Scaling theory of noisy nonlinear dynamical systems and dynamical localization in quantum systems

Abstract

The effects of additive colored noise near the onset of chaos have been studied for one--dimensional logistic map. Scaling form of Lyapunov exponent λ near the transition point was determined in terms of the noise amplitude σ and the correlation time τα, and the critical exponent α associated with τ was evaluated. The analytical study shows that a certain approximation, which we call the effective white noise approximation, followed by the renormalization group treatment, leads to essentially the same structure of scaling function as the one determined by the numerical method. We studied the quantum kicked rotator in order to investigate the role of the phase space structures, especially, the KAM tori in quantum dynamics. We found that the wave function show an exponential localization between the KAM tori and exponential tails and plateau structures outside of them in momentum space. The KAM tori have played a role as partial block of the probability flows in quantum dynamics. We also found numerically that the probability penetration through the KAM tori is exponential, and the average penetration depth scales in the limit $\hbar$ → 0 as λ ~ \hbar^{-0.685}$. Using the Wigner function formalism in terms of the density operator we evaluated analytically the scaling exponent of the penetration depth in the semiclassical limit. The destruction of localization, or decoherence, of the quantum kicked rotator coupled to external heat bath through the momentum variable has been studied. The heat bath is modeled as an external white noise source with Gaussian distribution. By inspecting the probability flows across the trace of the KAM tori in classical phase space and the change of the diffusion rates as the noise intensity is varied, we showed that the quantum coherence effect, localization, is destructed by external noise. A qualitative explanation of the destruction of localization by external noise is given. We attributed it to the appearance of continuous eigen...