2
β(ρ−ρc(β))L + · · · . (5)
Notice that representation of the limit (4) by the integral (see (1)) is valid only when λ2
β(ρ −
ρc(β)) < 2α. For ρ larger than the second critical density: ρm(β) := ρc(β) + 2α/λ2
β the correction
∆L(β, ρ) must converge to zero faster than e−2αL. Now to keep the difference ρ − ρm(β) > 0 we
have to return to the original sum representation (3) and (as for the standard BEC) to take into
account the impact of the ground state occupation density together with a saturated non-ground
state (i.e. generalized) condensation ρm(β) − ρc(β) as in (4). For this case the asymptotics of
∆L(β, ρ > ρm(β)) is altogether different from (5) and it is equal to ∆L(β, ρ) = [β(ρ−ρm(β))VL]−1.
Since VL = L3e2αL, we obtain:
lim
L→∞
∑
s=(s1>1,s2>1,1)
Ns(β, µ)
VL
= lim
L→∞
− 1
λ2
βL
ln[β∆L(β, ρ)] = 2α/λ2
β
= ρm(β) − ρc(β), (6)
and the ground-state term gives the macroscopic occupation:
ρ − ρm(β) = lim
L→∞
1
VL
1
eβ(ε(1,1,1)−µL(β,ρ)) − 1
. (7)
Notice that for ρc(β) < ρ < ρm(β) we obtain the vdBLP-GC (of the type III), i.e., none of the
single-particle states are macroscopically occupied, since by virtue of (1) and (5) for any s one has:
ρs(β, ρ) := lim
L→∞
1
VL
1
eβ(εs−µL(β,ρ)) − 1
= 0 . (8)
23003-3
M. Beau, V.A. Zagrebnov
On the other hand, the asymptotics ∆L(β, ρ > ρm(β)) = [β(ρ − ρm(β))VL]−1 implies
ρs6=(1,1,1)(β, ρ) := lim
L→∞
1
VL
1
eβ(εs−µL(β,ρ)) − 1
= 0 , (9)
i.e., for ρ > ρm(β) there is a coexistence of the saturated type III vdBLP-GC, with the constant
density (6), and the standard BEC (i.e., the type I vdBLP-GC) in the single state (7).
4. The second critical density for generalised BEC in beams and “cigar”
harmonic traps
It is curious to note that neither Casimir shaped boxes [10], nor the van den Berg boxes Λ =
LeαL×L×L, with one-dimensional anisotropy produce the second critical density ρm(β) 6= ρc(β).
To model the infinite squared beams with BEC transitions at two critical densities we propose
the one-particle Hamiltonian: T
(N=1)
Λ = −~
2∆/(2m) + mω2
1x
2
1/2, with the harmonic trap in the
direction x1 and, e.g., Dirichlet boundary conditions in the directions x2, x3. Then, the spectrum
is the set
εs := ~ω1(s1 + 1/2) +
~
2
2m
3∑
j=2
(πsj/Lj)
2
s∈N
. (10)
Here multi-index s = (s1, s2, s3) ∈ (N∪{0})×N
2, and the ground-state energy is ε(0,1,1). Then for
µL(β, %) := ε(0,1,1) − ∆L(β, %), the value of ∆L(β, %) > 0, is a solution of the equation:
% :=
∑
s=(s1,1,1)
ω1
Ns(β, µ)
L2L3
+
∑
s6=(s1,1,1)
ω1
Ns(β, µ)
L2L3
, (11)
where Ns(β, µ) = (eβ(εs−µ) − 1)−1.
Let ω1 := ~/(mL2
1) and L2 = L3 = L. Here L1 is the harmonic-trap characteristic size in the
direction x1. Then for any s1 > 0 and µ 6 0
%(β, µ) := lim
L1,L→∞
∑
s6=(s1,1,1)
ω1
Ns(β, µ)
L2L3
=
1
(2π)2
∞∫
0
dp
∫
R2
d2k
eβ(~p+~2k2/2m−µ) − 1
. (12)
Therefore, the first critical density is finite: %c(β) := supµ60 %(β, µ) = %(β, µ = 0) < ∞. If
% > %c(β), then the limit L → ∞ of the first sum in (11) is
lim
L1,L→∞
∑
s=(s1,1,1)
ω1
Ns(β, µL)
L2L3
= lim
L→∞
1
L2
∞∫
0
dp
eβ(~p+∆L(β,%)) − 1
= lim
L→∞
− 1
~βL2
ln[β∆L(β, %)] = % − %c(β). (13)
This means that the asymptotics of ∆L(β, ρ) is:
∆L(β, %) =
1
β
e−~β(%−%c(β))L2
+ · · · . (14)
Let L1 := LeγL2
, for γ > 0. Then, similar to our arguments in section 2, the representation of the
limit (13) by the integral is valid for ~β(%−%c(β)) < 2γ. For % larger than the second critical density:
%m(β) := %c(β) + 2γ/(~β) the chemical potential correction (14) must converge to zero faster than
e−2γL2
. By the same line of reasoning as in section 2, to keep the difference %−%m(β) > 0 we have to
use the original sum representation (11) and to take into account the input due to the ground state
occupation density together with a saturated non-ground state condensation %m(β) − %c(β) (13).
