The triangle map: a model of quantum chaos

We study an area preserving parabolic map which emerges from the Poincar\' e map of a billiard particle inside an elongated triangle. We provide numerical evidence that the motion is ergodic and mixing. Moreover, when considered on the cylinder, the motion appear to follow a gaussian diffusive process.Comment: 4 pages in RevTeX with 4 figures (in 6 eps-files

Quantum chaos and the double-slit experiment

We report on the numerical simulation of the double-slit experiment, where the initial wave-packet is bounded inside a billiard domain with perfectly reflecting walls. If the shape of the billiard is such that the classical ray dynamics is regular, we obtain interference fringes whose visibility can be controlled by changing the parameters of the initial state. However, if we modify the shape of the billiard thus rendering classical (ray) dynamics fully chaotic, the interference fringes disappear and the intensity on the screen becomes the (classical) sum of intensities for the two corresponding one-slit experiments. Thus we show a clear and fundamental example in which transition to chaotic motion in a deterministic classical system, in absence of any external noise, leads to a profound modification in the quantum behaviour.Comment: 5 pages, 4 figure

The Sato Grassmannian and the CH hierarchy

We discuss how the Camassa-Holm hierarchy can be framed within the geometry of the Sato Grassmannian.Comment: 10 pages, no figure

Asymmetric Wave Propagation in Nonlinear Systems

A mechanism for asymmetric (nonreciprocal) wave transmission is presented. As a reference system, we consider a layered nonlinear, non mirror-symmetric model described by the one-dimensional Discrete Nonlinear Schreodinger equation with spatially varying coefficients embedded in an otherwise linear lattice. We construct a class of exact extended solutions such that waves with the same frequency and incident amplitude impinging from left and right directions have very different transmission coefficients. This effect arises already for the simplest case of two nonlinear layers and is associated with the shift of nonlinear resonances. Increasing the number of layers considerably increases the complexity of the family of solutions. Finally, numerical simulations of asymmetric wavepacket transmission are presented which beautifully display the rectifying effect

A 3-component extension of the Camassa-Holm hierarchy

We introduce a bi-Hamiltonian hierarchy on the loop-algebra of sl(2) endowed with a suitable Poisson pair. It gives rise to the usual CH hierarchy by means of a bi-Hamiltonian reduction, and its first nontrivial flow provides a 3-component extension of the CH equation.Comment: 15 pages; minor changes; to appear in Letters in Mathematical Physic

Translationally invariant conservation laws of local Lindblad equations

We study the conditions under which one can conserve local translationally invariant operators by local translationally invariant Lindblad equations in one-dimensional rings of spin-1/2 particles. We prove that for any 1-local operator (e.g., particle density) there exist Lindblad dissipators that conserve that operator, while on the other hand we prove that among 2-local operators (e.g., energy density) only trivial ones of the Ising type can be conserved, while all the other cannot be conserved, neither locally nor globally, by any 2- or 3-local translationally invariant Lindblad equation. Our statements hold for rings of any finite length larger than some minimal length determined by the locality of Lindblad equation. These results show in particular that conservation of energy density in interacting systems is fundamentally more difficult than conservation of 1-local quantities.Comment: 15 pages, no fig

Directed deterministic classical transport: symmetry breaking and beyond

We consider transport properties of a double delta-kicked system, in a regime where all the symmetries (spatial and temporal) that could prevent directed transport are removed. We analytically investigate the (non trivial) behavior of the classical current and diffusion properties and show that the results are in good agreement with numerical computations. The role of dissipation for a meaningful classical ratchet behavior is also discussed.Comment: 10 pages, 20 figure

Steering Bose-Einstein condensates despite time symmetry

A Bose-Einstein condensate in an oscillating spatially asymmetric potential is shown to exhibit a directed current for unbiased initial conditions despite time symmetry. This phenomenon occurs only if the interaction between atoms, treated in mean-field approximation, exceeds a critical value. Our findings can be described with a three-mode model (TMM). These TMM results corroborate well with a many-body study over a time scale which increases with increasing atom number. The duration of this time scale probes the validity of the used mean-field approximation.Comment: 4 pages, 5 figure

Quantum Poincare Recurrences for Hydrogen Atom in a Microwave Field

We study the time dependence of the ionization probability of Rydberg atoms driven by a microwave field, both in classical and in quantum mechanics. The quantum survival probability follows the classical one up to the Heisenberg time and then decays algebraically as P(t) ~ 1/t. This decay law derives from the exponentially long times required to escape from some region of the phase space, due to tunneling and localization effects. We also provide parameter values which should allow to observe such decay in laboratory experiments.Comment: revtex, 4 pages, 4 figure

Entanglement between two subsystems, the Wigner semicircle and extreme value statistics

The entanglement between two arbitrary subsystems of random pure states is studied via properties of the density matrix's partial transpose, $\rho_{12}^{T_2}$. The density of states of $\rho_{12}^{T_2}$ is close to the semicircle law when both subsystems have dimensions which are not too small and are of the same order. A simple random matrix model for the partial transpose is found to capture the entanglement properties well, including a transition across a critical dimension. Log-negativity is used to quantify entanglement between subsystems and analytic formulas for this are derived based on the simple model. The skewness of the eigenvalue density of $\rho_{12}^{T_2}$ is derived analytically, using the average of the third moment over the ensemble of random pure states. The third moment after partial transpose is also shown to be related to a generalization of the Kempe invariant. The smallest eigenvalue after partial transpose is found to follow the extreme value statistics of random matrices, namely the Tracy-Widom distribution. This distribution, with relevant parameters obtained from the model, is found to be useful in calculating the fraction of entangled states at critical dimensions. These results are tested in a quantum dynamical system of three coupled standard maps, where one finds that if the parameters represent a strongly chaotic system, the results are close to those of random states, although there are some systematic deviations at critical dimensions.Comment: Substantially improved version (now 43 pages, 10 figures) that is accepted for publication in Phys. Rev.