Technical tools

Abstract

This chapter provides tools that are useful for the solution and handling of master equations. We start with simple analytic approaches including the equation of motion technique and the quantum regression theorem. As numerical techniques, we investigate a Runge–Kutta solver applied to a master equation and introduce the stochastic Schrödinger equation. For rate equations obeying local detailed balance, we treat the evolution of the Shannon entropy and connect it to the full counting statistics. We show how the statistics of energy and matter transfers can be extracted from the master equation. In particular, we demonstrate how the moments and cumulants of the corresponding distributions can be obtained. Finally, we relate symmetries in the respective generating functions with the fluctuation theorem for entropy production. The methods in this chapter may also be applied to Markovian master equations that are not in Lindblad form; only constant coefficients and a time-local evolution equation for the density matrix are required.