23003-4
The second critical density and anisotropic generalised condensation
The asymptotics of ∆L(β, % > %m(β)) is then equal to ∆L(β, %) = [βm(% − %m(β))L4e2γL2
/~]−1.
Hence,
lim
L→∞
∑
s=(s1>0,1,1)
~
m
Ns(β, µL)
L4e2γL2 = lim
L→∞
− 1
~βL2
ln[β∆L(β, %)] =
2γ
~β
= %m(β) − %c(β), (15)
and the ground-state term gives the macroscopic occupation:
% − %m(β) = lim
L→∞
~
mL4e2γL2
1
eβ(ε(0,1,1),L−µL(β,%)) − 1
. (16)
With this choice of boundary conditions and the one-dimensional anisotropic trap, our model
of the infinite squared beams manifests the BEC with two critical densities. Again for %c(β) < % <
%m(β) we obtain the type III vdBLP-GC, i.e., none of the single-particle states are macroscopically
occupied:
%s(β, %) := lim
L→∞
~
mL4e2γL2
1
eβ(εs−µL(β,%)) − 1
= 0 . (17)
When %m(β) < % there is a coexistence of the type III vdBLP-GC, with the constant density (15),
and the standard type I vdBLP-GC in the single state (16), since
%s6=(0,1,1)(β, %) := lim
L→∞
~
mL4e2γL2
1
eβ(εs−µL(β,%)) − 1
= 0 . (18)
Finally, it is instructive to study a “cigar”-type geometry ensured by the anisotropic harmonic
trap:
T
(N=1)
Λ = −~
2∆/(2m) +
∑
16j63
mω2
j x2
j/2 (19)
with ω1 = ~/(mL2
1), ω2 = ω3 = ~/(mL2). Here L1, L2 = L3 = L are the characteristic sizes of the
trap in three directions and ηs =
∑
16j63 ~ωj(sj +1/2) is the corresponding one-particle spectrum.
Then the same reasoning as in (12),(13), yields for µL(β, n) := η(0,0,0) − ∆L(β, n) and auxiliary
dimensionality factor κ > 0:
lim
L1,L→∞
∑
s=(s1,0,0)
κ3ω1ω2ω3Ns(β, µL) = lim
L→∞
− κ3
~
β(mL2)2
ln[β∆L(β, n)] = n − nc(β). (20)
Here the finite critical density nc(β) := n(β, µ = 0) is defined similarly to (12), where the particle
density is
n(β, µ) := lim
L1,L→∞
∑
s6=(s1,0,0)
κ3ω1ω2ω2Ns(β, µ) =
∫
R3
+
κ3dω1dω2dω3
eβ[~(ω1+ω2+ω3)−µ] − 1
. (21)
Equation (20) implies for ∆L(β, n) the asymptotics similar to (14):
∆L(β, n) =
1
β
e−β(n−nc(β))m2L4/(~κ3) + · · · . (22)
If we choose L1 := Leγ̂L4
, for γ̂ > 0, then the second critical density nm(β) := nc(β)+(γ̂~κ3)/(βm2).
