Quantum radiation force on a moving mirror with Dirichlet and Neumann boundary conditions for a vacuum, finite temperature, and a coherent state

Abstract

We consider a real massless scalar field in a two-dimensional spacetime, satisfying Dirichlet or Neumann boundary condition at the instantaneous position of a moving boundary. For a relativistic law of motion, we show that Dirichlet and Neumann boundary conditions yield the same radiation force on a moving mirror when the initial field state is invariant under time translations. We obtain the exact formulas for the energy density of the field and the radiation force on the boundary for vacuum, thermal, coherent, and squeezed states. In the nonrelativistic limit, our results coincide with those found in the literature.In the 1970s decade the first works investigating the quantum problem of the radiation emitted by moving mirrors in vacuum were published, motivated by the investigation of particle creation in the nonstationary universe (see ). Fulling and Davies [3] studied the moving mirror radiation problem in the context of a real scalar field in a two-dimensional spacetime. They obtained an exact formula for the finite physical part of the expected value of the energy-momentum tensor, assuming the initial state as the vacuum. Their results revealed that the radiation is originated at the mirror and propagates away from it. Ford and Vilenkin [8] developed a perturbative method which can be applied to mirrors moving in small displacements and with nonrelativistic velocities. In this approximation, they obtained, for a real scalar field in a twodimensional spacetime, that the radiation force is proportional to the third time derivative of the mirror law of motion, which is the nonrelativistic limit of the result obtained in Ref. [3]. Ford and Vilenkin also applied their method to a scalar field in four-dimensional spacetime, obtaining a force proportional to the fifth time derivative of the displacement of the mirror. Davies and Fulling [5] (see also Ref.[9]) found a class of hyperbolic trajectories of a mirror moving in a 1 þ 1 Minkowski spacetime, for which the emitted radiation corresponds to a thermal spectrum, that could be registered by an inertial detector. Moreover, several works have focused on the problem of moving mirrors placed in a thermal bath (considered as the ''in'' state, instead of the vacuum). Jaekel and Reynaud [10] obtained for the scalar field in 1 þ 1 dimensions the thermal contribution to the dissipative force proportional to the velocity of the mirror, valid in the nonrelativistic limit. Thermal effects have also been considered in Refs. [11][12][13][14][15][16]. The coherent state is another initial field state which has been considered [17,18], as well as the superposition of coherent states used to study decoherence via the dynamical Casimir effect [19]. Furthermore, attention has been given to the role that boundary conditions play on the dynamical Casimir effect. In the static Casimir effect, different boundary conditions can change the sign of the Casimir force [20]. Applications of this change, for instance, in the building of nanoelectromechanical systems have been discussed [21]....

Moving mirrors and the fluctuation-dissipation theorem

“…We should point out here that the procedure we have adopted and the complete relativistic result we have obtained for the radiation reaction force is different from another prior effort in this direction (see Ref. [12]). Nevertheless, we find that the results for the radiation reaction force match in the nonrelativistic limit [6,12,19], which is the domain of our primary interest in the latter part of this article.…”

“…These modes can be expressed in terms of the null coordinates u = t − x and v = t + x as follows [10,12,[17][18][19]:…”

“…[12]). Nevertheless, we find that the results for the radiation reaction force match in the nonrelativistic limit [6,12,19], which is the domain of our primary interest in the latter part of this article.…”

“…More recently, the radiation reaction force on the moving mirror at a finite temperature has been calculated as well (in this context, see Ref. [12]). However, to the best of our knowledge, this is the first time that the correlation function governing the radiation reaction force is being evaluated and the associated fluctuation-dissipation theorem is being explicitly established for the case of the moving mirror at a finite temperature (though we should clarify that the possibility has been briefly discussed in Ref.…”

“…[8][9][10][11][12][13][14]; also see the following reviews [15,16]). Our goal in this work is to reconsider the random motion of the mirror immersed in a thermal bath.…”