Dynamics of the Mean Photon Number: Time Series and Network Analysis

We present a self-contained survey of a relatively nascent approach towards understanding the dynamics of quantum observables such as the mean photon number in models of atom-light interaction. The procedure involves treating these observables as dynamical variables in an effective phase space, and applying tools from classical dynamical systems theory such as time series and network analysis to long data sets of these observables. Full revivals of the state of the system, when all observables return to their initial values, correspond to periodic orbits in the phase space. However, since the field subsystem undergoes dissipative evolution, it displays interesting ergodicity properties depending on the initial state, its degree of coherence, and the interplay between the couplings and nonlinearities in the model. The Lyapunov exponents, recurrence time distributions to cells in the coarse-grained phase space, network quantifiers, etc. provide insights into the effects of a finite versus infinite number of atomic levels, intensity-dependent couplings, and bipartite versus tripartite systems. This study also provides readily recognizable departures in the quantum case from known results on Poincaré recurrences in classical dynamical systems. This raises several new questions and opens up interesting avenues for the investigation of the quantum-classical divide.