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Nonclassical Stochastic Processes

F. Krumm, Jan Sperling, W. Vogel

Published Mar 31, 2016• 0 citations
OpenAlex

Abstract

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.

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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.
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.
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 ${g}^{(2)}(\ensuremath{\tau})$.
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.
In Quantum Optics, the widely accepted definition of nonclassicality is based on the P function of Glauber and Sudarshan. When it fails to be interpreted as a classical probability density, the corresponding quantum state is said to be a nonclassical one. Here we present the first reconstruction of a nonclassical P function of a single-photon added thermal state. We also consider the nonclassical properties of general spacegtime dependent correlation functions of radiation fields. For the detection of these correlation functions, a balanced homodyne correlation technique was proposed. It is shown that the measurable correlation functions also allow one to completely characterize bipartite entangled quantum states with a negative partial transposition. Finally, we present a method for identifying general bipartite entanglement for continuous variables.
Nonclassicality filters provide a universal method to visualize the nonclassicality of arbitrary quantum states of light through negativities of a regularized Glauber-Sudarshan PPP function, also denoted as nonclassicality quasiprobability. Such filters are introduced and analyzed for optimizing the experimental certification of nonclassical effects. An analytic filter is constructed which preserves the full information on the quantum state. For balanced homodyne detection, the number of data points is analyzed to get the negativities of the nonclassicality quasiprobability with high statistical significance. The method is applied to different scenarios, such as phase-randomized squeezed vacuum states, single-photon-added thermal states, and heralded state engineering with array detectors. The generalization to visualize quantum correlations of multimode radiation fields is also considered.