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

Piotr Deuar

Institute of Physics, Polish Academy of Sciences, Aleja Lotników 32/46, 02-668 Warsaw, Poland

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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.

Multi-time correlations are important for answering many physical questions. For example, the determination of lifetimes out-of-time-order correlations which are important indicators of quantum chaos, or finding the time resolution required to observe a transient effect. In general, however, they are more difficult to calculate in a quantum system than instantaneous correlations, and the difficulty grows with system size. Phase-space representations are a formulation of quantum mechanics in which the calculation of multi-time correlations has a particularly intuitive structure, and in which the difficulties of dealing with large systems are often alleviated.
In this work, the framework for calculating multi-time correlations with phase-space representations has been strongly extended to a much wider range of correlations and representations than before, facilitating future studies of large systems, including systems with dissipation.
The paper also describes a robust but simple algorithm for integration of phase space stochastic equations, something that has been difficult to find in the literature to date.

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Cited by

[1] J. A. Ross, P. Deuar, D. K. Shin, K. F. Thomas, B. M. Henson, S. S. Hodgman, and A. G. Truscott, "Survival of the quantum depletion of a condensate after release from a harmonic trap in theory and experiment", arXiv:2103.15283.

[2] Piotr Deuar, Alex Ferrier, Michał Matuszewski, Giuliano Orso, and Marzena H. Szymańska, "Fully quantum scalable description of driven dissipative lattice models", arXiv:2012.02014.

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