We report the direct observation of quantum dynamical tunneling of atoms between separated momentum regions in phase space. We study how the tunneling oscillations are affected as a quantum symmetry is broken and as the initial atomic state is changed. We also provide evidence that the tunneling rate is greatly enhanced by the presence of chaos in the classical dynamics. This tunneling phenomenon represents a dramatic manifestation of underlying classical chaos in a quantum system.
We report the direct observation of quantum dynamical tunneling of atoms between separated momentum regions in phase space. We study how the tunneling oscillations are affected as a quantum symmetry is broken and as the initial atomic state is changed. We also provide evidence that the tunneling rate is greatly enhanced by the presence of chaos in the classical dynamics. This tunneling phenomenon represents a dramatic manifestation of underlying classical chaos in a quantum system.
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.
We study quantum dynamical tunneling between two symmetry-related islands of stability in the phase space of a classically chaotic system. The setting for these experiments is the motion of carefully prepared samples of cesium atoms in an amplitude-modulated standing wave of light. We examine the dependence of the tunneling dynamics on the system parameters and indicate how the observed features provide evidence for chaos-assisted (three-state) tunneling. We also observe the influence of a noisy perturbation of the standing-wave intensity, which destroys the tunneling oscillations, and we show that the tunneling is more sensitive to the noise for a smaller value of the effective Planck constant.
In the presence of a complex classical dynamics associated with a mixed phase space, a quantum wave function can tunnel between two stable islands through the chaotic sea, an effect that has no classical counterpart. This phenomenon, referred to as chaos-assisted tunneling, is characterized by large fluctuations of the tunneling rate when a parameter is varied. To date, the full extent of this effect as well as the associated statistical distribution have never been observed in a quantum system. Here, we analyze the possibility of characterizing these effects accurately in a cold-atom experiment. Using realistic values of the parameters of an experimental setup, we examine through analytical estimates and extensive numerical simulations a specific system that can be implemented with cold atoms, the atomic modulated pendulum. We assess the efficiency of three possible routes to observe in detail chaos-assisted tunneling properties. Our main conclusion is that due to the fragility of the symmetry between positive and negative momenta as a function of quasimomentum, it is very challenging to use tunneling between classical islands centered on fixed points with opposite momentum. We show that it is more promising to use islands symmetric in position space, and characterize the regime where it could be done. The proposed experiment could be realized with current state-of-the-art technology.
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.
We study the interplay of chaos and tunneling between two weakly coupled Bose-Josephson junctions. The classical phase space of the composite system has a mixed structure including quasi-integrable self-trapping islands for particles and excitations, separated by a chaotic sea. We show that the many-body dynamical tunneling gap between macroscopic Schrödinger cat states supported by these islands is chaos-enhanced. The many-body tunneling rate fluctuates over several orders of magnitude with small variations of the system parameters or the particle number.
The interplay between classical chaos and quantum tunneling is examined in driven nonlinear systems, with emphasis on how semi classical phase space structures influence purely quantum transport phenomena. We show that, in the presence of external driving and stochastic perturbations, tunneling rates acquire an activated form determined by effective classical barriers, providing a transparent link between chaotic dynamics and quantum tunneling. Within this framework, chaos assisted tunneling and coherent destruction of tunneling emerge as closely related manifestations of the same underlying phase space restructuring and interference effects induced by driving. The results offer a unified perspective on tunneling control in non integrable systems and remain relevant for modern studies of driven quantum dynamics and decoherence resistant transport.