Chaos in the relativistic two-electron atom

Chaotic autoionization of the relativistic two-electron atom is investigated. A theoretical analysis of chaotic dynamics of the relativistic outer electron under the periodic perturbation due to the inner electron, assumed to be on a circular orbit, based on the Chirikov criterion, is given. The diffusion coefficient, the ionization rate, and time are calculated. (c) 2002 American Institute of Physics.     

The rotating two-electron atom in an external electric field

Three rigid-body solutions for a rotating two-electron atom under the influence of an electric field along the rotation axis have been obtained within a classical approach. One exact solution gives in the zero-field case a previous result known as a rotor, another exact solution in the zero-field case gives the Wannier unbound solution, and a numerical solution in the zero-field case gives the asymmetric top or Langmuir solution. The stability analysis of the linearized motions around each of the equilibrium configurations was made for different values of the electric field. We find critical values of the electric field beyond which no equilibrium exists. Values for the classical polarizability of rotating H^– and He are reported.     

Stochastic autoionization of a relativistic two-electron atom

On the basis of the analysis of absorption spectra of Er³⁺:PbMoO₄ crystals made for the transitions from the ground ⁴ I _15/2 state to excited states of Er³⁺ ions by the Judd-Ofelt method, the main spectroscopic characteristics of the crystals were obtained, including the transition probabilities and the radiative lifetimes.     

Quantum effects of periodic orbits for the kicked top

We analyze traces of powers of the time evolution operator of a periodically kicked top. Semiclassically, such traces are related to periodic orbits of the classical map. We derive the semiclassical traces in a coherent state basis and show how the periodic orbits can be recovered via a Fourier transform. A breakdown of the stationary phase approximation is detected. The quasi energy spectrum remains elusive due to lack of knowledge of sufficiently many periodic orbits. Divergencies of periodic orbit formulas are avoided by appealing to the finiteness of the quantum mechanical Hilbert space. The traces also enter the coefficients of the characteristic polynominal of the Floquet operator. Statistical properties of these coefficients give rise to a new criterion for the distinction of chaos and regular motion.     

No periodic orbits for the type A Bianchi's systems

It is known that the 6 models of Bianchi class A have no periodic solutions. In this article we provide a new, direct, unified and easier proof of this result.     

Nondispersive two-electron wave packets in a helium atom

We demonstrate the existence of stable nondispersing two-electron wave packets in the helium atom in combined magnetic and circularly polarized microwave fields. These packets follow circular orbits and we show that they can also exist in quantum dots. Classically the two electrons follow trajectories which resemble orbits discovered by Langmuir and which were used in attempts at a Bohr-like quantization of the helium atom. Eigenvalues of a generalized Hessian matrix are computed to investigate the classical stability of these states. Diffusion Monte Carlo simulations demonstrate the quantum stability of these two-electron wave packets in the helium atom and quantum-dot helium with an impurity center.     

Ionization of two-electron atom in ultrashort laser pulse

The method of numerical integration of the classical equations of motion was used to study interaction of a model two-electron atom with ultrashort laser pulses. Mechanisms and specific features of the ionization process were analyzed in a broad range of laser-field parameters.     

Quantum entanglement in a soluble two-electron model atom

We investigate the entanglement properties of bound states in an exactly soluble two-electron model, the Moshinsky atom. We present exact entanglement calculations for the ground, first and second excited states of the system. We find that these states become more entangled when the relative inter-particle interaction becomes stronger. As a general trend, we also observe that the entanglement of the eigenstates tends to increase with the states’ energy. There are, however, “entanglement level-crossings” where the entanglement of a state becomes larger than the entanglement of other states with higher energy. In the limit of weak interaction, we also compute (exactly) the entanglement of higher excited states. Excited states with anti-parallel spins are found to involve a considerable amount of entanglement even for an arbitrarily weak (but non zero) interaction. This minimum amount of entanglement increases monotonically with the state’s energy. Finally, the connection between entanglement and the Hartree-Fock approximation in the Moshinsky model is addressed. The quality of the ground-state Hartree-Fock approximation is shown to deteriorate, and the corresponding correlation energy to grow, as the entanglement of the (exact) ground state increases. The present work goes beyond previous related studies because we fully take into account the identical character of the two constituting particles in the entanglement calculations, and provide analytical, exact results both for the ground and the first few excited states.     

