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

Piotr Deuar

doi:10.22331/q-2021-05-10-455

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.

Highlights

Phase-space representations of quantum mechanics such as the Wigner, P, positive-P, Q and related approaches are a powerful tool for the study and understanding of quantum mechanics [14, 50, 62, 93, 118]
time reversal schemes like those used in experiment or theory could be attempted by changing the signs of constants
the prime advantage of the phase-space approach is that its computational cost typically grows

Summary

Introduction

Phase-space representations of quantum mechanics such as the Wigner, P, positive-P, Q and related approaches are a powerful tool for the study and understanding of quantum mechanics [14, 50, 62, 93, 118]. The determination of lifetimes in an equilibrium or stationary state, response functions, out-oftime-order correlations (OTOCs), or finding the time resolution required to observe a transient phenomenon. Their calculation in the phase-space framework has been restricted because few general results have been available. The Glauber-Sudarshan P representation results provide a intuitive framework, replacing Heisenberg operators a†(t) and a(t) with timedependent phase-space variables in normal-ordered observables such as a†(t1)a(t2). They apply to open systems [2, 62].

Methods

Results

Conclusion