Abstract
We discuss the dynamics of particles in one dimension in potentials that are random in both space and time. The results are applied to recent optics experiments on Anderson localization, in which the transverse spreading of a beam is suppressed by random fluctuations in the refractive index. If the refractive index fluctuates along the direction of the paraxial propagation of the beam, the localization is destroyed. We analyze this broken localization in terms of the spectral decomposition of the potential. When the potential has a discrete spectrum, the spread is controlled by the overlap of Chirikov resonances in phase space. As the number of Fourier components is increased, the resonances merge into a continuum, which is described by a Fokker–Planck equation. We express the diffusion coefficient in terms of the spectral intensity of the potential. For a general class of potentials that are commonly used in optics, the solutions to the Fokker–Planck equation exhibit anomalous diffusion in phase space, implying that when Anderson localization is broken by temporal fluctuations of the potential, the result is transport at a rate similar to a ballistic one or even faster. For a class of potentials which arise in some existing realizations of Anderson localization, atypical behavior is found.
Highlights
We discuss the dynamics of particles in one dimension in potentials that are random in both space and time
Some of the experiments that demonstrate an optical realization of Anderson localization involve an induction technique, where a change in the refractive index of a dielectric is induced by an interference pattern generated by external waves [28, 29]
In this paper, using the standard argument that at high energies quantum interference effects are negligible, we used effective particles to describe the dynamics, and argued that when Anderson localization is destroyed the long-time dynamics is determined by a semi-classical approximation
Summary
Some of the experiments that demonstrate an optical realization of Anderson localization (such as that described by Schwartz et al [3]) involve an induction technique, where a change in the refractive index of a dielectric is induced by an interference pattern generated by external waves (used strictly to induce the potentials) [28, 29]. In the limit of N → ∞ the Chirikov resonances become dense in momentum, which appears at first sight to complicate the problem In this limit the quasi-periodic potential is replaced by a random potential, and the change of the momentum in a time interval which is longer than the correlation time of this potential can be regarded as a stochastic variable. Using the assumptions (3) results in a translationally invariant correlation function in both space and time, σ2 C(x1 − x2, t1 − t2) = N km exp[i(km(x1 − x2) − ωm(t1 − t2))] + c.c. where P(k, ω) is the probability density of ω and k, which will dubbed, in what follows, the spectral content of the potential, introduced along with equation (3). An example of this type will be discussed
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
