Two-time second-order correlation function

TL;DR

This paper derives the two-time second-order correlation function using multiple approaches, providing a foundation for calculating quantum correlations over time, which is useful for understanding quantum systems.

Contribution

It introduces a comprehensive derivation of the two-time second-order correlation function using stochastic, propagator, and distribution function methods, facilitating future quantum correlation analyses.

Findings

Derivation methods are consistent and straightforward despite complex integrations.

Gaussian nature of c-number functions simplifies calculations.

Framework enables analysis of quantum correlations at different times.

Abstract

Derivation of two-time second-order correlation function by following approaches such as stochastic differential equation, coherent-state propagator, and quasi-statistical distribution function is presented. In the process, the time dependence of the operators is transferred to the density operator by making use of trace operation in which the coherent state propagator and $Q$-function that represent the quantum system under consideration are expressed in terms of different time parameters. Even though the number of resulting integrations are found to be large, the accompanying implementation turns out to be straightforward in view that the associated $c$-number functions are Gaussian by nature. In relation to the asserted possibility of rewriting the result of one of the approaches in terms of the other, the presented derivation is expected to lay a strong foundation for viable technique of calculating correlations of various moments at different times that can be deployed in revealing quantum correlations.