Title: Discrete Time-Crystalline Order in Cavity and Circuit QED Systems,
Abstract: Discrete time crystals are a recently proposed and experimentally observed
out-of-equilibrium dynamical phase of Floquet systems, where the stroboscopic
evolution of a local observable repeats itself at an integer multiple of the
driving period. We address this issue in a driven-dissipative setup, focusing
on the modulated open Dicke model, which can be implemented by cavity or
circuit QED systems. In the thermodynamic limit, we employ semiclassical
approaches and find rich dynamical phases on top of the discrete
time-crystalline order. In a deep quantum regime with few qubits, we find clear
signatures of a transient discrete time-crystalline behavior, which is absent
in the isolated counterpart. We establish a phenomenology of dissipative
discrete time crystals by generalizing the Landau theory of phase transitions
to Floquet open systems.