Non-equilibrium dynamics in the quantum Brownian oscillator and the second law of thermodynamics

We initially prepare a quantum linear oscillator weakly coupled to a bath in equilibrium at an arbitrary temperature. We disturb this system by varying a Hamiltonian parameter of the coupled oscillator, namely, either its spring constant or mass according to an arbitrary but pre-determined protocol in order to perform external work on it. We then derive a closed expression for the reduced density operator of the coupled oscillator along this non-equilibrium process as well as the exact expression pertaining to the corresponding quasi-static process. This immediately allows us to analytically discuss the second law of thermodynamics for non-equilibrium processes. Then we derive a Clausius inequality and obtain its validity supporting the second law, as a consistent generalization of the Clausius equality valid for the quasi-static counterpart, introduced in [1].