Quantum statistical properties of multiphoton hypergeometric coherent states and the discrete circle representation

TL;DR

This paper explores the mathematical and statistical properties of hypergeometric coherent states and their Schr"odinger cat states, introducing a circle representation and analyzing their quantum statistical features in various regimes.

Contribution

It introduces a discrete circle representation for hypergeometric coherent states and extends the analysis to finite-dimensional spaces, providing new insights into their quantum properties.

Findings

Hypergeometric Schr"odinger cat states can be expressed as superpositions of hypergeometric coherent states.

A continuous circle representation is developed for high $k$, linking number and coherent states.

The generalized Husimi $Q$-function reveals super- and sub-Poissonian statistics at different revival times.

Abstract

We review the definition of hypergeometric coherent states, discussing some representative examples. Then we study mathematical and statistical properties of hypergeometric Schr\"odinger cat states, defined as orthonormalized eigenstates of $k$-th powers of nonlinear $f$-oscillator annihilation operators, with $f$ of hypergeometric type. These "$k$-hypercats" can be written as an equally weighted superposition of hypergeometric coherent states $|z_l\rangle, l=0,1,\dots,k-1$, with $z_l=z e^{2\pi i l/k}$ a $k$-th root of $z^k$, and they interpolate between number and coherent states. This fact motivates a continuous circle representation for high $k$. We also extend our study to truncated hypergeometric functions (finite dimensional Hilbert spaces) and a discrete exact circle representation is provided. We also show how to generate $k$-hypercats by amplitude dispersion in a Kerr medium and analyze their generalized Husimi $Q$-function in the super- and sub-Poissonian cases at different fractions of the revival time.