Classical Orbits Research Articles

Nonuniversal Magnetic-Field Correlation of Conductance Fluctuations in Quantum Cavities Attached to a Superconductor

We investigate the magnetic-field-induced transmission resonance in two-dimensional-electron-gas–superconductor junctions. The magnetic-field correlation of the conductance fluctuations is nonuniversal at low magnetic fields provided the normal region has a certain kind of rectangular-shape. The anomalous magnetoconductance fluctuations are robust against the imperfection of the quantum cavity geometry and the occupation of multiple modes in the normal lead. However, the anomalous fluctuations are suppressed if the cavity size is not sufficiently large in comparison to the lead width. This indicates the importance of multiple Andreev reflection for the anomalous behavior. The wave function at the resonance in the anomalous fluctuations regime reveals scars by regular classical orbits. Hierarchical nature of the anomalous conductance fluctuations is evidenced by the distributions of the current and the charge.

Origin of degeneracies and self-similar structure in the spectrum of quantum dots

We show that the appearance of ‘accidental’ degeneracies (degeneracies which persist beyond the harmonic potential) in the single-particle spectrum of quantum dots in a magnetic field can be understood by analyzing the evolution of the structure of the classical orbits of the system as a function of magnetic field. We demonstrate that these degeneracies are associated with closed orbits, which occur at particular magic ratios of the cyclotron energy to the confinement energy in the harmonic case. The spectrum of degenerate energies forms an interesting self-similar structure with an associated complete ‘devil's staircase’. Since the degeneracies are not lifted by deviations from parabolicity of the radial confinement potential of the dot, the devil's staircase structure that we predict should be detectable experimentally using available techniques.

The linear baryon

We describe classical orbits and quantum states of a three particle system which is weakly chaotic. The quantum energies of the lowest 48 states which are invariant under permutations are given both in various approximations and to high accuracy. The quantum mechanical probability distributions are given in a two dimensional version of the system, the hexagonal pit. The quantum probabilities show strong resemblance to classical orbits of various types, including chaotic ones.

Diffractive Orbits in an Open Microwave Billiard

We demonstrate experimentally the existence and significance of diffractive orbits in an open microwave billiard. Orbits that diffract off a sharp edge strongly influence the conduction spectrum, especially in the regime where there are no stable classical orbits. On resonance, the wave functions are influenced by both classical and diffractive orbits. Off resonance, the wave functions are determined by the constructive interference of multiple transient orbits. The experimental results agree well with quantum computations and with a semiclassical theory that incorporates diffraction.

Exact Wave Functions for the Coulomb Problem from Classical Orbits

We present a simple formula that expresses energy eigenfunctions for the Coulomb problem exactly and entirely in terms of the corresponding classical orbits. Since both energy eigenvalues and eigenfunctions can be obtained exactly by a semiclassical procedure, the full quantum dynamics of the electron in a Coulomb field can be expressed completely in terms of its classical motion.

Excitation of a Three-Dimensionally Localized Atomic Electron Wave Packet

We have prepared an atomic electron in a three-dimensionally localized wave packet that travels along a highly eccentric orbit between the nucleus and a turning point approximately 1000 \AA{} away. State preparation consisted of two distinct steps. First the electron wave function was localized to the desired orbit by exciting a stationary extreme Stark state from the $n\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}30$ manifold of a sodium atom in a dc electric field. Then the dynamic wave packet was formed by applying a terahertz electromagnetic half-cycle pulse polarized along the direction of the dc field. The localized state is an ideal starting point for further quantum control proceeding along classical orbits.

Semiclassical mechanics of a nonintegrable spin cluster

We study detailed classical-quantum correspondence for a cluster system of three spins with single-axis anisotropic exchange coupling. With autoregressive spectral estimation, we find oscillating terms in the quantum density of states caused by classical periodic orbits: in the slowly varying part of the density of states we see signs of nontrivial topology changes happening to the energy surface as the energy is varied. Also, we can explain the hierarchy of quantum energy levels near the ferromagnetic and antiferromagnetic states with Einstein-Brillouin-Keller quantization to explain large structures and tunneling to explain small structures.

Classical-Quantum Correspondencein a Helium-Like System on Conic Sections

We present results of a study on the classical-quantum correspon- dence in a model of a three-body Coulomb system. The model simulates a helium atom, for which the motion of two electrons is restricted to move on coplanar ellipses. Some of classical periodic orbits of this model system are found analytically or numerically, including the well-known Wannier and Langmuir orbits. These periodic orbits play an important role in quantum solutions in the sense that they scar the probability amplitude distributions of some of the eigenfunctions of the system.

