R. Blümel

Excitation of molecular rotation by periodic microwave pulses. A testing ground for Anderson localization

We study the excitation of molecular rotation by microwave pulses of duration σ which occur periodically with frequency ω. We analyze the molecular dynamics both classically and quantum mechanically and consider situations where the coupling of the field to the molecule is strong. In both approaches, the angular momentum transmitted to the molecule is confined to a finite band of width ≊1/σ. But, while the classical dynamics displays chaotic features, the quantum treatment distinguishes clearly between two regimes. Resonance excitation occurs when ω is rationally related to the basic rotation frequency ω0. Off resonance (ω/ω0 irrational), the probability to transfer angular momentum to the molecule is small and the underlying mechanism for this effect is analogous to the Anderson model of localization in a one‐dimensional random lattice with a finite number of sites. We show that the conditions required by our analysis can be achieved with, e.g., PbTe or CsI molecules and conventional field strengths and ...

Microwave Ionization of Highly Excited Hydrogen Atoms

Within a 1-dimensional model we calculate quantum mechanically the probability to ionize a highly excited hydrogen atom by a monochromatic microwave field. Based on a detailed analysis of the ionization process we developed a computational scheme as well as a simple physical framework which are presented and discussed. Our calculations are in good agreement with the experimental results. We show that the experimentally measured ionization thresholds are due to a sharp transition between two localization regimes and that recently measured structures below the classical chaos border are due to unresolved clusters of Floquet pseudo crossings. We propose an experimental method by which one could measure the distance and distribution of crossing Floquet eigenvalues.

A simple model for chaotic scattering: II. Quantum mechanical theory

Abstract In this paper we present the quantum analysis of a scattering problem which displays chaotic (irregular) features when analyzed classically. We treat the problem both semi-classically and exactly and show that the “finger print” of the classical chaos on the quantum description in the appearance of universal fluctuations in the cross section. Their statistics is analogous to the one expected from a random matrix description.

Explicitly solvable cases of one-dimensional quantum chaos

We identify a set of quantum graphs with unique and precisely defined spectral properties called regular quantum graphs. Although chaotic in their classical limit with positive topological entropy, regular quantum graphs are explicitly solvable. The proof is constructive: we present exact, convergent periodic orbit expansions for individual energy levels, thus obtaining an analytical solution for the spectrum of regular quantum graphs that is complete, explicit, and exact.

On the transition to chaotic scattering

A classical scattering system is chaotic if it possesses a fractal set of trapped unstable orbits, resulting in singular deflection functions. A scattering system is regular if it supports only a countable set of trapped unstable orbits. Its deflection functions are piecewise smooth with at most a countable number of scattering singularities caused by the trapped orbits. Despite the simple structure of the deflection functions, the Poincare scattering mapping (PSM) may be regular, hyperbolic or display mixed dynamics. Thus, the degree of chaoticity of the PSM serves as a finer scale for the discussion of the transition to chaotic scattering in the classical domain. In the quantum domain the authors show that the properties of the PSM determine the statistics of the eigenphases of the S-matrix, and that, if the PSM is hyperbolic, the eigenphases follow the statistics predicted by random matrix theory.

Exact, convergent periodic-orbit expansions of individual energy eigenvalues of regular quantum graphs

We present exact, explicit, convergent periodic-orbit expansions for individual energy levels of regular quantum graphs in the paper. One simple application is the energy levels of a particle in a piecewise constant potential. Since the classical ray trajectories (including ray splitting) in such systems are strongly chaotic, this result provides an explicit quantization of a classically chaotic system.

Ray-Splitting Billiards

Ray splitting is a universal phenomenon that occurs with appreciable amplitude in all wave systems when the properties of the system change on a scale smaller than the wave length. We study the quantum implications of ray splitting theoretically and experimentally with the help of ray-splitting billiards in one and two dimensions. We show that Gutzwillers trace formula works even in the context of ray-splitting systems provided reflection and transmission of waves at ray-splitting boundaries is properly included.

Observation of dynamical localization in a rough microwave cavity

Abstract We measure the angular momentum content of modes in a flat, near-circular microwave cavity with a rough perimeter and demonstrate localization in angular momentum space. Because Schrodingers wave mechanics and Maxwells electrodynamics are equivalent for a 2d cavity, we compare our experimental results directly with the quantum theory of rough 2d cavities [K.M. Frahm and D.L. Shepelyansky, Phys. Rev. Lett. 78 (1997) 1440]. Introducing the concept of effective roughness we find good qualitative agreement.

Explicit analytical solution for scaling quantum graphs

We show that scaling quantum graphs with arbitrary topology are explicitly analytically solvable. This is surprising since quantum graphs are excellent models of quantum chaos and quantum chaotic systems are not usually explicitly analytically solvable.

Exact trace formulas for a class of one-dimensional ray-splitting systems

Using quantum graph theory we establish that the ray-splitting trace formula proposed by Couchman et al. [Phys. Rev. A 46, 6193 (1992)] is exact for a class of one-dimensional ray-splitting systems. Important applications in combinatorics are suggested.