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Optics Express

Non-classicality of optical fields as observed in photocount and photon-number distributions

Jan Peřina, Václav Michálek, Ondřej Haderka

Published Oct 6, 2020• 9 citations
Publisher

Abstract

Non-classicality criteria for optical fields based on the probabilities of photocount and photon-number distributions are derived. Relations among the criteria obtained by the applied methods are revealed. Redundant criteria are identified. The performance of the fundamental criteria is tested on a set of potentially sub-Poissonian fields generated by photon-number-resolved post-selection from a mesoscopic twin beam. The corresponding non-classicality depths are determined to quantitatively compare the used criteria.

DOI
10.1364/oe.405548

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Non-classicality criteria for optical fields based on the probabilities of photocount and photon-number distributions are derived. Relations among the criteria obtained by the applied methods are revealed. Redundant criteria are identified. The performance of the fundamental criteria is tested on a set of potentially sub-Poissonian fields generated by photon-number-resolved post-selection from a mesoscopic twin beam. The corresponding non-classicality depths are determined to quantitatively compare the used criteria.
Criteria defining higher-order sub-Poissonian-like fields are given using five different quantities: moments of (I) integrated intensity, (II) photon number, (III) integrated-intensity fluctuation, (IV) photon-number fluctuation, and (V) elements of photocount and photon-number distributions. Relations among the moment criteria are revealed. Performance of the criteria is experimentally investigated using a set of potentially sub-Poissonian fields obtained by post-selection from a twin beam. The criteria based on moments of integrated intensity and photon number and those using the elements of photocount distribution are found as the most powerful. States nonclassical up to the fifth order are experimentally reached in the former case, even the ninth-order nonclassicality is observed in the latter case.
Numerous nonclassicality criteria based on the probabilities of experimental photocount or theoretical photon-number distributions are derived using several approaches. Relations among the derived criteria are revealed and the fundamental criteria are identified. They are grouped into parametric systems that allow the analysis of the nonclassicality from different points of view (``local'' nonclassicality, pairwise character of photon correlations, etc.). Considering their structure, the criteria may be divided into groups that differ in the power to resolve the nonclassicality. Quantification of the nonclassicality using the Lee nonclassicality depth and the nonclassicality counting parameter is discussed. The number of field modes used is identified as an important parameter that may cause unexpected results. An appropriate linear transformation of a photocount (photon-number) distribution into its sss-ordered form, needed for the determination of the nonclassicality depth, is derived. For comparison, the derived criteria are applied both to an experimental photocount histogram of a twin beam and the reconstructed photon-number distributions with different levels of the noise.
Using a sub-Poissonian optical field generated from a weak twin beam by photon-number resolving post-selection we have simultaneously observed higher-order non-classicalities in photocount moments (sub-Poissonian statistics) and probabilities (witnessed by the Klyshko inequalities). Up to the seventh-order non-classicalities in photocount moments simultaneously with up to the eleventh-order non-classicalities in photocount probabilities have been experimentally observed. Non-classicality counting parameters of different orders as experimental counterparts of the theoretical Lee non-classicality depth have been suggested to quantify and also mutually compare the robustness of these non-classicalities against the noise.
Two twin beams with a shared signal beam and separated idler beams are used together with the photon-number-resolving postselection in the signal beam to arrive at two coupled beams with anticorrelations in photon-number fluctuations. Moreover, the beams exhibit the sub-Poissonian photon-number statistics in their marginal distributions under suitable conditions. The postselected fields with the increasing mean photon numbers are reconstructed from the experimental photocount histograms by the maximum likelihood approach. Also a suitable Gaussian fit of both original twin beams and simulation of the postselection process are applied to arrive at the corresponding photon-number distributions. Their nonclassical properties are analyzed by suitable nonclassicality criteria and quantified by the corresponding nonclassicality depths. Determining the appropriate quasidistributions of integrated intensities with negative values, the performance of different non-classicality criteria is judged. Properties of the postselected fields reached both by the used and ideal photon-number-resolved detectors are mutually compared.
Nonclassicality criteria for general NNN-dimensional optical fields are derived. They involve intensity moments, the probabilities of photon-number distributions, or combinations of both. The Hillery criteria for the sums of the probabilities of even or odd photon numbers are generalized to NNN-dimensional fields. As an example, the derived nonclassicality criteria are applied to an experimental three-mode optical field containing two types of photon-pair contributions. The accompanying nonclassicality depths are used to mutually compare their performance.
Different non-classicality criteria expressed in the form of inequalities among intensity moments and elements of photon-number distributions are applied to noisy twin beams and other two-mode states obtained from a twin beam by using a beam splitter. Their performance in revealing the non-classicality is judged in comparison with the exact results provided by suitable entanglement and local non-classicality quantifiers. Whereas the non-classicality of noisy twin beams is always revealed by these criteria, not all the nonclassical states obtained at the output of the beam splitter can be identified by these experimentally easily reachable criteria.