Understanding quantum dynamics away from equilibrium is an outstanding challenge in the modern physical sciences. It is well known that out-of-equilibrium systems can display a rich array of phenomena, ranging from self-organized synchronization to dynamical phase transitions1,2. More recently, advances in the controlled manipulation of isolated many-body systems have enabled detailed studies of non-equilibrium phases in strongly interacting quantum matter3-6. As a particularly striking example, the interplay of periodic driving, disorder, and strong interactions has recently been predicted to result in exotic "time-crystalline" phases7, which spontaneously break the discrete time-translation symmetry of the underlying drive8-11. Here, we report the experimental observation of such discrete time-crystalline order in a driven, disordered †

Insights into complex phenomena in quantum matter can be gained from simulation experiments with ultracold atoms, especially in cases where theoretical characterization is challenging. However, these experiments are mostly limited to short-range collisional interactions. Recently observed perturbative effects of long-range interactions were too weak to reach novel quantum phases [1,2]. Here we experimentally realize a bosonic lattice model with competing short-and infinite-range interactions, and observe the appearance of four distinct phases -a superfluid, a supersolid, a Mott insulator and a charge density wave. Our system is based on an atomic quantum gas trapped in an optical lattice inside a high finesse optical cavity. The strength of the shortranged on-site interactions is controlled by means of the optical lattice depth. The infinite-range interaction potential is mediated by a vacuum mode of the cavity [3,4] and is independently controlled by tuning the cavity resonance. When probing the phase transition between the Mott insulator and the charge density wave in real-time, we discovered a behaviour characteristic of a first order phase transition. Our measurements have accessed a regime for quantum simulation of manybody systems where the physics is determined by the intricate competition between two different types of interactions and the zero point motion of the particles.Experiments with cold atoms have contributed in many ways to elucidate fundamental behaviour of quantum matter [5]. An example is the realization of the BoseHubbard model, where the balance between the kinetic energy of particles moving in an optical lattice and the on-site collisional interactions drives a quantum phase transition from a superfluid to a Mott insulating phase [6,7]. Whilst collisions between atoms are naturally present in quantum gases and give rise to short-range interactions [8], longer ranged interactions are more elusive. In order to get a handle on the latter, ultracold gases of particles with large magnetic or electric dipole moments [9,10], atoms in Rydberg states [11], or cavitymediated interactions [3] have been studied. Indeed, already Hubbard models with additional nearest-neighbour interactions are predicted to show intriguing phases like charge and spin density waves, supersolids, topological phases or checkerboard and stripe phases [12][13][14][15][16][17][18].In our experiment, we achieve independent control over three energy scales by combining an optical lattice with cavity-mediated interactions, see Fig. 1. The underlying static lattices along all three directions are necessary to study the direct competition between short-and longrange interactions, as compared to the situation very recently investigated in [19]. Aspects of this scenario, in which on-site interactions compete with infinite-range interactions, have been theoretically studied in the context of self-consistent extended Hubbard models and various phases have been predicted [20][21][22]. The starting point is a Bose-Einstein condensate (...

Long-range interactions in quantum gases are predicted to give rise to an excitation spectrum of roton character, similar to that observed in superfluid helium. We investigated the excitation spectrum of a Bose-Einstein condensate with cavity-mediated long-range interactions, which couple all particles to each other. Increasing the strength of the interaction leads to a softening of an excitation mode at a finite momentum, preceding a superfluid-to-supersolid phase transition. We used a variant of Bragg spectroscopy to study the mode softening across the phase transition. The measured spectrum was in very good agreement with ab initio calculations and, at the phase transition, a diverging susceptibility was observed. The work paves the way toward quantum simulation of long-range interacting many-body systems.

We experimentally study the influence of dissipation on the driven Dicke quantum phase transition, realized by coupling external degrees of freedom of a Bose-Einstein condensate to the light field of a high-finesse optical cavity. The cavity provides a natural dissipation channel, which gives rise to vacuum-induced fluctuations and allows us to observe density fluctuations of the gas in real-time. We monitor the divergence of these fluctuations over two orders of magnitude while approaching the phase transition, and observe a behavior that deviates significantly from that expected for a closed system. A correlation analysis of the fluctuations reveals the diverging time scale of the atomic dynamics and allows us to extract a damping rate for the external degree of freedom of the atoms. We find good agreement with our theoretical model including dissipation via both the cavity field and the atomic field. Using a dissipation channel to nondestructively gain information about a quantum many-body system provides a unique path to study the physics of driven-dissipative systems.driven-dissipative phase transitions | critical behavior | Dicke model | quantum gas | cavity QED E xperimental progress in the creation, manipulation, and probing of atomic quantum gases has made it possible to study highly controlled many-body systems and to access their phase transitions. This unique approach to quantum many-body physics has substantiated the notion of quantum simulation for key models of condensed matter physics (1, 2). There has been increasing interest in generalizing such an approach to nonequilibrium zero-temperature or quantum phase transitions in driven-dissipative systems (3), as occurring in condensed matter systems coupled to light (4,5) or in open electronic systems (6, 7). Among the most tantalizing questions is how vacuum fluctuations from the environment influence the critical behavior at a phase transition via quantum backaction. Related to this question is whether driven-dissipative phase transitions give rise to new universal behavior, and under which conditions they exhibit classical critical behavior with an effective temperature (8-12).Coupling quantum gases to the field of an optical cavity is a particularly promising approach to realize a driven-dissipative quantum many-body system with a well-understood and controlled dissipation channel. A further advantage of this scheme is that the dissipation channel of the cavity mode can be directly used to investigate the system in a nondestructive way via the leaking cavity field (13). Combining the experimental setting of cavity quantum electrodynamics with that of quantum gases (14-18) led to the observation of quantum backaction heating caused by cavity dissipation (19,20), as well as to the realization of the nonequilibrium Dicke quantum phase transition (21). Here, we study the influence of cavity dissipation on the fluctuation spectrum at the Dicke phase transition by connecting these approaches. We nondestructively observe diverging fluctuations of the o...

Statistical mechanics underlies our understanding of macroscopic quantum systems. It is based on the assumption that out-of-equilibrium systems rapidly approach their equilibrium states, forgetting any information about their microscopic initial conditions. This fundamental paradigm is challenged by disordered systems, in which a slowdown or even absence of thermalization is expected. We report the observation of critical thermalization in a three dimensional ensemble of ∼10^{6} electronic spins coupled via dipolar interactions. By controlling the spin states of nitrogen vacancy color centers in diamond, we observe slow, subexponential relaxation dynamics and identify a regime of power-law decay with disorder-dependent exponents; this behavior is modified at late times owing to many-body interactions. These observations are quantitatively explained by a resonance counting theory that incorporates the effects of both disorder and interactions.