Semiclassical quantization of the scattering from a classically chaotic repellor

Path integration over closed loops and Gutzwiller's trace formula

We discuss a basic mathematical approach to Feynman path integrals as infinite-dimensional oscillatory integrals. We present new results on asymptotics of such integrals which exploit recently developed approximation techniques via finite dimensional oscillatory integrals. Applications are also given, namely to the study of the trace of the time evolution operator in quantum mechanics and to the interpretation of “Gutzwiller's trace formula” as a leading term in an asymptotic expansion around classical periodic orbits.

The connection between the wavefunctions and the classical periodic orbits in a 2D harmonic oscillator is analytically constructed by using the representation of SU(2) coherent states. It is found that the constructed wavefunction generally corresponds to an ensemble of classical trajectories and its localization is extremely efficient. With the constructed wavefunction, we also analyse the property of the probability current density associated with the classical periodic orbit. The appearance of vortex structure in the quantum flow is clearly found to arise from the wave interference.

Anderson localization is related to exponential localization of a particle in the configuration space in the presence of a disorder potential. Anderson localization can be also observed in the momentum space and corresponds to quantum suppression of classical diffusion in systems that are classically chaotic. Another kind of Anderson localization has been recently proposed, i.e. localization in the time domain due to the presence of {\it disorder} in time. That is, the probability density for the detection of a system at a fixed position in the configuration space is localized exponentially around a certain moment of time if a system is driven by a force that fluctuates in time. We show that an electron in a Rydberg atom, perturbed by a fluctuating microwave field, Anderson localizes along a classical periodic orbit. In other words the probability density for the detection of an electron at a fixed position on an orbit is exponentially localized around a certain time moment. This phenomenon can be experimentally observed.

We derive semiclassical trace formulae including Gutzwiller's trace formula using coherent states. This formulation has several advantages over the usual coordinate‐space formulation. Using a coherent‐state basis makes it immediately obvious that classical periodic orbits make separate contributions to the trace of the quantum‐mechanical time evolution operator. In addition, our approach is manifestly canonically invariant at all stages, and leads to the simplest possible derivation of Gutzwiller's formula.

An oscillatory pattern in the smoothed quantum spectrum, which is unique to single-particle motions in a reflection-asymmetric superdeformed oscillator potential, is investigated by means of the semiclassical theory of shell structure. Clear correspondence between the oscillating components of the smoothed level density and the classical periodic orbits is found. It is shown that an interference effect between two families of the short periodic orbits, called supershell effect, becomes more significant with increasing reflection-asymmetric deformations. Possible origins of this enhancement phenomena as well as quantum signatures of period-multipling bifurcations are discussed in connection with stabilities of the classical periodic orbits.

We first give a short outline of the semiclassical periodic orbit theory that allows one to relate quantum shell fluctuations in a mean-field system to the periodic orbits of the corresponding classical system. Earlier applications to gross-shell effects in nuclei (ground-state deformations), metal clusters (ground-state deformations and supershells), and quantum dots (conductance oscillations in weak magnetic fields) are briefly summarized. We then report on a recent simple and transparent semiclassical interpretation of the origin of the mass asymmetry in the fission of heavy nuclei. Using only a few classical periodic orbits and a cavity model for the nuclear mean field, we explain the onset of left-right asymmetric shapes at the fission isomer minimum by the principle of constant action of the shortest orbits.

The quantum form-factor K( tau )-the Fourier transform of the spectral auto-correlation function-may be represented semiclassically in terms of a sum over classical periodic orbits. We consider the problem of how this approximation behaves in the limit of long (scaled) time tau . It is shown that whilst K itself tends to unity, the periodic-orbit sum typically grows exponentially as tau to infinity . This behaviour is related to the fact that leading-order semiclassical quantization methods yield complex eigenvalues with imaginary parts that are of higher order in Planck's constant. Divergence from the quantum limit begins when tau = tau *(h(cross)), which, for typical two-degrees-of-freedom systems and maps, is shown to be independent of h(cross) as h(cross) to 0. In the case of the baker's map, however, quantum diffraction from the classical discontinuity instead causes the analogue of tau * to tend to zero like N-12/, Where N is the integer that corresponds to the inverse of Planck's constant. This is in agreement with recent numerical studies. Finally, we consider the implications of the semiclassical divergence studied here for the method developed by Argaman et al. (1993) of investigating correlations between the periodic orbits of chaotic systems.

