Classical Periodic Orbits Research Articles

Scar-Like States in Dynamical Electron-Wavepackets in Chaotic Billiard

The time-evolution of the wavepacket inside chaotic and integrable two-dimensional (2D) nanostructures is numerically studied. We have found the enhancement around the classical periodic orbits during the time-evolution in the stadium billiard. It is similar to the scars in the standing wave of the chaotic billiards. The initial position and velocity, and the shape of the wavepacket are crucial for the enhancement, but we can observe that the remnant of the initial wavepacket travels along the unstable periodic orbit. Then the wavepacket gradually diffuses around the structure. This behavior has close relation to the dynamical properties of electrons in the structure, e.g., the conductivity, the magneto-resistance etc. The quantum fidelity, which can measure the robustness of dynamical states inside the nanostructures, is also discussed. [DOI: 10.1380/ejssnt.2009.721]

Wavefunctional analysis of electron transfers in 2D nanostructures under external fields

AbstractThe time‐evolution of the wave packet inside chaotic and integrable two‐dimensional nanostructures is numerically studied. Recently the time‐evolution of quantum states is seriously considered to treat the dynamical property of nanostructures. By computer simulation, we have found the enhancement around the classical periodic orbits, which has been theoretically predicted. It is similar to the scars in the standing wave of the chaotic billiards and is also beyond the naive prediction of the random nodal patterns from the chaotic nanostructure. The initial position and velocity and the shape of the wave packet are crucial for the enhancement, however, we can observe that the remnant of the initial wave packet travels along the unstable periodic orbit. Then the wave packet gradually diffuses around the structure. This behavior has close relation to the dynamical properties of electrons in the structure, e.g., the conductivity, the magneto‐resistance etc. Copyright © 2008 John Wiley & Sons, Ltd.

Trace formula for dielectric cavities: General properties

The construction of the trace formula for open dielectric cavities is examined in detail. Using the Krein formula it is shown that the sum over cavity resonances can be written as a sum over classical periodic orbits for the motion inside the cavity. The contribution of each periodic orbit is the product of the two factors. The first is the same as in the standard trace formula and the second is connected with the product of reflection coefficients for all points of reflection with the cavity boundary. Two asymptotic terms of the smooth resonance counting function related with the area and the perimeter of a convex cavity are derived. The coefficient of the perimeter term differs from the one for closed cavities due to unusual high-energy asymptotics of the S matrix for the scattering on the cavity. Corrections to the leading semi-classical formula are briefly discussed. Obtained formulas agree well with numerical calculations for circular dielectric cavities.

Periodic orbit basis for the quantum baker map

A set of quantum states, dynamically related to the classical periodic orbits of a chaotic map, is used as a basis in which the description of the eigenstates of its quantum version is greatly simplified. This set can be improved with the inclusion of short time propagation along the stable and unstable manifolds of the periodic orbits resulting in a construction similar to the scar functions of Vergini [J. Phys. A 33, 4709 (2000)]. The average participation ratio is used to quantify the quality of the basis.

Worldline instantons, vacuum pair production and Gutzwiller's trace formula**Plenary talk at QFEXT07 (Quantum Fields Under External Conditions), Leipzig, September 2007.

In this talk, I describe a new application of the Gutzwiller trace formula formalism, to give a compact expression for the semiclassical vacuum pair production rate in quantum electrodynamics, for general inhomogeneous electromagnetic background fields. The semiclassical rate is expressed in terms of geometric properties of classical periodic phase space orbits for the associated Euclidean problem of propagation of a charged particle in the inhomogeneous background field, parametrized by the proper time.

High-energy localized eigenstates of an electronic resonator in a magnetic field

We present a semiclassical analysis of the high-energy eigenstates of an electron inside a closed resonator. An asymptotic method of the construction of the energy spectrum and eigenfunctions, localized in the small neighborhood of a periodic orbit, is developed in the presence of a homogeneous magnetic field and arbitrary scalar potential. The isolated periodic orbit is confined between two interfaces which could be planar, concave or even convex. Such a system represents a quantum electronic resonator, an analog of the well-known high-frequency optical or acoustic resonator with eigenmodes called ‘bouncing ball vibrations’. The first step in the asymptotic analysis involves constructing a solitary localized asymptotic solution to the Schrödinger equation (electronic Gaussian beam—wavepackage). Then, the stability of a closed continuous family of periodic trajectories confined between two reflecting surfaces of the resonator boundary was studied. The asymptotics of the eigenfunctions were constructed as a superposition of two electronic Gaussian beams propagating in opposite directions between two reflecting points of the periodic orbits. The asymptotics of the energy spectrum are obtained by the generalized Bohr–Sommerfeld quantization condition derived as a requirement for the eigenfunction asymptotics to be periodic. For one class of periodic orbits, localized eigenstates were computed numerically by the finite element method using FEMLAB and proved to be in a very good agreement with those computed semiclassically.

