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Resonance Photon Generation in a Vibrating Cavity

The problem of photon creation from vacuum due to the nonstationary Casimir effect in an ideal one-dimensional Fabry--Perot cavity with vibrating walls is solved in the resonance case, when the frequency of vibrations is close to the frequency of some unperturbed electromagnetic mode: $ω_w=p(πc/L_0)(1+δ)$, $|δ|\ll 1$, (p=1,2,...). An explicit analytical expression for the total energy in all the modes shows an exponential growth if $|δ|$ is less than the dimensionless amplitude of vibrations $ε\ll 1$, the increment being proportional to $p\sqrt{ε^2-δ^2}$. The rate of photon generation from vacuum in the (j+ps)th mode goes asymptotically to a constant value $cp^2\sin^2(πj/p)\sqrt{ε^2-δ^2}/[πL_0 (j+ps)]$, the numbers of photons in the modes with indices p,2p,3p,... being the integrals of motion. The total number of photons in all the modes is proportional to $p^3(ε^2-δ^2) t^2$ in the short-time and in the long-time limits. In the case of strong detuning $|δ|>ε$ the total energy and the total number of photons generated from vacuum oscillate with the amplitudes decreasing as $(ε/δ)^2$ for $ε\ll|δ|$. The special cases of p=1 and p=2 are studied in detail.