Imaginary potential as a counter of delay time for wave reflection from a one-dimensional random potential

Abstract

We show that the delay time distribution for wave reflection from a one-dimensional (1-channel) random potential is related directly to that of the reflection coefficient, derived with an arbitrarily small but uniform imaginary part added to the random potential. Physically, the reflection coefficient, being exponential in the time dwelt in the presence of the imaginary part, provides a natural counter for it. The delay time distribution then follows straightforwardly from our earlier results for the reflection coefficient, and coincides with the distribution obtained recently by Texier and Comtet [C.Texier and A. Comtet, Phys.Rev.Lett. 82, 4220 (1999)], with all moments infinite. Delay time distribution for a random amplifying medium is then derived . In this case, however, all moments work out to be finite.When a wavepacket centered at an energy E is scattered elastically from a scattering potential, it suffers a time delay before spreading out dispersively. This delay is related to the time for which the wave dwells in the interaction region. For the general case of a scatterer coupled to N open channels leading to the continuum, one defines the phase-shift time delays through the Hermitian energy derivative of the S-matrix, −ihS −1 ∂S/∂E, whose eigenvalues give the proper delay times. These delay times then averaged over the N-channels give the Wigner-Smith delay times introduced first by Wigner [1] for the 1-channel case, and generalized later by Smith [2] to the case of N open channels. Thus scattering delay time is the single most important quantity describing the time-dependent aspect i.e., physically, the reactive aspect of the scattering in open quantum systems, e.g. the chaotic microwave cavity and the quantum billiard (whose classical motion is chaotic) and the solid-state mesoscopic dots coupled capacitively to open leads terminated in the reservoir. The delay time is however not self averaging and one must have its full probability distribution over a statistical ensemble of random samples. The latter may be related ergodically to the ensembles generated parametrically e.g. by energy E variation over a sufficient interval. Thus we have the random matrix theory (RMT) for circular ensembles of the S-matrix giving delay times for all the three Dyson Universality classes for the case of a chaotic cavity connected to a single open channel [3]. Generalization to the case of N channels corresponded to the Laguarre ensemble [4] of RMT. The RMT approach has been treated earlier through the supersymmetric technique for the case of a quantum chaotic cavity having a few equivalent open channels [5]. However it has been suspected for quite sometime that the RMT based results and the universality claimed thereby may not extend to a strictly 1-dimensional random system where Anderson localization dominates, and that the 1D random system may constitute after all a different universality class [6]. This important problem has been re-examined recently by Texier and Comtet [7] who have derived the delay time distrib...

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