Wave chaos in a randomly inhomogeneous waveguide: Spectral analysis of the finite-range evolution operator

Abstract

The proplem of sound propagation in an oceanic waveguide is considered. Scattering on random inhomogeneity of the waveguide leads to wave chaos. Chaos reveals itself in spectral properties of the finite-range evolution operator (FREO). FREO describes transformation of a wavefield in course of propagation along a finite segment of a waveguide. We study transition to chaos by tracking variations in spectral statistics with increasing length of the segment. Analysis of the FREO is accompanied with ray calculations using the one-step Poincaré map which is the classical counterpart of the FREO. Underwater sound channel in the Sea of Japan is taken for an example. Several methods of spectral analysis are utilized. In particular, we approximate level spacing statistics by means of the Berry-Robnik and Brody distributions, explore the spectrum using the procedure elaborated by A. Relano with coworkers (Relano et al, Phys. Rev. Lett., 2002; Relano, Phys. Rev. Lett., 2008), and analyze modal expansions of the eigenfunctions. We show that the analysis of FREO eigenfunctions is more informative than the analysis of eigenvalue statistics. It is found that near-axial sound propagation in the Sea of Japan preserves stability even over distances of hundreds kilometers. This phenomenon is associated with the presence of a shearless torus in the classical phase space. Increasing of acoustic wavelength degrades scattering, resulting in recovery of localization near periodic orbits of the one-step Poincaré map. Relying upon the formal analogy between wave and quantum chaos, we suggest that the concept of FREO, supported by classical calculations via the one-step Poincaré map, can be efficiently applied for studying chaos-induced decoherence in quantum systems.

Order-to-chaos transition in the model of a quantum pendulum subjected to noisy perturbation

“…For relatively small values of τ , the dots corresponding to eigenfunctions form ordered patterns consisted of distinct slightly biased lines. Such patterns were earlier observed in [23], where they were called "stalagmites". Each "stalagmite" is formed by eigenfunctions localized near periodic orbits of the one-step Poincaré with the same location in the action space.…”

“…Moreover, analysis of level spacing statistics doesn't provide accurate estimate of the regular phase space area [23]. In this way, analysis of FTEO eigenfunctions seems to be more robust way.…”

“…FTEO was firstly utilized in [39] for the problem of noise-driven quantum diffusion. Wave analogue of the FTEO was used in [22,23,[40][41][42].…”

“…We use the quantum counterpart of the one-step Poincaré map, the so-called finite-time evolution operator, for exploring manifestations of finite-time stability in quantum motion. Mathematically equivalent approach was used for studying sound propagation in a randomly-inhomogeneous oceanic waveguide [22,23]. In the present paper, we use this approach for a purely quantum problem, namely for the quantum nonlinear pendulum subjected to broadband perturbation.…”

Random matrix theory for an adiabatically-varying oceanic acoustic waveguide

“…This property means that the propagator matrix belongs to the so-called circular ensemble of random matrices [8]. Scattering on random inhomogeneity reveals itself in statistics of level spacings [7,25]…”