Hierarchy of equations of multiple-time correlation functions

Abstract

In this paper we derive the evolution equations for non-Markovian multiple-time correlation functions of an open quantum system without using any approximation. We find that these equations conform an open hierarchy in which $N$-time correlation functions are dependent on $(N+1)$-time correlations. This hierarchy of equations is consistently obtained with two different methods: A first one based on Heisenberg equations of system operators, and a second one based on system propagators. The dependency on higher order correlations, and therefore the open hierarchy structure, only disappears in certain particular cases and when some hypothesis or approximations are considered in the equations. In this paper we consider a perturbative approximation and derive the general evolution equation for $N$-time correlations. This equation turns to depend only on $N$-time and lower order correlation functions, conforming a closed hierarchy structure that is useful for computational purposes.