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

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.