1. Introduction
Bose-Einstein condensate (BEC) has been attracting the attention of many researchers in both
theory and experiment. The creation of BEC in the laboratory and recent results has created a
great prospect for this promising field [1-4]. One of the basic problems for BEC researchers is to
determine the state of the system. In the non-relativistic case, based on approximation of the
average field, the BEC state is described by the nonlinear Schrodinger equation applied to the
multi-particle system [5]. This equation is also called the Gross-Pitaevski (GP) equation. In the
ground state, the wave function of the BEC is the solution of the GP equation. This is the
nonlinear differential equation and it only has the analytical solution for a special case. To find an
analytical solution for the basic state of the BEC system, there are many proposed approximate
approaches [6, 7].
For a single Bose-Einstein condensate, the Casimir effect has been considered in many
aspects. Using field theory in one-loop approximation, Schiefele and Henkel [8] expressed the
Casimir energy as an integral of density of state, their result shown that this energy decays.
At finite temperature, this effect was also investigated [9].
The Casimir force on an interacting Bose-Einstein condensate, which consists of surface tension
force and Casimir force, was calculated in [10]. In this paper, based on double parabola
approximation (DPA) proposed by Joseph et al. [7] we investigate the forces on a BEC with
constraint of both Dirichlet and periodic boundary conditions. Therefore the total force is also
taken into account.

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HNUE JOURNAL OF SCIENCE DOI: 10.18173/2354-1059.2018-0030
Natural Sciences 2018, Volume 63, Issue 6, pp. 66-73
This paper is available online at
CASIMIR FORCE ON A SINGLE INTERACTING BOSE-EINSTEIN CONDENSATE
IN THE DOUBLE-PARABOLA APPROXIMATION
Luong Thi Theu and Nguyen Van Thu
Faculty of Physics, Hanoi Pedagogical University 2
Abstract. The forces on a dilute single Bose-Einstein condensate confined between two
parallel palates consist of two kinds, namely surface tension force and Casimir force. Within
framework of double parabola approximation the surface tension force is investigated. The
Casimir force is studied by quantum field theory in one-loop approximation. The total force is
also obtained.
Keywords: Bose-Einstein condensate, double parabola approximation, Casimir force, surface
tension force.
1. Introduction
Bose-Einstein condensate (BEC) has been attracting the attention of many researchers in both
theory and experiment. The creation of BEC in the laboratory and recent results has created a
great prospect for this promising field [1-4]. One of the basic problems for BEC researchers is to
determine the state of the system. In the non-relativistic case, based on approximation of the
average field, the BEC state is described by the nonlinear Schrodinger equation applied to the
multi-particle system [5]. This equation is also called the Gross-Pitaevski (GP) equation. In the
ground state, the wave function of the BEC is the solution of the GP equation. This is the
nonlinear differential equation and it only has the analytical solution for a special case. To find an
analytical solution for the basic state of the BEC system, there are many proposed approximate
approaches [6, 7].
For a single Bose-Einstein condensate, the Casimir effect has been considered in many
aspects. Using field theory in one-loop approximation, Schiefele and Henkel [8] expressed the
Casimir energy as an integral of density of state, their result shown that this energy decays.
At finite temperature, this effect was also investigated [9].
The Casimir force on an interacting Bose-Einstein condensate, which consists of surface tension
force and Casimir force, was calculated in [10]. In this paper, based on double parabola
approximation (DPA) proposed by Joseph et al. [7] we investigate the forces on a BEC with
constraint of both Dirichlet and periodic boundary conditions. Therefore the total force is also
taken into account.
Received February 5, 2017. Revised July 3, 2018. Accepted July 10, 2018.
Contact Luong Thi Theu, e-mail address: luongtheu@gmail.com
Casimir force on a single interacting Bose-Einstein condenstate in the Double-Parabola
67
2. Content
2.1. The ground state in the double-prabola approximation
We consider a Bose gas of N identical particles confined between two large parallel plates
along the x y
plane and their separation along z direction be . For 0T , almost of all
particles 0N N
form the condensate. The positions of these slabs are 0z and z .
