Modulational instability of coupled ring waveguides with linear gain and nonlinear loss

Abstract

We consider a nanostructure of two coupled ring waveguides with constant linear gain and nonlinear absorption - the system that can be implemented in various settings including polariton condensates, optical waveguides or atomic Bose-Einstein condensates. It is found that, depending on the parameters, this simple configuration allows for observing several complex nonlinear phenomena, which include spontaneous symmetry breaking, modulational instability leading to generation of stable circular flows with various vorticities, stable inhomogeneous states with interesting structure of currents flowing between rings, as well as dynamical regimes having signatures of chaotic behavior.

Route to chaos in a coupled microresonator system with gain and loss

“…2. It was studied by some of us earlier revealing rich dynamical behavior such as the spontaneous symmetry breaking, modulational instability or complex and persistent flow of currents between the rings [27,28]. In the present work we concentrate on the route of chaos, which may be observed in this system as demonstrated numerically in Sec.…”

“…This interaction is characterized by the presence of a linear gain and nonlinear losses. We adopt the mean field model [27] described by a pair of Gross-Pitaevskii equations (GPE) coupled by a linear homogeneous term, characterized by a constant coupling c…”

“…All modes mentioned above are stable for specific ranges of parameters and in [27] we constructed a two dimensional map in γ/Γ and c to characterize the spectrum of stable solutions. In [27] we also discussed stability condition, applying linear stability analysis. In the present study we focus on possible instabilities describing in detail the "route to chaos" scenarios.…”

“…We have observed that asymptotically, for each set of control parameters, there is a clear behavior that can be classified within the limited set of categories: an antisymmetric state, a vortex state (as defined above), a limit cycle (with one or more frequencies), or finally, an irregular motion that reveals the features of the chaotic dynamics. In the previous study [27] we presented in a compact way the whole gallery of these states in some range of parameters. Here we focus on a small range of states to show possible "routes to chaos".…”

“…For the value of c below the lower bound the system admits regular solutions only. They can be antisymmetric or asymmetric with a constant homogeneous amplitude or even can be inhomogeneous with respect to the spatial coordinate x thus revealing a translational symmetry breaking [27]. All these solutions are treated within this paper as regular fixed points which are sinks (attractors).…”