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Physical Review A

Nonclassicality in phase by breaking classical bounds on statistics

Daniel Martín-Jiménez, Alfredo Luis

Published Sep 23, 2010• 4 citations
Publisher

Abstract

We derive upper bounds on the statistics of phase and phase difference that are satisfied by all classical states. They are obtained by finding the maximum projection of classical states on phase states. For a single-mode phase, meaningful bounds are obtained conditioned to a fixed mean number of photons. We also derive classical bounds for the projection on phase-coherent states, discussing their relation with phase-state bounds within the context of analytic representations. We find states with nonclassical phase properties disclosed by the violation of these classical bounds. These are quadrature and SU($2$) squeezed states and phase-coherent states.

DOI
10.1103/physreva.82.033829

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We derive upper bounds on the statistics of phase and phase difference that are satisfied by all classical states. They are obtained by finding the maximum projection of classical states on phase states. For a single-mode phase, meaningful bounds are obtained conditioned to a fixed mean number of photons. We also derive classical bounds for the projection on phase-coherent states, discussing their relation with phase-state bounds within the context of analytic representations. We find states with nonclassical phase properties disclosed by the violation of these classical bounds. These are quadrature and SU(222) squeezed states and phase-coherent states.
We derive exceedingly simple practical procedures revealing the quantum nature of states and measurements by the violation of classical upper bounds on the statistics of arbitrary measurements. Data analysis is minimum, and definite conclusions are obtained without evaluation of moments or any other more sophisticated procedures. These nonclassical tests are independent of other typical quantum signatures such as sub-Poissonian statistics, quadrature squeezing, or oscillatory statistics. This approach can be equally well applied to very diverse situations such as single- and two-mode fields, observables with continuous and discrete spectra, finite- and infinite-dimensional systems, and ideal and noisy measurements.
We derive a family of inequalities involving different phase-space distributions of a quantum state which have to be fulfilled by any classical state. The violation of these inequalities is a clear signature of nonclassicality. Our approach combines the characterization of nonclassical effects via negativities in phase-space distributions with inequality conditions usually being formulated for moments of physical observables. Importantly, the obtained criteria certify nonclassicality even when the involved phase-space distributions are non-negative. Moreover, we show how these inequalities are related to correlation measurements. The strength of the derived conditions is demonstrated by different examples, including squeezed states, lossy single-photon states, and even coherent states.
A new operator-based condition for distinguishing classical from nonclassical states of quantized radiation is developed. It exploits the fact that the normal ordering rule of correspondence to go from classical to quantum-dynamical variables does not in general maintain positivity. It is shown that the approach naturally leads to distinguishing several layers of increasing nonclassicality, with more layers as the number of modes increases. A generalization of the notion of sub-Poissonian statistics for two-mode radiation fields is achieved by analysing completely all correlations and fluctuations in quadratic combinations of mode annihilation and creation operators conserving the total photon number. This generalization is nontrivial and intrinsically two-mode as it goes beyond all possible single mode projections of the two-mode field. The nonclassicality of pair-coherent states, squeezed vacuum and squeezed thermal states is analysed and contrasted to one another, comparing the generalized sub-Poissonian statistics with extant signatures of nonclassical behaviour.
Coherent states are the quantum states, which give the closest description to classical states. Since their superpositions show quantum properties, research on these states has been of great interest. In addition, having nonclassical properties is necessary for quantum correlations. In this paper, we focus on two-mode entangled coherent states which are out of phase, and study the nonclassical properties such as squeezing of quadrature operators, antibunching and oscillatory photon statistics. Then we discuss about their entanglement, which is a quantum correlation in different conditions and compare the results.