For nc(β) < n < nm(β) we obtain the type III vdBLP-GC, i.e., none of the single-particle states
are macroscopically occupied:
ns(β, n) := lim
L→∞
κ3 ω1ω2ω3
eβ(ηs−µL(β,n)) − 1
= 0 . (23)
Although for nm(β) < n there is a coexistence of the type III vdBLP-GC, with the constant density
nm(β) − nc(β), and the standard type I vdBLP-GC in the ground-state:
n − nm(β) = lim
L→∞
κ3 ω1ω2ω3
eβ(η(0,0,0)−µL(β,n)) − 1
. (24)
23003-5
M. Beau, V.A. Zagrebnov
5. The second critical temperature and coexistence of condensates
In experiments with BEC, it is important to know the critical temperatures associated with
corresponding critical densities. The first critical temperatures: Tc(ρ), T̃c(ρ) or T̂c(ρ) are well-
known. For a given density ρ they verify the identities:
ρ = ρc(βc(ρ)) , % = %c(β̃c(%)) , n = nc(β̂c(n)) , (25)
respectively for our models of slabs, squared beams or “cigars”. Since the definition of the critical
densities yields the representations: ρc(β) =: T 3/2 Isl, %c(β) =: T 2 Ibl, nc(β) =: T 3 Icg, the expres-
sions for the second critical densities, one gets the following relations between the first and the
second critical temperatures:
T 3/2
m (ρ) + τ1/2 Tm(ρ) = T 3/2
c (ρ) (slab) ,
T̃ 2
m(%) + τ̃ T̃m(%) = T̃ 2
c (%) (beam) ,
T̂ 3
m(n) + τ̂2 T̂m(n) = T̂ 3
c (n) (cigar) .
Here τ = [αmkB/(π~
2Isl)]
2, τ̃ = 2γkB/(~Ibl) and τ̂ = [(γ̂~κ3kB)/(m2Icg)]
1/2 are “effective”
temperatures related to the corresponding geometrical shapes. Notice that the second critical
temperature modifies the usual law for the condensate fractions temperature dependence, since
now the total condensate density is ρ − ρc(β) := ρ0(β) = ρ0c(β) + ρ0m(β). Here ρ0m(β) :=
(ρ − ρm(β)) θ(ρ − ρm(β)).
Figure 2. The first (blue fit), The second (pink fit) and the total (green fit) condensate fractions
as a function of the temperature for 87Rb atoms in the slab geometry with Tc1 = 10−7 K and
τ = 4.43 × 10−7.
For example, in the case of the slab geometry the type III vdBLP-GC (i.e. the “quasi-conden-
sate”) ρ0c(β) behaves for a given ρ like (see figure 2)
ρ0c(β)
ρ
=
{
1 − (T/Tc)
3/2 , Tm 6 T 6 Tc ,√
τ T/T
3/2
c , T 6 Tm .
(26)
Similarly, for the type I vdBLP-GC in the ground state ρ0m(β) (i.e. the conventional BEC) we
obtain:
ρ0m(β)
ρ
=
{
0 , Tm 6 T 6 Tc ,
1 − (T/Tc)
3/2(1 +
√
τ/T ), T 6 Tm ,
(27)
see figure 2. The total condensate density ρ0(β) := ρ0c(β) + ρ0m(β) is the result of coexistence of
both of them: it gives the standard PBG expression ρ0(β)/ρ = 1 − (T/Tc)
3/2.
For the “cigars” geometry case the temperature dependence of the “quasi-condensate” r0c(β)
is
n0c(β)
n
=
{
1 − (T/T̂c)
3 , T̂m 6 T 6 T̂c ,
τ̂2 T/T̂ 3
c , T 6 T̂m .
(28)
23003-6
The second critical density and anisotropic generalised condensation
The corresponding ground state conventional BEC behaves as follows:
n0m(β)
n
=
{
0 , T̂m 6 T 6 T̂c ,
1 − (T/T̂c)
3(1 + τ̂2/T 2), T 6 T̂m ,
(29)
and again for the two coexisting condensates one gets n − nc(β) := n0(β) = n0c(β) + n0m(β) =
(1 − (T/Tc)
3/2)n.
Notice that for a given density, the difference between two critical temperatures for the slab
geometry can be calculated explicitly:
(Tc − Tm)/Tc = g(ρα/ρ) , (30)
where ρα := 8α3/ζ(3/2)2 and g(x) is an explicit algebraic function. For illustration consider a quasi-
2D PBG model of 87Rb atoms in trap with characteristic sizes L1 = L2 = 100 µm, L = 1µm and
with typical critical temperature Tc = 10−7 K. The anisotropy parameter is α = (1/L) ln(L1/L) =
4, 6 · 106 m−1. Then for τ = 4, 4 · 10−7 K we find Tm = 3, 7 · 10−8 K and (Tc − Tm)/Tc = 0, 63.
