N-disk microwave billiards, which are representative of open quantum systems, are studied experimentally. The transmission spectrum yields the quantum resonances which are consistent with semiclassical calculations. The spectral autocorrelation of the quantum spectrum is shown to be determined by the classical Ruelle-Pollicot resonances, arising from the complex eigenvalues of the Perron-Frobenius operator. This work establishes a fundamental connection between quantum and classical correlations in open systems.The quantum-classical correspondence for chaotic systems has been studied extensively in the context of universality and periodic orbit contributions. This approach has focussed on eigenvalues and eigenfunctions and their statistical properties. Universality has been shown to arise from Random Matrix Theory [1], while periodic orbit contributions have been analyzed in the semiclassical scheme for calculations of eigenvalue spectra [2] and constructions of eigenfunctions [3].An entirely different approach is to consider correlations of observables. In the classical context a probabilistic approach is best taken with Liouvillian dynamics. In certain classical systems these have been shown to lead to Ruelle-Pollicot (RP) resonances [4,5], arising from complex eigenvalues of the Perron-Frobenius operator. In open systems, this leads to a quantitative description of the timeevolution of classical observables, the most common being the particle density. In the quantum context, diffusive transport has been argued to be intimately connected with Liouvillian dynamics, not just in disordered systems where the correspondence is made with nonlinear σ−models of supersymmetry [6] but also in individual chaotic systems which represent a ballistic limit.In this paper we present a microwave experiment which demonstrates this deep connection between quantum properties and classical diffusion. Our experiment is a microwave realization of the wellknown n-disk geometry, which is a paradigm of an open quantum chaotic system, along with other systems such as the Smale horseshoe and the Baker map [7]. The classical scattering function of the chaotic n-disk system is nondifferentiable and has a selfsimilar fractal structure. 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 wave vector dependence of the spectral autocorrelation is shown to be completely described by the leading RP resonances of the corresponding classical system. The small κ (long time) behavior of the spectral autocorrelation provides a measure of the quantum escape rate, and is shown to be in good agr...

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 ...

We describe a wave-mechanical implementation of classically chaotic n-disk scattering based on thin two-dimensional microwave cavities. Two-, three-, and four-disk scatterings are investigated in detail. The experiments, which are able to probe the stationary Green's function of the system, yield both frequencies and widths of the low-lying quantum resonances. The observed spectra are found to be in good agreement with calculations based on semiclassical periodic orbit theory. Wave-vector autocorrelation functions are analyzed for various scattering geometries, the small wave-vector behavior allowing one to extract the escape rate from the quantum repeller. Quantitative agreement is found with the value predicted from classical scattering theory. For intermediate energies, nonuniversal oscillations are detected in the autocorrelation function, reflecting the presence of periodic orbits.

We report the first observation of bound-state proximity resonances in coupled dielectric resonators. The proximity resonances arise from the combined action of symmetry and dissipation. We argue that the large ratio between the widths is a distinctive signature of the multidimensional nature of the system. Our experiments shed light on the properties of 2D tunneling in the presence of a dissipative environment.Tunneling and dissipation are ubiquitous phenomena in physics. A detailed understanding of their combined action would be highly desirable given the relevance of the problem for atomic physics, condensed matter physics, chemistry and biology [1]. However, the incorporation of dissipative effects is by no means trivial. Due to limitations of the available analytical and computational methods, up-to-date descriptions are still restricted to a few manageable cases, the prototype situation involving a bistable potential in 1D [2].In this Letter we report the observation of novel aspects of tunneling in 2D potentials and its interplay with classical dissipation. In experiments utilizing microwave dielectric resonators, we find that symmetry not only plays a crucial role while shaping the eigenstates of the system, but also influences the way they couple to the external environment acquiring a finite width. In the observed resonance multiplets, we find that one of the members is extremely sharp due to the symmetry of the configuration. The large ratios of the observed widths appear to be a peculiar consequence of the multidimensional nature of the system. The experiments were carried out using M gT i dielectric cylinders placed between two parallel copper plates, 30 cm square, separated by a gap l = 6.38 mm (Fig. 1). The disks had diameter D = 12.65 mm and dielectric constant ε r = 16. After establishing input/output coupling to the near field of the resonators by inserting coax lines terminated by loops, measurements of the transmission amplitude as a function of the frequency were performed using an HP8510B network analyzer.The eigenvalue problem of a single dielectric resonator can be solved analytically by regarding the system as a waveguide along the direction z orthogonal to the plates [3]. The entire field configuration can be derived from * Electronic addresses: kpance@sagar-3.physics.neu.edu; vlorenza@mit.edu; srinivas@neu.edu. the knowledge of the longitudinal components {H z , E z } alone, that separately obey the Helmoltz equation:Here, r = (φ, ρ, z) in cylindrical coordinates and k = √ ε r (ω/c) denotes the medium wave number for a mode at frequency ω = 2πf (ε r = 1 outside the dielectric). For perfectly conducting walls, boundary conditions require k z = pπ/l, p integer. We have verified through explicit measurements of the field profile that p ≥ 1, and all modes are evanescent with a decay constant close to the expected value κ r = k 2 z − ω 2 /c 2 [4]. A generic mode of the dielectric is classified according to its azimuthal, radial and vertical quantum numbers (m, n, p). If m = 0, the mode has ...

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