Exact dynamics of finite Glauber-Fock photonic lattices

TL;DR

This paper provides an exact analytical solution for the dynamics of finite Glauber-Fock photonic lattices, revealing oscillatory behaviors and state revivals based on the spectral properties derived from Hermite polynomials.

Contribution

It introduces an exact diagonalization method for finite Glauber-Fock lattices, linking the spectra to Hermite polynomial roots and analyzing resulting coherent phenomena.

Findings

Spectra roots are given by Hermite polynomial roots.

Oscillatory dynamics with localized spectra are predicted.

Partial state revivals occur at specific waveguides.

Abstract

The dynamics of Glauber-Fock lattice of size N is given through exact diagonalization of the corresponding Hamiltonian; the spectra $\{\lambda_{k}\}$ is given as the roots of the $N$-th Hermite polynomial, $H_{N}(\lambda_k/\sqrt{2})=0$, and the eigenstates are given in terms of Hermite polynomials evaluated at these roots. The exact dynamics is used to study coherent phenomena in discrete lattices. Due to the symmetry and spacing of the eigenvalues $\{\lambda_{k}\}$, oscillatory behavior with highly localized spectra, that is, near complete revivals of the photon number and partial recovery of the initial state at given waveguides, is predicted.