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arXiv:2010.06927 · Physical Review A

Nonclassicality and entanglement criteria for bipartite optical fields characterized by quadratic detectors. II. Criteria based on probabilities

Jan Peřina, Ondřej Haderka, Václav Michálek

Published Oct 20, 2020• 18 citations
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Abstract

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 $s$-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.

DOI
10.1103/physreva.102.043713

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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.
Numerous inequalities involving moments of integrated intensities and revealing nonclassicality and entanglement in bipartite optical fields are derived using the majorization theory, nonnegative polynomials, the matrix approach, and the Cauchy-Schwarz inequality. Different approaches for deriving these inequalities are compared. Using the experimental photocount histogram generated by a weak noisy twin beam monitored by a photon-number-resolving intensified CCD camera, the performance of the derived inequalities is compared. A basic set of 10 inequalities suitable for monitoring entanglement of a twin beam is suggested. Inequalities involving moments of photocounts (photon numbers) as well as some containing directly the elements of photocount (photon-number) distributions are also discussed as a tool for revealing nonclassicality.
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.
Gradual loss of the entanglement of a twin beam containing around 25 photon pairs with the increasing external noise is experimentally investigated. The entanglement is quantified by the nonclassicality depths and the nonclassicality counting parameters related to several nonclassicality criteria. The reduction of intensity moments of the analyzed multimode twin beams to single-mode ones allows determination of the negativity as another quantifier of the entanglement. Both the raw photocount histograms and the reconstructed photon-number distributions are analyzed in parallel.
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.
Spatially-resolved photon counting of a twin beam performed by an iCCD camera allows for versatile tailoring the properties of the beams formed by parts of the original twin beam. Dividing the idler beam of the twin beam into three equally-intense parts and post-selecting by detecting a given number of photocounts in the whole signal beam we arrive at the idler fields exhibiting high degrees of nonclassicality and being endowed with tripartite quantum correlations. Nonclassicality is analyzed with the help of suitable nonclassicality witnesses and their corresponding nonclassicality depths. Suitable parameters are introduced to quantify quantum correlations. These parameters are analyzed as they depend on the field intensity. The experimental photocount histograms are reconstructed by the maximum-likelihood approach and the obtained photon-number distributions are compared with a suitable model in which the original twin beam is approximated by an appropriate multi-mode Gaussian field and undergoes the corresponding beams' transformations.
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.