Table of contents

Using the complex coordinate method, it is possible to compute the physical properties of atomic resonances above the ionization threshold. We show how to define and compute the electronic densities associated with these atomic resonances. The method is general and is illustrated for atomic Rydberg states in static and time-dependent external fields.

We compare quantum phase space distributions of individual quantum states of the diamagnetic hydrogen atom obtained by means of Wigner functions with those given by Husimi functions. The comparison is carried out at effective h(cross) approximately 0.035 and at a fixed scaled energy ( epsilon =-0.316) which corresponds to an especially interesting mixed classical phase-space structure. The object of the comparison is to establish which of the two distributions best correlates with the classical Poincare surface of section for a representative set of single states. As expected, the states investigated display the strongest positive intensity on a single local invariant phase-space structure (such as a scar or torus) and at this level there is little difference between the Husimi and the Wigner function. However the weak structures (fringes of the Wigner function or zeros of the Husimi function) are shown here to be radically different for the Wigner relative to the Husimi representation. In both cases the weak structures permeate all of phase space. In addition they show different character for integrable and chaotic dynamics and so reflect the global structure of phase-space to a much greater extent than the localized strong structure. The Wigner functions of selected states are shown to have additional dynamically significant structures which are not apparent in the Husimi functions. These include negative intensity features seen in the Wigner-but not the Husimi-functions which are very well correlated with structures of the classical Poincare surfaces of section. Despite its complex oscillatory nature the Wigner function delineates better classical features, for example showing the outline of islands of stability avoided by states supported by chaotic regions with greater sharpness than the Husimi function.

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.

Diamagnetism in a high Rydberg state is characterized by an adiabatic invariant that is quasi-conserved. It is expressed as an angle, sin^-1(1/ square root 5), that separates two types of localized states, along and perpendicular to the magnetic field. The value of the angle and the origins of localization are traced to the geometry and the dynamics of the interaction.

We present here a study of the quantum phase space localization (Wigner functions) in dependence of semiclassical quantization rules for the hydrogen atom in magnetic fields. We consider primarily two energy regions. In the first the corresponding classical system is near-integrable and in the second near fully chaotic. We study phase space localization (scars) close to stable and unstable trajectories, tori and the invariant manifold associated with the almost circular, unstable periodic trajectory. We find that these classical structures are an important element for the structure of the wavefunctions and the semiclassical quantization predictions are in good agreement with the quantal results.

The problem of computing the intrinsic properties of the system 'many-electron atomic state plus magnetic field' is treated by formulating and solving state-specific, non-Hermitian matrices of manageable sizes. These are created by appropriate choices and optimization of different function spaces representing the localized and the asymptotic part of the wavefunction of the field-dressed state which is made square-integrable through the use of r to re^{-i theta} for the coordinate of the outgoing electron. The localized parts of the wavefunction and of the Hamiltonian are expressed in terms of the real coordinates. Application was made to the closely lying H^- doubly excited states 2p²³P, '2s²+2p²' ¹S and 2p²¹D for field strengths gamma =0.0-0.03 au and results were obtained for the variation of the binding energies with respect to the H n=2 threshold, the autoionization widths and the geometrical shapes of the electron density.

We present a report on experiments performed at Imperial College on diamagnetic effects in many-electron atoms. These include both atomic-beam experiments with DC fields, for which the measurement technique involves detecting Rydberg atoms after laser excitation, and experiments on columns of atomic vapour, in which case we have studied Faraday rotation induced by a pulsed magnetic field. In the atomic-beam experiments, interesting differences between the sigma ⁺ and sigma ^- spectra have been observed. We also describe theoretical work related to the interpretation of the data for both sets of experiments.

We present a new method for computing the spectrum of one-electron Rydberg states of non-hydrogenic atoms in a magnetic field, at constant scaled energy. It is based on a variant of the R-matrix method allowing the computation of many energy levels in a single diagonalization. The results are compared with recently obtained high-resolution experimental spectra of the helium atom. The relation between peaks observed in the Fourier transform of scaled spectra and classical closed orbits is discussed. We show the existence of 'ghost' peaks not corresponding to any closed orbit, and also of peaks existing only in non-hydrogenic spectra, due the scattering of the electron by the ionic core.

