Staying positive: going beyond Lindblad with perturbative master equations

Abstract

The perturbative master equation (Bloch-Redfield) is extensively used to study dissipative quantum mechanics - particularly for qubits - despite the 25 year old criticism that it violates positivity (generating negative probabilities). We take an arbitrary system coupled to an environment containing many degrees-of-freedom, and cast its perturbative master equation (derived from a perturbative treatment of Nakajima-Zwanzig or Schoeller-Schon equations) in the form of a Lindblad master equation. We find that the equation's parameters are time-dependent. This time-dependence is rarely accounted for, and invalidates Lindblad's dynamical semigroup analysis. We analyze one such Bloch-Redfield master equation (for a two-level system coupled to an environment with a short but non-vanishing memory time), which apparently violates positivity. We show analytically that, once the time-dependence of the parameters is accounted for, positivity is preserved.