We derive a formula predicting dynamical tunneling rates from regular states to the chaotic sea in systems with a mixed phase space. Our approach is based on the introduction of a fictitious integrable system that resembles the regular dynamics within the island. For the standard map and other kicked systems we find agreement with numerical results for all regular states in a regime where resonance-assisted tunneling is not relevant. Tunneling of a quantum particle is one of the central manifestations of quantum mechanics. For simple 1D systems tunneling under a potential barrier is well understood and described, e.g. by using semiclassical WKB theory or the instanton approach [1]. For higherdimensional systems so-called "dynamical tunneling" [2] occurs between regions which are separated by dynamically generated barriers. Typically, such systems have a mixed phase space in which regions of regular motion and irregular dynamics coexist. Tunneling in these systems is barely understood as it generically cannot be reduced to the instanton or WKB approach. It has been studied theoretically [3,4,5,6,7,8,9,10,11,12,13,14] and experimentally, e.g. in cold atom systems [16,17] and semiconductor nanostructures [18]. A precise knowledge of tunneling rates is of current interest for e.g. eigenstates affected by flooding of regular islands [19,20], emission properties of optical microcavities [21] and spectral statistics in systems with a mixed phase space [22].There are different approaches for the prediction of tunneling rates depending on the ratio of Planck's constant h to the size A of the regular island. In the semiclassical regime, h ≪ A, small resonance chains inside the island dominate the tunneling process ("resonanceassisted tunneling") [11,12]. In contrast, we focus on the experimentally relevant regime of large h (while still h < A), where small resonance chains are expected to have no influence on the tunneling rates. This regime has been investigated in Ref. [14], however, the prediction does not seem to be generally applicable (see below). Other studies in this regime investigate situations [13,23], where dynamical tunneling can be described by 1D tunneling under a barrier, however, in our opinion they are non-generic. A generally applicable theoretical description of dynamical tunneling rates in systems with a mixed phase space is still an open question.In this paper we present a new approach to dynamical tunneling from a regular island to the chaotic sea. The central idea is the use of a fictitious integrable system resembling the regular island. This leads to a tunneling formula involving properties of this integrable system as well as its difference to the mixed system under consideration. It allows for the prediction of tunneling rates from any quantized torus within the regular island. We find excellent agreement with numerical data, see Fig. 1, for an example system where tunneling is not affected by phase-space structures like cantori at the border of the island. The applicability to more general sys...

We derive a prediction of dynamical tunneling rates from regular to chaotic phase-space regions combining the direct regular-to-chaotic tunneling mechanism in the quantum regime with an improved resonance-assisted tunneling theory in the semiclassical regime. We give a qualitative recipe for identifying the relevance of nonlinear resonances in a given variant Planck's over 2pi regime. For systems with one or multiple dominant resonances we find excellent agreement to numerics.

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...

Abstract. We study the number of bouncing ball modes N bb (E) in a class of two-dimensional quantized billiards with two parallel walls. Using an adiabatic approximation we show that asymptotically N bb (E) ∼ αE δ for E → ∞, where δ ∈] 1 2 , 1[ depends on the shape of the billiard boundary. In particular for the class of two-dimensional Sinai billiards, which are chaotic, one can get arbitrarily close (from below) to δ = 1, which corresponds to the leading term in Weyl's law for the mean behaviour of the counting function of eigenstates. This result shows that one can come arbitrarily close to violating quantum ergodicity. We compare the theoretical results with the numerically determined counting function N bb (E) for the stadium billiard and the cosine billiard and find good agreement.

Abstract:For a large class of quantized ergodic flows the quantum ergodicity theorem due to Shnirelman, Zelditch, Colin de Verdière and others states that almost all eigenfunctions become equidistributed in the semiclassical limit. In this work we first give a short introduction to the formulation of the quantum ergodicity theorem for general observables in terms of pseudodifferential operators and show that it is equivalent to the semiclassical eigenfunction hypothesis for the Wigner function in the case of ergodic systems. Of great importance is the rate by which the quantum mechanical expectation values of an observable tend to their mean value. This is studied numerically for three Euclidean billiards (stadium, cosine and cardioid billiard) using up to 6000 eigenfunctions. We find that in configuration space the rate of quantum ergodicity is strongly influenced by localized eigenfunctions like bouncing ball modes or scarred eigenfunctions. We give a detailed discussion and explanation of these effects using a simple but powerful model. For the rate of quantum ergodicity in momentum space we observe a slower decay. We also study the suitably normalized fluctuations of the expectation values around their mean, and find good agreement with a Gaussian distribution. †

