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

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