Fig. 4: Occurrence of a laser-absorber mode for evanescent waves.

Trajectories of the (a) pole and (b) zero of the generalized \(S\left(\hat{\omega }\right)\) matrix in the complex frequency plane, with the damping \(\gamma\) in the anti-PT symmetric dimer defect decreasing from \(\gamma /2{{{\rm{\pi }}}}=2\,{{{\rm{Hz}}}}\) to \(\gamma /2{{{\rm{\pi }}}}=-2\,{{{\rm{Hz}}}}\), as depicted by the solid curves. The pole and zero coincide on the real frequency axis at \(\omega /2{{{\rm{\pi }}}}=1.633\,{{{\rm{kHz}}}}\) (red and blue stars) when \(\gamma ={\gamma }_{{PZ}}=0\), indicating an evanescent wave laser-absorber mode. The pole’s location in the upper (lower) plane indicates the system’s transition to unstable (stable) states, as distinguished by light pink (light green) backgrounds. Dashed curves correspond to experimental scenarios with residual loss \({\gamma }_{c}/2{{{\rm{\pi }}}}=1\,{{{\rm{Hz}}}}\) in the port resonators. Density plot of the base-\(10\) logarithm of the normalized energy flux magnitude for (c) left (\({|}{\hat{J}}_{L}{|}\)) and (d) right (\({|}{\hat{J}}_{R}{|}\)) excitation, plotted against the detunings \({\varepsilon }_{\gamma }\) and \({\varepsilon }_{\omega }\) of the damping \(\gamma\) and normalized excitation frequency \(\hat{\omega }\), relative to the laser-absorber mode. The shaded regions indicate negative energy flux directed from the scatterer to the excitation source. (e, f) Correspond to (c, d), respectively, but display the experimental scenarios with observed energy flux between the scatterer and the excitation port. Common parameters include \(\triangle ={\Delta }_{{{{\rm{PZ}}}}}\), with other unspecified system parameters the same as in Fig. 3.