On the Redfield and Lindblad master equations

TL;DR

The paper addresses how Redfield and Lindblad master equations relate in open quantum systems and whether the Lindblad form can be obtained without the rotating wave approximation (RWA). Using a field-theoretical framework with a multi-oscillator bath, it identifies a discrepancy in the standard Redfield kernel tied to the quasi-particle approximation and resolves it by imposing energy-conservation on the Born level, effectively making the energy-conserving Redfield equivalent to Lindblad without RWA. This yields a precise mapping to the standard microscopic derivation and clarifies the conditions under which a quantum map is preserved, with implications for populations and coherences in steady states. Overall, the work provides a consistent, RWA-free route to Lindblad dynamics within a field-theoretical treatment of open quantum systems, improving foundational understanding and guiding future analyses of dissipative quantum dynamics.

Abstract

In a previous work we developed a field theoretical approach to open quantum systems using condensed matter methods. In the Born approximation we derived the Redfield equation on the basis of a multi-oscillator bath, a Dyson equation, a diagrammatic expansion and a quasi-particle approximation. In addition applying a rotating wave approximation we obtained the Lindblad equation describing a proper quantum map. The issue regarding the additional rotating wave approximation was left as an open problem. The present work addresses the open problem and presents new results. We identify a discrepancy in the popular and standard Redfield equation. The discrepancy is associated with the well-known fact that the Redfield equation does not represent a proper quantum map. The discrepancy is related to the diagrammatic expansion and a consistency requirement in the quasi-particle approximation. The explicit resolution of this discrepancy is obtained by imposing energy conservation on the Born level. As a result we obtain formal equivalence between the energy-conserving Redfield equation and the Lindblad equation without invoking the rotating wave approximation. We provide a detailed mapping of the field theoretical approach to the standard microscopic derivation in the theory of open quantum systems.

Figures (1)

• Figure 1: In Fig. a we depict the transmission matrix $T(t,t_i)_{pp',qq'}$ in (\ref{['density']}) showing the evolution of the density matrix $\rho_S(t)_{pp'}$ from the initial time $t=t_i$ to the final time $t$. In Fig. b we show the retarded and advanced Green's functions $G_R(t,t')_{pq}$ and $G_A(t,t')_{pq}$ and the bath correlation function $D^{\alpha\beta}(t,t')$ in (\ref{['greenret']}), (\ref{['greenadv']}), and (\ref{['bath']}), respectively. In Fig. c we depict the Dyson equation in (\ref{['dyson']}) satisfied by the transmission matrix $T$. Here $T^0$ is unperturbed transmission matrix in (\ref{['free']}) and $K$ the irreversible kernel. In Fig. d we show the inhomogeneous integral equation for $\rho_S(t)_{pp'}$ in (\ref{['intmaster']}) with inhomogeneous term $T^0\rho_S(t_i)$ and integral $T^0K\rho_S(t_i)$. In Fig. e we depict in (\ref{['kernelborn']}) the second order Born contribution to the irreversible kernel $K$. The terms (1) and (2) correspond to a renormalisation of the retarded and advanced Green's functions, respectively. In (3) and (4) correspond to cross correlations.