Quantum Resonances and Decay of a Chaotic Fractal Repeller Observed Using Microwaves

Abstract

The quantum resonances of classically chaotic n-disk geometries were studied experimentally utilizing thin 2D microwave geometries. The experiments yield the frequencies and widths of lowlying resonances, which are compared with semiclassical calculations. The long time or small energy behavior of the wave-vector autocorrelation gives information about the quantum decay rate, which is in good agreement with that obtained from classical scattering theory. The intermediate energy behavior shows nonuniversal oscillations determined by periodic orbits. [S0031-9007(99)09486-7] PACS numbers: 05.45.Mt, 03.65.Sq, 05.45.Ac, 84.40. -xThe n-disk scattering problem is one whose quantumclassical correspondence has received extensive theoretical attention [1,2], because it is a paradigm of an open quantum chaos system, much as the pendulum is for integrable systems. Furthermore, it is relevant to physical situations in diverse fields, such as the crossroads geometry for electron devices [3], unimolecular chemical reactions [2], and electromagnetic and acoustic scattering [4].The classical scattering function of n disks on a plane is nondifferentiable and forms a Cantor set. A central property is the exponential decay of an initial distribution of classical particles, hence the name (fractal) repeller. For closed quantum chaotic billiards, experimental and theoretical work is available on eigenvalues (which are purely real) and eigenfunctions, and the many features of universality and nonuniversal behavior are known [5]. In contrast, for open quantum systems, the eigenvalues are intrinsically complex and their universal behavior is a question of great interest [6]. Despite extensive theoretical treatment, there have been almost no real experiments on the n-disk geometry which exemplify this unique problem in quantum chaos.In this paper, we present a microwave realization of the chaotic n-disk problem. Experiments were carried out for n 1, 2, 3, 4, and 6, as well as for large n 20, the latter corresponding to the random Lorentz scatterer. In this paper, we focus on the case n 4. The experiments yield the frequencies and the widths of the low-lying quantum resonances of the four-disk repeller. We have also carried out semiclassical calculations of the resonances, which are shown to reproduce the resonances reasonably well. Our experiments enable us to explore the role of symmetry in a unique way by studying different irreducible representations. The experimental data are used to display the signatures of the classical chaos in the transmission spectra, through measures such as the spectral (wave-vector k) autocorrelation function. The small k (long time) behavior of this quantity provides a measure of the quantum escape rate, and is shown to be in good agreement with the corresponding classical escape rate. For large k (short time), the contribution of periodic orbits is observed as nonuniversal oscillations of the autocorrelation.The experiments are carried out in thin microwave structures consisting of two highly conducting Cu ...

Quantum Fingerprints of Classical Ruelle-Pollicott Resonances

“…A central property is the exponential decay of an initial distribution of classical particles, due to the unstable periodic orbits, which form a cantor set, hence the name fractal repeller. The experimental transmission spectrum directly yields the frequencies and the widths of the low lying quantum resonances of the system [8,9], which are in agreement with semiclassical periodic orbit calculations [10,11,12]. The same spectra are analyzed to obtain the spectral wave-vector autocorrelation C(κ) [8].…”

“…The above equation was used to fit the spectral autocorrelation for small κ and thus obtain the value of the experimental escape rate γ qm [8,9]. Good agreement of the escape rate is obtained between γ qm obtained from the experiments and γ cl of the classical theory [9].…”

“…For all metallic objects in the 2-D space between the plates, Dirichlet boundary conditions apply inside the metal. Similar microwave experiments, which exploit this QM-E&M mapping, have been used to study quantum chaos in closed [13,14] and open systems [8]. See [9] for details of the experiments.…”

Theoretical ingredients of a Casimir analog computer

“…For example, many analog computers have been developed to simulate fluid flow problems (34,35), electromagnetic and acoustic wavefields (36), cell electrolysis (37), and many of the problems governed by Laplace's equations (38). More recently, advances in the field of microwave electromagnetics (which offers unprecedented control over sources and detection of microwaves), have spurred the development of classical electromagnetic analog computations for studying complex quantum problems, including the calculation of the energy levels of Bloch electrons in magnetic fields (39) and the dynamics of certain classes of chaotic quantum systems (32), to name a few. Computing fluctuation-induced effects like the Casimir force is substantially different from previous analog-computation problems in quantum simulation, in that it involves not the evolution of a single quantum state but rather the combined effect of fluctuations over a broad (formally infinite) bandwidth.…”

“…Note that an experiment (centimeter-scale model) of the sort proposed here is not a Casimir "simulator," in that one is not measuring forces but rather a quantity that is mathematically related to the Casimir force-in this sense, it is an analog computer. The use of tabletop models and analog computers in physics, though previously unexplored in the context of quantum vacuum fluctuations, continues to play an important role in contemporary research areas like quantum evolution (32) and quantum information (33). For example, many analog computers have been developed to simulate fluid flow problems (34,35), electromagnetic and acoustic wavefields (36), cell electrolysis (37), and many of the problems governed by Laplace's equations (38).…”