Bifurcations, time-series analysis of observables, and network properties in a tripartite quantum system

In a tripartite system comprising a $\Lambda$-atom interacting with two radiation fields in the presence of field nonlinearities and an intensity-dependent field-atom coupling, striking features have been shown to occur in the dynamics of the mean photon number $\aver{N_{i}(t)}$ ($i = 1,\,2$) corresponding to either field. In this Letter, we carry out a detailed time-series analysis and establish an interesting correlation between the short-time and long-time dynamics of $\aver{N_{i}(t)}$. Lyapunov exponents, return maps, recurrence plots, recurrence-time statistics, as well as the clustering coefficient and the transitivity of networks constructed from the time series, are studied as functions of the intensity parameter $\kappa$. These are shown to carry signatures of a special value $\kappa = \bar{\kappa}$. Our work also exhibits how techniques from nonlinear dynamics help analyze the behavior of observables in multipartite quantum systems.