Non-Markovian trajectories involving future in semi-classical path integral expression

Abstract

Semi-classical path integral expression for a quantum system coupled to a harmonic bath is derived based on the stationary phase condition. It is discovered that the system path is non-Markovian. Most strikingly, the system path not only couples to its past, but also to its future, i.e. the equation of motion for the system is an integro-differential equation that involves all times. Numerical examples are given at the end. Because of the future-non-Markovian nature of the equation, the numerical solution cannot be obtained by iterative methods. Instead, root search algorithms must be employed.