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arXiv:1205.0994 · New Journal of Physics

Non-equilibrium many-body effects in driven nonlinear resonator arrays

Thomas Grujic, Stephen R. L. Clark, Dieter Jaksch, Dimitris G. Angelakis

Published Oct 16, 2012• 56 citations
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Abstract

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.

DOI
10.1088/1367-2630/14/10/103025

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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.
Strong effective photon-photon interactions mediated by atom-photon couplings have been routinely achievable in QED setups for some time now. Recently, there have been several proposals to push the physics of interacting photons into many- body distributed architectures. The essential idea is to coherently couple together arrays of QED resonators, such that photons can hop between resonators while interacting with each other inside each resonator. These proposed structures have attracted intense theoretical attention while simultaneously inspiring experimental efforts to realise this novel regime of strongly-correlated many-body states of light. A central challenge of both theoretical and practical importance is to understand the physics of such coupled resonator arrays (CRAs) beyond equilibrium, when unavoidable (or sometimes even desired) photon loss processes are accounted for. This thesis presents several studies whose purpose can roughly be divided in two aims. The first part studies just what constitutes a valid physical and computational representation of non-equilibrium driven-dissipative CRAs. Addressing these ques- tions constitutes essential groundwork for further investigations of CRA phenomena, as numerical experiments are likely to guide and interpret near-future experimen- tal array observations. The relatively small body of existing work on CRAs out of equilibrium has often truncated their full, rich physics. It is important to establish the effects and validity of these approximations. To this end we introduce powerful numerical algorithms capable of efficiently simulating the full dynamics of CRAs, and use them to characterise the non-equilibrium steady states of arrays reached under the combined influence of dissipation and pumping. Having established the rigour necessary to realistically describe CRAs, we exam- ine two novel phenomena observable in near-future small arrays. Firstly we relate a counter-intuitive ‘super bunching’ in the statistics of photons emitted from arrays engineered to demonstrate strong effective photon-photon repulsion at the single and two-photon level, to an interplay between the underlying eigen-structure and details of the non-equilibrium operation. Secondly we characterise a dynamical phenomenon in which domains of ‘frozen’ photons remain trapped in sufficiently nonlinear arrays. Finally we present a preliminary characterisation of a previously unexplored phase diagram of arrays under coherent two-photon pumping. Com- petition between the coherence injected by the pumping, photon interactions and delocalisation processes lead to interesting new physical signatures.
We analyze the nonequilibrium behavior of driven nonlinear photonic resonator arrays under the selective excitation of specific photonic many-body modes. Targeting the unit-filled ground state, we find a counterintuitive ``superbunching'' in the emitted photon statistics in spite of relatively strong on-site repulsive interaction. We consider resonator arrays with Kerr nonlinearities described by the Bose-Hubbard model, but also show that an analogous effect is observable in near-future experiments coupling resonators to two-level systems as described by the Jaynes-Cummings-Hubbard Hamiltonian. For the experimentally accessible case of a pair of coupled resonators forming a photonic molecule, we provide an analytical explanation for the nature of the effect.
We analyze in detail an open cavity array using mean-field description, where each cavity field is coupled to a number of three-level atoms. Such a system is highly tunable and can be described by a Jaynes–Cummings like Hamiltonian with additional nonlinear terms. In the single cavity case we provide simple analytic solutions and show, that the system features a bistable region. The extra nonlinear term gives rise to a rich dynamical behavior including occurrence of limit cycles through Hopf bifurcations. In the limit of large nonlinearity, the system exhibits an Ising like phase transition as the coupling between light and matter is varied. We then discuss how these results extend to the two-dimensional case.
We investigate Beliaev-Landau scattering in a gas of interacting photons in a coherently driven array of nonlinear dissipative resonators, as described by the one-dimensional driven-dissipative Bose-Hubbard model. Due to the absence of detailed balance in such an out-of-equilibrium setup, steady-state properties can be much more sensitive to the underlying microscopic dynamics. Because the popular truncated Wigner approximation dramatically fails in capturing this physics, we present an alternative approach, based on a systematic expansion beyond the Bogoliubov approximation, which includes the third-order correlation functions in the dynamics. As experimentally accessible signatures of Beliaev-Landau processes, we report a small but non-negligible correction to the Bogoliubov prediction for the steady-state momentum distribution, in the form of a characteristic series of peaks and dips, as well as non-Gaussian features in the statistics of the cavity output field.
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 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.