Multistable dissipative structures pinned to dual hot spots

Abstract

We analyze the formation of one-dimensional localized patterns in a nonlinear dissipative medium including a set of two narrow "hot spots" (HSs), which carry the linear gain, local potential, cubic self-interaction, and cubic loss, while the linear loss acts in the host medium. This system can be realized as a spatial-domain one in optics and also in Bose-Einstein condensates of quasiparticles in solid-state settings. Recently, exact solutions were found for localized modes pinned to the single HS represented by the δ function. The present paper reports analytical and numerical solutions for coexisting two- and multipeak modes, which may be symmetric or antisymmetric with respect to the underlying HS pair. Stability of the modes is explored through simulations of their perturbed evolution. The sign of the cubic nonlinearity plays a crucial role: in the case of the self-focusing, only the fundamental symmetric and antisymmetric modes, with two local peaks tacked to the HSs, and no additional peaks between them, may be stable. In this case, all the higher-order multipeak modes, being unstable, evolve into the fundamental ones. Stability regions for the fundamental modes are reported. A more interesting situation is found in the case of the self-defocusing cubic nonlinearity, with the HS pair giving rise to a multistability, with up to eight coexisting stable multipeak patterns, symmetric and antisymmetric ones. The system without the self-interaction, the nonlinearity being represented only by the local cubic loss, is investigated too. This case is similar to those with the self-focusing or defocusing nonlinearity, if the linear potential of the HS is, respectively, attractive or repulsive. An additional feature of the former setting is the coexistence of the stable fundamental modes with robust breathers.

Photon-pair generation in a lossy microring resonator. I. Theory

“…In the Appendix we examine G(ω) and H(ω) in the high cavity Q limit defined by ρ k ≡ e −γ k T k /2 → 1, ω T k 1 where the standard Langevin approximation [17,20] is valid, and compare our results with recent related work [10] using the later formulation.…”

“…The second term in Eq. (32c) e −iθp(ω) G ab (ω) G * ba (ω) ∝ |g α p T ab | 2 represents a higher order pump dependent correction to ψ (2) ab (ω) involving the product of Lorentzian lineshape factors √ γ k /(s + Γ k /2) (where s ≡ −iω can be considered as a Laplace transform solution variable [10]), relating the fields ( a) k inside the cavity to the input fields ( a in ) k .…”

“…The two-photon generation rate [7,10] is given by R ab (ω) = |r ab (ω)| 2 |ψ (2) ab (ω)| 2 . The second line of Eq.…”

“…, (35) and the output heralding efficiency [10] of say the an idler photon by the measurement of a signal photon as…”

“…Such high-Q mircoring resonators exhibit nonlinear optical properties allowing for biphoton generation arising from the χ (2) processes of spontaneous parametric down conversion (SPDC), and the χ (3) processes of spontaneous four-wave mixing (SFWM). Much theoretical research has been devoted to study the generation of entangled photon pair within cavities and mrr in the weak pump field driving limit [2,[7][8][9][10][11][12][13][14], and more recently in the strong pump field regime [15] where higher order nonlinear effects such as self-phase and cross-phase modulation become important.…”