Evolution of Gaussian mixed states under the Markovian master equation for a driven quantum oscillator

We study a generic quantum Markovian master equation for a linearly displaced or driven harmonic oscillator. It was known that the displacement dynamics of Gaussian mixed states depends on the unitary part of the Liouvillian, the decay rate of the system but not on the bath temperature. Here we further show that the fast-rotating modes do not affect the system's displacement dynamics under linear driving forces. Analytical solutions of the quantum master equation are obtained for displaced Gaussian mixed states. Because the non-driven and driven Liouvillians are related by a unitary displacement operator, they are expected to share the same exceptional points structure. At the exceptional points, the displacement of critically damped oscillator displays a characteristics polynomial-in-time prefactor multiplied by an exponential decay. We discuss how external time-dependent forces affect the displacement dynamics using impulsive force and harmonic force as examples. The results obtained for constant driving remain valid in the presence of time-dependent driving.