Alternate islands of multiple isochronous chains in wave-particle interactions

We analyze the dynamics of a relativistic particle moving in a uniform magnetic field and perturbed by a standing electrostatic wave. We show that a pulsed wave produces an infinite number of perturbative terms with the same winding number, which may generate islands in the same region of phase space. As a consequence, the number of isochronous island chains varies as a function of the wave parameters. We observe that in all the resonances, the number of chains is related to the amplitude of the various resonant terms. We determine analytically the position of the periodic points and the number of island chains as a function of the wave number and wave period. Such information is very important when one is concerned with regular particle acceleration, since it is necessary to adjust the initial conditions of the particle to obtain the maximum acceleration.Comment: Submitte

Decoherence of Semiclassical Wigner Functions

The Lindblad equation governs general markovian evolution of the density operator in an open quantum system. An expression for the rate of change of the Wigner function as a sum of integrals is one of the forms of the Weyl representation for this equation. The semiclassical description of the Wigner function in terms of chords, each with its classically defined amplitude and phase, is thus inserted in the integrals, which leads to an explicit differential equation for the Wigner function. All the Lindblad operators are assumed to be represented by smooth phase space functions corresponding to classical variables. In the case that these are real, representing hermitian operators, the semiclassical Lindblad equation can be integrated. There results a simple extension of the unitary evolution of the semiclassical Wigner function, which does not affect the phase of each chord contribution, while dampening its amplitude. This decreases exponentially, as governed by the time integral of the square difference of the Lindblad functions along the classical trajectories of both tips of each chord. The decay of the amplitudes is shown to imply diffusion in energy for initial states that are nearly pure. Projecting the Wigner function onto an orthogonal position or momentum basis, the dampening of long chords emerges as the exponential decay of off-diagonal elements of the density matrix.Comment: 23 pg, 2 fi

Semiclassical theory for small displacements

Characteristic functions contain complete information about all the moments of a classical distribution and the same holds for the Fourier transform of the Wigner function: a quantum characteristic function, or the chord function. However, knowledge of a finite number of moments does not allow for accurate determination of the chord function. For pure states this provides the overlap of the state with all its possible rigid translations (or displacements). We here present a semiclassical approximation of the chord function for large Bohr-quantized states, which is accurate right up to a caustic, beyond which the chord function becomes evanescent. It is verified to pick out blind spots, which are displacements for zero overlaps. These occur even for translations within a Planck area of the origin. We derive a simple approximation for the closest blind spots, depending on the Schroedinger covariance matrix, which is verified for Bohr-quantized states.Comment: 16 pages, 4 figures

Semiclassical Evolution of Dissipative Markovian Systems

A semiclassical approximation for an evolving density operator, driven by a "closed" hamiltonian operator and "open" markovian Lindblad operators, is obtained. The theory is based on the chord function, i.e. the Fourier transform of the Wigner function. It reduces to an exact solution of the Lindblad master equation if the hamiltonian operator is a quadratic function and the Lindblad operators are linear functions of positions and momenta. Initially, the semiclassical formulae for the case of hermitian Lindblad operators are reinterpreted in terms of a (real) double phase space, generated by an appropriate classical double Hamiltonian. An extra "open" term is added to the double Hamiltonian by the non-hermitian part of the Lindblad operators in the general case of dissipative markovian evolution. The particular case of generic hamiltonian operators, but linear dissipative Lindblad operators, is studied in more detail. A Liouville-type equivariance still holds for the corresponding classical evolution in double phase, but the centre subspace, which supports the Wigner function, is compressed, along with expansion of its conjugate subspace, which supports the chord function. Decoherence narrows the relevant region of double phase space to the neighborhood of a caustic for both the Wigner function and the chord function. This difficulty is avoided by a propagator in a mixed representation, so that a further "small-chord" approximation leads to a simple generalization of the quadratic theory for evolving Wigner functions.Comment: 33 pages - accepted to J. Phys.

Hyperbolic Scar Patterns in Phase Space

We develop a semiclassical approximation for the spectral Wigner and Husimi functions in the neighbourhood of a classically unstable periodic orbit of chaotic two dimensional maps. The prediction of hyperbolic fringes for the Wigner function, asymptotic to the stable and unstable manifolds, is verified computationally for a (linear) cat map, after the theory is adapted to a discrete phase space appropriate to a quantized torus. The characteristic fringe patterns can be distinguished even for quasi-energies where the fixed point is not Bohr-quantized. The corresponding Husimi function dampens these fringes with a Gaussian envelope centered on the periodic point. Even though the hyperbolic structure is then barely perceptible, more periodic points stand out due to the weakened interference.Comment: 12 pages, 10 figures, Submited to Phys. Rev.

