# ERC Consolidator grant: Pho(n)ton induced phase transitions

One of our dreams for the future is to control and manipulate complex materials and devices at will. This progress would revolutionize technology and influence many aspects of our everyday life. A promising direction is the control of material properties by electromagnetic radiation leading to photo-induced phase transitions. An example of such a transition is the reported dynamically induced superconductivity via a laser pulse. Whereas the theoretical description of the coupling of fermions to bosonic modes in equilibrium has seen enormous progress and explains highly non-trivial phenomena as the phonon-induced superconductivity, driven systems pose many puzzles. In addition to the inherent time-dependence of the external driving field, a multitude of possible excitation and relaxation mechanisms challenge the theoretical understanding. Recently in the field of quantum optics, a much cleaner realization of a photo-induced phase transition, the Dicke transition, has been observed for bosonic quantum gases loaded in an optical cavity. Above a critical pump strength of an external laser field, the ensemble undergoes a transition to an ordered phase.

In the project we advanced the general theoretical understanding of photo-induced phase transitions both in the field of solid state physics and quantum optics. In particular, we proposed how to realize photo-induced transitions to unconventional superconductivity and non-trivial topological phases. We further showed that for the occurring dissipative phase transitions fluctuations are crucial to identify stable steady states. Thus, we developed novel methods which go beyond the previously often employed mean-field description.

## Summary

Within the project we have proposed an experimentally realistic setup in order to dynamically generate and stabilize non-trivial topological states. This setup uses a quantum gas coupled in a novel way to an optical resonator. By the novel coupling a cavity-assisted movement of the atoms occurs which leads to the topologically non-trivial phases. The topologically non-trivial state is stabilized by the dissipative attractor dynamics due to cavity losses.

Further, we design a setup which would enable the dynamic generation of superconducting and charge density wave states and quantum devices which filter different kinds of particles.

We find interesting dynamics, e.g. critical dynamics or aging, in dissipative systems signaled by two-time correlations functions.

We have developed several novel methods which can treat the coupling of a bosonic mode (photons or phonons) to an interacting many body system (atoms or electrons) going beyond mean-field treatments. In particular, we devised a novel numerical method based on the matrix product states which has proven to be efficient in treating these complex hybrid systems. We discovered that the fluctuations determine which steady state is taken and that the dissipative transitions taking place are dominated by the fluctuations. This is in contrast to previous mean-field studies which proposed pure state transitions for the dissipative transitions. Additionally, we pointed out that the symmetries are crucial to obtain the correct phase diagram. Our results are disseminated by publications in internationally recognized journals and presentations on conferences and workshops.

We propose how a fermionic quantum gas confined to an optical lattice and coupled to an optical cavity can self-organize into a state where the spontaneously emerging cavity field amplitude induces an artificial magnetic field. The fermions form either a chiral insulator or a chiral liquid carrying chiral currents. The feedback mechanism via the dynamical cavity field enables robust and fast switching in time of the chiral phases, and the cavity output can be employed for a direct nondestructive measurement of the chiral current.

## Influence of symmetries in dissipative quantum systems

In dissipative quantum systems, strong symmetries can lead to the existence of conservation laws and multiple steady states. In this Letter we investigate a strong symmetry for bosonic atoms coupled to an optical cavity, an experimentally relevant system, generalizing the adiabatic elimination techniques and using numerically exact matrix product state methods. We show that for ideal bosons coupled to the cavity multiple steady states exist and in each symmetry sector a dissipative phase transition occurs at a different critical point. This implies that phases of very different natures can coexist. We find that the introduction of a slight breaking of the strong symmetry by a small interaction term leads to a direct transition from multiple steady states to a unique steady state. We point out the phenomenon of dissipative freezing, the breaking of the conservation law at the level of individual realizations in the presence of strong symmetry. For a small breaking of the strong symmetry we see that the behavior of the individual trajectories still shows some signs of this dissipative freezing before it fades out for larger symmetry breaking terms.

### Numerical exact treatment of atoms coupled to a cavity

We investigate the full quantum evolution of ultracold interacting bosonic atoms on a chain and coupled to an optical cavity. Extending the time-dependent matrix product state techniques and the many-body adiabatic elimination technique to capture the global coupling to the cavity mode and the open nature of the cavity, we examine the long time behavior of the system beyond the mean-field elimination of the cavity field.