The grand canonical Hamiltonian operator for such a system of interacting Bose gas reads.
,b
V
(1)
where
b
is Hamiltonian in bulk, without an external field, which has the form
*( ) ( ) ,b GP
m
z z V
2
2
2
(2)
in which
( ) ( ) ,GP
g
V z z
4
2
(3)
is GP potential. Here ( )z is wave function of the ground state, which plays the role of order
parameter, m is atomic mass, g is coupling constant. The connection of this coupling constant
with the
sa -wave scattering length s
a
g
m
2
4
. The chemical potential is read as gn 0
with n0 is bulk density. Minimizing the total Hamiltonian (2) leads to the time-independent GP
equation [11]
( ) ( ) ( ) ,r g r
m
r 2
3
2
2
0 . (4)
To seek simplicity we take on the form of the dimensionless equation by introducing
dimensionless quantity
z
, with
mgn
0
2
being healing length.
The wave function is scaled to bulk density n0 as
n
0
. Equation (4) reduces to
,
d
d
2
3
2
0
and GP potential has the form
4
.
2
V
(5)
(6)
When we study BEC system, an essential problem is solving equation (5). However, this is a
nonlinear second order differential equation and it is not easy to solve it. Therefore, we can not
find an exact analytical solution. Now one invokes the DPA to achieve our aim. Near the wall,
Luong Thi Theu and Nguyen Van Thu
68
the wave function of the system decreases from the bulk density value so that, in the first order
approximation, the order parameter can be expanded around the bulk density value
,a 1 (7)
with a is small real number. Inserting (7) in (5), we obtain
( ) .DPAV
2 1
2 1
2
(8)
Therefore, we obtain the Euler-Lagrange equation
( ) ,
d
d
2
2
2
1 0 (9)
where 2 . Using Dirichlet boundary conditions
( ) ( )L 0 0 (10)
where /L . It is easily to find the solution for (9) with constraint of (10). We obtain the wave
function of ground state
( ) cosh sinh tanh
L
1
2
. (11)
0 2 4 6 8 10
0.0
0.2
0.4
0.6
0.8
1.0
Figure 1. The order parameter ( ) depends on coordinates
2.2. Surface tension force
In order to consider the surface tension force we first calculate the surface energy. To do this,
one recast grand potential
,
V
bdV (12)
hence
Casimir force on a single interacting Bose-Einstein condenstate in the Double-Parabola
69
* ,DPAP A V d
2
0
0
2 (13)
with /P gn 2
0 0
2 being the bulk pressure. Combining (8) and (13) on has the excess energy
(or surface tension) per unit area
* ( ) .
PV
P d
A
2 20
0
0
2 2 1 (14)
Substituting (11) and (14) we obtain
sinh sinh
cosh
L L L
P
L
2
0
2
2 (15)
0 1 2 3 4 5
0
1
2
3
4
L
P
0
Figure 2. The L-dependence of surface tension
Figure 2 shows the L-dependence of surface tension. We see that, at 0L the surface
tension is zero and it increases as L increases and when the distance is large enough the surface
tension becomes a constant
0lim 4 .
L
P
(16)
The force corresponds to excess surface energy is defined as gradient of surface tension
energy. In grand canonical ensemble has form
.F
L
1
(17)
Putting (15) into (17) we get
cosh sinh
.
cosh
L L L
L
F P
0
2
2
1
4
1 2
1
(18)
Luong Thi Theu and Nguyen Van Thu
70
0 1 2 3 4 5
2.0
1.5
1.0
0.5
0.0
L
F P
0
Figure 3. The surface tension force in GCE versus L
Figure 3 shows the surface tension force in GCE versus L. It is obvious that this force is
attractive and its strength tends to zero at large-L region.
