Quantum many-body dynamics of cavity QED systems

Over the past 20 years the enormous advancement in experimental techniques in manipulating atomic matter at temperatures near absolute zero has allowed to realize Richard Feynmans dream of a quantum simulator. A quantum simulator is a simple and highly controllable system that can be used to experimentally mimic the behavior of other, more complex quantum systems. There are numerous instances of quantum simulators with ultracold atoms in a free space that can be used to study, for example, quantum magnetism, superconductivity or quark confinement. What is interesting, if atoms are not in a free space, but in a resonant cavity instead, then quantum properties of light become important. An optical cavity, which is basically a set of two or more high-quality mirrors, is able to confine light at specific resonant frequencies enhancing atom-photon coupling. The latter usually leads to atomic back-action on the field and non-local interactions. For these reasons, composite quantum gas optical cavity systems constitute a unique platform for quantum simulations by exploiting quantum properties of light and its interaction with ultracold matter. The scientific aim of this proposal is to utilize the greatest assets of the quantum gas cavity platforms in the theoretical investigation of continuous and higher-order dissipative time crystals, flat-band many-body physics with cavity mediated long-ranged interactions and systems coupled to synthetic dynamical gauge fields. The research objectives of this proposal belong to the modern trends in ultracold atomic and condensed matter physics and lie within the range of contemporary tabletop experiments. We anticipate that the results will not only directly influence future research but also contribute to the long term goals. In particular, while the robustness of discrete time crystals can be used in quantum computations, the higher-order time crystals can host condensed matter phenomena opening the door to multidimensional space-time electronics. Moreover, cavity mediated interaction-induced dynamics on flat bands can give rise to new materials, such as exotic highly correlated lattice supersolids. Finally, dynamical gauge fields are elementary building blocks of the fundamental physical theories and, therefore, efficient quantum simulation of gauge theories opens possibilities for deep understanding of fundamental problems like quark confinement, or strongly interacting counterparts of topological insulators.

Modern experimental platforms allow us to explore quantum systems both in and out of equilibrium. These platforms complement one another, strengthening our fundamental understanding about quantum realms wich drives progress toward new quantum technologies. In particular, ultracold gases in optical lattices and cavities provide clean, tunable environments for studying non-equilibrium dynamics and localization. Conversely, photonic systems, comprising coupled or multimode resonators, operate naturally in driven-dissipative settings. By testing many-body concepts under both particle-conserving and gain-loss conditions, these approaches link fundamental theory to practical implementation. In this project, we leveraged these distinct platforms to broaden our understanding of the quantum world, focusing on dynamics out of equilibrium. We demonstrated that complex quantum systems can settle into predictable regimes even when exposed to dissipation. Specifically, in arrays of optical resonators, we observed dynamical condensation of light into preferred states and stable limit cycles. Extending this control into the temporal domain, we proposed crystals in time within ring resonators, where temporal modulation can mimic solid-state spatial behaviors. Complementing these photonic studies, we utilized Rydberg-atom arrays to simulate vibration-assisted transport, identifying regimes where excitations travel nondispersively and remain robust against disorder. Furthermore, we introduced a trajectory-based protocol utilizing superradiance in lossy cavities to rapidly generate macroscopic entangled states. Finally, we identified a distinct topological signature of dynamical phase transitions in three-dimensional quantum matter. Rather than manifesting as isolated defects, the system forms closed loops in momentum space. Collectively, these results provide novel effects and control strategies for non-equilibrium quantum systems, with significant implications for future photonic devices and quantum sensing.