Acknowledgments For two-dimensional, symmetric double well potentials separated by a tunneling barrier, it has been shown theoretically that quantum mechanical tunneling rates are dependent on the shape of the well in the limit of small quantum wavelength. Shapes that support chaotic wave functions produce statistically smaller fluctuations in the tunneling rate than classical (integrable) wells. This effect is called “regularization” of tunneling rates and can be analyzed by examining the splitting in energy level between symmetric and antisymmetric wave functions. Utilizing the similarity in the wave natures of a quantum mechanical particle in a symmetric double well and transverse electromagnetic waves in a 2-D cavity, we investigate chaos regularization in coupled microwave cavities both numerically and experimentally. The difference between the antisymmetric and symmetric eigenstate frequencies squared and evanescent electromagnetic wave propagation in a cutoff tunnel is shown to be analogous to energy splitting and quantum mechanical tunneling, respectively. This analogy allows investigation of regularization in various microwave cavities and, by extension to a lab-scale model, the behavior of complex quantum mechanical systems.
Acknowledgments For two-dimensional, symmetric double well potentials separated by a tunneling barrier, it has been shown theoretically that quantum mechanical tunneling rates are dependent on the shape of the well in the limit of small quantum wavelength. Shapes that support chaotic wave functions produce statistically smaller fluctuations in the tunneling rate than classical (integrable) wells. This effect is called “regularization” of tunneling rates and can be analyzed by examining the splitting in energy level between symmetric and antisymmetric wave functions. Utilizing the similarity in the wave natures of a quantum mechanical particle in a symmetric double well and transverse electromagnetic waves in a 2-D cavity, we investigate chaos regularization in coupled microwave cavities both numerically and experimentally. The difference between the antisymmetric and symmetric eigenstate frequencies squared and evanescent electromagnetic wave propagation in a cutoff tunnel is shown to be analogous to energy splitting and quantum mechanical tunneling, respectively. This analogy allows investigation of regularization in various microwave cavities and, by extension to a lab-scale model, the behavior of complex quantum mechanical systems.
Quantum tunneling rates through a barrier separating two-dimensional, symmetric, double-well potentials are shown to depend on the classical dynamics of the billiard trajectories in each well and, hence, on the shape of the wells. For shapes that lead to regular (integrable) classical dynamics the tunneling rates fluctuate greatly with eigenenergies of the states sometimes by over two orders of magnitude. Contrarily, shapes that lead to completely chaotic trajectories lead to tunneling rates whose fluctuations are greatly reduced, a phenomenon we call regularization of tunneling rates. We show that a random-plane-wave theory of tunneling accounts for the mean tunneling rates and the small fluctuation variances for the chaotic systems.
Recent numerical experiments of Pecora et al. [Phys. Rev. E 83, 065201 (2011)] have investigated tunneling between two-dimensional symmetric double wells separated by a tunneling barrier. The wells were bounded by hard walls and by the potential barrier which was created by a step increase from the zero potential within a well to a uniform barrier potential within the barrier region, which is a situation potentially realizable in the context of quantum dots. Numerical results for the splitting of energy levels between symmetric and antisymmetric eigenstates were calculated. It was found that the splittings vary erratically from state to state, and the statistics of these variations were studied for different well shapes with the fluctuation levels being much less in chaotic wells than in comparable nonchaotic wells. Here we develop a quantitative theory for the statistics of the energy level splittings for chaotic wells. Our theory is based on the random plane wave hypothesis of Berry. While the fluctuation statistics are very different for chaotic and nonchaotic well dynamics, we show that the mean splittings of differently shaped wells, including integrable and chaotic wells, are the same if their well areas and barrier parameters are the same. We also consider the case of tunneling from a single well into a region with outgoing quantum waves.
We solve the Dirac equation in two spatial dimensions in the setting of resonant tunneling, where the system consists of two symmetric cavities connected by a finite potential barrier. The shape of the cavities can be chosen to yield both regular and chaotic dynamics in the classical limit. We find that certain pointer states about classical periodic orbits can exist, which suppress quantum tunneling, and the effect becomes less severe as the underlying classical dynamics in the cavity is chaotic, leading to regularization of tunneling dynamics even in the relativistic quantum regime. Similar phenomena have been observed in graphene. A physical theory is developed to explain the phenomenon based on the spectrum of complex eigenenergies of the non-Hermitian Hamiltonian describing the effectively open cavity system.
Since the early days of quantum mechanics, tunneling has been recognized as one of the hallmarks of the wave character of microscopic physics. The possibility of a quantum particle to penetrate an energetic barrier represents certainly one of the most spectacular implications of quantum theory and has led to various applications in nuclear, atomic, and molecular physics as well as in mesoscopic science. Typical scenarios in which tunneling manifests are the escape route of a quantum particle from a quasibounded region, the transition between two or more symmetry-related, but classically disconnected wells (which we shall focus on in the following), as well as scattering or transport through potential barriers. The spectrum of scenarios becomes even richer when the concept of tunneling is generalized to any kind of classically forbidden transitions in phase space, that is, to transitions that are not necessarily inhibited by static potential barriers but by some other constraints of the underlying classical dynamics (such as integrals of motion). Such “dynamical tunneling” processes arise frequently in molecular systems [1] and were realized with cold atoms propagating in periodically modulated optical lattices [2-4]. Moreover, the electromagnetic analog of dynamical tunneling was also obtained with microwaves in billiards [5].
In generic Hamiltonian systems that are neither completely integrable nor fully chaotic, phase space consists of a mixture of regular and chaotic components. In classical dynamics, transitions between different invariant sets in phase space are strictly forbidden, and these sets act as dynamical barriers to one another. In quantum mechanics, in contrast, wave effects allow transitions through such dynamical barriers. This process, known as dynamical tunneling, refers to penetration through dynamical barriers in phase space and was first recognized in the early 1980s. Since then, various aspects of dynamical tunneling have been elucidated, significantly advancing our understanding of such a novel quantum phenomenon. In this article, we provide an overview of several phenomenological perspectives of dynamical tunneling, including chaos-assisted and resonance-assisted tunneling, and also introduce approaches based on classical mechanics extended into the complex domain. In particular, we seek to clarify what is meant by the common claim that "chaos leads to an enhancement of the tunneling probability", which is often made when dynamical tunneling is dressed. We discuss what regime this refers to and, if such an enhancement occurs, what its likely origin is.
Two experiments with super- and normal conducting microwave cavities which are used as an analog to two-dimensional, infinitely deep quantum potentials are presented. The eigenvalues of such a potential can be measured directly through the resonance frequencies of the cavity, while the eigenvectors can be determined by measuring the field distributions inside the cavity. In the case of open systems another information – the imaginary part of the eigenvalues – is observable by measuring the widths of the resonances. As examples for this experimental approach to quantum chaos the semiclassical reconstruction of length spectra of billiards with varying chaoticity based of various trace formulas proposed by Gutzwiller, by Ullmo, Grinberg and Tomsovic and by Berry and Tabor for systems with chaotic, mixed and regular dynamics, respectively, and the first observation of square root singularities – so called exceptional points recently proposed by Heiss – in the energy spectrum of coupled cavities are presented.