Nature and origin of the operators entering the master equation of an open quantum system

TL;DR

This paper derives a generalized master equation for open quantum systems using a novel formalism, revealing multiple Lindblad-like operators in non-Markovian cases and recovering standard equations in classical limits.

Contribution

It introduces a non-Markovian master equation with explicit operators, extending the Lindblad formalism to more general quantum environments.

Findings

Operators depend on the environment's Hamiltonian.

Multiple Lindblad-like operators exist in non-Markovian regimes.

Standard master equations are recovered in the classical environment limit.

Abstract

By exploiting the peculiarities of a recently introduced formalism for describing open quantum systems (the Parametric Representation with Environmental Coherent States) we derive an equation of motion for the reduced density operator of an open quantum system that has the same structure of the celebrated Gorini-Kossakowski-Sudarshan-Lindblad equation, but holds regardless of markovianity being assumed. The operators in our result have explicit expressions in terms of the Hamiltonian describing the interactions with the environment, and can be computed once a specific model is considered. We find that, instead of a single set of Lindblad operators, in the general (non-markovian) case there one set of Lindblad-like operators for each and every point of a symplectic manifold associated to the environment. This intricacy disappears under some assumptions (which are related to markovianity and the classical limit of the environment), under which it is possible to recover the usual master equation formalism. Finally, we find such Lindblad-like operators for two different models of a qubit in a bosonic environment, and show that in the classical limit of the environment their renown master equations are recovered.