The quantum behavior of general time dependent quadratic systems linearly coupled to a bath

In this paper we solve for the quantum propagator of a general time dependent system quadratic in both position and momentum, linearly coupled to an infinite bath of harmonic oscillators. We work in the regime where the quantum optical master equation is valid. We map this master equation to a Schroedinger equation on Super-Hilbert space and utilize Lie Algebraic techniques to solve for the dynamics in this space. We then map back to the original Hilbert space to obtain the solution of the quantum dynamics. The Lie Algebraic techniques used are preferable to the standard Wei-Norman methods in that only coupled systems of first order ordinary differential equations and purely algebraic equations need only be solved. We look at two examples.