Some Rules for Polydimensional Squeezing

TL;DR

This paper derives explicit photon distribution functions for multimode Gaussian states, explores new uncertainty relations, and provides detailed photon statistics for various complex quantum states, advancing understanding of polydimensional quantum light.

Contribution

It introduces explicit formulas for photon distributions in multimode Gaussian states and reveals new uncertainty relations based on photon state mixing.

Findings

Photon distribution functions expressed via Hermite polynomials

New sum rules for multivariable Hermite polynomials

Photon statistics for complex multimode quantum states

Abstract

The review of the following results of the Refs. \cite{Sem} - \cite{Ans} is presented: For mixed state light of $N$-mode electromagnetic field described by Wigner function which has generic Gaussian form the photon distribution function is obtained and expressed expliciltly in terms of Hermite polynomials of $2N$-variables.The momenta of this distribution are calculated and expressed as functions of matrix invariants of the dispersion matrix.The role of new uncertainty relation depending on photon state mixing parameter is elucidated.New sum rules for Hermite polynomials of several variables are found.The photon statistics of polymode even and odd coherent light and squeezed polymode Schr\"odinger cat light is qiven explicitly.Photon distribution for polymode squeezed number states expressed in terms of multivariable Hermite polynomials is discussed.