  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].
