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arXiv:1704.08978 · Physical Review A

Phase diagram of incoherently driven strongly correlated photonic lattices

Alberto Biella, Florent Storme, José Lebreuilly, Davide Rossini, Rosario Fazio, Iacopo Carusotto, Cristiano Ciuti

Published Aug 17, 2017• 69 citations
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Abstract

We explore theoretically the nonequilibrium photonic phases of an array of coupled cavities in presence of incoherent driving and dissipation. In particular, we consider a Hubbard model system where each site is a Kerr nonlinear resonator coupled to a two-level emitter, which is pumped incoherently. Within a Gutzwiller mean-field approach, we determine the steady-state phase diagram of such a system. We find that, at a critical value of the intercavity photon hopping rate, a second-order nonequilibrium phase transition associated with the spontaneous breaking of the $\text{U}(1)$ symmetry occurs. The transition from an incompressible Mott-like photon fluid to a coherent delocalized phase is driven by commensurability effects and not by the competition between photon hopping and optical nonlinearity. The essence of the mean-field predictions is corroborated by finite-size simulations obtained with matrix product operators and corner-space renormalization methods.

DOI
10.1103/physreva.96.023839

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We explore theoretically the nonequilibrium photonic phases of an array of coupled cavities in presence of incoherent driving and dissipation. In particular, we consider a Hubbard model system where each site is a Kerr nonlinear resonator coupled to a two-level emitter, which is pumped incoherently. Within a Gutzwiller mean-field approach, we determine the steady-state phase diagram of such a system. We find that, at a critical value of the intercavity photon hopping rate, a second-order nonequilibrium phase transition associated with the spontaneous breaking of the U(1)\text{U}(1)U(1) symmetry occurs. The transition from an incompressible Mott-like photon fluid to a coherent delocalized phase is driven by commensurability effects and not by the competition between photon hopping and optical nonlinearity. The essence of the mean-field predictions is corroborated by finite-size simulations obtained with matrix product operators and corner-space renormalization methods.
In recent years, the control of photon-photon interactions in optical nonlinear media has led to the realization of quantum fluids of light. One of the current challenges is to increase the strength of these interactions and enter the so-called strongly correlated regime. To achieve this goal, arrays of nonlinear cavities are a very promising candidate. In this thesis, theoretical results on arrays of nonlinear cavities described by a driven-dissipative Bose-Hubbard model are presented. In particular, a general method to compute the mean-field phase diagram of this model is described. Due to the finite life time of photons, the system is intrinsically dissipative: cavity losses must be compensated by an external driving field. This nonequilibrium nature gives rise to interesting features, such as a transition between monostable and bistable phases induced by tunneling. In the limit of weak dissipation and weak driving, analytical results describing generalized Mott insulating phases are derived. These states survive up to a critical tunneling strength, above which a crossover to a classical coherent state takes place. Finally, the issue of how to go beyond the mean-field approximation is addressed by performing exact numerical simulations. Large arrays of cavities were simulated by implementing a new method specifically tailored for driven-dissipative systems.
We investigate steady-state properties of a two-dimensional incoherent-pumped dissipative Bose-Hubbard model, which describes a photon square lattice. This incoherent pumping exhibits an important environment-induced higher-order fluctuation effect, which induces a strong competition between the driven-dissipative channel, the photon-photon interaction, and the photon hopping in multi-photon processes. This new competition gives rise to a spontaneous breaking of the U(1) symmetry of system. As a result, we predict a many-body steady-state localized-delocalized phase transition and an anti-blockade effect, in which the increasing of the repulsive photon-photon interaction promotes the emergence of phase transition. These unconventional many-body steady-state phenomena can be understood by analyzing the single-cavity properties. Our results pave a new way to control many-body dynamics of driven-dissipative systems.
We study the nonequilibrium steady state of the driven-dissipative Bose-Hubbard model with Kerr nonlinearity. Employing a mean-field decoupling for the intercavity hopping JJJ, we find that the steep crossover between low and high photon-density states inherited from the single cavity transforms into a gas--liquid bistability at large cavity-coupling JJJ. We formulate a van der Waals--like gas--liquid phenomenology for this nonequilibrium setting and determine the relevant phase diagrams, including a new type of diagram where a lobe-shaped boundary separates smooth crossovers from sharp, hysteretic transitions. Calculating quantum trajectories for a one-dimensional system, we provide insights into the microscopic origin of the bistability.
We investigate the occurrence of a phase transition, characterized by the spontaneous breaking of a discrete symmetry, in a driven-dissipative Bose-Hubbard lattice in the presence of two-photon coherent driving. The driving term does not lift the original U(1)\text{U}(1)U(1) symmetry completely and a discrete Z2{\mathbb{Z}}_{2}Z2​ symmetry is left. When driving the bottom of the Bose-Hubbard band, a mean-field analysis of the steady state reveals a second-order transition from a symmetric phase to a quasicoherent state with a finite expectation value of the Bose field. For larger driving frequency, the phase diagram shows a third region, where both phases are stable and the transition becomes of first order.
We analyze the nonequilibrium dynamics of a gas of interacting photons in an array of coupled dissipative nonlinear cavities when driven by a pulsed external coherent field. Using a mean-field approach, we show that the response of the system is strongly sensitive to the underlying (equilibrium) quantum phase transition from a Mott insulator to a superfluid state at commensurate filling. We find that the coherence of the cavity emission after a quantum quench can be used to determine the phase diagram of an optical many-body system even in the presence of dissipation.
We study the properties of an array of QED cavities coupled by nonlinear elements in the presence of photon leakage and driven by a coherent source. The main effect of the nonlinear couplings is to provide an effective cross-Kerr interaction between nearest-neighbor cavities. Additionally, correlated photon hopping between neighboring cavities arises. We provide a detailed mean-field analysis of the steady-state phase diagram as a function of the system parameters, the leakage, and the external driving and show the emergence of a number of different quantum phases. A photon crystal associated with a spatial modulation of the photon blockade appears. The steady state can also display oscillating behavior and bistability. In some regions the crystalline ordering may coexist with the oscillating behavior. Furthermore, we study the effect of short-range quantum fluctuations by employing a cluster mean-field analysis. Focusing on the corrections to the photon crystal boundaries, we show that, apart for some quantitative differences, the cluster mean field supports the findings of the simple single-site analysis. In the last part of the paper we concentrate on the possibility of building up the class of arrays introduced here, by means of superconducting circuits of existing technology. We consider a realistic choice of the parameters for this specific implementation and discuss some properties of the steady-state phase diagram.