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Journal of Experimental and Theoretical Physics Letters

Hyper-Poisson Photon Statistics

Yu. I. Bogdanov, Nadezhda A. Bogdanova, Konstantin G. Katamadze, G. V. Avosopiants, Vladimir F. Lukichev

Published May 1, 2020• 2 citations
Publisher

Abstract

The statistical distribution for the number of bosons in a specified subsystem of a finite-dimensional multilevel system is obtained and investigated under conditions for which the total number of particles is a random variable characterized by the Poisson distribution. The resulting hyper-Poisson distribution is determined by the confluent hypergeometric function and contains as its particular limiting cases the Poisson distribution, as well as the single-mode and multimode thermal distributions. The developed model can be efficiently used to describe the thermal characteristics of bounded quantum systems, as well as in the problems of statistical reconstruction of optical quantum states, processes, and detectors.

DOI
10.1134/s0021364020100069

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The statistical distribution for the number of bosons in a specified subsystem of a finite-dimensional multilevel system is obtained and investigated under conditions for which the total number of particles is a random variable characterized by the Poisson distribution. The resulting hyper-Poisson distribution is determined by the confluent hypergeometric function and contains as its particular limiting cases the Poisson distribution, as well as the single-mode and multimode thermal distributions. The developed model can be efficiently used to describe the thermal characteristics of bounded quantum systems, as well as in the problems of statistical reconstruction of optical quantum states, processes, and detectors.
This paper describes the model of a compound Poisson distribution for photon statistics with regard to their bunching in Fock states, thermal states, and others. The method of generating functions is used to calculate the probability distributions, moments, and correlation functions. The parameters of conditional states arising from the subtraction of photons by splitting the beam are determined. The problem of state reconstruction with regard to quadrature quantum measurements is considered. The study is aimed at developing high-precision methods for generating and controlling optical quantum states.
Thermal states of light are widely used in quantum optics due to their correlation properties. As is well known, their correlation properties and the photon-number distribution as a whole are strongly dependent on the mode number selected by the detection scheme. The same changes can be caused by photon subtraction. Therefore, we describe the general case of the multimode thermal state after a multiphoton subtraction, when the photon-number statistics is registered by the detector selecting a part of the initial modes. We present an analytical form of the obtained photon-number distribution and its general properties and check them in the experiment.
We present a software package aimed at simulating photon-number probability distributions of a range of classical and non-classical states of light. This software can generate arbitrary probability distributions from user-defined mode structure of a light field. It can also solve the reverse problem, i.e., reconstructing the mode structure of a light field from a given probability distribution. The mode structure fully defines a light field. Thus it provides information about the source of light without having to directly access the source. The multimode fields simulated by this software include those comprised of a number of thermal modes, a Poisson mode, and single-photon modes. In addition, conjugated multimode sources (such as those created via parametric downconversion (PDC) or four-wave mixing (FWM)) can be simulated. This software provides for a nearly-perfect reconstruction of multimode fields comprised of several correlated and uncorrelated thermal and Poisson modes. The code is particularly effective at characterizing mesoscopic quantum states of light. This software can be modified to include other types of modes (i.e., governed by other statistical distributions), as needed.
In the field of continuous variables and particularly of Gaussian states an important role is played by the statistics of photon numbers. This issue has been considered in pioneering works for the one- and two-mode cases, but a systematic approach seems not to be available. In this paper, the most general multimode mixed Gaussian state, which is generated starting from a multimode thermal state and processed by the most general Gaussian unitary, is considered. In the NNN-mode the photon numbers are represented by NNN random variables, one for each mode, and the target is the evaluation of the means and of the covariances. The means describe statistically the amount of photon numbers in each mode, while the covariances give the correlations between the photon numbers in the different modes. For both means and covariances a closed-form result is obtained, expressed in simple and compact formulas. The main tools are provided by the representation of Gaussian unitaries (given by a cascade of an NNN-mode squeeze operator, an NNN-mode rotation operator, and a parallel set of NNN single-mode displacement operators) and by the corresponding Bogoliubov transformations.
In modern photon statistics, classical and quantum behavior can be distinguished by various quantum states of photon statistical distributions: Poisson (coherent/semi-classical wave behavior), and sub-Poisson (compressed state/particle behavior). Since this type of measurement mechanism is often associated with advanced laser/optical or photonic techniques, can this type of distribution model be modeled using discrete 0-1 sequences? In this paper, several sets of simulation modes are designed, and FFT transformation is used to extract relevant eigenvalues. Following the processing methods in the variant construction, special filters are constructed using the quantum random sequence provided by ANU (Australian national university), and conditional random sub-sequences are collected as input sequences. Multiple segments are separated from a random sequence, and relevant eigenvalues of FFT are selected to form a special set of eigenvalues. The shift operations are used to transform each sequence, showing obvious non-stationary random effects on various maps.
The fluctuations of the constituent populations in a multi-level system are considered as applied to a three and four level semiconductor model. It is shown that under certain conditions the fluctuations of the electron numbers in the conduction band can be super-poisson, defined as having a variance greater than the mean number. The general autocorrelation function for a three level system is obtained. For certain limits, this correlation function is the sum of two damped sinusoidal terms. This would indicate there is an oscillatory interaction between the population numbers. The photon distribution as a function of position within a one-dimensional active medium is derived. When two partially reflecting mirrors are situated at both ends of the medium, the stationary photon distribution obtained is a function of the cavity amplification and the mirror reflectivities. The distribution of photo-electrons emitted from a detector of area A and resolving time T when illuminated by an incident light beam is derived. By using a binomial rather tha a deterministic quantum efficiency an additional term is obtained in the autocorrelation function. The resulting spectral density of the photo-current fluctuations is shown to be the sum of a Poisson particle noise and wave interference term. Several examples of different spectral line shapes are discussed. Also considered is an intensity modulated light beam.