6. Localisation of anisotropic generalised BEC and coherence length
Another physical observable to characterize this second critical temperature is the condensate
coherence length or the global spacial particle density distribution. The usual criterion is the
ODLRO, which is going back to Penrose and Onsager [19]. For a fixed particle density ρ it is
defined by the kernel:
K(x, y) := lim
L→∞
KΛ(x, y) = lim
L→∞
∑
s
φs,Λ(x)φs,Λ(y)
eβ(εs−µL(β,ρ)) − 1
. (31)
The limiting diagonal function ρ(x) := K(x, x) is local x-independent particle density.
To detect a trace of the geometry (or the second critical temperature) impact on the spatial
density distribution we follow a recent scaling approach to the generalized BEC developed in [17]
(see also [10, 14]) and introduce a scaled global particle density:
ξL(u) :=
∑
s
|φs,Λ(L1u1, L2u2, L3u3)|2
eβ(εs−µ) − 1
, (32)
with the scaled distances {uj = xj/Lj ∈ [0, 1]}j=1,2,3.
For a given ρ the scaled density (32) in the slab geometry is
ξsl
ρ,L(u) :=
∑
s
1
eβ(εs−µL(β,ρ)) − 1
d=3∏
j=1
2
Lj
[sin(πsjuj)]
2. (33)
Since 2[sin(πsjuj)]
2 = 1−cos{(2πsj/Lj)ujLj} and limL→∞ µL(β, ρ < ρc(β)) < 0, by the Riemann-
Lebesgue lemma we obtain that limL→∞ ξsl
ρ,Λ(u) = ρ for any u ∈ (0, 1)3. If ρ > ρc(β), one has to
proceed as in (3)–(5). Then for any u ∈ (0, 1)3:
lim
L→∞
∑
s=(s1,s2,1)
1
eβ(εs−µL(β,ρ)) − 1
d=3∏
j=1
2
Lj
[sin(πsjuj)]
2 =
= lim
L→∞
2[sin(πu3)]
2
(2π)2L
∫
R2
∏2
j=1(1 − cos(2kjujLj)d
2k
eβ(~2k2/2m+∆L(β,ρ)) − 1
= (ρ − ρc(β)) 2[sin(πu3)]
2 , (34)
lim
L→∞
∑
s6=(s1,s2,1)
1
eβ(εs−µL(β,ρ)) − 1
d=3∏
j=1
2
Lj
[sin(πsjuj)]
2 = ρc(β)) . (35)
23003-7
M. Beau, V.A. Zagrebnov
Then the limit of (33) is equal to
ξsl
ρ (u) = (ρ − ρc(β)) 2[sin(πu3)]
2 + ρc(β) . (36)
It manifests a space anisotropy of the type III vdBLP-GC for ρc(β) < ρ < ρm(β) in u3 direction.
For ρ > ρm(β) one has to use representation (3) and asymptotics (6), (7). Then following the
arguments developed above we obtain
ξsl
ρ (u) = (ρ − ρm(β))
3∏
j=1
2[sin(πuj)]
2 + (ρm(β) − ρc(β)) 2[sin(πu3)]
2 + ρc(β) . (37)
So, the anisotropy of the space particle distribution is still in u3 direction due to the type III
vdBLP-GC.
