Bifurcation analysis of nonlinear coupled photonic nanocavities

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.