State Dependent Pseudoresonances and Excess Noise

We show that strong response to nonresonant modulations and excess noise are state dependent in generic nonlinear systems; i.e., they affect some output states but are absent from others. This is demonstrated in complex Swift-Hohenberg models relevant to optics, where it is caused by the non-normality of the linearized stability operators around selected output states, even though the cavity modes are orthogonal. In particular, we find the effective parameters that control excess noise and the response to modulations and show cases where these phenomena are enhanced by an order of magnitude.

Figure 1

Response of (2, 3) to a monochromatic modulation $\exp [i(k_{f}x+{\omega }_{f}t)]$ of the field $\tilde{\psi }$ as a function of ${\omega }_f$ at $k_f=4$. The lines are: maximum response of the linear system based on the norm of the resolvent (thick solid line) or on the distance from the eigenvalues (thick dashed line). The other lines represent the expected response of ${\varphi }_k$ (solid line), ${\varphi }_{-k}$ (dashed line) and of ${\nu }_k$ (dot-dashed line). The crosses, pluses, and circles are the response measured from direct integration of (2, 3) with added forcing term. Parameter values: $\alpha =0.2$, $\rho =1.5$, $\eta =-0.09$, $k_f=4$.

Figure 2

Response off (2, 3) to a modulation $\cos (k_{f}x+{\omega }_{f}t)$ of the population inversion $n$ as a function of ${\omega }_f$ at $k_f=4$. The lines are as in Fig. 1. Parameter values: $\alpha =0.2$, $\rho =1.5$, $\eta =-0.09$, $k_f=4$.

Figure 3

Example of transient growth: plot of $\parallel \exp (L_{k}t)\parallel$ for $k=4$. Parameter values: $\alpha =2$, $\eta =0.09$, $\rho =0.6$. The insert shows the pseudospectrum of $L_k$ for $-20\le k\le 20$. The contour lines correspond to $ϵ={10}^{\{0,-1,-2,-3\}}$ from outside inward.