Uniform semiclassical approximations on a topologically non-trivial configuration space: The hydrogen atom in an electric field

Semiclassical periodic-orbit theory and closed-orbit theory represent a quantum spectrum as a superposition of contributions from individual classical orbits. Close to a bifurcation, these contributions diverge and have to be replaced with a uniform approximation. Its construction requires a normal form that provides a local description of the bifurcation scenario. Usually, the normal form is constructed in flat space. We present an example taken from the hydrogen atom in an electric field where the normal form must be chosen to be defined on a sphere instead of a Euclidean plane. In the example, the necessity to base the normal form on a topologically non-trivial configuration space reveals a subtle interplay between local and global aspects of the phase space structure. We show that a uniform approximation for a bifurcation scenario with non-trivial topology can be constructed using the established uniformization techniques. Semiclassical photo-absorption spectra of the hydrogen atom in an electric field are significantly improved when based on the extended uniform approximations

Fictitious time wave packet dynamics: I. Nondispersive wave packets in the quantum Coulomb problem

Nondispersive wave packets in a fictitious time variable are calculated analytically for the field-free hydrogen atom. As is well known by means of the Kustaanheimo-Stiefel transformation the Coulomb problem can be converted into that of a four-dimensional harmonic oscillator, subject to a constraint. This regularization makes use of a fictitious time variable, but arbitrary Gaussian wave packets in that time variable in general violate that constraint. The set of "restricted Gaussian wave packets" consistent with the constraint is constructed and shown to provide a complete basis for the expansion of states in the original three-dimensional coordinate space. Using that expansion arbitrary localized Gaussian wave packets of the hydrogen atom can be propagated analytically, and exhibit a nondispersive periodic behavior as functions of the fictitious time. Restricted wave packets with and without well defined angular momentum quantum n umbers are constructed. They will be used as trial functions in time-dependent variational computations for the hydrogen atom in static external fields in the subsequent paper [T. Fab\v{c}i\v{c} et al., submitted].Comment: 12 pages, 3 figure

Closed orbits and their bifurcations in the crossed-fields hydrogen atom

A systematic study of closed classical orbits of the hydrogen atom in crossed electric and magnetic fields is presented. We develop a local bifurcation theory for closed orbits which is analogous to the well-known bifurcation theory for periodic orbits and allows identifying the generic closed-orbit bifurcations of codimension one. Several bifurcation scenarios are described in detail. They are shown to have as their constituents the generic codimension-one bifurcations, which combine into a rich variety of complicated scenarios. We propose heuristic criteria for a classification of closed orbits that can serve to systematize the complex set of orbits

Semiclassical quantization of the hydrogen atom in crossed electric and magnetic fields

The S-matrix theory formulation of closed-orbit theory recently proposed by Granger and Greene is extended to atoms in crossed electric and magnetic fields. We then present a semiclassical quantization of the hydrogen atom in crossed fields, which succeeds in resolving individual lines in the spectrum, but is restricted to the strongest lines of each n-manifold. By means of a detailed semiclassical analysis of the quantum spectrum, we demonstrate that it is the abundance of bifurcations of closed orbits that precludes the resolution of finer details. They necessitate the inclusion of uniform semiclassical approximations into the quantization process. Uniform approximations for the generic types of closed-orbit bifurcation are derived, and a general method for including them in a high-resolution semiclassical quantization is devised

Exceptional Points in Atomic Spectra

We report the existence of exceptional points for the hydrogen atom in crossed magnetic and electric fields in numerical calculations. The resonances of the system are investigated and it is shown how exceptional points can be found by exploiting characteristic properties of the degeneracies, which are branch point singularities. A possibility for the observation of exceptional points in an experiment with atoms is proposed.Comment: 4 pages, 4 figures, 1 table, to be published in Physical Review Letter

The hydrogen atom in an electric field: Closed-orbit theory with bifurcating orbits

Closed-orbit theory provides a general approach to the semiclassical description of photo-absorption spectra of arbitrary atoms in external fields, the simplest of which is the hydrogen atom in an electric field. Yet, despite its apparent simplicity, a semiclassical quantization of this system by means of closed-orbit theory has not been achieved so far. It is the aim of this paper to close that gap. We first present a detailed analytic study of the closed classical orbits and their bifurcations. We then derive a simple form of the uniform semiclassical approximation for the bifurcations that is suitable for an inclusion into a closed-orbit summation. By means of a generalized version of the semiclassical quantization by harmonic inversion, we succeed in calculating high-quality semiclassical spectra for the hydrogen atom in an electric field

Photoabsorption spectra of the diamagnetic hydrogen atom in the transition regime to chaos: Closed orbit theory with bifurcating orbits

With increasing energy the diamagnetic hydrogen atom undergoes a transition from regular to chaotic classical dynamics, and the closed orbits pass through various cascades of bifurcations. Closed orbit theory allows for the semiclassical calculation of photoabsorption spectra of the diamagnetic hydrogen atom. However, at the bifurcations the closed orbit contributions diverge. The singularities can be removed with the help of uniform semiclassical approximations which are constructed over a wide energy range for different types of codimension one and two catastrophes. Using the uniform approximations and applying the high-resolution harmonic inversion method we calculate fully resolved semiclassical photoabsorption spectra, i.e., individual eigenenergies and transition matrix elements at laboratory magnetic field strengths, and compare them with the results of exact quantum calculations.Comment: 26 pages, 9 figures, submitted to J. Phys.

Long-range, critical-point dynamics in oil field flow rate data

Earthquake triggering data exhibit long-range spatio-temporal correlations of the power-law form C(l) ∼ l−α and anomalously-slow temporal diffusion of the mean triggering distance of the form: 〈l〉 ∼ tH, with H < 0.5. We examine spatio-temporal correlations in subsurface effective stress state caused by fluid injection and extraction at well sites in a hydrocarbon reservoir using a multivariate statistical regression model, and observe long-range correlations in flow rate that cannot be caused by Darcy flow alone. Significantly-correlated well pairs align with the directions of incipient horizontal-displacement tensile and shear failure in the present-day stress field, while the contours of the first principal component of the regression matrix closely follow the macroscopic fault pattern in the main producing horizon. The correlation function for well pairs has a power-law form with α ≈ 0.5, and the mean correlation distance increases with H ≈ 0.33, implying a similar critical-point response to perturbations in effective stress as the earthquake data