Non-Markovian unravelling for the harmonic oscillator

Based on a seminal paper by Feynman and Vernon [1], in the 1980s formally exact expressions for the dynamics of the reduced density matrix of the system have been derived within the path integral representation [2]. Central ingredient there is the influence functional which captures the correlations between system and reservoir via self-interactions non-local in time. Recently, it has been shown that the influence functional can be exactly recovered from a stochastic averaging procedure without non-local memory terms [3, 4] by reformulating the nonequilibrium quantum dynamics using stochastic Schrödinger equations with complex valued noise forces. The drawback though is that for nonlinear systems the convergence of the stochastic average for relatively long times is still an unsolved problem. This issue can be attacked by combining the exact stochastic Schrödinger formulation with a proper semiclassical real-time technique based on a time-dependent initial value representation of the quantum mechanical propagator. Recent evidence has shown that the most powerful semiclassical representation for the real-time propagator of non-dissipative systems is the so-called Hermann-Kluk (HK) propagator [5], which since the pioneering works [6] has seen an impressive number of applications ranging from atomic [7] to chemical physics [8]. In order to model dissipation, we employ the standard decomposition of the total Hamiltonian H ̂ = ĤS + ĤB + ĤI (1) as a sum of a system part, that for reasons of simplicity shall here depend on one degree of freedom x, a bath part consisting of an infinity of harmonic oscillators together with a bilinear interaction between the two systems. In case of a factorized initial density with a bath residing in thermal equilibrium at temperature T a path integral expression for the time-evolved reduced density matrix of the form [2] ρ(xf, x f, t) = dxidx iρ(xi,