The Casimir Effect in a Weakly Interacting Bose Gas
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The Casimir Effect in a Weakly Interacting Bose Gas Pham The Song1 · Nguyen Van Thu2 Received: 8 May 2020 / Accepted: 24 September 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract By means of the quantum field theory, the Casimir effect in an interacting Bose gas confined between two parallel plates is considered in the one-loop approximation. The Casimir effect due to the quantum fluctuations and thermal fluctuations is calculated associated with the periodic boundary condition applied at the plates. Our results show that the Casimir force is short-ranged in every ranges of the temperature. Keywords Finite-size effect · Casimir effect · Interacting Bose gas · Finite temperature
1 Introduction Ten years after H.B.G. Casimir predicted that there is an attractive force acting on two parallel neutral plates in the electromagnetic vacuum [1, 2], the experimental observations achieved the results appropriate to his theoretical calculations [3, 4]. According to it, the Casimir force acting on a unit of area of plate, so-called Casimir force, is proportional to 𝓁 −4 with 𝓁 being the inter-distance between two plates. These results encouraged physicists in finding many interesting properties of the Casimir effect in many scopes of physics, such as molecular interacting [5], quantum field theory [6], gravitation and cosmology physics [7], quark matter [8] and so on. In field of Bose–Einstein condensate (BEC), the Casimir effect occurs when the Bose gas is confined by a parallel plate geometry. Motivated by experiments on the critical Casimir force [9, 10] and the Casimir–Polder force [11–13], the studies on Casimir effect have been rapidly developed in recent years at both zero and nonzero * Nguyen Van Thu [email protected] Pham The Song [email protected] 1
Faculty of Natural Sciences and Technology, Tay Bac University, Son La, Vietnam
2
Department of Physics, Hanoi Pedagogical University 2, Hanoi 10000, Vietnam
13
Vol.:(0123456789)
Journal of Low Temperature Physics
temperature. Depending on the boundary condition applied at the plates confined the BEC, the Casimir force is either attractive or repulsive. Many authors have pointed out that the Casimir force is attractive associated with the periodic, Dirichlet or Neumann boundary condition, for example [14–16]. In contrast, this force is recently reported to be repulsive corresponding to Zaremba and anti-periodic boundary conditions [17, 18]. At zero temperature, the Casimir effect is only caused by the quantum fluctuations on top of the ground state, which is associated with the phononic excitation [19–21]. It has been investigated by many authors in the one-loop approximation. Using Abel–Plana formula, the authors of Ref. [19] expressed the Casimir force as an integral of the density of state. In another way, employing the Euler–Maclaurin formula, the Casimir energy density and the resulting Casimir force are written as a function of the distance 𝓁 between plates [14, 20, 22],
FC = −
𝜋 2 ℏvs , 480 𝓁 4
(1)
in which ℏ
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