Dicke time crystals in driven-dissipative quantum many-body systems

The Dicke model { a paradigmatic example of superradiance in quantum optics { describes an<br> ensemble of atoms which are collectively coupled to a leaky cavity mode. As a result of the cooperative<br> nature of these interactions, the system's dynamics are captured by the behaviour of a single<br> mean- eld, collective spin. In this mean- eld limit, it has recently been shown that the interplay<br> between photon losses and periodic driving of light-matter coupling can lead to time-crystalline-like<br> behaviour of the collective spin [1]. In this work, we investigate whether such a Dicke time crystal<br> is stable to perturbations that explicitly break the mean- eld solvability of the conventional Dicke<br> model. In particular, we consider the addition of short-range interactions between the atoms which<br> breaks the collective coupling and leads to complex many-body dynamics. In this context, the interplay<br> between periodic driving, dissipation and interactions yields a rich set of dynamical responses<br> including long-lived and metastable Dicke time crystals, where losses can cool down the many-body<br> heating resulting from the continuous pump of energy from the periodic drive. Speci cally, when<br> the additional short-range interactions are ferromagnetic, we observe time crystalline behaviour at<br> non-perturbative values of the coupling strength, suggesting the possible existence of stable dynamical<br> order in a driven-dissipative quantum many-body system. These ndings illustrate the rich<br> nature of novel dynamical responses with many-body character in quantum optics platforms.