The new method recently presented by Halley et al., combining R-matrix and complex coordinate rotation methods for the calculation of near-threshold resonance dominated spectra, is extended and applied to the calculation of spectra in electric and parallel electric/magnetic fields. Results obtained for the Stark spectra of the sodium atom above the classical ionization threshold are in excellent agreement with experimental findings and those from a previous calculation using a semiclassical method. Results obtained for the sodium atom in the same energy region but in parallel electric and magnetic fields represent the first obtained for any non-hydrogenic atom in such combined fields above the classical ionization threshold.

The degree of linear polarization P_L for two-photon absorption of alkali atoms in intermediate strength magnetic fields has been computationally investigated. Polarization spectra were obtained for n_as ²S_1/2 to n_bs ²S_1/2 transitions with the photons in near resonance with intermediate ²P_j levels. Previous experiments and calculations in the absence of a magnetic held revealed a complete polarization reversal (P_L approximately -100%) for frequencies near resonance between the ²P_1/2 and ²P_3/2 levels due to fine-structure coherence. Application of weak to intermediate strength fields induce additional depolarization features as a result of anomalous Zeeman splitting. The fine-structure depolarization feature disappears for strong fields (Paschen-Back regime), while the calculations predict that Zeeman depolarization persists though the transitions become forbidden. Magnetic-field-dependent polarization spectra can be measured in the laboratory with high precision and it should prove interesting to trace through the weak, intermediate, and strong field regimes.

The study of quantum manifestations of chaos in a generalized van der Waals potential can be carried out using the SO(4, 2) group theoretical approach. We discuss certain salient features of using the method. By analysing some of the recently introduced measures such as the cumulative spacing distribution, T-function and U-function, we demonstrate the efficacy of using the Brody distribution in this problem.

For the hydrogen atom in crossed, perpendicular, static electric and magnetic fields we show the existence of three elementary Kepler ellipses, two in the plane perpendicular to the magnetic field and one, three-dimensional, largely perpendicular to the electric field. They are determined from the symmetry properties in the low field limit, occurring for all external field combinations. Their parameter dependences and stability properties are analysed in detail. In the case that the orbits are stable the corresponding maximum orientated quantum states are identified. For states with sufficient low principal quantum numbers the total splitting of a given n-shell is described by semiclassical quantization of these three Kepler ellipses.

We examine the quantum spectra of hydrogen atoms in external magnetic and electric fields above the ionization threshold with respect to signatures of classical chaos characteristics of open systems. The spectra are obtained by calculating wavefunctions and photoionization cross sections in the continuum region with the aid of the complex-coordinate-rotation method. We find that the photoionization cross sections exhibit strong Ericson fluctuations, a quantum feature characteristic of classically chaotic scattering, in energy-field regions where classical trajectory calculations reveal a fractal dependence of the classical ionization time on the initial conditions. We also compare the nearest-neighbour-spacing distributions of complex resonance energies with predictions of random-matrix theories and and that our results are well reproduced by a Ginibre distribution.

A report on experiments performed in Munich on Rydberg atoms in strong static fields is presented. These include high-resolution laser spectroscopy and wavepacket generation and detection. The results are mainly interpreted by semiclassical techniques. Using the Fourier analysis of high-resolution scaled spectra in strong crossed electric and magnetic fields, quasi-Landau resonances in a particular parameter range are investigated. The problem of singularities in the semiclassical models describing the quasi-Landau spectra is discussed, and a qualitative solution to this problem is given. Rydberg wavepackets in strong magnetic fields were investigated. For this purpose Rydberg atoms are excited by pairs of time-separated short UV laser pulses. In these experiments the superior performance of a recently proposed technique is demonstrated. The obtained results for strong magnetic fields are interpreted semiclassically. Wavepacket propagation along a particular quasi-Landau orbit is clearly identified. In the last part of the paper, properties of high angular momentum Rydberg atoms in strong crossed electric and magnetic fields are discussed. The classically regular motion in that regime corresponds to situations well known from the motion of charged particles in Penning traps. A simple method for the preparation and isolation of this type of Rydberg atom is presented.