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arXiv:1603.09555 · Physical Review A

Multitime correlation functions in nonclassical stochastic processes

F. Krumm, Jan Sperling, W. Vogel

Published Jun 27, 2016• 13 citations
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Abstract

A general method is introduced for verifying multitime quantum correlations through the characteristic function of the time-dependent $P$ functional that generalizes the Glauber-Sudarshan $P$ function. Quantum correlation criteria are derived which identify quantum effects for an arbitrary number of points in time. The Magnus expansion is used to visualize the impact of the required time ordering, which becomes crucial in situations when the interaction problem is explicitly time dependent. We show that the latter affects the multi-time-characteristic function and, therefore, the temporal evolution of the nonclassicality. As an example, we apply our technique to an optical parametric process with a frequency mismatch. The resulting two-time-characteristic function yields full insight into the two-time quantum correlation properties of such a system.

DOI
10.1103/physreva.93.063843

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A general method is introduced for verifying multitime quantum correlations through the characteristic function of the time-dependent PPP functional that generalizes the Glauber-Sudarshan PPP function. Quantum correlation criteria are derived which identify quantum effects for an arbitrary number of points in time. The Magnus expansion is used to visualize the impact of the required time ordering, which becomes crucial in situations when the interaction problem is explicitly time dependent. We show that the latter affects the multi-time-characteristic function and, therefore, the temporal evolution of the nonclassicality. As an example, we apply our technique to an optical parametric process with a frequency mismatch. The resulting two-time-characteristic function yields full insight into the two-time quantum correlation properties of such a system.
A general method is introduced for verifying multi-time quantum correlations through the characteristic function of the time-dependent P functional, that generalizes the Glauber-Sudarshan P function. Quantum-correlation criteria are derived which identify quantum effects for an arbitrary number of points in time. The Magnus expansion is used to visualize the impact of the required time-ordering. We show that the latter affects the multi-time characteristic function and, therefore, the temporal evolution of the nonclassicality. As an example, we apply our technique to an optical parametric process with a frequency mismatch. The resulting two-time characteristic function yields full insight into the two-time quantum correlation properties of such a system.
General quasiprobabilities are introduced to visualize time-dependent quantum correlations of light in phase space. They are based on the generalization of the Glauber-Sudarshan PPP function to a time-dependent PPP functional [W. Vogel, Phys. Rev. Lett. 100, 013605 (2008)], which fully describes temporal correlations of radiation fields on the basis of continuous phase-space distributions. This approach is nontrivial, as the PPP functional itself is highly singular for many quantum states and nonlinear processes. In general, it yields neither a well-behaved nor an experimentally accessible description of quantum stochastic processes. Our regularized version of this multitime-dependent quasiprobability is a smooth function and applies to stronger divergences compared to the single-time and multimode scenario. The technique is used to characterize an optical parametric process with frequency mismatch and a strongly nonlinear evolution of the quantized center-of-mass motion of a trapped ion. A measurement scheme, together with a sampling approach, is provided which yields direct experimental access to the regularized PPP functional from measured data.
We use the exact calculation of the quantum mechanical, temporal characteristic function $\ensuremath{\chi}(\ensuremath{\eta})$ and the degree of second-order coherence ${g}^{(2)}(\ensuremath{\tau})$ for a single-mode, degenerate parametric amplifier for a system in the Gaussian state, viz., a displaced-squeezed thermal state, to study the different criteria for nonclassicality. In particular, we contrast criteria that involve only one-time functions of the dynamical system, for instance, the quasiprobability distribution $P(\ensuremath{\beta})$ of the Glauber-Sudarshan coherent or P representation of the density of state and the Mandel ${Q}_{M}(\ensuremath{\tau})$ parameter, versus the criteria associated with the two-time correlation function ${g}^{(2)}(\ensuremath{\tau})$.
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.
Experiments performed on quantum systems often measure multitime correlation functions. When quantum systems are weakly coupled to their environment, the time evolution of such correlation functions can be reduced to that of the reduced density matrix by the quantum regression theorem (QRT). While no QRT is available for general non-Markovian open quantum systems, we show that for time-independent Hamiltonians and finite memory times τcτ_cτc​, an exact factorization rule exists that relates higher-order multitime correlations to products of lower-order correlations. Consequently, all information needed to reconstruct nnn-time correlations is contained in a temporal volume of O(τcn)\mathcal{O}(τ_c^n)O(τcn​). On the example of quantum dots coupled to phonons, we demonstrate that this factorization makes numerical calculations of multitime correlations extremely efficient and even enables semianalytical solutions in systems where the standard QRT breaks down.
We devise a method to certify nonclassical features via correlations of phase-space distributions by unifying the notions of quasiprobabilities and matrices of correlation functions. Our approach complements and extends recent results that were based on Chebyshev's integral inequality \cite{BA19}. The method developed here correlates arbitrary phase-space functions at arbitrary points in phase space, including multimode scenarios and higher-order correlations. Furthermore, our approach provides necessary and sufficient nonclassicality criteria, applies to phase-space functions beyond s -parametrized ones, and is accessible in experiments. To demonstrate the power of our technique, the quantum characteristics of discrete- and continuous-variable, single- and multimode, as well as pure and mixed states are certified only employing second-order correlations and Husimi functions, which always resemble a classical probability distribution. Moreover, nonlinear generalizations of our approach are studied. Therefore, a versatile and broadly applicable framework is devised to uncover quantum properties in terms of matrices of phase-space distributions.