Quantum effects of periodic orbits for the kicked top

We analyze traces of powers of the time evolution operator of a periodically kicked top. Semiclassically, such traces are related to periodic orbits of the classical map. We derive the semiclassical traces in a coherent state basis and show how the periodic orbits can be recovered via a Fourier transform. A breakdown of the stationary phase approximation is detected. The quasi energy spectrum remains elusive due to lack of knowledge of sufficiently many periodic orbits. Divergencies of periodic orbit formulas are avoided by appealing to the finiteness of the quantum mechanical Hilbert space. The traces also enter the coefficients of the characteristic polynominal of the Floquet operator. Statistical properties of these coefficients give rise to a new criterion for the distinction of chaos and regular motion.     

Exact solution of the Schrödinger equation for a particle in a regular N-simplex

We present the exact solution for the Schrödinger equation for a particle inside an N-dimensional regular simplex shaped enclosure. This result extends and unifies the earlier results for equilateral triangle and K-tetrahedron billiards.     

Renormalization group analysis of boundary conditions in potential scattering

We analyze how a short distance boundary condition for the Schrödinger equation must change as a function of the boundary radius by imposing the physical requirement of phase shift independence on the boundary condition. The resulting equation can be interpreted as a variable phase equation of a complementary boundary value problem. We discuss the corresponding infrared fixed points and the perturbative expansion around them generating a short distance modified effective range theory. We also discuss ultraviolet fixed points, limit cycles, and attractors with a given fractality which take place for singular attractive potentials at the origin. The scaling behavior of scattering observables can analytically be determined and is studied with some emphasis on the low energy nucleon-nucleon interaction via singular pion exchange potentials. The generalization to coupled channels is also studied.     

An alternative quantum fidelity for mixed states of qudits

We give an alternative definition of quantum fidelity for two density operators on qudits in terms of their Hilbert-Schmidt inner product and their purity. It can be regarded as the well-defined operator fidelity for the two operators and satisfies all Jozsa's four axioms up to a normalization factor. This fidelity is not computationally demanding.     

Excitonic-vibronic coupled dimers: A dynamic approach

The dynamical properties of exciton transfer coupled to polarization vibrations in a two site system are investigated in detail. A fixed point analysis of the full system of Bloch-oscillator equations representing the coupled excitonic-vibronic flow is performed. For overcritical polarization a bifurcation converting the stable bonding ground state to a hyperbolic unstable state which is basic to the dynamical properties of the model is obtained. The phase space of the system is generally of a mixed type: Above bifurcation chaos develops starting from the region of the hyperbolic state and spreading with increasing energy over the Bloch sphere leaving only islands of regular dynamics. The behaviour of the polarization oscillator accordingly changes from regular to chaotic.     

Incipience of quantum chaos in the spin-boson model

The peculiar spectral properties of the spinboson model make it suitable for an investigation of quantum nonintegrability effects and level statistics from a new perspective. For fixed spin quantum numbers, its energy spectrum consists of 2s+1 sequences of levels with no upper bound. These sequences are identified and labelled consecutively by means of a quantum invariant calculated from the time average of a non-stationary operator. For integrable cases, level repulsion (on the energy axis) is limited to states within each sequence. From the observed spectral properties, we infer a series ofs-dependent level-spacing distributions. They converge towards a Poisson distribution fors. For nonintegrable cases, level repulsion becomes a universal phenomenon, but the amount of repulsion between two states decreases with increasing separation (in label) of the two sequences to which they belong. For smalls, the quantum nonintegrability effects are compelling but not at all chaotic. Nevertheless, they contain all the ingredients necessary to produce the symptoms commonly described as indicators of quantum chaos. In this model, we can observe quantum chaos in the making under very controllable conditions.     

Semiclassical Calculations of Autoionization Rate for Lithium Atoms in Parallel Electric and Magnetic Fields

By using a semiclassical method, we present theoretical computations of the ionization rate of Rydberg lithium atoms in parallel electric and magnetic fields with different scaled energies above the classical saddle point. The yielded irregular pulse trains of the escape electrons are recorded as a function of emission time, which allows for relating themselves to the terms of the recurrence periods of the photoabsorption. This fact turns to illustrate the dynamic mechanism how the electron pulses are stochastically generated. Comparing our computations with previous investigation results, we can deduce that the complicated chaos under consideration here consists of two kinds of seff-similar fractal structures which correspond to the contributions of the applied magnetic feld and the core scattering events. Furthermore, the effect of the magnetic field plays a major role in the profile of the autoionization rate curves, while the contribution of the core scattering is critical for specifying the positions of the pulse peaks.     

