Computation of the Arnol'd web for the hydrogen atom in crossed electric and magnetic fields

Dynamical Tunneling in More than Two Degrees of Freedom

Recent progress towards understanding the mechanism of dynamical tunneling in Hamiltonian systems with three or more degrees of freedom (DoF) is reviewed. In contrast to systems with two degrees of freedom, the three or more degrees of freedom case presents several challenges. Specifically, in higher-dimensional phase spaces, multiple mechanisms for classical transport have significant implications for the evolution of initial quantum states. In this review, the importance of features on the Arnold web, a signature of systems with three or more DoF, to the mechanism of resonance-assisted tunneling is illustrated using select examples. These examples represent relevant models for phenomena such as intramolecular vibrational energy redistribution in isolated molecules and the dynamics of Bose-Einstein condensates trapped in optical lattices.

Many-Body Interference at the Onset of Chaos

We unveil the signature of many-body interference across dynamical regimes of the Bose-Hubbard model. Increasing the particles' indistinguishability enhances the temporal fluctuations of few-body observables, with a dramatic amplification at the onset of quantum chaos. By resolving the exchange symmetries of partially distinguishable particles, we explain this amplification as the fingerprint of the initial state's coherences in the eigenbasis.

Chaos in the three-site Bose-Hubbard model: Classical versus quantum

We consider a quantum many-body system-the Bose-Hubbard system on three sites-which has a classical limit, and which is neither strongly chaotic nor integrable but rather shows a mixture of the two types of behavior. We compare quantum measures of chaos (eigenvalue statistics and eigenvector structure) in the quantum system, with classical measures of chaos (Lyapunov exponents) in the corresponding classical system. As a function of energy and interaction strength, we demonstrate a strong overall correspondence between the two cases. In contrast to both strongly chaotic and integrable systems, the largest Lyapunov exponent is shown to be a multivalued function of energy.

Spectroscopy characteristics and decay properties of plasma-embedded atoms in external electric and magnetic fields

This paper describes a computational approach within the framework of relativity theory for explaining the spectral and decay properties of atoms and ions embedded in a plasma and also in the presence of applied external electric and magnetic fields. It uses the configuration interaction approximation and the analytical potential derived from general ion-sphere theory to represent the atomic interactions within the plasma. In the model, the Dirac–Coulomb–Hamiltonian is reconstructed and the effects of the weak electric and magnetic field are treated as small perturbations. The eigenvalues for the orbitals and (radial) wave functions are obtained through the Dirac equations. A diagonalization is performed to include these plasma, electric, and magnetic terms. As an example, an environment with a hot-dense plasma and weak electric and magnetic fields is considered, where the interaction is much weaker than the Coulomb interaction, yet much stronger than the spin–orbit interaction. The atomic structures and spectra of a selected hydrogen atom are presented for a wide range of electron densities, temperatures, and electric and magnetic fields. The behavior of the energy and radiative transitions with respect to these interactions is analyzed in detail. The results obtained from the proposed approach are critically compared with other available results. The present study not only advances our understanding of the electronic structures and radiation characteristics of atomic systems in external fields but may also be relevant for astrophysics and laboratory experiments, especially on the solar corona, laser-produced plasmas, and so on.

Intramolecular vibrational energy redistribution and the quantum ergodicity transition: a phase space perspective

The onset of facile intramolecular vibrational energy flow can be related to features in the connected network of anharmonic resonances in the classical phase space.

Dynamical tunneling versus fast diffusion for a non-convex Hamiltonian

This paper attempts to resolve the issue of the nature of the 0.01-0.1 cm(-1) peak splittings observed in high-resolution IR spectra of polyatomic molecules. One hypothesis is that these splittings are caused by dynamical tunneling, a quantum-mechanical phenomenon whereby energy flows between two disconnected regions of phase-space across dynamical barriers. However, a competing classical mechanism for energy flow is Arnol'd diffusion, which connects different regions of phase-space by a resonance network known as the Arnol'd web. The speed of diffusion is bounded by the Nekhoroshev theorem, which guarantees stability on exponentially long time scales if the Hamiltonian is steep. Here we consider a non-convex Hamiltonian that contains the characteristics of a molecular Hamiltonian, but does not satisfy the Nekhoroshev theorem. The diffusion along the Arnol'd web is expected to be fast for a non-convex Hamiltonian. While fast diffusion is an unlikely competitor for longtime energy flow in molecules, we show how dynamical tunneling dominates compared to fast diffusion in the nearly integrable regime for a non-convex Hamiltonian, as well as present a new kind of dynamical tunneling.

