Electromagnetic fields in a 3D cavity and in a waveguide with oscillating walls

We consider classical and quantum electromagnetic fields in a threedimensional (3D) cavity and in a waveguide with oscillating boundaries of the frequency Ω. The photons created by the parametric resonance are distributed in the wave number space around Ω/2 along the axis of the oscillation. When classical waves propagate along the waveguide in the one direction, we observe the amplification of the original waves and another wave generation in the opposite direction by the oscillation of side walls. This can be understood as the classical counterpart of the photon production. In the case of two opposite walls oscillating with the same frequency but with a phase difference, the interferences are shown to occur due to the phase difference in the photon numbers and in the intensity of the generated waves. 03.65.Ca, 03.65.-w, 42.50.Dv Typeset using REVTEX 1 The Casimir effect [1] is a macroscopic manifestation of the change in the zero-point electromagnetic energy due to the walls. The time-varying boundary conditions induce the change of the vacuum states for the quantum electromagnetic fields and the difference between initial and final vacuum states results in the photon production. This dynamical Casimir effect provides the possibility to observe experimentally the vacuum change of quantum fields. This phenomenon has been extensively studied when the one of the walls oscillates [2–5]: For an almost sinusoidal movement of the mirror, using the formalism invented by Moore [6] and developed by Fulling and Davies [7], the quantum energy density has been calculated [8,9]. For a harmonic oscillation of the mirror Méplan and Gignoux have shown that a set of frequencies of the oscillating walls leads to an exponential growth of the energy of a wave [10] and the exponential growth of the number of generated photons can be easily understood from the Floquet’s theorem [11]. The scattering approach is used in analyzing the motion induced radiation from a vibrating cavity with partly transmitting mirror(s) [12,13]. For the small oscillation of the walls, the perturbation approach [14] have been developed to calculate the time evolution of the electromagnetic field in the instantaneous basis [15–18]. The aim of this Letter is to examine the photon production in a 3D cavity and to consider the classical electromagnetic fields propagating in a waveguide with oscillating walls. The photons created by the parametric resonance are distributed in the wave number space around the half of the oscillation frequency along the axis of the oscillating motion. We shall show that if we transmit the classical waves into the waveguide with oscillating walls, the waves are amplified and there are generated waves propagating in the opposite direction, which corresponds to the photon production in the quantum theory. When two walls oscillate we find the interference phenomena in the photon numbers and in the intensity of the generated waves. Assuming that the electric field E(r, t) is polarized in the z direction, we may write [6] A = A(x, y, t)ẑ, E = Eẑ = − ∂t ẑ, (1) B = ∂A ∂y x̂− ∂A ∂x ŷ. Consider a rectangular cavity with sides qx(t), Ly and Lz , where the one of the wall oscillates for a time interval 0 < t < T with a small amplitude (ǫ ≪ 1) according to qx(t) = Lx(1 + ǫ sinΩt). (2) In this cavity the field operator can be expanded A = ∑ n [bnψn + b † nψ ∗ n] (3) using the following instantaneous basis ψn(x, y|qx(t)) = ∑ k Qnkφk(x, y, t) (4)