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Revista Mexicana de Física

Quantum damped nonstationary oscillator and Dynamical Casimir Effect

V. V. Dodonov

Published Jan 1, 2011• 2 citations
OpenAlex

Abstract

"I consider the model of quantum damped oscillator with time-dependent frequency and damping coefficients, which can be used to predict the results of current experiments aimed at the discovering the so-called Dynamical Casimir Effect. It is based on the Heisenberg–Langevin approach to the description of open quantum systems. The main emphasis is made on the comparison of the exactly solvable special cases with general approximate solutions found earlier. The examples considered demonstrate that simple approximate formulas are quite reliable for the analysis of realistic experimental situations."

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"I consider the model of quantum damped oscillator with time-dependent frequency and damping coefficients, which can be used to predict the results of current experiments aimed at the discovering the so-called Dynamical Casimir Effect. It is based on the Heisenberg–Langevin approach to the description of open quantum systems. The main emphasis is made on the comparison of the exactly solvable special cases with general approximate solutions found earlier. The examples considered demonstrate that simple approximate formulas are quite reliable for the analysis of realistic experimental situations."
In this paper, we treat the so-called dynamical Casimir effect in a cavity with a two-level atom and give an analytic approximate solution under the general setting. The aim of the paper is to show another approach based on Mathematical Physics to the paper [Approximate analytical results on the cavity Casimir effect in the presence of a two-level atom, Phys. Rev. A85 (2012) 063804, arXiv: 1112.0523 [quant-ph]] by A. V. Dodonov and V. V. Dodonov. We believe that our method is simple and beautiful.
Using the exact path integral solution for the damped harmonic oscillator it is shown that in general there does not exist an exact dissipative Liouville operator describing the dynamics of the oscillator for arbitrary initial bath preparations. Exact nonstationary Liouville operators can be found only for particular preparations. Three physically meaningful examples are examined. An exact master equation is derived for thermal initial conditions. Second, the Liouville operator governing the time evolution of equilibrium correlations is obtained. Third, factorizing initial conditions are studied. Additionally, one can show that there are approximate Liouville operators independent of the initial preparation describing the long-time dynamics under appropriate conditions. The general form of these approximate master equations is derived and the coefficients are determined for special cases of the bath spectral density including the Ohmic, Drude, and weak coupling cases. The connection with earlier work is discussed.
An atom interacting with a quantized electromagnetic field in a cavity with time-dependent parameters is considered. Variation of the cavity parameters results in nonstationary dynamics of the field which leads, in turn, to excitation of the atom, even if photons were initially absent in the cavity. We distinguish three mechanisms of such excitation: excitation due to absorption of real photons created by the dynamical Casimir effect, excitation due to absorption of virtual photons during the transient process, and excitation due to nonadiabatic parametric modulation of the atomic Lamb shift. The last mechanism has no relation to the dynamical Casimir effect and thus should be considered as a new vacuum QED effect. Normally all these three mechanisms give a contribution to the amplitude of the atom excitation and are accompanied by the creation of photons. Therefore the presence of an atom in the cavity alters the average number of created photons in comparison with the case of an empty non-stationary cavity. Our consideration is based mainly on a simple model of a two-level atom interacting with a single mode of quantized electromagnetic field. However, our results are qualitatively valid for more realistic models.