It is instructive to compare this anisotropy with a coherence length analysis within the scaling
approach [17] to the BEC space distribution. To this end let us center the box Λ at the origin of
coordinates: xj = x̃j + Lj/2 and yj = ỹj + Lj/2. Then the ODLRO kernel (31) is:
KΛ(x̃, ỹ) =
∞∑
l=1
elβµL(β,ρ) R
(2)
l R
(1)
l , (38)
where after the shift of coordinates and using (1) we put
R
(2)
l (x̃(2), ỹ(2)) =
∑
s=(s1,s2)
e−lβεs1,s2 φs1,s2,Λ(x̃1, x̃2) φs1,s2,Λ(ỹ1, ỹ2), (39)
R(1)
s (x̃3, ỹ3) =
∑
s=(s3)
e−lβεs3
√
2
L3
sin
(
πs3
L3
(
x̃3 +
L3
2
))√
2
L3
sin
(
πs3
L3
(
ỹ3 +
L3
2
))
. (40)
Similar to (3), for ρc(β) < ρ < ρm(β) we must split the sum over s = (s1, s2, s3) in (38) into
two parts. Since by the generalized Weyl theorem one gets:
lim
L→∞
R
(2)
l (x̃(2), ỹ(2)) =
1
lλ2
β
e−π‖x̃(2)−ỹ(2)‖2/lλ2
β ,
by (38) for the first part we obtain the representation:
lim
L→∞
∞∑
l=1
elβµL(β,ρ)
∑
s=(s1,s2,1)
e−lβεs1,s2,1φs1,s2,1Λ(x̃) φs1,s2,1Λ(ỹ) =
= lim
L→∞
∞∑
l=1
e−lβ∆L(β,ρ) 1
lλ2
β
e−π‖x̃(2)−ỹ(2)‖2/lλ2
β
2
L
sin
(
π
L
(
x̃3 +
L
2
))
sin
(
π
L
(
ỹ3 +
L
2
))
. (41)
For the second part we apply the Weyl theorem for the 3-dimensional Green function:
lim
L→∞
∞∑
l=1
elβµL(β,ρ)
∑
s6=(s1,s2,1)
e−lβεsφs,Λ(x̃) φs,Λ(ỹ) =
∞∑
l=1
1
lλ3
β
e−π‖x̃−ỹ‖2/lλ2
β . (42)
If in (41) we change l → l ∆L(β, ρ), then it gets the form of the integral Darboux-Riemann sum,
where ‖x̃(2) − ỹ(2)‖2 is scaled as ‖x̃(2) − ỹ(2)‖2 ∆L(β, ρ). Therefore, the coherence length Lch in
the direction perpendicular to x3 is Lch(β, ρ)/L := 1/
√
∆L(β, ρ). A similar argument is valid for
ρ > ρm(β) with obvious modifications due to BEC for s = (1, 1, 1) (7) and to another asymptotics
(6) for ∆L(β, ρ). To compare the coherence length with the scale L1,2 = LeαL, let us define the
critical exponent γ(T, ρ) such that limL→∞(Lch(β, ρ)/L)(L1/L)−γ(T,ρ) = 1. Then we get:
γ(T, ρ) = λ2
β (ρ − ρc(β))/2α , ρc(β) < ρ < ρm(β) = λ2
β (ρm(β) − ρc(β))/2α , ρm(β) 6 ρ . (43)
23003-8
The second critical density and anisotropic generalised condensation
For a fixed density, taking into account (26) we find the temperature dependence of the exponent
γ(T ) := γ(T, ρ), see figure 3:
γ(T ) =
{ √
T/τ{(Tc/T )3/2 − 1} , Tm < T < Tc ,
1, T 6 Tm .
(44)
Notice that in the both cases the ODLRO kernel is anisotropic due to impact of the type III
condensation (41) in the states s = (s1, s2, 1), whereas the other states give a symmetric part of
correlations (42), which includes a constant term ρc(β).
Figure 3. Exponent γ(T ) for evolution of the coherence length for the quasi-condensate with
temperature corresponding to 87Rb atoms in the slab geometry with Tc = 10−7 K and τ =
4.43 × 10−7 K.
Numerically, for L1 = L2 = 100 µm, L3 = 1 µm and Tm < T = 0.75Tc the coherence length of
the condensate is equal to 2.8 µm � 100 µm. This decrease of the coherence length is experimentally
observed in [6].
7. Concluding remarks
In conclusion, we add several remarks about a possible impact of particle interaction. Since the
“quasi-condensate” is observed in extremely anisotropic traps [6–8], we think that the geometry
of the vessels is predominant. So, the study of the PBG is able to catch the phenomenon and so
it seems to be relevant. Next, in this letter we did not enter the details of the phase-fluctuations
[6, 7], although we suppose that for the vdBLP-GC it can be studied by switching different Bogoli-
ubov quasi-average sources in condensed modes. Finally, since a repulsive interaction is capable of
transforming the conventional one-mode BEC (type I) into the vdBLP-GC of type III, [20, 21], it is
important to combine the study of this interaction with the results already obtained for interacting
gases in [6–8] and in [18].
The pioneer calculations of a crossover in a trapped 1D PBG are due to [22]. It is similar to
the vdBLP-GC in our exact calculations for the “cigars” geometry and it apparently persists for
a weakly interacting Bose-gas as argued in [8]. Although the ultimate aim is to understand the
relevance of these quasi-1D calculations for the Lieb-Liniger exact analysis of a strongly interacting
gas [23]. We return to these issues in our next papers.
