Stochastic unravellings of non-Markovian completely positive and trace preserving maps

TL;DR

This paper develops a framework for representing non-Markovian open quantum system dynamics through stochastic Schrödinger equations, introducing new methods for Gaussian unravellings and explicit solutions for quadratic bosonic Hamiltonians.

Contribution

It introduces a novel operatorial approach to non-Markovian unravellings, replacing functional derivatives with recursive operators, and provides explicit solutions for quadratic bosonic systems.

Findings

Derived conditions for CPTP dynamics via stochastic Schrödinger equations.

Replaced functional derivatives with recursive operatorial structures in Gaussian unravellings.

Explicit operatorial dependence obtained for quadratic bosonic Hamiltonians.

Abstract

We consider open quantum systems with factorized initial states, providing the structure of the reduced system dynamics, in terms of environment cumulants. We show that such completely positive (CP) and trace preserving (TP) maps can be unraveled by linear stochastic Schr\"odinger equations (SSEs) characterized by sets of colored stochastic processes (with n-th order cumulants). We obtain both the conditions such that the SSEs provide CPTP dynamics, and those for unraveling an open system dynamics. We then focus on Gaussian non-Markovian unravellings, whose known structure displays a functional derivative. We provide a novel description that replaces the functional derivative with a recursive operatorial structure. Moreover, for the family of quadratic bosonic Hamiltonians, we are able to provide an explicit operatorial dependence for the unravelling.