We study the nonequilibrium steady-state of interacting photons in cavity arrays as described by the driven-dissipative Bose-Hubbard and spin-1/2 XY model. For this purpose, we develop a selfconsistent expansion in the inverse coordination number of the array ( z 1 ) to solve the Lindblad master equation of these systems beyond the mean-field approximation. Our formalism is compared and benchmarked with exact numerical methods for small systems based on an exact diagonalization of the Liouvillian and a recently developed corner-space renormalization technique. We then apply this method to obtain insights beyond mean-field in two particular settings: (i) we show that the gasliquid transition in the driven-dissipative Bose-Hubbard model is characterized by large density fluctuations and bunched photon statistics. (ii) We study the antibunching-bunching transition of the nearest-neighbor correlator in the driven-dissipative spin-1/2 XY model and provide a simple explanation of this phenomenon.
The driven, dissipative Bose-Hubbard model (BHM) provides a generic description of collective phases of interacting photons in cavity arrays. In the limit of strong optical nonlinearities (hard-core limit), the BHM maps on the dissipative, transverse-field XY model (XYM). The steady-state of the XYM can be analyzed using mean-field theory, which reveals a plethora of interesting dynamical phenomena. For example, strong hopping combined with a blue-detuned drive, leads to an instability of the homogeneous steady-state with respect to antiferromagnetic fluctuations. In this paper, we address the question whether such an antiferromagnetic instability survives in the presence of quantum correlations beyond the mean-field approximation. For that purpose, we employ a self-consistent 1/z expansion for the density matrix, where z is the lattice coordination number, i.e., the number of nearest neighbours for each site. We show that quantum fluctuations stabilize a new homogeneous steady-state with antiferromagnetic correlations in agreement with exact numerical simulations for finite lattices. The latter manifests itself as short-ranged oscillations of the first and second-order spatial coherence functions of the photons emitted by the array. arXiv:1806.11067v1 [cond-mat.quant-gas]
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