Quantum monodromy for diatomic molecules in combined electrostatic and pulsed nonresonant laser fields

It was recently shown (Keating J P and Prado S D 2001 Proc. R. Soc. A 457 1855–72) that, in the semiclassical limit, the scarring of quantum eigenfunctions by classical periodic orbits in chaotic systems may be dramatically enhanced when the orbits in question undergo bifurcation. Specifically, a bifurcating orbit gives rise to a scar with an amplitude that scales as ℏα and a width that scales as ℏω, where α and ω are bifurcation-dependent scar exponents whose values are typically smaller than those (α = ω = ½) associated with isolated and unstable periodic orbits. We here analyse the influence of bifurcations on the autocorrelation function of quantum eigenstates, averaged with respect to energy. It is shown that the length-scale of the correlations around a bifurcating orbit scales semiclassically as ℏ1−α, where α is the corresponding scar amplitude exponent. This imprint of bifurcations on quantum autocorrelations is illustrated by numerical computations for a family of perturbed cat maps.

We analytically study the vortex formation associated with the classical periodic orbits in a nonseparable quantum billiard. The influence of vortex formation on the quantum flux in the mesoscopic region is clearly demonstrated. Remarkably, the high-order vortex structures in an equilateral triangle billiard display the triangular, swirl, and kagomè vortex lattices. The relationship between various vortex lattices is found to rely on the relative phase factor in the coherent state.

A reflection-asymmetric deformed oscillator potential is analyzed from the classical and quantum mechanical point of view. The connection between occurrence of shell structures and classical periodic orbits is studied using the "removal of resonances method" in a classical analysis. In this approximation, the effective single particle potential becomes separable and the frequencies of the classical trajectories are easily determined. It turns out that the winding numbers calculated in this way are in good agreement with the ones found from the corresponding quantum mechanical spectrum using the particle number dependence of the fluctuating part of the total energy. When the octupole term is switched on it is found that prolate shapes are stable against chaos and can exhibit shells where spherical and oblate cases become chaotic. An attempt is made to explain this difference in the quantum mechanical context by looking at the distribution of exceptional points which results from the matrix structure of the respective Hamiltonians. In a similar way we analyze the modified Nilsson model and discuss its consequences for metallic clusters.

The arbitrary trajectory quantization method (ATQM) is a time dependent approach to quasiclassical quantization based on the approximate dual relationship that exists between the quantum energy spectra and classical periodic orbits. It has recently been shown however, that, for polygonal billiards, the periodicity criterion must be relaxed to include closed almost-periodic (CAP) orbit families in this relationship. In light of this result, we reinvestigate the ATQM and show that at finite energies, a smoothened quasiclassical kernel corresponds to the modified formula that includes CAP families while the delta function kernel corresponding to the periodic orbit formula is recovered as E-->infinity. Several clarifications are also provided.

The localization of eigenfunctions around classical periodic orbits is studied numerically for the H-atom in a strong magnetic field by calculating their Husimi distribution in phase space. In contrast to the configuration space representation, the phase space distributions are simply structured: about 90% of eigenstates may be unambigously related to fixed points and invariant manifolds of periodic orbits, indicating that scars are the rule rather than the exception. In order to measure the influence of one particular orbit, we calculate the integrals of the energetically lowest 500 Husimi distributions along the orbit. Their incoherent superposition defines the scar strength distribution for the particular periodic orbit which is analyzed by Fourier transformation. The Husimi distribution at (q, p) in phase space may be represented as a scalar product of the wavefunction with a coherent state of the unperturbed system, i.e., a radial Gaussian wave packet located at (q, p) in the (regularized) Coulomb system. This simplifies the actual calculation of the Husimi distribution and allows to treat their incoherent superposition within Gutzwillers theory extended to matrix elements of an operator A, if we choose A to be the projector on a coherent state.

D${}_{2}$ and H${}_{2}$ Stark recurrence spectra have been measured at scaled energies of $\ensuremath{\epsilon}=\ensuremath{-}2.3$ and $\ensuremath{\epsilon}=\ensuremath{-}3.2$. An isotope shift in the scaled-action locations of recurrence peaks is found and explained by the increased density of states associated with perturbing rotational series. This reinforces the insight gained in previous work on the correspondence between the structures of a quantum spectrum and its classical periodic orbits.