Semiclassical theory of transport in antidot lattices

Motivated by a recent experiment by Weiss et al. [Phys. Rev. Lett. 70, 4118 (1993)], we present a detailed study of quantum transport in large antidot arrays whose classical dynamics is chaotic. We calculate the longitudinal and Hall conductivities semiclassically starting from the Kubo formula. The leading contribution reproduces the classical conductivity. In addition, we find oscillatory quantum corrections to the classical conductivity which are given in terms of the periodic orbits of the system. These periodic-orbit contributions provide a consistent explanation of the quantum oscillations in the magnetoconductivity observed by Weiss et al. We find that the phase of the oscillations with Fermi energy and magnetic field is given by the classical action of the periodic orbit. The amplitude is determined by the stability and the velocity correlations of the orbit. The amplitude also decreases exponentially with temperature on the scale of the inverse orbit traversal time/T . The Zeeman splitting leads to beating of the amplitude with magnetic field. We also present an analogous semiclassical derivation of Shubnikov-de Haas oscillations where the corresponding classical motion is integrable. We show that the quantum oscillations in antidot lattices and the Shubnikov-de Haas oscillations are closely related. Observation of both effects requires that the elastic and inelastic scattering lengths be larger than the lengths of the relevant periodic orbits. The amplitude of the quantum oscillations in antidot lattices is of a higher power in Planck's constant and hence smaller than that of Shubnikov-de Haas oscillations. In this sense, the quantum oscillations in the conductivity are a sensitive probe of chaos.This paper is dedicated to Prof. H. Wagner on the occasion of his 60th birthday     

The dynamic Kingdon trap: A novel design for the storage and crystallization of laser-cooled ions

On the basis of analytical pseudo-potential calculations and extensive numerical simulations it is proved that the dynamic Kingdon trap is a working design for the permanent storage and the crystallization of laser-cooled ions.Dedicated to H. Walther on the occasion of his 60th birthday     

Darboux Transformation and Multi-Solitons for Complex mKdV Equation

An explicR N-fold Darboux transformation with multi-parameters for coupled mKdV equation is constructed with the help of a gauge transformation of the Ablowitz-Kaup-Newell-Segur （AKNS） system spectral problem. By using the Darboux transformation and the reduction technique, some multi-soliton solutions for the complex mKdV equation are obtained.     

A one-dimensional model of resonances with a delta barrier and mass jump

In this Letter, we present a one-dimensional model that includes a hard core at the origin, a Dirac delta barrier at a point in the positive semiaxis and a mass jump at the same point. We study the effect of this mass jump in the behavior of the resonances of the model. We obtain an infinite number of resonances for this situation, showing that for the case of a mass jump the imaginary part of the resonance poles tend to a fixed value depending on the quotient of masses, and demonstrate that none of these resonances is degenerated.     

s-wave bound and scattering state wave functions for a velocity-dependent Kisslinger potential

A relation linking the normalized s-wave scattering and the corresponding bound state wave functions at bound state poles is derived. This is done in the case of a non-local, velocity-dependent Kisslinger potential. Using formal scattering theory, we present two analytical proofs of the validity of the theorem. The first tackles the non-local potential directly, while the other transforms the potential to an equivalent local but energy-dependent one. The theorem is tested both analytically and numerically by solving the Schr?dinger equation exactly for the scattering and bound state wave functions when the Kisslinger potential has the form of a square well. A first order approximation to the deviation from the theorem away from bound state poles is obtained analytically. Furthermore, a proof of the analyticity of the Jost solutions in the presence of a non-local potential term is also given. Received: 3 March 2001 / Accepted: 9 June 2001     

Structure identification and adaptive synchronization of uncertain general complex dynamical networks

This Letter proposes an approach to identify the topological structure and unknown parameters for uncertain general complex networks simultaneously. By designing effective adaptive controllers, we achieve synchronization between two complex networks. The unknown network topological structure and system parameters of uncertain general complex dynamical networks are identified simultaneously in the process of synchronization. Several useful criteria for synchronization are given. Finally, an illustrative example is presented to demonstrate the application of the theoretical results.     

