Wave Chaos

High frequency asymptotics – often also called semiclassical approximations in a quantum context – deals with understanding the transition from a wave to a ray description of linear wave equations. Examples are the relation between ray and wave optics or identifying the classical equations of motion (Hamilton’s equations) as a limiting case of quantum mechanics. In the last few decades, the interest shifted here towards relating the solutions of linear wave equations to the dynamical properties of the underlying ray dynamics. It could be shown that dynamical features ranging from regular to purely chaotic behaviour leave distinct fingerprints in the solutions of associated wave equations. These can be divided into systems specific contributions relating a ray dyanmics to particular wave solutions and universal features linking wav fluctuations to Random Matrix statistics. The theory has initially mainly advanced in a quantum context giving rise to the name quantum chaos. Schrödinger’s equation is a scalar, linear wave equation and is in that respect not very different from other physically relevant wave equations such as in optics or linear elasticity. The ‘weirder’ properties of quantum mechanics, as for example, the measurement process or many-particle effects and the violation of Bell’s inequalities have in fact rarely been considered in semiclassical approaches to quantum mechanics until recently. In that sense, techniques and insights from quantum chaos can and have been applied to classical wave equations such as in optics, elastodynamics or acoustics giving rise to a much broader field often referred to as wave chaos. In fact, many of the methods and concepts used and developed in quantum chaos have been considered independently in the optics and engineering community, and developments have run in parallel often with little cross fertilization. 

Random Coupling Model And Wave Statistics

Increasingly complex scenarios in electronics and telecommunications make the detail ofstructures and circuitry ever more difficult to model. In addition, it is often found in optics, electronics, and acoustics that, the need for higher bit-rates pushes wave sources to emit at very short wavelengths compared to the characteristics size of the excited system. In this regime, the scattering process can be very sensitive to details. The complex boundary shape of an enclosure can lead to the phenomenon of ray chaos in which a ray trajectory inside the enclosure shows strong sensitivity to initial conditions. From experimental observations, it is known that this results in a very high variability of wave properties of the system. A statistical approach to understanding the short-wavelength behavior of the system then becomes appropriate. Specifically, one can ask what the statistics of quantities of interest are relative to a suitable random choice of the system, and this is the goal of the RCM. The Random Coupling Model (RCM) is a statistical method that represents scattering and impedance matrixes of ports immersed in highly resonating cavities. The method has been successfully applied and tested in structures whose geometry is irregular and for which the wavelength of the excitation field is small compared to characteristic lengths scale of the cavity. Direct illumination or specular components between ports can be included into the RCM through Phase Space Methods. Antenna and aperture coupling, multiply connected chaotic environments, as well as through the wall transmission has been merged with universal statistics within the RCM, thus extending canonical power balance methods. This allows for tackling extremely complicated scenarios involving the coupling of an external radiation inside cavities with arbitrary losses. Current research efforts are devoted to model the radiation of statistical sources operating inside complex cavities through the RCM

Quantum Graphs And Quantum Search

Quantum walks can provide polynomial and even exponential speed-up compared to classical random walks and may serve as a universal computational primitive for quantum computation. One of the most fascinating applications of quantum walks is their use in spatial quantum search. Like Grover’s search algorithm for searching an unstructured database, quantum walk search algorithms can achieve up to quadratic speed-up compared to the corresponding classical search. The group has worked both on continuous discrete quantum walks and quantum search algorithms. It has been demonstrate that quantum searching is related to an avoided crossing problem with a particular low density of state near the crossing, such is for example present near the Dirac point in graphene. Performing quantum search and quantum state transfer on graphene provides a new way of channeling energy and information across lattices and between distinct sites.

Wave Chaos In Acoustics And Elasticity

Interpreting wave phenomena in terms of an underlying ray dynamics adds a new dimension to the analysis of linear wave equations. Forming explicit connections between spectra and wavefunctions on the one hand and the properties of a related ray dynamics on the other hand is a comparatively new research area, especially in elasticity and acoustics. The theory has indeed been developed primarily in a quantum context; it is increasingly becoming clear, however, that important applications lie in the field of mechanical vibrations and acoustics.

We provide an overview over basic concepts in this emerging field of wave chaos. This ranges from ray approximations of the Green function to periodic orbit trace formulae and random matrix theory and summarizes the state of the art in applying these ideas in acoustics—both experimentally and from a theoretical/numerical point of view.