Time-reversed imaging as a diagnostic of wave and particle chaos
Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics
In the presence of multiple scattering, waves and particles behave fundamentally differently. As a model for the stability of the temporal evolution of particle and wave propagation, a scattering system is presented in which particle propagation is strongly unstable while wave propagation is significantly more stable. Both analytical and numerical evidence for the different stability properties of wave and particle propagation is presented; the exponential divergence of particle trajectories
... cle trajectories leads to a critical length scale for the stability of particle propagation that depends exponentially on time ͓exp(Ϫt)͔, whereas the critical length scale for the stability of wave propagation decreases with time only as 1/ͱt. This fundamental difference is due to wave suppression of classical chaos that is intimately related to the concept of ray splitting. ͓S1063-651X͑98͒04611-X͔ PACS number͑s͒: 05.45.ϩb, 03.40.Kf, 03.20.ϩi I. TIME-REVERSED IMAGING AS A DIAGNOSTIC The relation of classical chaotic motion and the corresponding behavior of waves that propagate in the same system has been an active field of research. The words "quantum chaos" suggest that quantum systems can exhibit chaotic behavior. Classical chaotic systems display a fractal structure in phase space. Such a fractal structure in phase space is precluded in quantum mechanics by Heisenberg's uncertainty principle. In addition, closed quantum systems have discrete states that correspond to periodic motion, whereas classical chaos is characterized in the frequency domain by a continuous spectrum ͓1͔. It is thus not clear what the imprint of chaos on quantum-mechanical systems is. For this reason Berry ͓2͔ introduced the phrase "quantum chaology." The relation between classical chaos and quantum chaos is not trivial ͓3͔. For classical systems the Kolmogor-Arnold-Moser ͑KAM͒ tori form impenetrable barriers, but waves can tunnel through these barriers. Conversely, cantori ͑broken up KAM tori͒ can be penetrated by classical trajectories but the finite extent of a wave in phase space practically blocks the waves from crossing a cantorus. In addition, classical trajectories that nearly touch each other are fundamentally different from a classical point of view, but for the corresponding quantum system these touching trajectories lead to new phenomena ͓4͔. Although many aspects of the relation between classical chaos and quantum chaos are not completely understood it is clear that wave effects suppress the chaotic character of systems; one can speak of a quantum suppression of classical chaos ͓3͔. This notion has been formulated in the following way by Gutzwiller ͓5͔: "Quantum mechanics mitigates the destructive influence of classical chaos on simple physical processes. Indeed, quantum mechanics is sorely needed to save us from the bizarre aspects of classical mechanics; but most paradoxically this process of softening the many rough spots is entirely in our grasp as soon as the nature of the roughness is understood." It is the goal of this work to obtain a better understanding of the imprint of classical chaos on wave systems. The stability of wave and particle propagation is studied here using time-reversed imaging ͑TRI͒. The concept of TRI relies on the invariance of Newton's law or the wave equation under time reversal. Consider a wave or particle system that evolves forward in time from a source at time tϭ0 to a later time t. When the motion of the particles or the wave vector of the waves are reversed at this time, the particles and waves will retrace their original trajectories and return to the source where they originally started. However, when the system is perturbed before the reverse propagation the particles or waves do not necessarily return to their original source. The inability to return to the original source position is related to the stability of the wave or particle propagation to perturbations. Ballentine and Zibin ͓6͔ used reverse time propagation to study the stability of wave and particle propagation for the driven quartic oscillator and the periodically kicked rotator when the systems were perturbed by a uniform translation. This study has been motivated by recent laboratory experiments of TRI of acoustic and elastic waves ͓͑7-9͔͒. In these experiments TRI is achieved by driving one or more piezoelectric transducers with a time-reversed version of the recorded wave field. The process has proven to be surprisingly stable, even for an experiment involving a medium with 2000 strong scatterers ͓7͔. In ͓9͔ the ergodicity of stadium boundaries has been exploited to achieve TRI of elastic waves experimentally with only a single receiver.