Quantum Regression theorem for multi-time correlators : A detailed analysis in the Heisenberg Picture

TL;DR

This paper extends the quantum regression theorem to multi-time and out-of-time-ordered correlators using the Heisenberg picture, including non-Markovian dynamics, with applications to the spin-boson model.

Contribution

It introduces a Heisenberg picture derivation of the quantum regression theorem for multi-time and out-of-time-ordered correlators, including non-Markovian effects.

Findings

Regression theorem for multi-time correlators resembles the two-time case with a time ordering restriction.

Derived analogue of regression theorem for out-of-time-ordered correlators.

Extended the theorem to non-Markovian dynamics with specific modifications.

Abstract

Quantum regression theorem is a very useful result in open quantum system and extensively used for computing multi-point correlation functions. Traditionally it is derived for two-time correlators in the Markovian limit employing the Schr\"odinger picture. In this paper we make use of the Heisenberg picture to derive quantum regression theorems for multi-time correlation functions which in the special limit reduce to the well known two-time regression theorem. For multi-time correlation function we find that the regression theorem takes the same form as it takes for two-time correlation function with a mild restriction that one of the times should be greater than all the other time variables. Interestingly, the Heisenberg picture also allows us to derive analogue of regression theorem for out-of-time-ordered correlators (OTOCs). We further extend our study for the case of non-Markovian dynamics and report the modifications to the standard quantum regression theorem. We illustrate all of the above results using the paradigmatic dissipative spin-boson model.