For generic 4d symplectic maps we propose the use of 3d phase-space slices which allow for the global visualization of the geometrical organization and coexistence of regular and chaotic motion. As an example we consider two coupled standard maps. The advantages of the 3d phase-space slices are presented in comparison to standard methods like 3d projections of orbits, the frequency analysis, and a chaos indicator. Quantum mechanically, the 3d phase-space slices allow for the first comparison of Husimi functions of eigenstates of 4d maps with classical phase space structures. This confirms the semi-classical eigenfunction hypothesis for 4d maps.

We introduce a criterion for the existence of regular states in systems with a mixed phase space. If this condition is not fulfilled chaotic eigenstates substantially extend into a regular island. Wave packets started in the chaotic sea progressively flood the island. The extent of flooding by eigenstates and wave packets increases logarithmically with the size of the chaotic sea and the time, respectively. This new effect is observed for the example of island chains with just 10 islands.PACS numbers: 05.45. Mt, 03.65.Sq One of the cornerstones in the understanding of the structure of eigenstates in quantum systems is the semiclassical eigenfunction hypothesis [1]: in the semiclassical limit the eigenstates concentrate on those regions in phase space which a typical orbit explores in the longtime limit. For integrable systems these are the invariant tori. For ergodic dynamics the eigenstates become equidistributed on the energy shell [2]. Typical systems have a mixed phase space, where regular islands and chaotic regions coexist. In this case the semiclassical eigenfunction hypothesis implies that the eigenstates can be classified as being either regular or chaotic according to the phase-space region on which they concentrate. Note, that this may fail for an infinite phase space [3].In this paper we study mixed systems with a compact phase space, but away from the semiclassical limit. Here the properties of eigenstates depend on the size of phasespace structures compared to Planck's constant h. In the case of 2D maps this can be very simply stated [4]: a regular state with quantum number m = 0, 1, ... will concentrate on a torus enclosing an area (m + 1/2)h, as can be seen in Fig. 1(c).We will show that this WKB-type quantization rule is not a sufficient condition. We find a second criterion for the existence of a regular state on the m-th quantized torus,Here τ H = h/∆ ch is the Heisenberg time of the chaotic sea with mean level spacing ∆ ch and γ m is the decay rate of the regular state m if the chaotic sea were infinite. Quantized tori violating this condition will not support regular states. Instead, chaotic states will flood these regions, see Fig. 1(a). In terms of dynamics we find that wave packets started in the chaotic sea progressively flood the island as time evolves. Partial and even complete flooding is possible, depending on system properties. These findings are relevant for islands surrounded by a large chaotic sea. We numerically demonstrate the flooding and the disappearance of regular states for the important case of island chains. In typical Hamiltonian systems they appear around any regular island. On larger scales they are relevant for Hamiltonian ratchets [5], the kicked rotor with accelerator modes [6], and the experimentally [7,8,9] and theoretically [10] studied kicked atom systems. The flooding of regular islands by chaotic states is a new quantum signature of a classically mixed phase space. This phenomenon shows that not only local phasespace structures, but also global properties ...

The spectral statistics in the strongly chaotic cardioid billiard are studied. The analysis is based on the rst 11000 quantal energy levels for odd and even symmetry respectively. It is found that the level-spacing distribution is in good agreement with the GOE distribution of random-matrix theory. In case of the number variance and rigidity we observe agreement with the random-matrix model for short-range correlations only, whereas for long-range correlations both statistics saturate in agreement with semiclassical expectations. Furthermore the conjecture that for classically chaotic systems the normalized mode uctuations have a universal Gaussian distribution with unit variance is tested and found to be in very good agreement for both symmetry classes. By means of the Gutzwiller trace formula the trace of the cosinemodulated heat kernel is studied. Since the billiard boundary is focusing there are conjugate points giving rise to zeros at the locations of the periodic orbits instead of exclusively Gaussian peaks.