In the context of quantum chaos, both theory and numerical analysis predict large fluctuations of the tunneling transition probabilities when irregular dynamics is present at the classical level. Here we consider the nondissipative quantum evolution of cold atoms trapped in a time-dependent modulated periodic potential generated by two laser beams. We give some precise guidelines for the observation of chaos-assisted tunneling between invariant phase space structures paired by time-reversal symmetry.
In the context of quantum chaos, both theory and numerical analysis predict large fluctuations of the tunneling transition probabilities when irregular dynamics is present at the classical level. Here we consider the nondissipative quantum evolution of cold atoms trapped in a time-dependent modulated periodic potential generated by two laser beams. We give some precise guidelines for the observation of chaos-assisted tunneling between invariant phase space structures paired by time-reversal symmetry.
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.
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.
In the presence of a complex classical dynamics associated with amixed 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.
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.
We study the motion of a cold atom in a frequency-modulated standing laser wave. If the detuning between the atomic electronic transition and the field is large, the atom moves in a modulated optical potential demonstrating known classical nonlinear effects such as chaos and nonlinear resonances. If the atom-field detuning is small, then two optical potentials emerge in the system, and the atom performs Landau-Zener (LZ) tunnelings between them. It is a radically non-classical behavior. However, we show that classical nonlinear structures in system's phase space (KAM-tori and chaotic layers) survive. Quantum effect of LZ tunnelings only induces small random jumps of trajectories between these structures (dynamical tunneling).
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.