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Physical Review A

Nonclassicality criteria in terms of moments

Evgeny V Shchukin, Th. Richter, W. Vogel

Published Jan 26, 2005• 102 citations
Publisher

Abstract

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 $P$ function to be a probability density. A possibility to measure the required moments is proposed.

DOI
10.1103/physreva.71.011802

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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.
Necessary and sufficient conditions are derived that allow one to completely characterize the nonclassicality of an arbitrary quantum state of a harmonic oscillator. Starting from a theorem by Bochner we derive an infinite hierarchy of conditions for the characteristic functions of the quantum state. Following the hierarchy step by step, the nonclassicality can be classified with respect to increasing orders. The full hierarchy of conditions is completely equivalent to the most commonly accepted criterion for nonclassicality: the failure of the Glauber–Sudarshan P -function to be a probability density. Our reformulated version of the criterion is directly related to measurable quantities that can be observed in phase-sensitive measurements such as homodyne detection. It directly generalizes a criterion for nonclassicality (Vogel 2000 Phys. Rev. Lett. 84 1849) that has already been applied in experiments (Lvovsky and Shapiro 2002 Phys. Rev. A 65 033830). Examples of phase-insensitive quantum states are considered that exhibit no first-order nonclassical effects, but that are nonclassical in second order.
The characterization of nonclassicality of quantum states in terms of measurable quadrature distributions is studied, on the basis of recently derived necessary and sufficient conditions for the failure of the Glauber–Sudarshan P -function to be a probability density. When the noise-subtracted quadrature distribution, that is the marginal of the P -function, is not a probability density, the same is true for the Glauber–Sudarshan P -function. Inversely, for a complete characterization of a nonclassical P -function one needs phase-sensitive criteria that connect quadrature distributions for different phases. Even for phase-insensitive quantum states, phase-sensitive nonclassicality conditions are needed, as is illustrated for one-photon added thermal states.
Practically applicable criteria for the nonclassicality of quantum states are formulated in terms of different types of moments. For this purpose the moments of the creation and annihilation operators, of two quadratures, and of a quadrature and the photon number operator turn out to be useful. It is shown that all the required moments can be determined by homodyne correlation measurements. An example of a nonclassical effect that is easily characterized by our methods is amplitude-squared squeezing.
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.
A method is introduced for the verification of nonclassicality in terms of moments of nonclassicality quasiprobability distributions. The latter are easily obtained from experimental data and will be denoted as nonclassicality moments. Their relation to normally ordered moments is derived, which enables us to verify nonclassicality by using well established criteria. Alternatively, nonclassicality criteria are directly formulated in terms of nonclassicality moments. The latter converge in proper limits to the usually used criteria, as is illustrated for squeezing and sub-Poissonian photon statistics. Our theory also yields expectation values of any observable in terms of nonclassicality moments.