In order to model realistic quantum devices it is necessary to simulate quantum systems strongly coupled to their environment. To date, most understanding of open quantum systems is restricted either to weak system–bath couplings or to special cases where specific numerical techniques become effective. Here we present a general and yet exact numerical approach that efficiently describes the time evolution of a quantum system coupled to a non-Markovian harmonic environment. Our method relies on expressing the system state and its propagator as a matrix product state and operator, respectively, and using a singular value decomposition to compress the description of the state as time evolves. We demonstrate the power and flexibility of our approach by numerically identifying the localisation transition of the Ohmic spin-boson model, and considering a model with widely separated environmental timescales arising for a pair of spins embedded in a common environment.

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.

We develop a nonequilibrium model of condensation and lasing of photons in a dye filled microcavity. We examine in detail the nature of the thermalization process induced by absorption and emission of photons by the dye molecules, and investigate when the photons are able to reach a thermal equilibrium Bose-Einstein distribution. At low temperatures, or large cavity losses, the absorption and emission rates are too small to allow the photons to reach thermal equilibrium and the behavior becomes more like that of a conventional laser. DOI: 10.1103/PhysRevLett.111.100404 PACS numbers: 03.75.Hh, 42.55.Mv, 67.85.Hj, 71.38.Àk Bose-Einstein condensation (BEC) has been observed in a wide variety of systems, from ultracold atomic gases [1,2] to quasiparticles in solid state systems such as polaritons [3][4][5][6], excitons [7], and magnons [8]. Recently experiments have shown convincing evidence of a BoseEinstein distribution, and macroscopic occupation of the lowest mode for a gas of photons confined in a dye-filled optical microcavity [9][10][11][12]. In these experiments, the thermal equilibrium distribution of photons arises because of phonon dressing of the absorption and emission by the dye molecules, and the rapid thermalization of rovibrational modes of the dye molecules by their collisions with the solvent. This leads to the accumulation of low-energy photons, closely following a Bose-Einstein distribution, as is clearly seen experimentally [10].Such a system is very closely related to a dye laser [13], but differs in the near-thermal emission spectrum that is observed below and near the threshold density and in the fact that the macroscopic population occurs at the minimum energy mode of the cavity and is not related to the gain maximum of the dye [10]. There are also close connections to microlasers [14]. However microlasers, having strong coupling between the gain medium and cavity, display thresholdless lasing [15]. In contrast, the observed behavior in the photon condensate [10] is that there is a sharp threshold which occurs far below inversion.In the context of polariton condensation [3][4][5][6] there has been much debate [16,17] about the extent to which the lack of true thermal equilibrium in experiments means the system should be called a condensate or a laser. However, various calculations for polaritons, from quantum kinetics [18,19] to Schwinger-Keldysh path integrals [20], have found a relatively smooth crossover between behavior typical of a laser, and that typical of an equilibrium condensate. Both lasers and condensates involve a spontaneous phase-symmetry breaking, and a transition to a macroscopically occupied mode, and so their connection has long been recognized [21]. The photon condensate system provides a further example of a system in which the distinction between Bose condensation and lasing must be carefully examined.The nature of the thermalization process in the photon condensate differs significantly from that found in other systems which exhibit BEC. There are no direct photon...