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arXiv:0707.0846 · New Journal of Physics

Solitonic behaviour in coupled multi atom–cavity systems

Mauro Paternostro, G. S. Agarwal, M S Kim

Published Jan 30, 2009• 25 citations
arXivHTMLPDF

Abstract

In view of current research effort on semiconducting photonic-crystal defect microcavities, we consider the dynamics of an array of coupled optical cavities, each containing an ensemble of qubits. By concentrating on the strong coupling regime, we analytically prove that the nonlinearity inherent in the dynamics of each ensemble coupled to the respective cavity field allows the formation of solitons. We further show how the use of the Holstein–Primakoff transformation and the large-detuning limit with the cavity allows one to recover the Bose–Hubbard model.

DOI
10.1088/1367-2630/11/1/013059

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In view of current research effort on semiconducting photonic-crystal defect microcavities, we consider the dynamics of an array of coupled optical cavities, each containing an ensemble of qubits. By concentrating on the strong coupling regime, we analytically prove that the nonlinearity inherent in the dynamics of each ensemble coupled to the respective cavity field allows the formation of solitons. We further show how the use of the Holstein–Primakoff transformation and the large-detuning limit with the cavity allows one to recover the Bose–Hubbard model.
We reveal the existence of polariton soliton solutions in the array of weakly coupled optical cavities, each containing an ensemble of interacting qubits. An effective, complex Ginzburg-Landau equation is derived in the continuum limit, taking into account the effects of cavity field dissipation and qubit dephasing. We have shown that an enhancement of the induced nonlinearity can be achieved by two orders of magnitude with negative interaction strength, which implies a large negative qubit-field detuning as well. Bright solitons are found to be supported under perturbations only in the upper (optical) branch of polaritons, for which the corresponding group velocity is controlled by tuning the interacting strength. With the help of perturbation theory for solitons, we also demonstrate that the group velocity of these polariton solitons is suppressed by the diffusion process.
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.
Recently, various research groups have studied spontaneous symmetry breaking and bistable behaviour in both active and passively driven coupled photonic nanocavities[1]. This work is of interest both in itself and because it naturally extends to coupled nonlinear quantum systems described by the Bose-Hubbard model [2]. Here we present a theoretical study of the dynamics in such a system by considering the bifurcation behaviour as the different parameters are varied. In normalised units the model can be written as: dA/dt = i(δ τ + |A|2)A-A+(iκ + γ)τB+f (1) dB/dt = i(δ τ + |B|2)B-B+(iκ + γ)τA+f (2) where A and B are the slowly varying normalised amplitudes of the fields in each nano-cavity. Further, δ is the detuning from the cavity resonance, τ is the photon lifetime, κ and γ describe the linear coupling between the cavities (if γ = 0 and κ < 0 the model is identical to the Bose-Hubbard model [2] in the semiclassical regime), while f represents a coherent driving term. In deriving the above equations it is assumed that the cavities are identical, and so a focus is the switching between the symmetrical state [A(t) = B(t)] and asymmetrical states, which must come in pairs. Fig. 1(a) shows a typical boundary curve dividing the (δ, f) plane into the regions where the symmetric and asymmetric solutions exist. Further analysis shows that more complicated dynamics arise from the asymmetric state. Fig. 1(b) shows the field intensity for the symmetric and asymmetric steady-states as a function of f, where stable solutions are represented by solid and unstable ones by dashed curves. Hopf bifurcations, marked by H, cause the asymmetric solution to change stability. A typical bifurcating periodic solution is shown in Fig. 1(c) for f = 6.
We examine the problem of polariton soliton formation in the array of weakly coupled optical cavities, each containing an ensemble of ultracold non-interacting two-level atoms. An effective complex Ginzburg-Landau equation (GLE) is derived in the continuum limit taking into account the effects of cavity field dissipation and atomic polarization dephasing. Bright solitons are found to be supported under perturbations only in the upper (optical) branch of polaritons, for which the corresponding group velocity is controlled by tuning the atom-light detuning. With the help of perturbation theory for solitons, we also demonstrate that the group velocity of these polariton solitons is suppressed by the diffusion process.
Abstract We investigate a hybrid quantum system consisting of two coupled optical cavities, each containing an ensemble of coherently pumped artificial or real two-level atoms. The system is modelled within the dressed-state formalism, accounting for coherent driving and dissipative interactions with independent electromagnetic reservoirs. Focusing on the regime where the Rabi frequency greatly exceeds the collective spontaneous emission and cavity decay rates, respectively, we derive the equations of motion characterizing the system and analyse its steady-state behaviours. Special attention is given to the scenario in which only one cavity contains a pumped multi-atom ensemble, and as a consequence, coherent excitation transfer to the second cavity occurs via cavity–cavity coupling. In this case, the mean photon number in the second cavity mode scales proportional to the squared number of atoms, i.e., N2 from the first cavity. The proposed model offers insight into light–matter interaction in composite systems and suggests viable mechanisms for engineering collective interactions and non-local excitation transport in quantum optical platforms.
Nowadays the light-matter interaction phenomena in spatially periodic structure have been studied extensive, i.e. photonic crystals, cavity array. In this thesis, we consider coherent properties of coupled atom-field systems with adding the interaction between atoms and discuss the quantum phase transition and envelope function method for this system. We further show that beyond widely used low excitation density limit and Holstein-Primakoff transformation. We have shown that the phase transition diagram of the normal to superfluid (BEC) state of low branch polaritons in the critical temperature domain. Finally, a multiple-scale technique has been used to analyze the electromagnetic properties of the nonlinear dielectric periodic thin-film stacks. We have found a solitary–wave equation behavior in the array which includes the atom-atom interaction.