Arnold diffusion in a driven optical lattice

The effect of time-periodic forces on matter has been a topic of growing interest since the advent of lasers. It is known that dynamical systems with 2.5 or more degrees of freedom are intrinsically unstable. As a consequence, time-periodic driven systems can experience large excursions in energy. We analyze the classical and quantum dynamics of rubidium atoms confined to a time-periodic optical lattice with 2.5 degrees of freedom. When the laser polarizations are orthogonal, the system consists of two 1.5 uncoupled dynamical systems. When laser polarizations are turned away from orthogonal, an Arnold web forms and the dynamics undergoes a fundamental change. For parallel polarizations, we find huge random excursions in the rubidium atom energies and significant entanglement of energies in the quantum dynamics.

Classical dynamics and localization of resonances in the high-energy region of the hydrogen atom in crossed fields

When superimposing the potentials of external fields on the Coulomb potential of the hydrogen atom, a saddle point (called the Stark saddle point) appears. For energies slightly above the saddle point energy, one can find classical orbits that are located in the vicinity of this point. We follow those so-called quasi-Penning orbits to high energies and field strengths, observing structural changes and uncovering their bifurcation behavior. By plotting the stability behavior of those orbits against energy and field strength, the appearance of a stability apex is reported. A cusp bifurcation, located in the vicinity of the apex, will be investigated in detail. In this cusp bifurcation, another orbit of similar shape is found. This orbit becomes completely stable in the observed region of positive energy, i.e., in a region of parameter space, where the Kepler-like orbits located around the nucleus are already unstable. By quantum mechanically exact calculations, we prove the existence of signatures in quantum spectra belonging to those orbits. Husimi distributions are used to compare quantum-Poincaré sections with the extension of the classical torus structure around the orbits. Since periodic orbit theory predicts that each classical periodic orbit contributes an oscillating term to photoabsorption spectra, we finally give an estimation for future experiments, which could verify the existence of the stable orbits.

Dynamical traps lead to the slowing down of intramolecular vibrational energy flow

The phenomenon of intramolecular vibrational energy redistribution (IVR) is at the heart of chemical reaction dynamics. Statistical rate theories, assuming instantaneous IVR, predict exponential decay of the population with the properties of the transition state essentially determining the mechanism. However, there is growing evidence that IVR competes with the reaction timescales, resulting in deviations from the exponential rate law. Dynamics cannot be ignored in such cases for understanding the reaction mechanisms. Significant insights in this context have come from the state space model of IVR, which predicts power law behavior for the rates with the power law exponent, an effective state space dimensionality, being a measure of the nature and extent of the IVR dynamics. However, whether the effective IVR dimensionality can vary with time and whether the mechanism for the variation is of purely quantum or classical origins are issues that remain unresolved. Such multiple power law scalings can lead to surprising mode specificity in the system, even above the threshold for facile IVR. In this work, choosing the well-studied thiophosgene molecule as an example, we establish the anisotropic and anomalous nature of the quantum IVR dynamics and show that multiple power law scalings do manifest in the system. More importantly, we show that the mechanism of the observed multiple power law scaling has classical origins due to a combination of trapping near resonance junctions in the network of classical nonlinear resonances at short to intermediate times and the influence of weak higher-order resonances at relatively longer times.

Mapping the Arnold web with a graphic processing unit

The Arnold diffusion constitutes a dynamical phenomenon which may occur in the phase space of a non-integrable Hamiltonian system whenever the number of the system degrees of freedom is M ≥ 3. The diffusion is mediated by a web-like structure of resonance channels, which penetrates the phase space and allows the system to explore the whole energy shell. The Arnold diffusion is a slow process; consequently, the mapping of the web presents a very time-consuming task. We demonstrate that the exploration of the Arnold web by use of a graphic processing unit-supercomputer can result in distinct speedups of two orders of magnitude as compared with standard CPU-based simulations.

