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

TL;DR

This work develops a unifying framework for non-Markovian open quantum dynamics by applying Wick's theorem on the Keldysh contour to systems coupled to Gaussian environments. It yields a compact deterministic reduced-dynamics expression and an exact stochastic unraveling that reproduces the stochastic von Neumann equation (SVNE) after a Keldysh rotation, while allowing extensions to initial system-environment correlations and thermodynamic energy statistics via contour-extended generating functions. A key result is the reduction to a single physical-time noise, along with conditions under which environmental measurements can be interpreted as classical noise or measurement records in a semiclassical regime. The approach also covers quantum-thermodynamic quantities through a two-point measurement framework and suggests potential numerical and conceptual benefits for studying a wide class of open quantum systems on arbitrary contours.

Abstract

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.

Figures (4)

• Figure 1: Schematic depiction of the Keldysh contour used in this paper. The contour $\gamma(t)$ is composed by a backward branch $\gamma_+(t)$ running from $t$ to $0$ and a forward branch $\gamma_-(t)$ running from $0$ to $t$, in this order. For the sake of clarity, we added some of the integration variables appearing in Eq. \ref{['eq:againdens']}. The variables $z_1 = \tau_{1 -},z_n = \tau_{n -}$ and $z_1'= \tau_{1 +}', z_m' = \tau_{m +}'$ belong to the $\gamma_-$ and $\gamma_+$ branches, respectively, since they originally represent contributions coming from the time-ordered and anti time-ordered parts of Eq. \ref{['eq:vNeu_sol_expanded']}.
• Figure 2: A summary of the results obtained from Sec. \ref{['sec:keldysh']} to Sec. \ref{['sec:svne']}. The equations are divided in deterministic and stochastic ones (left and right in the figure, respectively). The domain of definition of the time variables of the operator/noises appearing in the equations give an alternative way to classify them. This domain can coincide with either the Keldysh contour or the physical time domain (up and down in the figure, respectively). Working in the Keldysh formalism, we have shown that the reduced dynamics of a system in contact with a Gaussian bath is described by Eq. \ref{['eq:gaussian_exact']} and in Sec. \ref{['sec:svne']} we proved that this can be arranged as the stochastic operator in Eq. \ref{['eq:R_solution']}. The connection between the Keldysh contour approach and the physical time approach is always obtained by introducing the Keldysh components (i.e., by splitting $\gamma(t)$ in its forward and backward branches). When applied to Eq. \ref{['eq:gaussian_exact']} one recovers the influence functional reported in Ref. diosi2014gaussian; see Eq. \ref{['eq:Ferialdi']} for the explicit expression of the superoperator $\hat{\mathcal{W}}$. The connection between the two-state unraveling \ref{['eq:twostate_unrav']} and the SVNE \ref{['eq:stocvon']} is known tanimura2006reviewtilloy2017unraveling: here we showed that it is realized by a Keldysh rotation. For completeness, notice that the connection between Eq. \ref{['eq:Ferialdi']} and the two-state unraveling can be done via a Hubbard-Stratonovich technique tanimura2006review or a stochastic ansatz with two noises tilloy2017unraveling. Finally, to address cases where the stability condition does not hold, Eq. \ref{['eq:gaussian_exact']} and the SVNE must be replaced with Eq. \ref{['eq:gaussian_exact_shifted']} and Eq. \ref{['eq:shiftedeq']}, respectively.
• Figure 3: The contour $\gamma(t,b)$, obtained by $\gamma(t)$ after shifting $\gamma_\pm(t)$ by $\pm i b$ and after adding a vertical branch $\gamma^M(b)$ running from $ib$ to $-ib$. The key insight for understanding the emergence of the new branch is to interpret the initial correlations between $S$ and $E$ as an effective interaction acting in the imaginary-time (temperature) domain. This interaction, $V$, originates from Eq. \ref{['eq:initial_canonical']} and leads to the interaction-picture operator $\mathcal{V}$ defined in Eqs. \ref{['eq:exp_corr']}, \ref{['eq:interaction_mod']} with arguments taken along $\gamma(t,b)$.
• Figure 4: The contour $\gamma(t,\lambda)$, obtained by $\gamma(t)$ after adding two additional horizontal branches of length $\lambda$ before $\gamma_+$ and $\gamma_-$. This contour is a powerful tool to study the statistics in TPEM schemes funo2018pathint.