Robust Scattered Fields from Adiabatically Driven Targets around Exceptional Points

Scattering processes are typically sensitive to the incident wave properties and to interference effects generated via wave-matter interactions with the target. We challenge this general belief in the case of targets that undergo time-periodic modulations encircling quasiadiabatically an exceptional point in a given parameter space. When the scattering dwell time is above a critical value ${\tau }_c$, the scattered field is surprisingly insensitive to the properties of the incoming wave and local operational details of the driving. Instead, it reaches a fixed point attractor that can be controlled by the direction of the driving cycle. For dwell times below ${\tau }_c$, the unusual robustness is abruptly suppressed. Such protocols may become useful tools in control engineering, including the management of thermal and quantum fluctuations.

Figure 1

(a) A scattering setup where an EP degeneracy is adiabatically encircled via the variation of two parameters. Magenta arrows indicate a fixed outgoing scattering field which is independent from incident excitations with random amplitudes and phases (yellow arrows of different width); (b) CW encirclement leads to a scattered field that has a “fixed-point” attractor determined by the mode $|{\mu }_1^t⟩$ (i.e., the normalized weight amplitude ${\tilde{b}}_2=0$) for $\mathrm{\Gamma }<{\mathrm{\Gamma }}_c$. For $\mathrm{\Gamma }>{\mathrm{\Gamma }}_c$, the scattered field is random, i.e., both ${\tilde{b}}_{1,2}$ depend on the input. (c) The same as in (b) but now for a CCW encirclement. In this case the scattered signal is dominated by $|{\mu }_2^t⟩$, i.e., ${\tilde{b}}_1=0$ for $\mathrm{\Gamma }<{\mathrm{\Gamma }}_c$. The sharp transition at $\mathrm{\Gamma }={\mathrm{\Gamma }}_c$ is indicated with the transparent green plane. We assumed that $\gamma =1$ (unit of frequency).

Figure 2

(a) An ensemble of initial waves $|S^{+}⟩$ is sent into a quasiadiabatically modulated linear target when the coupling to the continuum [indicated by the green plane in the $\mathcal{I}m(\lambda )$ axis of the spectrum of $H_0$] is weak in comparison with the cumulative amplification that the field experiences during one driving cycle. The solid trajectories on the eigenfrequencies surfaces represent the measure $p(t)=[|a_1(t){|}^2{\lambda }_1^t+|a_2(t){|}^2{\lambda }_2^t]/[|a_1(t){|}^2+|a_2(t){|}^2]$, which quantifies the relative weight with which each instantaneous eigenvalue and the corresponding eigenvector $|{\lambda }_{1,2}^t⟩$ participate in the internal evolution. The weights $a_{1,2}$ are evaluated via the decomposition of the evolved state $|\mathrm{\Psi }⟩$ in the instantaneous basis of $H_0$. At the end of the CW (CCW) loop around an EP (upper [lower] subfigures) the field inside the scatterer is dominated by only one eigenmode, residing at the blue (red) Riemann surface. This internal dynamics overwhelms the outgoing signal, $|S^{-}⟩=(S_1^{-},S_2^{-}{)}^T$, whose ratio $S_2^{-}/S_1^{-}$ reaches a fixed-point attractor (see red and blue points at the outgoing panel). (b) The same as in (a) but for a strong coupling to the continuum. In this case, the outgoing signal is largely influenced by the properties of the incident wave.

Figure 3

Electric-circuit scattering setup where a quasiadiabatic encirclement of an EP degeneracy can lead to a chiral control of the scattered field when the coupling to TLs is smaller than a critical value. The outgoing scattered field, $|S_V^{-}⟩=(S_1^{-},S_2^{-}{)}^T$, has an attractor determined by the direction of encircling (bottom left). When the coupling to the continuum is large, the outgoing field is sensitive to the specific features of the incident field.