arXiv:2011.10107 · Quantum

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

Piotr Deuar

Published May 10, 2021• 9 citations

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.

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

Related papers

Quantum-field-theoretical approach to phase-space techniques: Generalizing the positive-Prepresentation

Lev I. Plimak, Michael Fleischhauer, M. K. Olsen +1 more

We present an introduction to phase-space techniques (PST) based on a quantum-field-theoretical (QFT) approach. In addition to bridging the gap between PST and QFT, our approach results in a number of generalizations of the PST. First, for problems where the usual PST do not result in a genuine Fokker-Planck equation (even after phase-space doubling) and hence fail to produce a stochastic differential equation (SDE), we show how the system in question may be approximated via stochastic difference equations $(\mathrm{S}\ensuremath{\Delta}\mathrm{E}).$ Second, we show that introducing sources into the SDE's (or $\mathrm{S}\ensuremath{\Delta}\mathrm{E}\mathrm{'}\mathrm{s})$ generalizes them to a full quantum nonlinear stochastic response problem (thus generalizing Kubo's linear reaction theory to a quantum nonlinear stochastic response theory). Third, we establish general relations linking quantum response properties of the system in question to averages of operator products ordered in a way different from time normal. This extends PST to a much wider assemblage of operator products than are usually considered in phase-space approaches. In all cases, our approach yields a very simple and straightforward way of deriving stochastic equations in phase space.

Physical Review AJan 29, 200323 citations

Phase-space distributions

P. D. Drummond, Mark S. Hillery

In this chapter, we will develop methods for mapping operator equations to equivalent c-number equations. This results in a continuous phase-space representation of a many-body quantum system, using phase-space distributions instead of density matrices. In order to do this, we will first find c-number representations of operators. We have already seen one such representation, the Glauber–Sudarshan P -representation for the density matrix. We now introduce several more such representations. The main focus will be on the truncated Wigner representation, valid at large photon number, and the positive P -representation, which uses a double-dimensional phase space and exists as a positive probability for all quantum density matrices. Phase-space techniques have a great advantage over conventional matrix-type solutions to the Schrödinger equation, in that they do not have an exponential growth in complexity with mode and particle number. Instead, the equations that describe the dynamics of these c-number representations of the density matrix are Fokker–Planck equations. These have equivalent stochastic differential equations, which behave as c-number analogs of Heisenberg-picture equations of motion. This means that problems that would be essentially impossible to solve using conventional number-state representations can be transformed into readily soluble differential equations. In many cases, no additional approximations, such as perturbation theory or factorization assumptions, are needed.

Cambridge University Press eBooksMar 27, 20140 citations

Stochastic wave-function approach to the calculation of multitime correlation functions of open quantum systems

Heinz‐Peter Breuer, Bernd Kappler, Francesco Petruccione

Within the framework of probability distributions on projective Hilbert space, a scheme for the calculation of multitime correlation functions is developed. The starting point is the Markovian stochastic wave-function description of an open quantum system coupled to an environment consisting of an ensemble of harmonic oscillators in arbitrary pure or mixed states. It is shown that matrix elements of reduced Heisenberg picture operators and general time-ordered correlation functions can be expressed by time-symmetric expectation values of extended operators in a doubled Hilbert space. This representation allows the construction of a stochastic process in the doubled Hilbert space which enables a determination of arbitrary matrix elements and correlation functions. The numerical efficiency of the resulting stochastic simulation algorithm is investigated and compared with an alternative Monte Carlo wave function method proposed first by Dalibard, Castin, and M\o{}lmer [Phys. Rev. Lett. 68, 580 (1992)]. By means of a standard example the suggested algorithm is shown to be more efficient numerically and to converge faster. Finally, some specific examples from quantum optics are presented in order to illustrate the proposed method, such as the coupling of a system to a vacuum, a squeezed vacuum within a finite solid angle, and a thermal mixture of coherent states.

arXiv:quant-ph/9807079Physical Review ASep 1, 199733 citations

Phase Space Distributions

Bryan J. Dalton, John Jeffers, Stephen Mark Barnett

Abstract This chapter formally presents the representation of the quantum density operator by phase space distribution functions of phase space variables, leading to expressions for quantum correlation functions in terms of phase space integrals in which phase space variables for each mode—c-numbers for bosons, Grassmann numbers for fermions—replace mode annihilation and creation operators. The emphasis is on double-phase-space normally ordered (positive P type) normalised distribution functions, with distinct phase space variables for annihilation and creation operators; however, symmetrically ordered (Wigner type) distribution functions are also considered, along with unnormalised B distribution functions, which lead to phase space integrals for Fock state populations and coherences. Characteristic functions are first defined and then shown to be related to distribution functions via phase space integrals. The existence and symmetry properties of distribution functions—which are non-unique and non-analytic for bosons—is demonstrated using Bargmann coherent-state projectors.

Nov 13, 20141 citation

Higher-order stochastic differential equations and the positive Wigner function

P. D. Drummond

General higher-order stochastic processes that correspond to any diffusion-type tensor of higher than second order are obtained. The relationship of multivariate higher-order stochastic differential equations with tensor decomposition theory and tensor rank is explained. Techniques for generating the requisite complex higher-order noise are proved to exist either using polar coordinates and $\ensuremath{\gamma}$ distributions, or from products of Gaussian variates. This method is shown to allow the calculation of the dynamics of the Wigner function, after it is extended to a complex phase space. The results are illustrated physically through dynamical calculations of the positive Wigner distribution for three-mode parametric downconversion, widely used in quantum optics. The approach eliminates paradoxes arising from truncation of the higher derivative terms in Wigner function time evolution. Anomalous results of negative populations and vacuum scattering found in truncated Wigner quantum simulations in quantum optics and Bose-Einstein condensate dynamics are shown not to occur with this type of stochastic theory.

Physical Review ADec 4, 20177 citations

Quantum-classical correspondences for atomic operators: A doubled atomic-space approach

Li You, J. Cooper

An operational method is presented that can avoid some of the difficulties associated with numerically simulating the Langevin operator equations for a dissipative quantum optical system. The approach is based on a set of quantum-classical correspondence rules that relate atomic operators in the Heisenberg picture to their corresponding double-dimensioned classical atomic variables. With this approach, the Langevin operator equations can be transformed into a doubled set of classical stochastic differential equations (SDE's). The relationship to the quantum regression theory up to the two-time correlation function and similarities to the positive P representation are discussed. As an illustrative example, we show that both the resonance fluorescence spectrum (Mollow spectrum) as well as the absorption spectrum for a two-level atom interacting with a near-resonance field can be simulated from the doubled set of classical SDE's without explicit reference to the quantum regression theorem.

Physical Review ADec 1, 19940 citations