Using a model Hamiltonian for a single-mode electromagnetic field interacting with a nonlinear medium, we show that quantum expectation values of subsystem observables can exhibit remarkably diverse ergodic properties even when the dynamics of the total system is regular. The time series of the mean photon number is studied over a range of values of the ratio of the strength $gamma$ of the nonlinearity to that of the inter-mode coupling $g$. We obtain the power spectrum, estimate the embedding dimension of the reconstructed phase space and the maximal Liapunov exponent $lambda_{rm max}$, and find the recurrence-time distribution of the coarse-grained dynamics. The dynamical behavior ranges from quasiperiodicity (for $gamma/g ll 1$) to chaos as characterized by $lambda_{rm max} > 0$ (for $gamma/g gtrsim 1$), and is interpreted.
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