Local predictability and nonhyperbolicity through finite Lyapunov exponent distributions in two-degrees-of-freedom Hamiltonian systems

By using finite Lyapunov exponent distributions, we get insight into both the local and global properties of a dynamical flow, including its nonhyperbolic behavior. Several distributions of finite Lyapunov exponents have been computed in two prototypical four-dimensional phase-space Hamiltonian systems. They have been computed calculating the growth rates of a set of orthogonal axes arbitrarily pointed at given intervals. We analyze how such distributions serve or not for tracing the orbit nature and local flow properties such as the unstable dimension variability, as the axes are allowed or not to tend to the largest stretching direction. The relationship between the largest and closest to zero exponent distribution is analyzed. It shows a linear dependency at short intervals, related to the number of degrees of freedom of the system. Finally, the hyperbolicity indexes, associated to the shadowing times, are calculated. They provide interesting information at very local scales, even when there are no Gaussian distributions and the values cannot be regarded as random variables.

Extracting multidimensional phase space topology from periodic orbits

We establish a hierarchical ordering of periodic orbits in a strongly coupled multidimensional Hamiltonian system. Phase space structures can be reconstructed quantitatively from the knowledge of periodic orbits alone. We illustrate our findings for the hydrogen atom in crossed electric and magnetic fields.

Ericson fluctuations in an open deterministic quantum system: theory meets experiment

We provide numerically exact photoexcitation cross sections of rubidium Rydberg states in crossed, static electric, and magnetic fields, in quantitative agreement with recent experimental results. Their spectral backbone underpins a clear transition towards the Ericson regime, associated with a universal, fluctuating behavior of the cross section of strongly coupled, fragmenting quantum systems.

Analyzing intramolecular dynamics by fast Lyapunov indicators

We report an analysis of intramolecular dynamics of the highly excited planar carbonyl sulfide below and at the dissociation threshold by the fast Lyapunov indicator method. By mapping out the variety of dynamical regimes in the phase space of this molecule, we obtain the degree of regularity of the system versus its energy. We combine this stability analysis with a periodic orbit search, which yields a family of elliptic periodic orbits in the regular part of phase space and a family of in-phase collinear hyperbolic orbits associated with the chaotic regime.

Quantum Arnol'd diffusion in a simple nonlinear system

We study the fingerprint of the Arnol'd diffusion in a quantum system of two coupled nonlinear oscillators with a two-frequency external force. In the classical description, this peculiar diffusion is due to the onset of a weak chaos in a narrow stochastic layer near the separatrix of the coupling resonance. We have found that global dependence of the quantum diffusion coefficient on model parameters mimics, to some extent, the classical data. However, the quantum diffusion happens to be slower than the classical one. Another result is the dynamical localization that leads to a saturation of the diffusion after some characteristic time. We show that this effect has the same nature as for the studied earlier dynamical localization in the presence of global chaos. The quantum Arnol'd diffusion represents a new type of quantum dynamics and can be observed, for example, in two-dimensional semiconductor structures (quantum billiards) perturbed by time-periodic external fields.

Manifestation of Arnol'd diffusion in quantum systems

We study an analog of the classical Arnol'd diffusion in a quantum system of two coupled nonlinear oscillators one of which is governed by an external periodic force with two frequencies. In a classical model this very weak diffusion happens in a narrow stochastic layer along the coupling resonance and leads to an increase of the total energy of the system. We show that quantum dynamics of wave packets mimics, up to some extent, global properties of the classical Arnol'd diffusion. This specific diffusion represents a new type of quantum dynamics and may be observed, for example, in 2D semiconductor structures (quantum billiards) perturbed by time-periodic external fields.

A method for visualization of invariant sets of dynamical systems based on the ergodic partition

We provide an algorithm for visualization of invariant sets of dynamical systems with a smooth invariant measure. The algorithm is based on a constructive proof of the ergodic partition theorem for automorphisms of compact metric spaces. The ergodic partition of a compact metric space A, under the dynamics of a continuous automorphism T, is shown to be the product of measurable partitions of the space induced by the time averages of a set of functions on A. The numerical algorithm consists of computing the time averages of a chosen set of functions and partitioning the phase space into their level sets. The method is applied to the three-dimensional ABC map for which the dynamics was visualized by other methods in Feingold et al. [J. Stat. Phys. 50, 529 (1988)]. (c) 1999 American Institute of Physics.