Nonadiabaticity of Quantum harmonic oscillators.

We study a frequency-modulated quantum harmonic oscillator as a thermodynamic system. For this purpose, we introduce an `invariant' thermal state by using Ermakov-Lewis-Riesenfeld invariant in place of an initial state. This prescription enables us to analyze the thermodynamics of the oscillator system regardless of whether the process is slowly varying (adiabatic) or not (nonadiabatic). We introduce a quantity $\mathscr{S}$ that describes the `nonadiabaticity' contribution satisfactorily. We write down the thermodynamics of the oscillator system by using this quantity in addition to the ordinary thermodynamical ones. As a result, we extend the first law of thermodynamics to nonadiabatic processes. We discuss universality for the method and some possible applications. In short, we suggest a schematic procedure for obtaining a measure of the `degree of nonadiabaticity' and present an application to the thermodynamics of the squeezed quantum oscillators.