&Bullet; *physics* 14, 104

Experiments successfully record signatures of a discrete-time crystal phase in an open quantum many-body system.

Time crystals, as proposed by Frank Wilczek in 2012, are temporal analogues to conventional space crystals [1] . Just as conventional crystals require the breaking of the space translation symmetry, time crystals require the breaking of the time translation symmetry (see Point of View: Crystals of Time). These exotic, dynamic phases of matter were realized on different experimental platforms, but in all cases the time crystal phases were recorded by closed systems that are subject to coherent manipulations (see Viewpoint: How to Create a Time Crystal) [2] . Hans Keßler from the University of Hamburg and his colleagues have now reported on the first observation of time-crystalline behavior in an open quantum system [3] .

Time crystals can break the continuous symmetry if they are realized in a time-independent (energy-saving) system, and break the discrete time translation if they are realized in a periodically driven (Floquet) system. The former, as conceived by Wilczek, turned out to be impossible in ground states or thermal equilibrium states of short-range interacting systems [4] . The latter, however, in which the constituents adopt a repeating spatial configuration with a period that is a multiple (typically twice) the driving period, has been demonstrated in some closed spin systems with strong disorder and interactions [5] . In order to distinguish these “discrete time crystals” from other dynamic phenomena such as Rabi oscillations, this period multiplication must have “stiffness”; that is, it has to be robust against small disturbances in system parameters or driving logs. A widely used recipe for creating time-discrete crystals with the necessary rigidity is to drive a symmetrical system in such a way that it changes from one symmetry-broken state to another per drive period.

All experimental demonstrations of time-discrete crystals so far have used closed systems, which leaves the question open whether they can be achieved in the presence of dissipation and decoherence. This fundamental question has practical relevance because real systems can never be completely isolated from their environment. While dissipation generally destroys the order of a time crystal, there are situations in which order is maintained if the system-environment coupling can be tailored appropriately. In quantum computing and quantum state engineering, such tailoring already enables dissipation to be used as a useful resource [6] . The latest developments in experimental atomic, molecular and optical physics have now made it possible to use this approach to implement new dynamic orders in open quantum systems [7] .

Kessler and colleagues take advantage of these developments and provide strong evidence that a discrete-time crystal can exist in a powered, open atomic cavity system. Their experiment follows theoretical predictions of the time-crystalline order in a driven open thickness model [8] . In its basic (non-time-crystalline) form, the open thickness model describes an ensemble of two-stage atoms that are driven by a pump laser and coupled to a single lossy photon mode. This system shows a dissipative, super-radiating phase transition, which means that from a certain pumping threshold the atomic ensemble is spatially ordered and the atoms emit photons coherently. As with the first experimental demonstration of this phase transition [9] , Keßler and colleagues use a Bose-Einstein condensate (BEC) in a cavity, whereby the two levels are manifested in the degrees of freedom of movement of the atoms. But while this first demonstration was static, Kessler and colleagues control their system by varying the power of the pump laser, which induces the coupling between atoms and cavity photons, over time.

The team measured the dynamics of the system using two methods. First, they continuously monitored the phase and number of photons in the cavity by detecting the photons emerging from the cavity. Second, they measured the momentum distribution of the atoms at a given point in time by releasing the BEC and observing its expansion (i.e. the time-of-flight image).

These measurements show that, given correctly selected parameters, the system periodically switches between two symmetry-broken super-radian states, with the BEC adopting a different checkerboard pattern in each state. The complete cycle of switching between the states is twice the driving period and is robust against aperiodic temporal disturbances in this driving period. This robustness differs from the usual stiffness of crystals with discrete time, in which the perturbed system is strictly periodic [5] , it is analogous to the robustness of space crystals, which can tolerate spatially inhomogeneous impurities.

The period doubling behavior observed by the team is reminiscent of a phenomenon in nonlinear optics, in which a subharmonic generator generates output photons whose frequency is an integral fraction (typically half) of the input photons. There is a conceptual difference, however: discrete, time-crystalline oscillations are a dynamic property of the system itself, while the subharmonic generation is directly related to the output signal and not to the system (where the “system” is the medium that carries the input photons). In addition, the crucial condition for time crystals that they are robust against disturbances is not necessary for the subharmonic generation. Nevertheless, it might be interesting to consider whether findings from the manufacture of time-discrete crystals could help to achieve efficient subharmonic generation.

One of the limitations of the time crystal pointed out by Keßler and his colleagues is that the oscillations decay after a few periods. This decay is mainly due to the loss of atoms from the BEC, which effectively reduces the collective atom-photon coupling in the system and displaces it from the super-radiating phase. In order to extend the life of this time crystal, an important future task is to overcome this atomic loss. It would also be interesting to coordinate both the loss of photons from the cavity and the interatomic interactions (the influence of which is overshadowed by the loss of atoms in the present experiment). Such tuning could reveal new phenomena not predicted by the standard semi-classical description (middle field) [10] and confirm that the time-crystalline order is indeed stabilized by dissipation [8] . Apart from the periodically driven system used by Keßler and his colleagues, the researchers still have to demonstrate a continuous time crystal in a time-independent open system [11] . Although it has been shown that such time crystals cannot be reached in equilibrium systems [4] , they should be achievable in a system that is distant from thermal equilibrium.

From a broader perspective, the experiment provides a prototypical example of studying a rich interplay between propulsion, interaction, and dissipation in a quantum many-body system under continuous monitoring. We expect that this work will stimulate further theoretical and experimental studies on the largely unexplored phase structures and dynamic phenomena in non-equilibrium open quantum systems.

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