Derivation of the quantum-optical master equation based on coarse-graining of time

Abstract

This is a derivation of the quantum-optical master equation using coarse-graining of time, which aims at exploring Markovian evolution from a microscopic perspective. Here, coarse-graining refers to finding an approximate evolution operator by integrating over a time step Δt, and considering the field states to be approximated by a similarly coarse-grained basis. By using these ideas and keeping the field states coarse-grained throughout the derivation, I explore the past-future independence of the quantum white noise formalism explicitly. Namely, that a quantum-optical system interacting with a vacuum field may be viewed as sequentially interacting with a series of initially uncorrelated and temporally coarse-grained vacuum states—this process leads to correlations only between the system and the time bins which have interacted with the system in the past, along with a Markovian evolution of the reduced system state. The derivation here may also be viewed in terms of a type of quantum-mechanical path integral.