Multi-time correlations in the positive-P, Q, and doubled phase-space representations

TL;DR

This paper develops a comprehensive framework for calculating multi-time quantum correlations in phase-space representations, extending existing methods to various orderings and representations, and demonstrating their effectiveness through stochastic simulations.

Contribution

It introduces new derivative-free operator identities and conversion rules that enable non-perturbative, scalable calculations of multi-time observables across multiple phase-space representations.

Findings

Extended theory of multi-time observables in phase-space representations.

Validated methods through stochastic simulations of photon blockade and Bose-Hubbard systems.

Provided a simple algorithm for stochastic equation integration.

Abstract

A number of physically intuitive results for the calculation of multi-time correlations in phase-space representations of quantum mechanics are obtained. They relate time-dependent stochastic samples to multi-time observables, and rely on the presence of derivative-free operator identities. In particular, expressions for time-ordered normal-ordered observables in the positive-P distribution are derived which replace Heisenberg operators with the bare time-dependent stochastic variables, confirming extension of earlier such results for the Glauber-Sudarshan P. Analogous expressions are found for the anti-normal-ordered case of the doubled phase-space Q representation, along with conversion rules among doubled phase-space s-ordered representations. The latter are then shown to be readily exploited to further calculate anti-normal and mixed-ordered multi-time observables in the positive-P, Wigner, and doubled-Wigner representations. Which mixed-order observables are amenable and which are not is indicated, and explicit tallies are given up to 4th order. Overall, the theory of quantum multi-time observables in phase-space representations is extended, allowing non-perturbative treatment of many cases. The accuracy, usability, and scalability of the results to large systems is demonstrated using stochastic simulations of the unconventional photon blockade system and a related Bose-Hubbard chain. In addition, a robust but simple algorithm for integration of stochastic equations for phase-space samples is provided.