Fig. 5: Limit cycles and temporal dynamics in nonlinear, non-Hermitian multimode systems.
From: Non-reciprocal frequency conversion in a non-Hermitian multimode nonlinear system
a For low quality factors of the idler bath mode (QT), the system initially features damped relaxation oscillations (ROs) and the amplitude modulation drives the system towards a stable steady state that features non-reciprocal frequency conversion. As QT increases, the system eventually transitions into a regime of stable limit cycles that feature periodic oscillations in modal energy. Finally, larger QT cause a return back to damped ROs and Bloch oscillations, with the nonlinearity and amplitude modulation enabling a gain-loss equilibrium. Oscillations in modal energy in the limit cycle regime have a GHz repetition rate, roughly corresponding to the amplitude modulation strength κ. b Phase diagram showing steady state (SS) and limit cycle (LC) regimes as a function of the nonlinear and amplitude modulation strengths. For small κ, a symmetric comb in steady state is produced (ξ, η ≈ 0, with reciprocal frequency conversion (RFC SS)). For larger κ, an asymmetric comb in steady state is produced by the non-reciprocal frequency conversion (NRFC SS, ξ, η > 0). For the largest κ, LCs are present. Along the transition boundary, the temporal dynamics feature long-lasting Bloch solitons where energy bounces back and forth in the frequency space cavity. c Representative time traces of the power in the outcoupled field \({s}_{{{{\rm{out}}}}}(t)={\sum }_{n}{s}_{n,{{{\rm{out}}}}}(t){e}^{-i{\omega }_{n}t}\) corresponding to the phases in (b). Simulation parameters used in (a) are Q0 = 1.1 × 107, κ = 2π ⋅ 100 MHz, β0 = 10−4 J−1/2. The pump power is 1 kW and the comb length is 2N + 1 = 21.