References
1. Bogoliubov N.N., Phys. Abhandl. Sowijetunion, 1962, 6, 1–110; ibid., 1962, 6, 113–229.
2. Hohenberg P.C., Phys. Rev., 1967, 158, 383–386.
3. Bouziane M., Martin Ph.A., J. Math. Phys., 1976, 17, 1848–1851.
4. Bogoliubov N.N., Selected Works, vol.II: Quantum Statistical Mechanics. Gordon and Breach, N.Y.,
1991.
23003-9
M. Beau, V.A. Zagrebnov
5. Bogoliubov N.N., Collection of Scientific Papers in 12 vols.: Statistical Mechanics, vol.6, Part II. Nauka,
Moscow, 2006.
6. Gerbier F., Thywissen J.H., Richard S., Hugbart M., Bouyer P., Aspect A., Phys. Rev. A, 2003, 67,
051602(R).
7. Petrov D.S., Shlyapnikov G.V., Walraven J.T.M., Phys. Rev. Lett., 2001, 87, 050404.
8. Petrov D.S., Shlyapnikov G.V., Walraven J.T.M., Phys. Rev. Lett., 2000, 85, 3745.
9. Hadzibabic Z., Krüger P., Cheneau M., Battelier B., Dalibard J., Nature, 2006, 441, 1118.
10. van den Berg M., Lewis J.T., Pulé J., Helv. Phys. Acta, 1986, 59, 1273–1288.
11. Casimir H.B.G., On Bose-Einstein condensation – In: Fund. Probl. in Stat. Mech. v. III, Ed.
E.G.D.Cohen. North-Holland Publ.Comp., Amsterdam, 1968, p. 188–196.
12. Girardeau M., J. Math. Phys., 1960, 1, 516–523.
13. van den Berg M., J. Stat. Phys., 1983, 31, 623–637.
14. van den Berg M., Lewis J.T., Lunn M., Helv. Phys. Acta, 1986, 59, 1289–1310.
15. Gough J., Pulé J., Helv. Phys. Acta, 1993, 66, 17.
16. Patrick A., J. Stat. Phys., 1994, 75, 253.
17. Beau M., J. Phys. A, 2009, 42, 235204.
18. Mullin W.J., Holzmann M., Laloë F., J. Low Temp. Phys., 2000, 121, 263.
19. Penrose O., Onsager L., Phys. Rev., 1956, 104, 576.
20. Michoel T., Verbeure A., J. Math. Phys., 1999, 40, 1268.
21. Bru J.-B., Zagrebnov V.A., Physica A, 1999, 268, 309.
22. Ketterle W., van Druten N.J., Phys. Rev. A, 1996, 54, 656.
23. Lieb E.H., Liniger W., Phys. Rev., 1963, 130, 1605; Lieb E.H., Phys. Rev., 1963, 130, 1616.
Друга критична густина та анiзотропна узагальнена
конденсацiя
М. Бо, В.А. Загребнов
Середземноморський унiверситет i Центр теоретичної фiзики, Марсель, Францiя
У цiй статтi ми обговорюємо важливiсть конденсацiї 3D iдеального Бозе газу (PBG) в екстремально
видовжених посудинах для вивчення когерентностi анiзотропного конденсату i “квазiконденсату”.
Для цього ми аналiзуємо випадок експоненцiйно анiзотропних (ван ден Берг) боксiв, коли є двi кри-
тичнi густини ρc < ρm, для узагальненої конденсацiї Бозе-Ейнштейна (BEC). Тут ρc – це стандартна
критична густина для PBG. Ми розглядаємо три приклади анiзотропної геометрiї: щiлини, квадратнi
бруси i “сигари” з метою продемонструвати, що “квазiконденсат”, який iснує в областi ρc < ρ < ρm, є
фактично узагальненою конденсацiєю ван ден Берга-Левiса-Пулє (vdBLP-GC) типу III, що макроско-
пiчно не займає жодної моди. Ми показуємо, що для геометрiї щiлини друга критична густина ρm
є порогом мiж квазiдвовимiрним (квазi-2D) конденсатом i тривимiрним (3D) режимом, коли є спiв-
iснування “квазiконденсату” iз стандартною одномодовою BEC. З iншого боку, у випадку квадратних
брусiв i “сигар”, критична густина ρm вiдокремлює режими квазi-1D i 3D. Ми обчислюємо значення
рiзницi мiж ρc, ρm (та мiж вiдповiдними критичними температурами Tm, Tc) для того, щоб показати,
що спостережена просторова анiзотропiя когерентностi конденсату може бути описана критичним
iндексом γ(T ), пов’язаним iз анiзотропним ODLRO. Ми порiвнюємо нашi обчислення iз фiзичними
результатами для екстремально подовжених пасток, що демонструють “квазiконденсат”.
