Dynamical Tunneling in Mushroom Billiards

Abstract

We study the fundamental question of dynamical tunneling in generic two-dimensional Hamiltonian systems by considering regular-to-chaotic tunneling rates. Experimentally, we use microwave spectra to investigate a mushroom billiard with adjustable foot height. Numerically, we obtain tunneling rates from high precision eigenvalues using the improved method of particular solutions. Analytically, a prediction is given by extending an approach using a fictitious integrable system to billiards. In contrast to previous approaches for billiards, we find agreement with experimental and numerical data without any free parameter. Typical Hamiltonian systems have a mixed phase space in which regular and chaotic motion coexist. While classically these regions are separated, quantum mechanically they are coupled by tunneling. This process has been called "dynamical tunneling" [1] as it occurs across a dynamically generated barrier in phase space. Tunneling has been studied between symmetry related regular regions (chaos-assisted tunneling) [2,3,4,5,6,7] and from a single regular region to the chaotic sea [8,9,10,11,12]. In contrast to the well understood 1D tunneling through a barrier, the quantitative prediction of dynamical tunneling is a major challenge. Results have been found for specific systems or system classes only, e.g. recently for 2D quantum maps with an approach using a fictitious integrable system [12]. However, a precise knowledge of tunneling rates is of great importance. Recent examples are spectral statistics in systems with a mixed phase space [13], eigenstates affected by flooding of regular islands [14], and emission properties of optical micro-cavities [15].Billiards are an important class of Hamiltonian systems. Classically, a point particle moves along straight lines inside a domain with elastic reflections at its boundary. Quantum-mechanical approaches for dynamical tunneling rates have so far escaped a full quantitative prediction as they required fitting by a factor of 6 for the annular billiard [3] and by a factor of 100 (see below) for the mushroom billiard [16].In this paper we present a combined experimental, theoretical, and numerical investigation of dynamical tunneling rates in mushroom billiards [17], which are of great current interest [13,16,18,19,20] due to their sharply divided phase space. Experiments are performed using a microwave cavity. Extending the approach using a fictitious integrable system [12] to billiards, we find quantitative agreement in the experimentally accessible regime, see Fig. 1, without a free parameter. In addition, numerical computations verify the predictions over 18 orders of magnitude with errors typically smaller than a factor of 2, see Fig. 4. The theoretical approach thus provides unprecedented agreement for tunneling rates in billiards.We consider the desymmetrized mushroom billiard, i.e. the 2D autonomous system H(p, q) = p 2 /2M + V (q) shown in Fig. 2b, characterized by the radius of the quarter circle R, the foot width a and the foot height l. The pote...

Dielectric microcavities: Model systems for wave chaos and non-Hermitian physics

“…An example is the tunneling from a regular to the chaotic region in the phase space of the mushroom billiard, see Fig. 4, as studied theoretically and experimentally in a microwave mushroom billiard by Bäcker et al (2008a).…”

Tractable Problems in Multidimensional Tunneling

“…This results in a smooth exponential-like decay of the average tunneling rate with 1/ , which was indeed found to be in very good agreement with the exact tunneling rates for quantum maps and billiards [22,23]. Those systems, however, were designed such as to yield a "clean" mixed regular-chaotic phase space, containing a regular island and a chaotic region which both do not exhibit appreciable substructures [22,23].…”

“…Bäcker, Ketzmerick, Löck, and coworkers [22,23] recently undertook the effort to derive more rigorously the regular-to-chaotic coupling rate governing chaos-assisted tunneling. Their approach is based on the construction of an integrable approximation for the nonintegrable system, designed to accurately describe the motion within the regular islands under consideration.…”

Energy-dependent correlations in the S-matrix of chaotic systems

“…This situation can be realized in microwave scattering in metallic cavities [1,2,3,4,5,6] and in electron scattering in condensed matter systems, [7,8,9,10,11] among other possibilities. We assume there is a well defined decay rate for the ray dynamics, Γ; this means that the total energy inside the system decays exponentially in time as e −Γt .…”