Vibrational analysis of HOCl up to 98% of the dissociation energy with a Fermi resonance Hamiltonian

We have analyzed the vibrational energies and wave functions of HOCl obtained from previous ab initio calculations [J. Chem. Phys. 109, 2662 (1998); 109, 10273 (1998)]. Up to approximately 13 000 cm−1, the normal modes are nearly decoupled, so that the analysis is straightforward with a Dunham model. In contrast, above 13 000 cm−1 the Dunham model is no longer valid for the levels with no quanta in the OH stretch (v1=0). In addition to v1, these levels can only be assigned a so-called polyad quantum number P=2v2+v3, where 2 and 3 denote, respectively, the bending and OCl stretching normal modes. In contrast, the levels with v1⩾2 remain assignable with three vi quantum numbers up to the dissociation (D0=19 290 cm−1). The interaction between the bending and the OCl stretch (ω2≅2ω3) is well described with a simple, fitted Fermi resonance Hamiltonian. The energies and wave functions of this model Hamiltonian are compared with those obtained from ab initio calculations, which in turn enables the assignment of many additional ab initio vibrational levels. Globally, among the 809 bound levels calculated below dissociation, 790 have been assigned, the lowest unassigned level, No. 736, being located at 18 885 cm−1 above the (0,0,0) ground level, that is, at about 98% of D0. In addition, 84 “resonances” located above D0 have also been assigned. Our best Fermi resonance Hamiltonian has 29 parameters fitted with 725 ab initio levels, the rms deviation being of 5.3 cm−1. This set of 725 fitted levels includes the full set of levels up to No. 702 at 18 650 cm−1. The ab initio levels, which are assigned but not included in the fit, are reasonably predicted by the model Hamiltonian, but with a typical error of the order of 20 cm−1. The classical analysis of the periodic orbits of this Hamiltonian shows that two bifurcations occur at 13 135 and 14 059 cm−1 for levels with v1=0. Above each of these bifurcations two new families of periodic orbits are created. The quantum counterpart of periodic orbits are wave functions with “pearls” aligned along the classical periodic orbits. The complicated sequence of ab initio wave functions observed within each polyad is nicely reproduced by the wave functions of the Fermi resonance Hamiltonian and by the corresponding shapes of periodic orbits. We also present a comparison between calculated and measured energies and rotational constants for 25 levels, leading to a secure vibrational assignment for these levels. The largest difference between experimental and calculated energies reaches 22 cm−1 close to D0.

Uniform Approximations for Non-generic Bifurcation Scenarios Including Bifurcations of Ghost Orbits

Gutzwiller's trace formula allows interpretation of the density of states of a classically chaotic quantum system in terms of classical periodic orbits. It diverges when periodic orbits undergo bifurcations and must be replaced with a uniform approximation in the vicinity of the bifurcations. As a characteristic feature, these approximations require the inclusion of complex “ghost orbits”. By studying an example taken from the Diamagnetic Kepler Problem, viz. the period-quadrupling of the balloon orbit, we demonstrate that these ghost orbits themselves can undergo bifurcations, giving rise to non-generic complicated bifurcation scenarios. We extend classical normal form theory so as to yield analytic descriptions of both bifurcations of real orbits and ghost orbit bifurcations. We then show how the normal form serves to obtain a uniform approximation taking the ghost orbit bifurcation into account. We find that the ghost bifurcation produces signatures in the semiclassical spectrum in much the same way as a bifurcation of real orbits does.

Spectrum of stochastic evolution operators: local matrix representation approach.

A matrix representation of the evolution operator associated with a nonlinear stochastic flow with additive noise is used to compute its spectrum. In the weak noise limit a perturbative expansion for the spectrum is formulated in terms of local matrix representations of the evolution operator centered on classical periodic orbits. The evaluation of perturbative corrections is easier to implement in this framework than in the standard Feynman diagram perturbation theory. The results are perturbative corrections to a stochastic analog of the Gutzwiller semiclassical spectral determinant computed to several orders beyond what has so far been attainable in stochastic and quantum-mechanical applications.

Planar cyclotron motion in unidirectional superlattices defined by strong magnetic and electric fields: Traces of classical orbits in the energy spectrum

We compare the quantum and the classical description of the two-dimensional motion of electrons subjected to a perpendicular magnetic field and a one-dimensional lateral superlattice defined by spatially periodic magnetic and electric fields of large amplitudes. We explain in detail the complicated energy spectra, consisting of superimposed branches of strong and of weak dispersion, by the correspondence between the respective eigenstates and the ``channeled'' and ``drifting'' orbits of the classical description.

Nonaxial octupole deformations and shell phenomena

Manifestations of pronounced shell effects are found when combining nonaxial octupole deformations with a three-dimensional harmonic oscillator model. The contributions of the main classical periodic orbits of this intrinsically chaotic three degrees of freedom system are obtained from a frequency analysis. An adiabatic approximation based on the removal of resonance method yields explicit expressions for winding numbers from which the quantum spectrum can be predicted. The shell structure obtained from this quadrupole+octupole combination turns out to be equivalent to the shell structure of a plain but more strongly deformed quadrupole potential for about the first 100 levels.

Semiclassical spectra and diagonal matrix elements by harmonic inversion of cross-correlated periodic orbit sums.