Ключовi слова: узагальнена анiзотропна Бозе-конденсацiя, конденсацiя ван ден
Берга-Левiса-Пулє типу III, вплив експоненцiйної анiзотропiї, друга критична точка для конденсацiї
Бозе-Ейнштейна
23003-10
Introduction
Conventional BEC of the Perfect Bose-Gas
The second critical density for generalised BEC in slabs
The second critical density for generalised BEC in beams and ``cigar'' harmonic traps
The second critical temperature and coexistence of condensates
Localisation of anisotropic generalised BEC and coherence length
Concluding remarks
|
| id | nasplib_isofts_kiev_ua-123456789-32089 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1607-324X |
| language | English |
| last_indexed | 2025-12-07T15:40:37Z |
| publishDate | 2010 |
| publisher | Інститут фізики конденсованих систем НАН України |
| record_format | dspace |
| spelling | Beau, M. Zagrebnov, V.A. 2012-04-08T15:32:04Z 2012-04-08T15:32:04Z 2010 The second critical density and anisotropic generalised condensation / M. Beau, V.A. Zagrebnov // Condensed Matter Physics. — 2010. — Т. 13, № 2. — С. 23003: 1-10. — Бібліогр.: 23 назв. — англ. 1607-324X PACS: 05.30.Jp, 03.75.Hh, 67.40.-w https://nasplib.isofts.kiev.ua/handle/123456789/32089 In this letter we discuss the relevance of the 3D Perfect Bose gas (PBG) condensation in extremely elongated vessels for the study of anisotropic condensate coherence and the "quasi-condensate". To this end we analyze the case of exponentially anisotropic (van den Berg) boxes, when there are two critical densities ρc<ρm for a generalised Bose-Einstein Condensation (BEC). Here ρc is the standard critical density for the PBG. We consider three examples of anisotropic geometry: slabs, squared beams and "cigars" to demonstrate that the "quasi-condensate" which exists in domain ρc<ρ<ρm is in fact the van den Berg-Lewis-Pulé generalised condensation (vdBLP-GC) of the type III with no macroscopic occupation of any mode. We show that for the slab geometry the second critical density ρm is a threshold between quasi-two-dimensional (quasi-2D) condensate and the three dimensional (3D) regime when there is a coexistence of the "quasi-condensate" with the standard one-mode BEC. On the other hand, in the case of squared beams and "cigars" geometries, critical density ρm separates quasi-1D and 3D regimes. We calculate the value of the difference between ρc, ρm (and between corresponding critical temperatures Tm, Tc) to show that the observed space anisotropy of the condensate coherence can be described by a critical exponent γ(T) related to the anisotropic ODLRO. We compare our calculations with physical results for extremely elongated traps that manifest "quasi-condensate". en Інститут фізики конденсованих систем НАН України Condensed Matter Physics The second critical density and anisotropic generalised condensation Друга критична густина та анізотропна узагальнена конденсація Article published earlier |
| spellingShingle | The second critical density and anisotropic generalised condensation Beau, M. Zagrebnov, V.A. |
| title | The second critical density and anisotropic generalised condensation |
| title_alt | Друга критична густина та анізотропна узагальнена конденсація |
| title_full | The second critical density and anisotropic generalised condensation |
| title_fullStr | The second critical density and anisotropic generalised condensation |
| title_full_unstemmed | The second critical density and anisotropic generalised condensation |
| title_short | The second critical density and anisotropic generalised condensation |
| title_sort | second critical density and anisotropic generalised condensation |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/32089 |
| work_keys_str_mv | AT beaum thesecondcriticaldensityandanisotropicgeneralisedcondensation AT zagrebnovva thesecondcriticaldensityandanisotropicgeneralisedcondensation AT beaum drugakritičnagustinataanízotropnauzagalʹnenakondensacíâ AT zagrebnovva drugakritičnagustinataanízotropnauzagalʹnenakondensacíâ AT beaum secondcriticaldensityandanisotropicgeneralisedcondensation AT zagrebnovva secondcriticaldensityandanisotropicgeneralisedcondensation |