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.

Using periodic-orbit theory beyond the diagonal approximation we investigate the form factor, K(tau), of a generic quantum graph with mixing classical dynamics and time-reversal symmetry. We calculate the contribution from pairs of self-intersecting orbits that differ from each other only in the orientation of a single loop. In the limit of large graphs, these pairs produce a contribution -2tau(2) to the form factor which agrees with random-matrix theory.

We present a comprehensive account of directed transport in one-dimensional Hamiltonian systems with spatial and temporal periodicity. They can be considered as Hamiltonian ratchets in the sense that ensembles of particles can show directed ballistic transport in the absence of an average force. We discuss general conditions for such directed transport, like a mixed classical phase space, and elucidate a sum rule that relates the contributions of different phase-space components to transport with each other. We show that regular ratchet transport can be directed against an external potential gradient while chaotic ballistic transport is restricted to unbiased systems. For quantized Hamiltonian ratchets we study transport in terms of the evolution of wave packets and derive a semiclassical expression for the distribution of level velocities which encode the quantum transport in the Floquet band spectra. We discuss the role of dynamical tunneling between transporting islands and the chaotic sea and the breakdown of transport in quantum ratchets with broken spatial periodicity.

We present the first quantum system where Anderson localization is completely described within periodic-orbit theory. The model is a quantum graph analogous to an a-periodic Kronig-Penney model in one dimension. The exact expression for the probability to return of an initially localized state is computed in terms of classical trajectories. It saturates to a finite value due to localization, while the diagonal approximation decays diffusively. Our theory is based on the identification of families of isometric orbits. The coherent periodic-orbit sums within these families, and the summation over all families are performed analytically using advanced combinatorial methods.Anderson localization is a genuine quantum phenomenon. So far, attempts to study this effect within a semiclassical (periodic-orbit) theory seemed to be doomed to fail from the outset: It is not clear whether the leading semiclassical approximation for the amplitude associated with a single classical orbit is sufficiently accurate. Even more seriously, there is no method available to add coherently the contributions from the exponentially large number of contributing orbits. Here, we address the second problem and develop a method to perform the coherent periodic-orbit (PO) sums in a standard model-a quantum graph analogous to the Kronig-Penney model in 1D-for which the PO theory is exact. For a list of references on the long history of graph models see [1].For investigating Anderson localization we consider the quantum return probability (RP). It is defined as the mean probability that a wave packet initially localized at a site is at the same site after a given time. We show that the long-time RP approaches a positive constant, which proves that the spectrum has a point-like component with normalizable eigenstates. The asymptotic RP is the inverse participation ratio, which is a standard measure of the degree of localization. The RP can also be seen as 2-point form factor of the local spectrum [2]. As such, it belongs to the class of quantities which can be expressed as double sums over PO's of the underlying classical dynamics [3]. Because of the exponential proliferation of the PO's in chaotic systems, the resulting sums are hard to perform. Consequently, most semiclassical approaches to spectral two-point correlations were restricted to the diagonal approximation where the interference between different PO's is neglected [3,2,[4][5][6]. While this method is very successful for short-time correlations, it fails to reproduce long-time effects such as Anderson localization which are due to quantum interferences. In [7] the universal long-time behavior of the form factor was related to universal classical action correlations between PO's of a chaotic system. A deeper understanding of how quantum universality is encoded in classical correlations is highly desirable but still lacking, despite some recent progress [8,9]. This context is our motivation for developing the first PO theory of 1D Anderson localization, although the phenomenon as such is...

We derive an explicit expression for the coupling constants of individual eigenstates of a closed billiard which is opened by attaching a waveguide. The Wigner time delay and the resonance positions resulting from the coupling constants are compared to an exact numerical calculation. Deviations can be attributed to evanescent modes in the waveguide and to the finite number of eigenstates taken into account. The influence of the shape of the billiard and of the boundary conditions at the mouth of the waveguide are also discussed. Finally we show that the mean value of the dimensionless coupling constants tends to the critical value when the eigenstates of the billiard follow random-matrix theory.