Nonadiabaticity of Quantum harmonic oscillators

We propose a quantity, $\mathcal{A}\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}\mathrm{/}$, as a measure describing the nonadiabaticity of a thermodynamic process. For this purpose, we use a schematic method to find the measure of the degree of nonadiabaticity'. The method utilizes an invariant’ thermal state constructed from the Ermakov-Lewis-Riesenfeld invariant. Specifically, we study a frequency-modulated quantum harmonic oscillator as a thermodynamic system. Naturally, we write the first law of thermodynamics with $\mathcal{A}\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}\mathrm{/}$ as a measurable quantity. We discuss universality for the method and some possible applications.