Fate of Discrete Time Crystals in Finite-Size Cavity QED Systems

It has been shown that Discrete Time Crystal (DTC) behavior may arise in the periodically driven, open Dicke model in the thermodynamic limit ($N\to \mathrm{\infty }$). In this work, we demonstrate that finite $N$ non-mean-field fluctuations in this model can lead to decay rates and amplitudes of transient DTCs that display a non-trivial non-monotonic behavior with an increasing number of atoms. We show that the second-order cumulant expansion fails to capture the finite-size dynamics, while a stochastic version of the Truncated Wigner Approximation (TWA) is remarkably accurate, particularly in the large atom ensemble limit. Using the TWA and the Monte Carlo Wave Function (MCWF) approach, we identify three distinct dynamical regimes of transient DTC behavior in the finite-size Dicke model. We outline the mechanism behind the non-monotonic finite size dynamical decay of DTC order and discuss the implications for experiments performed on finite-sized atomic assemblies.