Wave excitation and dynamics in non-Hermitian disordered systems

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 fundamental wave properties with loss and gain, and their dependence upon dimensionality and spectral overlap. We begin with the demonstrations that the transmission time in random 1D media is equal to the density of states even in the presence of ultrastrong absorption and that its ensemble average is independent of the strengths of scattering and absorption. In contrast, the Wigner time becomes imaginary in the presence of loss, with real and imaginary parts that fall with absorption. In multichannel media, the transmission time remains equal to the density of states and is independent of the scattering strength in unitary systems but falls with absorption to a degree that increases with the strengths of absorption and scattering, and the number of channels coupled to the medium. We show that the relationships between key propagation variables in non-Hermitian systems can be understood in terms of the singularities of the phase of the determinant of the transmission matrix. The poles of the transmission matrix are the same as those of the scattering matrix, but the transmission zeros are fundamentally different. Whereas the zeros of the scattering matrix are the complex conjugates of the poles, the transmission zeros are topological: in unitary systems they occur only singly on the real axis or as conjugate pairs. We follow the evolution and statistics of zeros in the complex plane as random samples are deformed. The sensitivity of the spacing of zeros in the complex plane with deformation of the sample has a square-root singularity at a zero point at which two single zeros and a complex pair interconvert. The transmission time is a sum of Lorentzian functions associated with poles and zeros. The sum over poles is the density of states with an average that is independent of scattering and dissipation. But the sum over zeros changes with loss, gain, scattering strength and the number of channels in ways that make it possible to control ultranarrow spectral features in transmission and transmission time. We show that 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 energy excited may be expressed in terms of the resonances of the medium and is equal to the dwell time even in the presence of loss or gain.