The exploration of non-equilibrium phases of matter has emerged as a captivating frontier in contemporary research, offering novel insights into the behavior of physical systems far from equilibrium. Our group has investigated a broad range of phenomena including time crystals, which challenge our traditional understanding of time translation symmetry. Time crystals exhibit temporal order that spontaneously breaks the symmetry of continuous time translations, presenting a new paradigm in the study of phase transitions and fundamental symmetries. Prethermalization is another captivating concept, wherein a system temporarily behaves as if it were in equilibrium before evolving towards a different non-equilibrium steady state. This phenomenon, observed in both classical and quantum systems, plays a crucial role in understanding the dynamics of driven systems and thermalization processes. Exploring these non-equilibrium phases of matter opens up exciting avenues for understanding the intricate dynamics and behavior of physical systems beyond the constraints of thermal equilibrium.

Just like their spatial counterparts, the ground state of time crystals breaks translational invariance. This intriguing property was conceptualized only within the last decade and sparked the interest of many groups – including ours. By harnessing the interplay between periodic driving and many-body interactions, we have successfully engineered systems that display this remarkable behavior in experimental platforms.

Prethermalization behavior arises from the interplay of initial conditions, system parameters, and the nature of interactions. Experimental and theoretical studies have shed light on the emergence of prethermalization phases in various physical systems, providing insights into the dynamics and relaxation processes of driven systems. Our research focuses on understanding and characterizing these classical prethermalization phases and their implications for controlling and manipulating non-equilibrium dynamics.

Long-range measurement-induced phase transitions have recently emerged as a fascinating area of research in the field of quantum many-body systems. These phase transitions occur when a quantum system undergoes a sudden change in its ground state due to measurements performed over long distances. Such transitions defy conventional notions of local interactions and exhibit non-local correlations. We are explored the theoretical foundations and experimental realizations of long-range measurement-induced phase transitions, opening up avenues for studying complex classical and quantum phenomena and potentially developing novel quantum technologies based on non-local measurements.