Wave excitation and dynamics in non-Hermitian disordered systems

Abstract

Dynamic and steady state aspects of wave propagation are deeply connected in lossless open systems in which the scattering matrix is unitary. There is then an equivalence among the energy excited within the medium through all channels, the Wigner time delay, which is the sum of dwell times in all channels coupled to the medium, and the density of states. But these equivalences fall away in the presence of material loss or gain. In this paper, we use microwave measurements, numerical simulations, and theoretical analysis to discover the changing relationships among the internal field, transmission, transmission time, dwell time, total excited energy, and the density of states in with loss and gain, and their dependence upon dimensionality and spectral overlap. The field, including the contribution of the still coherent incident wave, is a sum over modal partial fractions with amplitudes that are independent of loss and gain. The total energy excited is equal to the dwell time. When modes are spectrally isolated, the energy is proportional to the sum of products of the modal contribution to the density of states and the ratio of the linewidth without and with absorption. When modes overlap, however, the total energy is further reduced by loss and enhanced by gain. In 1D, the average transmission time is independent of loss, gain, and scattering strength. In higher dimensions, however, the average transmission time falls with loss, scattering strength and channel number, and increases with gain. It is the sum over Lorentzian functions associated with the zeros as well as the poles of the transmission matrix. This endows transmission zeros with robust topological characteristics: in unitary media, zeros occur either singly on the real axis or as conjugate pairs in the complex frequency plane. The transmission zeros are moved down or up in the complex plane by an amount equal to the internal rate of decay or growth of the field. In weakly absorbing media, the spectrum of the transmission time of the lowest transmission eigenchannel is the sum of Lorentzians due to transmission zeros plus a background due to far-off-resonance poles. As the scattering strength increases, conjugate pairs of zeros are brought to the real axis and converted to two single zeros which are constrained to move on the real axis until they combine and leave the real axis as a conjugate pair. The average density of transmission zeros in the complex plane is found from the fall of the average transmission time with absorption as zeros are swept into the lower half of the complex frequency plane. As a sample is deformed, two single zeros and a conjugate pair of zeros may interconvert on the real axis with a square root singularity in the sensitivity of the spacing between transmission zeros to structural change. Thus, the disposition of poles and zeros in the complex frequency plane provides a framework for understanding and controlling wave propagation in non-Hermitian systems.

Reflectionless programmable signal routers

“…Zeros of scattering coefficients that lie on the real frequency axis lead to phase singularities and anomalously long diverging delay times (7,25,(46)(47)(48)(49). The interpretation and statistical properties of complex Wigner time delays in subunitary (and possibly overmoded) scattering systems, as well as their relation to the singularities (poles and zeros) of the associated wave transport matrix, is currently an active area of research (26,(48)(49)(50)(51). In general, away from such singularities, it is well established that longer dwell times increase the sensitivity to minute perturbations, with important implications for precision sensing (52,53).…”

Quantum echo route towards exceptional points in Anderson localized lasers

“…The ability of precise nanofabrication and on-demand gain/loss has allowed for the realization of several photonic platforms for the observation of EPs [33,34]. However, to our knowledge, there are no experimental reports of observations of EPs in disordered mesoscopic systems despite several investigations on non-Hermiticity in such systems [35][36][37][38][39][40][41].…”

Time delay in 1D disordered media with high transmission

“…To gain some insight into the evolution of the statistics of the time delay with the system length, we study the first and second moments of τ r as a function of the system length. The average τ r is given by [16,39,42]…”

“…This result agrees with a general property: the average time delay is determined essentially by the boundary of the system and it is independent of the details of the disorder. This interesting property has been studied experimentally and theoretically [39][40][41][42][43], and it has been deduced based on the direct relation of the average of the total time delay, τ = T τ t + Rτ r , with the density of states inside the sample [43][44][45]. Notice that in our 1D systems, as a consequence of the unitarity of the scattering matrix, τ t = τ r [11].…”