A compound tunneling mechanism from one integrable region to another mediated by a delocalized state in an intermediate chaotic region of phase space was recently introduced to explain peculiar features of tunneling in certain two-dimensional systems. This mechanism is known as chaos-assisted tunneling. We study its consequences for the distribution of the level splittings and obtain a general analytical form for this distribution under the assumption that chaos assisted tunneling is the only operative mechanism. %The validity of this form can then in %principle be checked either numerically or even experimentally. We have checked that the analytical form we obtain agrees with splitting distributions calculated numerically for a model system in which chaos-assisted tunneling is known to be the dominant mechanism. The distribution depends on two parameters: The first gives the scale of the splittings and is related to the magnitude of the classically forbidden processes, the second gives a measure of the efficiency of possible barriers to classical transport which may exist in the chaotic region. If these are weak, this latter parameter is irrelevant; otherwise it sets an energy scale at which the splitting distribution crosses over from one type of behavior to another. The detailed form of the crossover is also obtained and found to be in good agreement with numerical results for models for chaos-assisted tunneling.
A compound tunneling mechanism from one integrable region to another mediated by a delocalized state in an intermediate chaotic region of phase space was recently introduced to explain peculiar features of tunneling in certain two-dimensional systems. This mechanism is known as chaos-assisted tunneling. We study its consequences for the distribution of the level splittings and obtain a general analytical form for this distribution under the assumption that chaos assisted tunneling is the only operative mechanism. %The validity of this form can then in %principle be checked either numerically or even experimentally. We have checked that the analytical form we obtain agrees with splitting distributions calculated numerically for a model system in which chaos-assisted tunneling is known to be the dominant mechanism. The distribution depends on two parameters: The first gives the scale of the splittings and is related to the magnitude of the classically forbidden processes, the second gives a measure of the efficiency of possible barriers to classical transport which may exist in the chaotic region. If these are weak, this latter parameter is irrelevant; otherwise it sets an energy scale at which the splitting distribution crosses over from one type of behavior to another. The detailed form of the crossover is also obtained and found to be in good agreement with numerical results for models for chaos-assisted tunneling.
We discuss and give evidence for the existence of a new mechanism for quantum dynamical tunneling. It may occur when a quantum system's underlying classical dynamics are far from integrability. In these circumstances, we show that the dominant tunneling contributions arise through chaos-assisted processes. This leads to behavior that is drastically different from that found in integrable and quasi-integrable systems. In particular, one can observe a marked crossing mechanism when a chaotic level passes nearby the tunneling ones, and the distributions of splitting (due to tunneling) can be modeled using properly designed ensembles of random matrices. Such tunneling should be amenable to experimental detection.
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.
The study of quantum tunneling dates back to the early days of quantum mechanics, and has been a major topic of investigation in both theory and experiment. In recent years, there has been significant effort to understand tunneling phenomena for systems that exhibit chaos in their classical counterpart [13,15,18,29,42]. In particular, tunneling was predicted for mixed systems where chaotic and stable regions are intermingled [14]. The tunneling in this case is between two stable regions (referred to as nonlinear resonances, or islands of stability) in the classical phase space. The classical transport between these islands is forbidden by “dynamical barriers” in phase space. In contrast, quantum tunneling can couple the two islands so that a wave packet oscillates coherently between the two symmetry-related regions. This phenomenon is called dynamical tunneling because it occurs in phase space, without a fixed potential barrier. In the case of a mixed phase space, the presence of the chaotic region can enhance the rate of dynamical tunneling, a phenomenon known as chaos-assisted tunneling [53]. In this chapter, we review our experimental observation of chaosassisted tunneling with ultracold cesium atoms [48-51]. We first give a brief overview of some basic concepts and of earlier work in order to provide a context for our results.
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.
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].
We develop the semiclassical method of complex trajectories in application to chaotic dynamical tunneling. First, we suggest a systematic numerical technique for obtaining complex tunneling trajectories by the gradual deformation of the classical ones. This provides a natural classification of the tunneling solutions. Second, we present a heuristic procedure for sorting out the least suppressed trajectory. As an illustration, we apply our technique to the process of chaotic tunneling in a quantum mechanical model with two degrees of freedom. Our analysis reveals rich dynamics of the system. At the classical level, there exists an infinite set of unstable solutions forming a fractal structure. This structure is inherited by the complex tunneling paths and plays a central role in the semiclassical study. The process we consider exhibits the phenomenon of optimal tunneling: the suppression exponent of the tunneling probability has a local minimum at a certain energy which is thus (locally) the optimal energy for tunneling. We test the proposed method by comparison of the semiclassical results with the results of the exact quantum computations and find a good agreement.