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

TL;DR

This paper derives an exact solution for the quantum dynamics of a general time-dependent quadratic system coupled to a bath, using Lie algebraic methods to simplify the solution process.

Contribution

It introduces a novel approach employing Lie algebraic techniques to solve the quantum propagator for time-dependent quadratic systems coupled to a bath, improving on standard methods.

Findings

Successfully mapped the master equation to a Schrödinger equation in Super-Hilbert space.

Applied Lie algebraic techniques to solve for the quantum dynamics.

Provided two example applications demonstrating the method's effectiveness.

Abstract

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.