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

Abstract

A number of physically intuitive results for the calculation of multi-time correlations in phasespace 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-and 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 in the Q and doubled phase-space Q representations, 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 observables in the positive-P, Wigner, and doubled-Wigner representations. Which mixed-order observables are amenable and which not is indicated, and explicit tallies are given up to 4th order. Overall then, 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.

On the survival of the quantum depletion of a condensate after release from a magnetic trap

“…Here the ket ψ B (x, t) and bra ψ B (x, t) amplitudes provide the positive-P representation of the Bogoliubov field in 3d space. We used the robust stochastic integration procedure described in [69]. The ξ(x, t) and ξ(x, t) are independent white Gaussian noise fields of zero mean and variance…”

Fully quantum scalable description of driven dissipative lattice models

“…Using this example, we can also show how to calculate multi-time correlations with positive-P. Any multitime correlation function that is normally-and timeordered can be calculated in the positive-P representation in a simple way, by averaging the corresponding product of phase space variables over the trajectories [72,73]. This follows by straightforward extension of the derivation found in Gardiner [73] for the Glauber-P representation.…”