The Dicke model describes the coupling between a quantized cavity field and a large ensemble of two‐level atoms. When the number of atoms tends to infinity, this model can undergo a transition to a superradiant phase, belonging to the mean‐field Ising universality class. The superradiant transition was first predicted for atoms in thermal equilibrium and was recently realized with a quantum simulator made of atoms in an optical cavity, subject to both dissipation and driving. This progress report offers an introduction to some theoretical concepts relevant to the Dicke model, reviewing the critical properties of the superradiant phase transition and the distinction between equilibrium and nonequilibrium conditions. In addition, it explains the fundamental difference between the superradiant phase transition and the more common lasing transition. This report mostly focuses on the steady states of atoms in single‐mode optical cavities, but it also mentions some aspects of real‐time dynamics, as well as other quantum simulators, including superconducting qubits, trapped ions, and using spin–orbit coupling for cold atoms. These realizations differ in regard to whether they describe equilibrium or nonequilibrium systems.

The Dicke model is a fundamental model of quantum optics that describes the coupling between many two level systems and a single cavity model. One of the key features of the model is a phase transition between a normal phase, in which the atoms are mostly unexcited, and a superradiant phase, in which the atoms emit light coherently into the cavity. For further information see the Progress Report by Emanuele G. Dalla Torre and co‐workers, article number https://doi.org/10.1002/qute.201800043. (Cover image design: Mor M. Roses.)

We present a fermionic description of non-equilibrium multi-level systems. Our approach uses the Keldysh path integral formalism and allows us to take into account periodic drives, as well as dissipative channels. The technique is based on the Majorana fermion representation of spin-1/2 models which follows earlier applications in the context of spin and Kondo systems. We apply this formalism to problems of increasing complexity: a dissipative two-level system, a driven-dissipative multi-level atom, and a generalized Dicke model describing many multi-level atoms coupled to a single cavity. We compare our theoretical predictions with recent QED experiments and point out the features of a counter-lasing transition. Our technique provides a convenient and powerful framework for analyzing driven-dissipative quantum systems, complementary to other approaches based on the solution of Lindblad master equations.

The Dicke model is a fundamental model of quantum optics, which describes the interaction between light and matter. In the Dicke model, the light component is described as a single quantum mode, while the matter is described as a set of two-level systems. When the coupling between the light and matter crosses a critical value, the Dicke model shows a mean-field phase transition to a superradiant phase. This transition belongs to the Ising universality class and was realized experimentally in cavity quantum electrodynamics experiments. Although the superradiant transition bears some analogy with the lasing instability, these two transitions belong to different universality classes.

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