Universal and nonuniversal level statistics in a chaotic quantum spin chain

We study the level statistics of an interacting multiqubit system, namely the kicked Ising spin chain, in the regime of quantum chaos. Long range quasienergy level statistics show effects analogous to the ones observed in semiclassical systems due to the presence of short classical periodic orbits, while short range level statistics display perfect statistical agreement with random matrix theory. Even though our system possesses no classical limit, our results suggest existence of an important nonuniversal system specific behavior at short time scale, which clearly goes beyond finite size effects in random matrix theory.

Vacuum Energy as Spectral Geometry

Quantum vacuum energy (Casimir energy) is reviewed for a mathematical audience as a topic in spectral theory. Then some one-dimensional systems are solved exactly, in terms of closed classical paths and periodic orbits. The relations among local spectral densities, energy densities, global eigenvalue densities, and total energies are demonstrated. This material provides background and motivation for the treatment of higher-dimensional systems (self-adjoint second-order partial differential operators) by semiclassical approxi- mation and other methods.

A trace formula for the nodal count sequence

The sequence of nodal count is considered for separable drums. A recently derived trace formula for this sequence stores geometrical information of the drum. This statement is demonstrated in detail for the Laplace-Beltrami operator on simple tori and surfaces of revolution. The trace formula expresses the cumulative sum of nodal counts This sequence is expressed as a sum of two parts: a smooth (Weyl like) part which depends on global geometrical parameters, and a fluctuating part which involves the classical periodic orbits on the torus and their actions (lengths). The geometrical context of the nodal sequence is thus explicitly revealed.

Lissajous curves and semiclassical theory: The two-dimensional harmonic oscillator

The semiclassical treatment of the two-dimensional harmonic oscillator provides an instructive example of the relation between classical motion and the quantum mechanical energy spectrum. We extend previous work on the anisotropic oscillator with incommensurate frequencies and the isotropic oscillator to the case with commensurate frequencies for which the Lissajous curves appear as classical periodic orbits. Because of the three different scenarios depending on the ratio of its frequencies, the two-dimensional harmonic oscillator offers a unique way to explicitly analyze the role of symmetries in classical and quantum mechanics.

Nodes of entangledN-particle wave functions

In a recent paper [Bressanini et al. Phys. Rev. Lett. 95, 110201 (2005)] it was pointed out that ``the nodes of even simple wave functions are largely unexplored.'' Here we show that for $N$-particle wave functions nodal surfaces arise from the spin and orbital entanglement of constituent two-particle wave functions and derive, for two-electron atoms, 11 exact nodal rules applicable in $LS$ coupling. In addition, the ``higher symmetry'' identified numerically in the above paper is shown to be an approximate dynamical symmetry described by a molecular model or a classical unstable periodic orbit. We show that the analysis is readily extended to four-particle wave functions and consider the case of the hydrogen molecule, both fixed in space and rotating.

Wave functions with localizations on classical periodic orbits in weakly perturbed quantum billiards

We analytically investigate quantum wave functions with localizations on classical periodic orbits (POs) in the circular billiards. We construct the coherent states which are the coherent superpositions of nearly degenerate eigenstates and found to be localized around single classical POs. With the constructed coherent states, we can analytically express the mesoscopic eigenstates in the weakly perturbed systems, which are found to be localized on the ensemble of classical POs. Furthermore, these coherent states can be utilized to study the quantum vortex structures in quantum billiards.

Can One Count the Shape of a Drum?

Sequences of nodal counts store information on the geometry (metric) of the domain where the wave equation is considered. To demonstrate this statement, we consider the eigenfunctions of the Laplace-Beltrami operator on surfaces of revolution. Arranging the wave functions by increasing values of the eigenvalues, and counting the number of their nodal domains, we obtain the nodal sequence whose properties we study. This sequence is expressed as a trace formula, which consists of a smooth (Weyl-like) part which depends on global geometrical parameters, and a fluctuating part, which involves the classical periodic orbits on the torus and their actions (lengths). The geometrical content of the nodal sequence is thus explicitly revealed.