Measuring scars of periodic orbits

The phenomenon of periodic orbit scarring of eigenstates of classically chaotic systems is attracting increasing attention. Scarring is one of the most important "corrections" to the ideal random eigenstates suggested by random matrix theory. This paper discusses measures of scars and in so doing also tries to clarify the concepts and effects of eigenfunction scarring. We propose a universal scar measure which takes into account an entire periodic orbit and the linearized dynamics in its vicinity. This measure is tuned to pick out those structures which are induced in quantum eigenstates by unstable periodic orbits and their manifolds. It gives enhanced scarring strength as measured by eigenstate overlaps and inverse participation ratios, especially for longer orbits. We also discuss off-resonance scars which appear naturally on either side of an unstable periodic orbit.     

Stability of minimal periodic orbits

Symplectic twist maps are obtained from a Lagrangian variational principle. It is well known that nondegenerate minima of the action correspond to hyperbolic orbits of the map when the twist is negative definite and the map is two-dimensional. We show that for more than two dimensions, periodic orbits with minimal action in symplectic twist maps with negative definite twist are not necessarily hyperbolic. In the proof we show that in the neighborhood of a minimal periodic orbit of period n, the nth iterate of the map is again a twist map. This is true even though in general the composition of twist maps is not a twist map.     

Basins of periodic orbits for elliptic maps of the torus

The behavior of basins of periodic orbits, for families of elliptic maps in the 2D torus depending on a parameter, is studied. We give an explicit formula for periodic orbits (i.e., central points of basins), considering also the occurrence of singular situations. Such a formula describes the evolution of basins, showing that onset and disappearance of periodic orbits cannot be reduced to a simple bifurcation scheme. Also, the stochastic features of the strange attractor at the border of ellipticity may be related to the dynamics of collapsing basins.     

The periodic orbits of an area preserving twist map

We study the oscillation properties of periodic orbits of an area preserving twist map. The results are inspired by the similarity between the gradient flow of the associated action-function, and a scalar parabolic PDE in one space dimension. The Conley-Zehnder Morse theory is used to construct orbits with prescribed oscillatory behavior.     

The growth rate for the number of singular and periodic orbits for a polygonal billiard

For any simply connected polygon in the plane, the number of billiard orbits which begin and end at a vertex grows subexponentially with respect to the length or to the number of reflections. This implies that the numbers of isolated periodic orbits and of families of parallel periodic orbits do grow subexponentially. The main technical device is a calculation showing that the topological entropy of the Poincaré map for the billiard flow is equal to zero.Supported in part by NSF Grant #DMS-8414400     

Recurrence plots and unstable periodic orbits

A recurrence plot is a two-dimensional visualization technique for sequential data. These plots are useful in that they bring out correlations at all scales in a manner that is obvious to the human eye, but their rich geometric structure can make them hard to interpret. In this paper, we suggest that the unstable periodic orbits embedded in a chaotic attractor are a useful basis set for the geometry of a recurrence plot of those data. This provides not only a simple way to locate unstable periodic orbits in chaotic time-series data, but also a potentially effective way to use a recurrence plot to identify a dynamical system. (c) 2002 American Institute of Physics.     

Dynamical tracking of unstable periodic orbits

Tracking unstable periodic orbits and its stabilization by large periodic modulation of a control parameter are studied numerically in the Hénon map and laser equations. Some important scaling relations linking the tracking range to the modulation amplitude and frequency are deduced. The results obtained with both models are compared. Experimental realization of dynamical tracking is demonstrated in a loss-driven CO₂ laser where cavity detuning or losses are periodically modulated.