Field-theoretical approach to open quantum systems and the Lindblad equation

We develop a systematic field-theoretical approach to open quantum systems based on condensed-matter many-body methods. The time evolution of the reduced density matrix for the open quantum system is determined by a transmission matrix. Developing diagrammatic perturbation theory, invoking Wick's theorem in connection with a Caldeira-Leggett quantum oscillator environment in thermal equilibrium, the transmission matrix satisfies a Dyson equation characterized by an irreducible kernel. Unlike the Nakajima-Zwanzig and standard approaches, the Dyson equation is equivalent to a general non-Markovian master equation for the reduced density matrix, incorporating secular effects and independent of the initial preparation. The kernel is determined by a systematic diagrammatic expansion in powers of the interaction. We consider the Born approximation for the kernel. Applying a condensed-matter pole or, equivalently, a quasiparticle-type approximation, equivalent to the usual assumption of a timescale separation, we derive a master equation of the Markov type. Furthermore, imposing the rotating-wave approximation, we obtain a Markov master equation of the Lindblad form. To illustrate the method, we consider the standard example of a single qubit coupled to a thermal heat bath.

Figure 1

Here we depict the transmission matrix $T(t,t_i)$ shown as a shaded box describing the evolution of the reduced density operator ${\rho }_S\left(t\right)$ from the initial time $t_i$ to the final time $t$. The legs on the reducible kernel $M$, and the retarded and advanced Green's functions $G_R$ and $G_A$, are denoted by directed arrows.

Figure 2

Here we depict the Dyson equation for the transmission matrix. $T(t,t^{\prime })$ and the irreducible kernel $K(t^{\prime },t^{\prime \prime })$ characterized by shaded boxes. $T^0(t,t^{\prime })$ denotes the unperturbed transmission matrix, and the retarded and advanced Green's functions $G_R(t,t^{\prime })$ and $G_A(t^{\prime },t)$ are denoted by directed arrows.

Figure 3

Here we depict the irreducible kernel $K(t,t^{\prime })$ in the Born approximation. The retarded and advanced Green's functions $G_R(t,t^{\prime })$ and $G_A(t^{\prime },t)$ are denoted by directed lines. The bath correlation function $D(t,t^{\prime })$ is denoted by a dotted line. The vertices $\mathit{S}$ are denoted by dots. Diagrams (a) and (b) correspond to populations, diagrams (c) and (d) to coherences.