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arXiv:0812.3015 · Physical Review A

Experimental test of nonclassicality criteria for phase-diffused squeezed states

T. Kiesel, W. Vogel, Boris Hage, James DiGuglielmo, Aiko Samblowski, Roman Schnabel

Published Feb 27, 2009• 34 citations
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Abstract

We experimentally examine the nonclassical character of a class of non-Gaussian states known as phase-diffused squeezed states. These states may show no squeezing effect at all and therefore provide an interesting example to test nonclassicality criteria. The characteristic function of the Glauber-Sudarshan representation ($P$ function) proves to be a powerful tool to detect nonclassicality. Using this criterion we find that phase-diffused squeezed states are always nonclassical, even if the squeezing effect vanishes. Testing other criteria of nonclassicality based on higher-order squeezing and the positive semidefiniteness of special matrices of normally ordered moments, it is found that these criteria fail to reveal the nonclassicality for some of the prepared phase-diffused squeezed states.

DOI
10.1103/physreva.79.022122

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We experimentally examine the nonclassical character of a class of non-Gaussian states known as phase-diffused squeezed states. These states may show no squeezing effect at all and therefore provide an interesting example to test nonclassicality criteria. The characteristic function of the Glauber-Sudarshan representation (PPP function) proves to be a powerful tool to detect nonclassicality. Using this criterion we find that phase-diffused squeezed states are always nonclassical, even if the squeezing effect vanishes. Testing other criteria of nonclassicality based on higher-order squeezing and the positive semidefiniteness of special matrices of normally ordered moments, it is found that these criteria fail to reveal the nonclassicality for some of the prepared phase-diffused squeezed states.
In this paper, we use the characteristic function, i.e., the Fourier transform of the Glauber-Sudarshan phase-space distribution, to determine the degree of nonclassicality of a given state. This degree of nonclassicality quantifies the nonclassicality in terms of quantum superpositions. We demonstrate two ways to exactly find or to lower-bound the degree of nonclassicality by studying the properties of the characteristic functions. The developed criteria are applied to two examples of squeezed states undergoing a classical mixing or a nonclassical superposition with vacuum.
In this work we generalize the Bochner criterion addressing the characteristic function, i.e., the Fourier transform, of the Glauber-Sudarshan phase-space function. For this purpose we extend the Bochner theorem by including derivatives of the characteristic function. The resulting necessary and sufficient nonclassicality criteria unify previously known moment-based criteria with those based on the characteristic function. For applications of the generalized nonclassicality probes, we provide direct sampling formulas for balanced homodyne detection. A squeezed vacuum state is experimentally realized and characterized with our method. This complete framework---theoretical unification, sampling approach, and experimental implementation---presents an efficient toolbox to characterize quantum states of light for applications in quantum technology.
Criteria for the nonclassicality of a quantum state of the harmonic oscillator are formulated in terms of normally ordered moments of quadratures. Inequalities are obtained that are equivalent to the failure of the Glauber-Sudarshan PPP function to be a probability density. A possibility to measure the required moments is proposed.
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 theoretically propose and experimentally demonstrate a nonclassicality test of a single-mode field in phase space, which has an analogy with the nonlocality test proposed by Banaszek and Wódkiewicz [Phys. Rev. Lett. 82, 2009 (1999)]. Our approach to deriving the classical bound draws on the fact that the Wigner function of a coherent state is a product of two independent distributions as if the orthogonal quadratures (position and momentum) in phase space behave as local realistic variables. Our method detects every pure nonclassical Gaussian state, which can also be extended to mixed states. Furthermore, it sets a bound for all Gaussian states and their mixtures, thereby providing a criterion to detect a genuine quantum non-Gaussian state. Remarkably, our phase-space approach with invariance under Gaussian unitary operations leads to an optimized test for a given non-Gaussian state. We experimentally show how this enhanced method can manifest quantum non-Gaussianity of a state by simply choosing phase-space points appropriately, which is essentially equivalent to implementing a squeezing operation on a given state.
While negativity in phase space is a well-known signature of nonclassicality, a wide variety of nonclassical states require their characterization beyond negativity. We establish a framework of nonclassicality in phase space that addresses nonclassical states comprehensively with a direct experimental evidence. This includes the negativity of phase-space distribution as a special case and further analyzes quantum states with positive distributions effectively. We prove that it detects all nonclassical Gaussian states and all non-Gaussian states of arbitrary dimension remarkably by examining three phase-space points only. Our formalism also provides an experimentally accessible lower bound for a nonclassicality measure based on trace distance. Importantly, this foundational approach can be further adapted to constitute practical tests in two directions looking into particle and wave nature of bosonic systems, via an array of nonideal on-off detectors and coarse-grained homodyne measurement, respectively. All these tests are practically powerful in characterizing nonclasssical states reliably against noise, making a versatile tool for a broad range of quantum systems in quantum technologies.