Exact dynamics of finite Glauber-Fock photonic lattices

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.