Photodetachment ofH−by a short laser pulse in crossed static electric and magnetic fields

We present a detailed quantum mechanical treatment of the photodetachment of ${\mathrm{H}}^{\ensuremath{-}}$ by a short laser pulse in the presence of crossed static electric and magnetic fields. An exact analytic formula is presented for the final state electron wave function (describing an electron in both static electric and magnetic fields and a short laser pulse of arbitrary intensity). In the limit of a weak laser pulse, final state electron wave packet motion is examined and related to the closed classical electron orbits in crossed static fields predicted by Peters and Delos [Phys. Rev. A 47, 3020 (1993)]. Owing to these closed orbit trajectories, we show that the detachment probability can be modulated, depending on the time delay between two laser pulses and their relative phase, thereby providing a means to partially control the photodetachment process. In the limit of a long, weak pulse (i.e., a monochromatic radiation field) our results reduce to those of others; however, for this case we analyze the photodetachment cross section numerically over a much larger range of electron kinetic energy (i.e., up to $500\phantom{\rule{0.3em}{0ex}}{\mathrm{cm}}^{\ensuremath{-}1}$) than in previous studies and relate the detailed structures both analytically and numerically to the above-mentioned, closed classical periodic orbits.

Orbits, shapes and currents

Shapes and current distributions of nuclei and alkali clusters are discussed in terms of the single particle motion of fermions in the average potential. For small particle number, an interpretation in terms of the lowest spherical harmonics is presented. For large particle number, an interpretation in terms of classical periodic orbits is presented.

Duality between quantum and classical dynamics for integrable billiards

We establish a duality between the quantum wave vector spectrum and the eigenmodes of the classical Liouvillian dynamics for integrable billiards. Signatures of the classical eigenmodes appear as peaks in the correlation function of the quantum wave vector spectrum. A semiclassical derivation and numerical calculations are presented in support of the results. These classical eigenmodes can be observed in physical experiments through the autocorrelation of the transmission coefficient of waves in quantum billiards. Exact classical trace formulas of the resolvent are derived for the rectangle, equilateral triangle, and circle billiards. We also establish a correspondence between the classical periodic orbit length spectrum and the quantum spectrum for integrable polygonal billiards.

Quantum signatures of nonlinear resonances in mesoscopic systems: Efficient extension of localized wave functions

We investigate the quantum signatures of classical nonlinear resonances by making the analytic connection between the quantum wave functions and the classical periodic orbits for the uncoupled systems. It is found that the highly efficient extension of the localized coherent states within the classical caustics is an intriguing phenomenon in mesoscopic systems with nonlinear resonances. With the theoretical analysis, we experimentally demonstrate that the laser resonator with an intracavity saturable absorber can be employed to visualize the wave patterns analogous to the quantum wave functions associated with Fermi resonance.

Bound states in Andreev billiards with soft walls

The energy spectrum and the eigenstates of a rectangular quantum dot containing soft potential walls in contact with a superconductor are calculated by solving the Bogoliubov--de Gennes equation. We compare the quantum mechanical solutions with a semiclassical analysis using a Bohr-Sommerfeld (BS) quantization of periodic orbits. We propose a simple extension of the BS approximation which is well suited to describe Andreev billiards with parabolic potential walls. The underlying classical periodic electron-hole orbits are directly identified in terms of ``scar''-like features engraved in the quantum wave functions of Andreev states which we determine here explicitly.

Classical and quantum mechanics of diatomic molecules in tilted fields

We investigate the classical and quantum mechanics of diatomic molecules in noncollinear (tilted) static electric and nonresonant linearly polarized laser fields. The classical diatomic in tilted fields is a nonintegrable system, and we study the phase space structure for physically relevant parameter regimes for the molecule KCl. While exhibiting low-energy (pendular) and high-energy (free-rotor) integrable limits, the rotor in tilted fields shows chaotic dynamics at intermediate energies, and the degree of classical chaos can be tuned by changing the tilt angle. We examine the quantum mechanics of rotors in tilted fields. Energy-level correlation diagrams are computed, and the presence of avoided crossings quantified by the study of nearest-neighbor spacing distributions as a function of energy and tilting angle. Finally, we examine the influence of classical periodic orbits on rotor wave functions. Many wave functions in the tilted field case are found to be highly nonseparable in spherical polar coordinates. Localization of wave functions in the vicinity of classical periodic orbits, both stable and unstable, is observed for many states.

A large number of higher-energy eigenvalues of a huge dimensional matrix for a quantum chaotic study of a quartic potential

A system of two quartic oscillators coupled by a quartic perturbation is numerically studied for quantum mechanical eigenvalues and classical periodic orbits. The coupling strength serves as a control parameter to simulate the transition from integrable to chaotic regimes. In order to obtain higher-energy eigenvalues of a huge dimensional matrix, the Lanczos method and the equi-energy method are investigated for a practical use.