Multidimensional Hermite polynomials and photon distribution for polymode mixed light

TL;DR

This paper derives explicit formulas for photon distribution functions in N-mode Gaussian and squeezed states using multidimensional Hermite polynomials, providing new analytical tools for quantum optics analysis.

Contribution

It introduces explicit expressions for photon distributions in N-mode mixed states using Hermite polynomials, extending previous methods to multidimensional cases.

Findings

Explicit photon distribution formulas for Gaussian states

Photon number mean and dispersion calculations

Representation of distributions for squeezed states

Abstract

For N-mode light described by the Wigner function of generic Gaussian form the photon distribution function is obtained explicitly and expressed in terms of Hermite polynomial of $2N$ variables with equal pairs of indices.The mean values and dispersions of photon numbers are obtained for this generic mixed state.Generating function for photon distribution is obtained explicitly. The expression for $N$-mode photon distribution function for squeezed photon number states in terms of Hermite polynomials of $2N$ variables and for squeezed coherent states in terms of Hermite polynomials of $N$ variables is discussed.