Arnd Bäcker

Visualization and comparison of classical structures and quantum states of four-dimensional maps.

For generic 4D symplectic maps we propose the use of 3D phase-space slices, which allow for the global visualization of the geometrical organization and coexistence of regular and chaotic motion. As an example, we consider two coupled standard maps. The advantages of the 3D phase-space slices are presented in comparison to standard methods, such as 3D projections of orbits, the frequency analysis, and a chaos indicator. Quantum mechanically, the 3D phase-space slices allow for the comparison of Husimi functions of eigenstates of 4D maps with classical phase-space structures. This confirms the semiclassical eigenfunction hypothesis for 4D maps.

Regular-to-chaotic tunneling rates using a fictitious integrable system.

We derive a formula predicting dynamical tunneling rates from regular states to the chaotic sea in systems with a mixed phase space. Our approach is based on the introduction of a fictitious integrable system that resembles the regular dynamics within the island. For the standard map and other kicked systems we find agreement with numerical results for all regular states in a regime where resonance-assisted tunneling is not relevant.

Flooding of chaotic eigenstates into regular phase space islands.

We introduce a criterion for the existence of regular states in systems with a mixed phase space. If this condition is not fulfilled chaotic eigenstates substantially extend into a regular island. Wave packets started in the chaotic sea progressively flood the island. The extent of flooding by eigenstates and wave packets increases logarithmically with the size of the chaotic sea and the time, respectively. This new effect can be observed for island chains with just 10 islands.

Behaviour of boundary functions for quantum billiards

We study the behaviour of the normal derivative of eigenfunctions of the Helmholtz equation inside billiards with Dirichlet boundary condition. These boundary functions are of particular importance because they uniquely determine the eigenfunctions inside the billiard and also other physical quantities of interest. Therefore, they form a reduced representation of the quantum system, analogous to the Poincare section of the classical system. For the normal derivatives we introduce an equivalent to the standard Green function and derive an integral equation on the boundary. Based on this integral equation we compute the first two terms of the mean asymptotic behaviour of the boundary functions for large energies. The first term is universal and independent of the shape of the billiard. The second one is proportional to the curvature of the boundary. The asymptotic behaviour is compared with numerical results for the stadium billiard, different limacon billiards and the circle billiard, and good agreement is found. Furthermore, we derive an asymptotic completeness relation for the boundary functions.

Global structure of regular tori in a generic 4D symplectic map

For the case of generic 4d symplectic maps with a mixed phase space, we investigate the global organization of regular tori. For this, we compute elliptic 1-tori of two coupled standard maps and display them in a 3d phase-space slice. This visualizes how all regular 2-tori are organized around a skeleton of elliptic 1-tori in the 4d phase space. The 1-tori occur in two types of one-parameter families: (α) Lyapunov families emanating from elliptic-elliptic periodic orbits, which are observed to exist even far away from them and beyond major resonance gaps, and (β) families originating from rank-1 resonances. At resonance gaps of both types of families either (i) periodic orbits exist, similar to the Poincaré-Birkhoff theorem for 2d maps, or (ii) the family may form large bends. In combination, these results allow for describing the hierarchical structure of regular tori in the 4d phase space analogously to the islands-around-islands hierarchy in 2d maps.

Computational Physics Education with Python

Educators at an institution in Germany have started using Python to teach computational physics. The author describes how graphical visualizations also play an important role, which he illustrates here with a few simple examples

Hierarchical Fractal Weyl Laws for Chaotic Resonance States in Open Mixed Systems

In open chaotic systems the number of long-lived resonance states obeys a fractal Weyl law, which depends on the fractal dimension of the chaotic saddle. We study the generic case of a mixed phase space with regular and chaotic dynamics. We find a hierarchy of fractal Weyl laws, one for each region of the hierarchical decomposition of the chaotic phase-space component. This is based on our observation of hierarchical resonance states localizing on these regions. Numerically this is verified for the standard map and a hierarchical model system.

Quality factors and dynamical tunneling in annular microcavities

The key characteristic of an optical mode in a microcavity is its quality factor describing the optical losses. The numerical computation of this quantity can be very demanding for present-day devices. Here we show for a certain class of whispering-gallery cavities that the quality factor is related to dynamical tunneling, a phenomenon studied in the field of quantum chaos. We extend a recently developed approach for determining dynamical tunneling rates to open cavities. This allows us to derive an analytical formula for the quality factor which is in very good agreement with full solutions of Maxwell’s equations.

Quantum Chaos in Billiards

An important class of systems - billiards - can show a wide variety of dynamical behavior. Using tools developed in Python, researchers can interactively study the complexity of these dynamics. Such behavior is directly reflected in properties of the corresponding quantum systems, such as eigenvalue statistics or the structure of eigenfunctions

Universal Quantum Localizing Transition of a Partial Barrier in a Chaotic Sea

Generic 2D Hamiltonian systems possess partial barriers in their chaotic phase space that restrict classical transport. Quantum mechanically, the transport is suppressed if Plancks constant h is large compared to the classical flux, h>>Φ, such that wave packets and states are localized. In contrast, classical transport is mimicked for h<<Φ. Designing a quantum map with an isolated partial barrier of controllable flux Φ is the key to investigating the transition from this form of quantum localization to mimicking classical transport. It is observed that quantum transport follows a universal transition curve as a function of the expected scaling parameter Φ/h. We find this curve to be symmetric to Φ/h=1, having a width of 2 orders of magnitude in Φ/h, and exhibiting no quantized steps. We establish the relevance of local coupling, improving on previous random matrix models relying on global coupling. It turns out that a phenomenological 2×2 model gives an accurate analytical description of the transition curve.