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arXiv:1408.1330 · Physical Review A

Bose-Hubbard model: Relation between driven-dissipative steady states and equilibrium quantum phases

Alexandre Le Boité, Giuliano Orso, Cristiano Ciuti

Published Dec 15, 2014• 72 citations
arXivHTMLPDF

Abstract

We present analytical solutions for the mean-field master equation of the driven-dissipative Bose-Hubbard model for cavity photons, in the limit of both weak pumping and weak dissipation. Instead of pure Mott-insulator states, we find statistical mixtures with the same second-order coherence ${g}^{(2)}(0)$ as a Fock state with $n$ photons, but a mean photon number of $n/2.$ These mixed states occur when $n$ pump photons have the same energy as $n$ interacting photons inside the nonlinear cavity and survive up to a critical tunneling coupling strength, above which a crossover to a classical coherent state takes place. We also explain the origin of both antibunching and superbunching predicted by P-representation mean-field theory at higher pumping and dissipation. In particular, we show that the strongly correlated region of the associated phase diagram cannot be described within the semiclassical Gross-Pitaevskii approach.

DOI
10.1103/physreva.90.063821

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We present analytical solutions for the mean-field master equation of the driven-dissipative Bose-Hubbard model for cavity photons, in the limit of both weak pumping and weak dissipation. Instead of pure Mott-insulator states, we find statistical mixtures with the same second-order coherence g(2)(0){g}^{(2)}(0)g(2)(0) as a Fock state with nnn photons, but a mean photon number of n/2.n/2.n/2. These mixed states occur when nnn pump photons have the same energy as nnn interacting photons inside the nonlinear cavity and survive up to a critical tunneling coupling strength, above which a crossover to a classical coherent state takes place. We also explain the origin of both antibunching and superbunching predicted by P-representation mean-field theory at higher pumping and dissipation. In particular, we show that the strongly correlated region of the associated phase diagram cannot be described within the semiclassical Gross-Pitaevskii approach.
We determine the steady-state phases of a driven-dissipative Bose-Hubbard model, describing, e.g., an array of coherently pumped nonlinear cavities with a finite photon lifetime. Within a mean-field master equation approach using exact quantum solutions for the one-site problem, we show that the system exhibits a tunneling-induced transition between monostable and bistable phases. We characterize the corresponding quantum correlations, highlighting the essential differences with respect to the equilibrium case. We also find collective excitations with a flat energy-momentum dispersion over the entire Brillouin zone that trigger modulational instabilities at specific wave vectors.
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 study the non-equilibrium behavior of optically driven dissipative coupled resonator arrays. Assuming each resonator is coupled with a two-level system via a Jaynes–Cummings interaction, we calculate the many-body steady state behavior of the system under coherent pumping and dissipation. We propose and analyze the many-body phases using experimentally accessible quantities such as the total excitation number, the emitted photon spectra and photon coherence functions for different parameter regimes. In parallel, we also compare and contrast the expected behavior of this system assuming the local nonlinearity in the cavities is generated by a generic Kerr effect as described by the Bose–Hubbard (BH) model rather than a Jaynes–Cummings interaction. We find that the behavior of the experimentally accessible observables produced by the two models differs for realistic regimes of interactions even when the corresponding nonlinearities are of similar strength. We analyze in detail the extra features available in the Jaynes–Cummings–Hubbard (JCH) model originating from the mixed nature of the excitations and investigate the regimes where the BH approximation would faithfully match the JCH physics. We find that the latter is true for values of the light–matter coupling and losses beyond the reach of current technology. Throughout the study we operate in the weak pumping, fully quantum mechanical regime where approaches such as mean field theory fail, and instead use a combination of quantum trajectories and the time evolving block decimation algorithm to compute the relevant steady state observables. In our study we have assumed small to medium size arrays (from 3 up to 16 sites) and values of the ratio of coupling to dissipation rate g / γ ∼ 20, which makes our results implementable with current designs in circuit QED and with near future photonic crystal set ups.
This dissertation discusses a number of theoretical models of coupled bosonic modes, all closely related to the Bose-Hubbard dimer. In studying these models, we will repeatedly return to two unifying themes: the classical structure underlying quantum dynamics and the impact of weakly coupling a system to an environment. Or, more succinctly, semiclassical methods and open quantum systems. \n \nOur primary motivation for studying models such as the Bose-Hubbard is their relevance to ongoing ultracold atom experiments. We review these experiments, derive the Bose-Hubbard model in their context and briefly discuss its limitations in the first half of Chapter 1. In its second half, we review the theory of open quantum systems and the master equation description of the dissipative Bose-Hubbard model. This opening chapter constitutes a survey of existing results, rather than original work. \n \nIn Chapter 2, we turn to the mean-field limit of the Bose-Hubbard model. After reviewing the striking localization phenomena predicted by the mean-field (and confirmed by experiment), we identify the first corrections to this picture for the dimer. The most interesting of these is the dynamical tunneling between the self-trapping points of the mean-field. We derive an accurate analytical expression for the tunneling rate using semiclassical techniques. \n \nWe continue studying the dynamics near the self-trapping fixed points in Chapter 3, focusing on corrections to the mean-field that arise at larger nonlinearities and on shorter time scales than dynamical tunneling. We study the impact of dissipation on coherence and entanglement near the fixed points, and explain it in terms of the structure of the classical phase space. \n \nThe last chapter of the dissertation is also devoted to a dissipative bosonic dimer model, but one arising in a very different physical context. Abandoning optical lattices, we consider the problem of formulating a quantum model of operation of the cylindrical anode magnetron, a vacuum tube crossed-field microwave amplifier. We derive an effective dissipative dimer model and study its relationship to the classical description. Our dimer model is a first step towards the analysis of solid-state analogs of such devices.
The driven-dissipative Bose-Hubbard model can be experimentally realized with either negative or positive onsite detunings, inter-site hopping energies, and onsite interaction energies. Here we use one-dimensional matrix product density operators to perform a fully quantum investigation of the dependence of the non-equilibrium steady states of this model on the signs of these parameters. Due to a symmetry in the Lindblad master equation, we find that simultaneously changing the sign of the interaction energies, hopping energies, and chemical potentials leaves the local boson number distribution and inter-site number correlations invariant, and the steady-state complex conjugated. This shows that all driven-dissipative phenomena of interacting bosons described by the Lindblad master equation, such as "fermionization" and "superbunching", can equivalently occur with attractive or repulsive interactions.
We investigate the non-equilibrium dynamics and steady-state properties of a driven-dissipative Bose-Hubbard chain using a self-consistent Gutzwiller mean-field (GMF) approach. By employing a robust Picard iteration scheme, we solve the non-linear master equation for the non-equilibrium steady state (NESS) in the presence of strong Kerr nonlinearity. We identify two distinct dynamical regimes governed by the interplay between coherent drive, dissipation, and interaction: a regular quasilinear regime and a chaotic regime. Linear stability analysis reveals that the transition to the chaotic regime is triggered by parametric instabilities arising from the drive-induced coherence. Furthermore, we characterize the onset of quantum chaos by calculating the out-of-time-order correlator (OTOC). Our results show that in the strong coupling regime, the OTOC exhibits rapid exponential growth and saturation, providing a clear signature of information scrambling in this open quantum system. The proposed numerical framework offers an efficient pathway to explore many-body correlations in larger photonic lattices.