Casimir force on a single interacting Bose-Einstein condensate in the double-parabola approximation

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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66 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 REFERENCES [1] Joseph O Indekeu and Bert Van Schaeybroeck, 2004. Extraordinary wetting phase diagram for mixtures of Bose-Einstein condensates. Physical review letters 93, 210402. [2] Nguyen Van Thu, Tran Huu Phat and Pham The Song, 2017. 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