Surface-integral formulation of scattering theory

We formulate scattering theory in the framework of a surface-integral approach utilizing analytically known asymptotic forms of the two-body and three-body scattering wavefunctions. This formulation is valid for both short-range and long-range Coulombic interactions. New general definitions for the potential scattering amplitude are presented. For the Coulombic potentials, the generalized amplitude gives the physical on-shell amplitude without recourse to a renormalization procedure. New post and prior forms for the Coulomb three-body breakup amplitude are derived. This resolves the problem of the inability of the conventional scattering theory to define the post form of the breakup amplitude for charged particles. The new definitions can be written as surface-integrals convenient for practical calculations. The surface-integral representations are extended to amplitudes of direct and rearrangement scattering processes taking place in an arbitrary three-body system. General definitions for the wave operators are given that unify the currently used channel-dependent definitions.     

Exact scattering length for a potential of Lennard-Jones type

For a modified Lennard-Jones interaction potential of the form ∼[(r₀/r)^2n-2-(r₀/r)ⁿ], an exact and simple expression for the s-wave scattering length is presented, and discussed in some detail. For heavy alkali atoms, which nowadays are routinely being employed to produce Bose-Einstein condensates, this potential is well compatible with known experimental data when n = 6.     

Stability of two-ion crystals in the Paul trap: A comparison of exact and pseudopotential calculations

The stability of two-ion crystals in a Paul trap with a dc component in the quadrupole potential has been studied with the use of the monodromy matrix. The pseudopotential model predicts crystals with the ions at rest either along the trap axis or in the radial plane. The solutions of the full equations of motion disagree with the predictions of the pseudopotential model when the radial and axial secular frequencies are nearly degenerate: the crystal is either unstable (as first noted by Emmertet al.) or exists in a previously unanticipated configuration in which the ions lie at an angle to the trap axes. A bifurcation diagram near the edge of the crystalline stability range does not support a frequency-doubling route to chaos.Dedicated to H. Walther on the occasion of his 60th birthday     

A neutral atom and a wire: towards mesoscopic atom optics

A large variety of trapping and guiding potentials can be designed by bringing cold atoms close to charged or current-carrying material objects. Using a current-carrying wire we demonstrate how to build guides and traps for neutral atoms and using a charged wire we study a 1/r ² singularity. The simplicity and versatility of the principles demonstrated in our experiments will allow for miniaturization and integration of atom optical elements into matter-wave quantum circuits. Received: 13 December 1998 / Revised version: 8 July 1999 / Published online: 8 September 1999     

Intertwining operator method and supersymmetry for effective mass Schrödinger equations

By application of the intertwining operator method to Schrödinger equations with position-dependent (effective) mass, we construct Darboux transformations, establish the supersymmetry factorization technique and show equivalence of both formalisms. Our findings prove equivalence of the intertwining technique and the method of point transformations.     

Complex Ginzburg-Landau equation for nonlinear travelling waves in extrinsic semiconductors

We study the nonlinear state of a travelling-wave instability occurring close to the onset of impact ionization in extrinsic semiconductors. Our investigations are based on a complex Ginzburg-Landau equation (CGLE). For a simple generation-recombination function including impact ionization and thermal recombination of the charge carriers, we find a supercritical bifurcation of stable travelling waves for most parameter values. The results are compared with a numerical solution of the basic equations of motion. Furthermore, we expect that weak turbulence phenomena should be observed in semiconductors if their specific generation-recombination kinetics leads to a CGLE with appropriate coefficients.     

Scanning probe microscopy on superconductors: Achievements and challenges

A dislocation dynamical model of the reaction-diffusion type is used to describe the spatio-temporal dynamics of Lüders band propagation in polycrystals. The diffusive nature of dislocation glide is traced back to the random crystallographic orientation of the active slip systems. The role of pile-ups in dislocation multiplication is accounted for by a dynamical generalization of the Hall-Petch law. It is argued that Lüders bands in polycrystals are related to a bistable dynamics of mobile dislocations. Further results obtained cover the dependences on material parameters and deformation conditions of (1) the occurrence, (2) the strain, propagation velocity and width of Lüders bands, and (3) the upper and lower yield stresses. These results are in good agreement with experimental findings.