Analytically solvable Hamiltonian in invariant subspaces

Abstract

We study a generic time-dependent Jaynes–Cummings model, discovering universal features of its time evolution on a dynamical time scale which is here defined as an integral over time of the field-atom coupling strength modulus. Using the total excitation number as a constant of motion and the symmetry properties of the Hamiltonian, we decompose the infinite-dimensional Hilbert space of the system in a one-dimensional and an infinite number of two-dimensional dynamically invariant subspaces, in which the dynamical problem is solved using the Messina–Nakazato parameterization method. Therefore, we obtain a solution of the dynamics over the full Hilbert space. Its application to the evolution of the atomic population inversion over the dynamical time, under different initial conditions, highlights Rabi-like oscillations and revivals, which do not depend on the functional dependence of the Hamiltonian parameters on time and can be controlled by a crucial parameter in the solution. The universal dynamical features brought to light in this study assume particular significance in this quantum computing era, where general quantum simulators are of great use.