A Magnus approximation approach to harmonic systems with time-dependent frequencies

Abstract

We use a Magnus approximation at the level of the equations of motion for a harmonic system with a time-dependent frequency, to find an expansion for its in-out effective action, and a unitary expansion for the Bogoliubov transformation between in and out states. The dissipative effects derived therefrom are compared with the ones obtained from perturbation theory in powers of the time-dependent piece in the frequency, and with those derived using multiple scale analysis in systems with parametric resonance. We also apply the Magnus expansion to the in-in effective action, to construct reality and causal equations of motion for the external system. We show that the nonlocal equations of motion can be written in terms of a "retarded Fourier transform" evaluated at the resonant frequency. observable, namely, the dynamical equations for the external degrees of freedom. They may be obtained from the in-in effective action, and exhibit a back-reaction due to emission of quantum field modes [5,6].Except for rather special cases, it is not possible to obtain closed expressions for the effective action, even in the case of a single harmonic mode with a time-dependent frequency. Indeed, the problem of evaluating the effective action for a system like this, may be posed in terms of a functional determinant, an object which may be obtained from the solution of a linear secondorder differential equation: the classical equation of motion. Since, except for rather special cases, closed solutions for the latter are not known, it is natural to implement approximate treatments. Assuming that the time dependent piece of the frequency is small in comparison with the constant (average) part, an expansion in powers of the former seems natural. The alternative we follow here, corresponds to writing the (formal) solution to the classical equation of motion in phase space, and implementing the Magnus expansion [7,8,9] to solve the latter. This preserves, order by order, the time evolution as a canonical transformation, both at the classical and quantum levels (since the equations of motion are linear). This is a higher desirable feature when considering an expansion for the Bogoliubov transformations, Thus, our approach may be interpreted as an alternative expansion in the time-dependent part of the frequency, which preserves unitarity, and would correspond to a resummation of infinite terms on the usual expansion.The Magnus approach has been used in a large number of works and is of interest in many branches of quantum mechanics. For example, it han been used in the open quantum systems theory as a tool to investigate how to manipulate the irreversible component of open-system evolutions (decoherence and dissipation) through the application of external controllable interactions [10]. Also in the study of periodically driven systems, by means of an expansion both in the driving term and the inverse of the driving frequency, applicable to isolated or dissipative systems. In [11], a systematic Magnus expansion is used to derive explicit e...