Classical and quantum regimes of the inhomogeneous Dicke model and its Ehrenfest time

Abstract

We show that in the few-excitation regime, the classical and quantum time evolution of the inhomogeneous Dicke model for $N$ two-level systems coupled to a single boson mode agree for $N⪢1$. In the presence of a single excitation only, the leading term in an $1/N$ expansion of the classical equations of motion reproduces the result of the Schr\"odinger equation. For a small number of excitations, the numerical solutions of the classical and quantum problems become equal for $N$ sufficiently large. By solving the Schr\"odinger equation exactly for two excitations and a particular inhomogeneity, we obtain $1/N$ corrections which lead to a significant difference between the classical and quantum solutions at a new time scale which we identify as an Ehrenferst time, given by ${\ensuremath{\tau}}_{E}=\sqrt{N/⟨{g}^{2}⟩}$, where $\sqrt{⟨{g}^{2}⟩}$ is an effective coupling strength between the two-level systems and the boson.