Semiclassical spectra weighted with products of diagonal matrix elements of operators A(alpha), i.e., g(alphaalpha')(E)= summation operator(n)<nA(alpha)n><nA(alpha')n>/(E-E(n)), are obtained by harmonic inversion of a cross-correlation signal constructed of classical periodic orbits. The method provides highly resolved semiclassical spectra even in situations of nearly degenerate states, and opens the way to reducing the required signal lengths to shorter than the Heisenberg time. This implies a significant reduction of the number of orbits required for periodic orbit quantization by harmonic inversion.

The correspondence principle and quantum dipole matrix element between parabolic Rydberg states

We derive a quantum formula of the dipole matrix element for parabolic Rydberg states in terms of the Bessel functions without appealing to any quasiclassical methods. We exploit this quantum formula to explain the discrepancy between the other quasi-classical expressions available in the literature in terms of the Airy functions. We use the quantum formula to examine critically the region of validity of the corresponding formula derived by Born using the classical electron orbit and correspondence principle.

The quantum behavior of an electron in a uniform magnetic field

The total probability current of an electron in a uniform magnetic field is calculated in the symmetric gauge. Even though the eigen motion of the electron in this gauge is similar to the classical orbit motion, the total probability current is found to be equal to zero when the canonical angular momentum is less than zero. The reason is that in this case, the origin of the coordinate system lies outside the circular orbit of corresponding classical motion, in addition to the absence of radial component of the probability current density because of the indeterminacy of the location of the orbit center.

Full length article

Quantum chaos is generally referred to as the study of quantum manifestations or fingerprints of nonlinear dynamical and chaotic behaviors in the corresponding classical system, an interdisciplinary field that has been active for about four decades. In closed quantum systems, for example, the basic phenomena studied include energy level-spacing statistics and quantum scarring. In open Hamiltonian systems, quantum chaotic scattering has been investigated extensively. Previous works were almost exclusively for nonrelativistic quantum systems described by the Schrödinger equation. Recent years have witnessed a rapid growth of interest in Dirac materials such as graphene, topological insulators, molybdenum disulfide and topological Dirac semimetals. A common feature of these materials is that their physics is described by the Dirac equation in relativistic quantum mechanics, generating phenomena that do not usually emerge in conventional semiconductor materials. This has important consequences. In particular, at the level of basic science, a new field has emerged: Relativistic Quantum Chaos (RQC), which aims to uncover, understand, and exploit relativistic quantum manifestations of classical nonlinear dynamical behaviors including chaos. Practically, Dirac materials have the potential to revolutionize solid-state electronic and spintronic devices, and have led to novel device concepts such as valleytronics. Exploiting manifestations of nonlinear dynamics and chaos in the relativistic quantum regime can have significant applications.The aim of this article is to give a comprehensive review of the basic results obtained so far in the emergent field of RQC. Phenomena to be discussed in depth include energy level-spacing statistics in graphene or Dirac fermion systems that exhibit various nonlinear dynamical behaviors in the classical limit, relativistic quantum scars (unusually high concentrations of relativistic quantum spinors about classical periodic orbits), peculiar features of relativistic quantum chaotic scattering and quantum transport, manifestations of the Klein paradox and its effects on graphene or 2D Dirac material based devices, regularization of relativistic quantum tunneling by chaos, superpersistent currents in chaotic Dirac rings subject to a magnetic flux, and exploitation of relativistic quantum whispering gallery modes for applications in quantum information science. Computational methods for solving the Dirac equation in various situations will be introduced and physical theories developed so far in RQC will be described. Potential device applications will be discussed.

Magnetic-Field-Induced Intensity Revivals in Harmonic Generation

We demonstrate control of high harmonic generation by a linearly polarized laser field using a uniform static magnetic field parallel to the laser polarization. We predict that particular values of the magnetic field can increase harmonic intensities by orders of magnitude. Classical orbit calculations show that these magnetic-field-induced intensity revivals occur when the return time for laser-driven motion of the electron back to the origin is a multiple of the cyclotron period for motion perpendicular to the laser polarization direction.

The semiclassical regime of intramolecular vibrational dynamics

We report on a study of intramolecular dynamics after an ultrashort excitation. We show that a semiclassical regime of the vibrational dynamics exists on an intermediate time scale between the initial dephasing on the ultrashort time of the excitation pulse and the Heisenberg time scale associated with the mean level spacing. In this semiclassical regime, intramolecular dynamics presents time recurrences at the periods of the classical orbits. The amplitudes of these semiclassical time recurrences are evaluated for classically integrable and chaotic systems using Gutzwiller and the Berry–Tabor trace formulas.

Ballistic transport and scarring effects in coupled quantum dots

We have performed numerical simulations of coupled, open quantum dots, connected by contacts supporting a number of modes. In single dots, the resonant wave functions can be scarred by classical periodic orbits. Coupling between dots allows for the two-dimensional analog of molecular states. With these molecular states, we find that the wave functions can be scarred by orbits that are clearly shared between dots. In a finite chain of dots, we find that the formation of energy bands and gaps can be a by-product of complicated scarred states involving orbits that can extend over many dots. These orbits appear to be variations of those that contribute to the atomic and molecular states.