2.3. Casimir force
We consider the Casimir force caused by the quantum fluctuations on top of ground state,
which corresponds to phononic excitations [12]. The Bogoliubov dispersion law for element
excitation read as
( ) ,
k k
k g
m m
2 2 2 2
2
2 2
in dimensionless form
( ) , 2 2 2 (19)
with dimensionless wave vector k . The density of free energy has the form
.
( )
gn d
3
2 2 20
3 3
2 2
(20)
Because of the confinement along z-axis, the wave vector is quantized as follows:
,jk k k
2 2 2
in which the wave vector component k perpendicular to z0 -axis and
j
j
k
is parallel with z0 -axis. In dimensionless form one has
.j
2 2 2
(21)
with ,j
j j L
L
L L
and .L
We only consider here at zero temperature, i.e. only quantum fluctuation is taken into
account hence the density of free energy (20) can be rewritten as
Casimir force on a single interacting Bose-Einstein condenstate in the Double-Parabola
71
2
2 2 2 2 20
3 2
( )( ).
2 (2 )
j j
j
gn d
(22)
Equation (22) can be read
.
( )n
gn d
L j M j
L
3
2 2 2 2 20
3 2 2
12 2
(23)
with M L
2 2 . Using is a momentum cut-off for , Eq. (23) rewritten
.
n
gn
d L j M j
L
2 2 2 2 20
3 2
00
4
(24)
To calculate the Casimir energy (24), we use the Euler-Maclaurin formula [13] and take a
limit ,
' ''' ( )( ) ( ) ( ) ( ) ( ) ...,n
n
F n F n dn F F F
5
0 0
1 1 1
0 0 0
12 720 30240
(25)
One finds the Casimir energy which is the finite part of density of free energy (22),
,
gn
L
2
0
2 3
1440
(26)
In GCE, because the bulk density of condenstate is a constant, we obtain the density of
Casimir force in GCE
.C
gn
F
L
2
0
2 4
480
(27)
Figure 4 shows the evolution of Casimir force versus distance L. Based on Eq. (27) and
Figure4 one can give several comments:
- The Casimir force is always attractive, thus it enhances the strength of total force acting on
the palates.
- There is a divergence at L = 0, this is typical characteristic of distort of vacuum energy.
- The strength decays sharply as distance increases, which obeys law of L
- 4
. This means that
Casimir force is noticeable at small distance.
0 1 2 3 4 5
0.12
0.10
0.08
0.06
0.04
0.02
0.00
L
F
C
g
n
0 2
Figure 4. The L-dependence of the Casimir force density in GCE
Luong Thi Theu and Nguyen Van Thu
72
To compare to the surface tension force, Eq. (27) should be rewritten in form of pressure
.C
mg
F P
L
2
0 2 4
4
480
(28)
Using the speed of sound /sv gn m 0 , Eq (28) can be read
.sC
v
F
4
The total force acts on palates is defined as
.total CF F F (29)
Experimentally, consider for rubidi87 [14] with . , .sm kg a m
25 91 44 10 5 05 10 .
We obtained the L-dependence of the total force
1 2 3 4 5
1.5
1.0
0.5
0.0
L
F
to
ta
l
Figure 5. The total force depends on L
The L-dependence of the total force is plotted in Figure 5. In small-L region, expanding in
power series of the distance, the total force has the form
total
g
F
m
L
2
2
2 4
2 2
120
.
Eq. (30) shows that in small-L region, the Casimir force is dominant.
(30)
3. Conclusion
The main results of our work are as follows:
- The wave function for ground state was found by mean of double parabola approximation.
It is quite simple form and useful for our aims.
- From analytical form of the surface energy one obtained formula for surface tension.
- The Casimir force was considered and obtained the analytical solution. Based on these, the
total force can be invoked. The numerical computation was made for rubidium.
Note that our result for Casimir force is improved in [10], in which instead of expanding
wave vector and keeping up to fourth order, we used momentum cut-off and taking limit of
infinite.
Casimir force on a single interacting Bose-Einstein condenstate in the Double-Parabola
73
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