Isochronous island bifurcations driven by resonant magnetic perturbations in Tokamaks

Recent evidences show that heteroclinic bifurcations in magnetic islands may be caused by the amplitude variation of resonant magnetic perturbations in tokamaks. To investigate the onset of these bifurcations, we consider a large aspect ratio tokamak with an ergodic limiter composed of two pairs of rings that create external primary perturbations with two sets of wave numbers. An individual pair produces hyperbolic and elliptic periodic points, and its associated islands, that are consistent with the Poincar\'e-Birkhoff fixed point theorem. However, for two pairs producing external perturbations resonant on the same rational surface, we show that different configurations of isochronous island chains may appear on phase space according to the amplitude of the electric currents in each pair of the ergodic limiter. When one of the electric currents increases, isochronous bifurcations take place and new islands are created with the same winding number as the preceding islands. We present examples of bifurcation sequences displaying (a) direct transitions from the island chain configuration generated by one of the pairs to the configuration produced by the other pair, and (b) transitions with intermediate configurations produced by the limiter pairs coupling. Furthermore, we identify shearless bifurcations inside some isochronous islands, originating nonmonotonic local winding number profiles with associated shearless invariant curves

Isochronous bifurcations in a two-parameter twist map

Isochronous islands in phase space emerge in twist Hamiltonian systems as a response to multiple resonant perturbations. According to the Poincar\'e-Birkhoff theorem, the number of islands depends on the system characteristics and the perturbation. We analyze, for the two-parameter standard map, also called two-harmonic standard map, how the island chains are modified as the perturbation amplitude increases. We identified three routes for the transition from one chain, associated with one harmonic, to the chain associated with the other harmonic, based on a combination of pitchfork and saddle-node bifurcations. These routes can present intermediate island chains configurations. Otherwise, the destruction of the islands always occurs through the pitchfork bifurcation

Classical orbit bifurcation and quantum interference in mesoscopic magnetoconductance

We study the magnetoconductance of electrons through a mesoscopic channel with antidots. Through quantum interference effects, the conductance maxima as functions of the magnetic field strength and the antidot radius (regulated by the applied gate voltage) exhibit characteristic dislocations that have been observed experimentally. Using the semiclassical periodic orbit theory, we relate these dislocations directly to bifurcations of the leading classes of periodic orbits.Comment: 4 pages, including 5 figures. Revised version with clarified discussion and minor editorial change

Noise models for superoperators in the chord representation

We study many-qubit generalizations of quantum noise channels that can be written as an incoherent sum of translations in phase space. Physical description in terms of the spectral properties of the superoperator and the action in phase space are provided. A very natural description of decoherence leading to a preferred basis is achieved with diffusion along a phase space line. The numerical advantages of using the chord representation are illustrated in the case of coarse-graining noise.Comment: 8 pages, 5 .ps figures (RevTeX4). Submitted to Phys. Rev. A. minor changes made, according to referee suggestion

Semiclassical spatial correlations in chaotic wave functions

We study the spatial autocorrelation of energy eigenfunctions $\psi_n({\bf q})$ corresponding to classically chaotic systems in the semiclassical regime. Our analysis is based on the Weyl-Wigner formalism for the spectral average $C_{\epsilon}({\bf q^{+}},{\bf q^{-}},E)$ of $\psi_n({\bf q}^{+})\psi_n^*({\bf q}^{-})$, defined as the average over eigenstates within an energy window $\epsilon$ centered at $E$. In this framework $C_{\epsilon}$ is the Fourier transform in momentum space of the spectral Wigner function $W({\bf x},E;\epsilon)$. Our study reveals the chord structure that $C_{\epsilon}$ inherits from the spectral Wigner function showing the interplay between the size of the spectral average window, and the spatial separation scale. We discuss under which conditions is it possible to define a local system independent regime for $C_{\epsilon}$. In doing so, we derive an expression that bridges the existing formulae in the literature and find expressions for $C_{\epsilon}({\bf q^{+}}, {\bf q^{-}},E)$ valid for any separation size $|{\bf q^{+}}-{\bf q^{-}}|$.Comment: 24 pages, 3 figures, submitted to PR