Recent experiments have demonstrated an open system realization of the Dicke quantum phase transition in the motional degrees of freedom of an optically driven Bose-Einstein condensate in a cavity. Relevant collective excitations of this light-matter system are polaritonic in nature, allowing access to the quantum critical behavior of the Dicke model through light leaking out of the cavity. This opens the path to using photodetection-based quantum optical techniques to study the dynamics and excitations of this elementary quantum critical system. We first discuss the photon flux observed at the cavity face and find that it displays a different scaling law near criticality than that obtained from the mean-field theory for the equivalent closed system. Next, we study the second-order correlation measurements of photons leaking out of the cavity. Finally, we discuss a modulation technique that directly captures the softening of polaritonic excitations. Our analysis takes into account the effect of the finite size of the system, which may result in an effective symmetry-breaking term.

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Systems of strongly interacting atoms and photons, that can be realized wiring up individual cavity QED systems into lattices, are perceived as a new platform for quantum simulation. While sharing important properties with other systems of interacting quantum particles here we argue that the nature of light-matter interaction gives rise to unique features with no analogs in condensed matter or atomic physics setups. By discussing the physics of a lattice model of delocalized photons coupled locally with two-level systems through the elementary light-matter interaction described by the Rabi model, we argue that the inclusion of counter rotating terms, so far neglected, is crucial to stabilize finite-density quantum phases of correlated photons out of the vacuum, with no need for an artificially engineered chemical potential. We show that the competition between photon delocalization and Rabi non-linearity drives the system across a novel Z2 parity symmetry-breaking quantum criticality between two gapped phases which shares similarities with the Dicke transition of quantum optics and the Ising critical point of quantum magnetism. We discuss the phase diagram as well as the low-energy excitation spectrum and present analytic estimates for critical quantities. Introduction -Interaction between light and matter is one of the most basic processes in nature and represents a cornerstone in our understanding of a broad range of physical phenomena. In the study of strongly correlated systems and collective phenomena, light has traditionally assumed the role of a spectroscopic probe. The increasing level of control over light-matter interactions with atomic and solid-state systems [1-3] has brought forth a new class of quantum many body systems where light and matter play equally important roles in emergent phenomena: photon lattices [4][5][6][7][8][9][10][11][12][13][14][15]. The basic building block of such systems is the elementary Cavity QED (CQED) system formed by a two-level system (TLS) interacting with a single mode of an electromagnetic resonator. When CQED systems are coupled to form a lattice, the interplay between photon blockade [17][18][19] and inter-cavity photon tunnelling leads to phenomenology akin to those of Hubbard models of massive bosons as realized e.g. by ultracold atoms in optical lattices [20]. The possibility of quantum phase transitions of light between Mott-like insulating and superfluid phases has stimulated a great deal of discussion recently [4][5][6][7][8][9][11][12][13][14]. The excitement about these systems stems from their potential as dissipative quantum simulators that provide full access to individual sites through continuous weak measurements [16].

We explore the phase diagram of the dissipative Rabi-Hubbard model, as could be realized by a Ramanpumping scheme applied to a coupled cavity array. There exist various exotic attractors, including ferroelectric, antiferroelectric, and incommensurate fixed points, as well as regions of persistent oscillations. Many of these features can be understood analytically by truncating to the two lowest lying states of the Rabi model on each site. We also show that these features survive beyond mean field, using matrix product operator simulations. DOI: 10.1103/PhysRevLett.116.143603 Introduction.-A number of recent experimental breakthroughs [1][2][3][4] have spurred the investigation of nonequilibrium properties of hybrid quantum many-body systems of interacting matter and light. Characterized by excitations with a finite lifetime, when sustained by finiteamplitude optical drives they display steady-state phases that are generally far richer [5][6][7][8][9][10] than their equilibrium counterparts [11,12]. Critical phenomena in these open driven-dissipative systems often come with genuinely new properties and novel dynamic universality classes, even when an effective temperature can be identified [13][14][15][16][17], a statement that can be made robust in renormalization group calculations [18,19]. Coupled cavity QED systems [20][21][22] have emerged as natural platforms to study many-body physics of open quantum systems. The current fabrication and control capabilities in solid-state quantum optics allows us to probe lattice systems [23][24][25][26][27][28][29][30][31] in the mesoscopic regime, providing a first glimpse into how macroscopic quantum behavior may arise far from equilibrium. It is therefore of interest to identify a physical system where a nonequilibrium phase transition (i) can be studied-at least in principle-in the thermodynamic limit, (ii) can be compared to an equilibrium analogue through a proper limiting procedure, and (iii) can be easily realized in an architecture that is currently available.The Rabi-Hubbard (RH) model [32] represents the minimal description of coupled cavity QED systems, explicitly containing terms which do not conserve the particle number. These terms are relevant for the lowfrequency behavior of the coupled system and their inclusion lead, in equilibrium, to a Z 2 -symmetry breaking phase transition between a quantum disordered paraelectric phase and an Ising ferroelectric [32,33]. The equilibrium RH transition requires a sizable intercavity hopping or light-matter interaction, of the order of the transition

We discuss the physics of the Rabi-Hubbard model describing large arrays of coupled cavities interacting with two level atoms via a Rabi non-linearity. We show that the inclusion of counterrotating terms in the light-matter interaction, often neglected in theoretical descriptions based on Jaynes-Cumming models, is crucial to stabilize finite-density quantum phases of correlated photons with no need for an artificially engineered chemical potential. We show that the physical properties of these phases and the quantum phase transition occurring between them is remarkably different from those of interacting bosonic massive quantum particles. The competition between photon delocalization and Rabi non-linearity drives the system across a novel Z2 parity symmetry-breaking quantum phase transition between two gapped phases, a Rabi insulator and a delocalized superradiant phase.