Dynamical Casimir effect with cylindrical waveguides

I consider the quantum electromagnetic field in a coaxial cylindrical waveguide, such that the outer cylindrical surface has a time-dependent radius. The field propagates parallel to the axis, inside the annular region between the two cylindrical surfaces. When the mechanical frequency and the thickness of the annular region are small enough, only Transverse Electromagnetic (TEM) photons may be generated by the dynamical Casimir effect. The photon emission rate is calculated in this regime, and compared with the case of parallel plates in the limit of very short distances between the two cylindrical surfaces. The proximity force approximation holds for the transition matrix elements in this limit, but the emission rate scales quadratically with the mechanical frequency, as opposed to the cubic dependence for parallel plates. The reflection of vacuum fluctuations by an oscillating mirror generates frequency sidebands. This frequency modulation mixes up positive and negative field frequencies below the mechanical frequency. Because of the association between positive (negative) frequencies and annihilation (creation) operators, this mixing leads to the creation of real low-frequency photon pairs out of the vacuum field state. The very small orders of magnitude involved in this “dynamical Casimir” effect have so far ruled out its experimental verification. For a single plate moving in vacuum, the photon emission rate is of the order of a few microwave photons per day even for mechanical frequencies as high as 10 GHz [1]. Much higher orders of magnitude are obtained if the mechanical frequency is tuned into parametric resonance with a microwave high-Q closed cavity [2]. Another possibility is to consider the dynamical counterpart of the original configuration proposed by Casimir [3], with two parallel plates moving along the direction perpendicular to the plates (z direction) [4]. Assume that one of the plates oscillates with frequency ω0: δz1(t) = δz0 cosω0t, and the second plate is at rest at z2 = a. Since this ‘cavity’ is laterally open, the spectrum of field modes is continuous, and there is no re-cycling of photons (except in the very particular case of propagation along the z direction). As a consequence, one may employ a perturbative