We explain the mechanism leading to directed chaotic transport in Hamiltonian systems with spatial and temporal periodicity. We show that a mixed phase space comprising both regular and chaotic motion is required and we derive a classical sum rule which allows one to predict the chaotic transport velocity from properties of regular phase-space components. Transport in quantum Hamiltonian ratchets arises by the same mechanism as long as uncertainty allows one to resolve the classical phase-space structure. We derive a quantum sum rule analogous to the classical one, based on the relation between quantum transport and band structure.

It is shown that conductance fluctuations due to phase coherent ballistic transport through a chaotic cavity generically are fractals. The graph of conductance vs externally changed parameter, e.g., magnetic field, is a fractal with dimension Dϭ2Ϫ␤/2 between 1 and 2. It is governed by the exponent ␤ (р2) of the power-law distribution P(t)ϳt Ϫ␤ for a classically chaotic trajectory to stay in the cavity up to time t, which is typical for chaotic systems with a mixed ͑chaotic and regular͒ phase space. The phenomenon should be observable in semiconductor nanostructures and microwave billiards.

We show that chaos and nonlinear resonances are clearly reflected in the magnetotransport in lateral surface superlattices and thereby explain a series of magnetoresistance peaks observed recently in "antidot" arrays on semiconductor heterojunctions. We find a mechanism of cyclotron-orbit pinning in an electric field resulting from Kolmogorov-ArnoFd-Moser tori. An experimental verification is suggested in terms of an enhanced cyclotron frequency associated with an anomalously reduced cyclotron radius.

Although the statistical mechanics of periodically driven ("Floquet") systems in contact with a heat bath has some formal analogy with the traditional statistical mechanics of undriven systems, closer examination reveals radical differences. In Floquet systems all quasienergies epsilon_{j} can be placed in a finite frequency interval 0< or =epsilon_{j}

Conductance fluctuations have been studied in a soft wall stadium and a Sinai billiard defined by electrostatic gates on a high mobility semiconductor heterojunction. These reproducible magnetoconductance fluctuations are found to be fractal confirming recent theoretical predictions of quantum signatures in classically mixed (regular and chaotic) systems. The fractal character of the fluctuations provides direct evidence for a hierarchical phase space structure at the boundary between regular and chaotic motion.

We prove that the temporal autocorrelation function C(t) for quantum systems with Cantor spectra has an algebraic decay C(t)~t ~s, where 8 equals the generalized dimension Di of the spectral measure and is bounded by the Hausdorff dimension DQ>8. We study various incommensurate systems with singular continuous and absolutely continuous Cantor spectra and find extremely slow correlation decays in singular continuous cases (5=0.14 for the critical Harper model and 0<£<0.84 for the Fibonacci chains). In the kicked Harper model we demonstrate that the quantum mechanical decay is unrelated to the existence of classical chaos. PACS numbers: 03.65.-w, 05.45.+b, 73.20.DxThe temporal decay of correlation functions plays an important role in classical physics, as it can be used in ergodic theory to define mixing, a somewhat weaker property than chaotic behavior. As quantum mechanics precludes sensitive dependence on initial conditions, the possibility that quantum systems might exhibit mixing behavior through their decay of correlations has attracted considerable attention in recent years. The situation is far more complex than in classical physics and some of the investigations have arrived at controversial conclusions [1]. The complication is due to the absence of one-to-one relations between the nature of the decay and the spectral type (absolutely continuous, singular continuous, pure point, or any mixture). Only decays faster than any power law can be uniquely related to an absolutely continuous spectrum. Slow power-law decays, however, can be compatible with a singular continuous spectrum, as well as an absolutely continuous spectrum. Therefore, in order to relate the decay to the spectral type, Avron and Simon [2] had to introduce a distinction between "transient" and "recurrent" absolutely continuous spectra. Many problems, however, still remain to be solved. In this situation it is useful to investigate the correlation decay of various systems for which the spectral types are known and to develop a new general concept for a quantitative determination of the correlation decay.In this Letter we first analyze the decay of the correlation function C(t) in the unkicked and kicked Harper model for localized, critical, and extended states, as well as in the Fibonacci chains. Numerically we find slow algebraic decays C(t)~~t~s with 0<£<0.84. This is the first quantitative determination of the correlation decay in these systems and confirms the conjecture of anomalous transport. For the regime of extended states of the Harper model the power-law decay has an exponent £=0.84 ±0.01 reflecting a recurrent absolutely continuous spectrum, whereas the singular continuous spectrum in the critical case gives rise to an extremely slow decay with 5=0.14 ±0.01. The singular continuous spectrum of the Fibonacci chains shows variable exponents 0<5<0.84 with 8 approaching 0.84 as V-* 0, where the spectrum becomes absolutely continuous. In the kicked Harper model, which is classically chaotic, we demonstrate that the decay of the quantum...

We derive a formula predicting dynamical tunneling rates from regular states to the chaotic sea in systems with a mixed phase space. Our approach is based on the introduction of a fictitious integrable system that resembles the regular dynamics within the island. For the standard map and other kicked systems we find agreement with numerical results for all regular states in a regime where resonance-assisted tunneling is not relevant. Tunneling of a quantum particle is one of the central manifestations of quantum mechanics. For simple 1D systems tunneling under a potential barrier is well understood and described, e.g. by using semiclassical WKB theory or the instanton approach [1]. For higherdimensional systems so-called "dynamical tunneling" [2] occurs between regions which are separated by dynamically generated barriers. Typically, such systems have a mixed phase space in which regions of regular motion and irregular dynamics coexist. Tunneling in these systems is barely understood as it generically cannot be reduced to the instanton or WKB approach. It has been studied theoretically [3,4,5,6,7,8,9,10,11,12,13,14] and experimentally, e.g. in cold atom systems [16,17] and semiconductor nanostructures [18]. A precise knowledge of tunneling rates is of current interest for e.g. eigenstates affected by flooding of regular islands [19,20], emission properties of optical microcavities [21] and spectral statistics in systems with a mixed phase space [22].There are different approaches for the prediction of tunneling rates depending on the ratio of Planck's constant h to the size A of the regular island. In the semiclassical regime, h ≪ A, small resonance chains inside the island dominate the tunneling process ("resonanceassisted tunneling") [11,12]. In contrast, we focus on the experimentally relevant regime of large h (while still h < A), where small resonance chains are expected to have no influence on the tunneling rates. This regime has been investigated in Ref. [14], however, the prediction does not seem to be generally applicable (see below). Other studies in this regime investigate situations [13,23], where dynamical tunneling can be described by 1D tunneling under a barrier, however, in our opinion they are non-generic. A generally applicable theoretical description of dynamical tunneling rates in systems with a mixed phase space is still an open question.In this paper we present a new approach to dynamical tunneling from a regular island to the chaotic sea. The central idea is the use of a fictitious integrable system resembling the regular island. This leads to a tunneling formula involving properties of this integrable system as well as its difference to the mixed system under consideration. It allows for the prediction of tunneling rates from any quantized torus within the regular island. We find excellent agreement with numerical data, see Fig. 1, for an example system where tunneling is not affected by phase-space structures like cantori at the border of the island. The applicability to more general sys...

We derive a prediction of dynamical tunneling rates from regular to chaotic phase-space regions combining the direct regular-to-chaotic tunneling mechanism in the quantum regime with an improved resonance-assisted tunneling theory in the semiclassical regime. We give a qualitative recipe for identifying the relevance of nonlinear resonances in a given variant Planck's over 2pi regime. For systems with one or multiple dominant resonances we find excellent agreement to numerics.