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Exact non-Markovian master equations: a generalized derivation for Gaussian systems

Antonio D'Abbruzzo, Vittorio Giovannetti, Vasco Cavina

TL;DR

The paper derives an exact non-Markovian master equation, the Gaussian Master Equation (GME), for quadratic quantum systems bilinearly coupled to Gaussian environments of matching statistics. It applies to both bosonic and fermionic baths and remains valid in the presence of initial system–environment correlations, avoiding the common factorized-initial-state assumption. A Schwinger–Keldysh contour-based derivation yields a Redfield-like operator form with a single dressed kernel that captures all virtual system–environment interactions. The authors demonstrate numerical practicality with a fermionic system featuring superconducting pairing and discuss potential simplifications and extensions.

Abstract

We derive an exact master equation that captures the dynamics of a quadratic quantum system linearly coupled to a Gaussian environment of the same statistics: the Gaussian Master Equation (GME). Unlike previous approaches, our formulation applies universally to both bosonic and fermionic setups, and remains valid even in the presence of initial system-environment correlations, allowing for the exact computation of the system's reduced density matrix across all parameter regimes. Remarkably, the GME shares the same operatorial structure as the Redfield equation and depends on a single kernel - a dressed environment correlation function accounting for all virtual interactions between the system and the environment. This simple structure grants a clear physical interpretation and makes the GME easy to simulate numerically, as we show by applying it to an open system based on two fermions coupled via superconductive pairing.

Exact non-Markovian master equations: a generalized derivation for Gaussian systems

TL;DR

The paper derives an exact non-Markovian master equation, the Gaussian Master Equation (GME), for quadratic quantum systems bilinearly coupled to Gaussian environments of matching statistics. It applies to both bosonic and fermionic baths and remains valid in the presence of initial system–environment correlations, avoiding the common factorized-initial-state assumption. A Schwinger–Keldysh contour-based derivation yields a Redfield-like operator form with a single dressed kernel that captures all virtual system–environment interactions. The authors demonstrate numerical practicality with a fermionic system featuring superconducting pairing and discuss potential simplifications and extensions.

Abstract

We derive an exact master equation that captures the dynamics of a quadratic quantum system linearly coupled to a Gaussian environment of the same statistics: the Gaussian Master Equation (GME). Unlike previous approaches, our formulation applies universally to both bosonic and fermionic setups, and remains valid even in the presence of initial system-environment correlations, allowing for the exact computation of the system's reduced density matrix across all parameter regimes. Remarkably, the GME shares the same operatorial structure as the Redfield equation and depends on a single kernel - a dressed environment correlation function accounting for all virtual interactions between the system and the environment. This simple structure grants a clear physical interpretation and makes the GME easy to simulate numerically, as we show by applying it to an open system based on two fermions coupled via superconductive pairing.

Paper Structure

This paper contains 2 sections, 8 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Contour dynamics

Figures (1)

  • Figure 1: Schematic depiction of the contours considered in this paper. On the left, $\gamma_+(t)$ goes from $t$ to $0$ while $\gamma_-(t)$ goes from $0$ to $t$: this is used for factorized initial states. On the right, $\gamma_\pm(t)$ is shifted by $\pm i \beta/2$ and a vertical track is added that runs from $i\beta/2$ to $-i\beta/2$: this is used for the generic Gaussian initial state in Eq. \ref{['eq:rho_0']}.