Unifying quantum stochastic methods using Wick's theorem on the Keldysh contour

We present a method, based on the Keldysh formalism, for deriving stochastic master equations that describe the non-Markovian dynamics of a quantum system coupled to a Gaussian environment. This approach yields a compact expression for the system's propagator, which we show to be equivalent to existing formulations, such as the stochastic von Neumann equation (SVNE). A key advantage of our method is its generality: It can be extended to describe any open-system evolution defined on a suitable ordering contour. As a result, we adapt it to derive generalized versions of the SVNE that account for initial system-environment correlations, as well as stochastic equations that incorporate information about the statistics of energy flows in the environment. The insights offered by our technique further allow us to examine the nature of the noise processes appearing in the SVNE. We prove that its solution can be expressed in terms of a single physical noise, without any loss of information. Finally, we propose a semiclassical scenario in which this noise can be interpreted as arising from an initial measurement process on the environment.