We experimentally explore the dynamic optical hysteresis of a semiconductor microcavity as a function of the sweep time. The hysteresis area exhibits a double power law decay due to the shot noise of the driving laser, which triggers switching between metastable states. Upon increasing the average photon number and approaching the thermodynamic limit, the double power law evolves into a single power law. This algebraic behavior characterizes a dissipative phase transition. Our findings are in good agreement with theoretical predictions, and the present experimental approach is promising for the exploration of critical phenomena in photonic lattices.Optical bistability -the existence of two stable states with different photon numbers for the same driving conditions -is a general feature of driven nonlinear systems described within the mean-field approximation (MFA) [1]. Beyond the MFA, a quantum treatment predicts that the steady-state of a nonlinear cavity is unique at any driving condition [2]. The origin of this apparent contradiction was noted by Bonifacio and Lugiato [3], and by Drummond and Walls [4]: quantum fluctuations (the lost feature in the MFA) trigger switching between states and the exact solution corresponds to a weighted average over the two metastable states. Experiments in the 80's with two-mode lasers evidenced extremely long switching times [5], which were predicted to diverge for weak fluctuations and/or large photon numbers [6]. Already in these early works, this dramatic slowing down of the system dynamics was linked to a first order phase transition [5][6][7].The physics of nonlinear resonators is receiving renewed interest in connection to predictions of quantum many-body phases [8][9][10][11][12][13], critical phenomena [5,[12][13][14][16][17][18], and dissipative phase transitions [4]. Impressive progress is being made in building lattices of nonlinear resonators, such as photonic crystal cavities [20,21], waveguides [22], superconducting microwave resonators [23,24], or optomechanical resonators [25,26]. In this context, semiconductor microcavities operating in the exciton-photon strong coupling regime provide a versatile platform where photon hopping and the pumping geometry can be controlled [27]. Lattices of different dimensionalities can be engineered [28,29], and the hybrid light-matter nature of their elementary excitations, namely cavity polaritons, provide a strong and tunable Kerr nonlinearity via the exciton component [1,[30][31][32].Recently, it was predicted that even in a single resonator, critical exponents could be retrieved from dynamical hysteresis measurements [17]. More precisely, when the driving power is swept at a finite speed across a bistability, the area of the hysteresis cycle is expected to close following a double power-law as a function of the sweep time [5, 6]. The long-time decay arises from quantum fluctuations, and presents a universal −1 exponent [5]. In the thermodynamic limit wherein the photon number in the bistability tends to infinity and fluctua...
We present a theoretical method to study driven-dissipative correlated systems on lattices with two spatial dimensions (2D). The steady-state density-matrix of the lattice is obtained by solving the master equation in a corner of the Hilbert space. The states spanning the corner space are determined through an iterative procedure, using eigenvectors of the density-matrix of smaller lattice systems, merging in real space two lattices at each iteration and selecting M pairs of states by maximizing their joint probability. Accuracy of the results is then improved by increasing M , the number of states of the corner space, until convergence is reached. We demonstrate the efficiency of such an approach by applying it to the driven-dissipative 2D Bose-Hubbard model, describing, e.g., lattices of coupled cavities with quantum optical nonlinearities. PACS numbers:Simulating large quantum systems is a challenging task because their complexity grows exponentially with their size. Indeed, the dimension of the Hilbert space for a multipartite system consisting of m subsystems, each of them described by a space of dimension N , is N m . Furthermore, for open systems the physics can no longer be described only in terms of the eigenstates of an Hamiltonian, requiring instead the knowledge of the densitymatrix. In this case, the number of variables to be determined scales as N 2m , namely the square of the size of the Hilbert space.In the last decades, several methods have been proposed to reduce the complexity of this problem. The first attempt in this direction is the renormalization group technique, proposed by Wilson [1] and successfully applied to the Kondo problem. Numerical implementations of this approach are based on the solution of a system with a smaller Hilbert space, where only the relevant physical states with the lowest energies are retained. Ideally, this procedure can be iterated by arbitrarily growing the size of a block system step by step, for instance by doubling the size of the block at each iteration. However, such a numerical implementation of the real-space renormalization group can yield inaccurate results for the system ground state, because the boundary conditions imposed while solving the smaller system might be inappropriate to describe the doubled one [2]. In the case of one-dimensional systems, a powerful method is represented by the density-matrix renormalization group (DMRG) [3], which is based on the selection of the most probable states of the reduced density-matrix of a block, obtained by determining the ground state of the Hamiltonian of a larger section of the lattice. The generalization to two spatial dimensions is challenging and currently under intense study [4,5]: one approach exploits the artificial description in terms of one-dimensional systems with * Electronic address: cristiano.ciuti@univ-paris-diderot.fr long-range interactions [6], while another is based on the generalization of matrix product states [7] to projected entangled-pair states [8]. These theoretical methods have bee...
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