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Quantum Electronics and Laser Science Conference

Probing mixed phase space dynamics using atom optics

D. Steck, Windell H. Oskay, Mark G. Raizen

Published Nov 13, 2002• 0 citations
Publisher

Abstract

Summary form only given. Atom optics has proven to be a fruitful testbed for the experimental study of quantum dynamics in classically chaotic systems. However, the quantum behavior in a mixed classical phase space, where both stable and chaotic regions are present, remains an interesting and challenging subject, with much left to understand. To probe local dynamics in mixed phase space, the atomic sample must be prepared such that it is localized in phase space. To accomplish this, cesium atoms are first prepared in a standard magneto-optic trap. They are then further cooled in a 3-D far-detuned optical lattice. A model system for the study of mixed phase space is realized when the atoms are subjected to an amplitude-modulated standing wave, where the intensity varies as sin/sup 2/ (/spl pi/t/T) (T is the pulse duration). There are three primary resonances in this system; one is at zero momentum, and the other two are located symmetrically about p = 0. As the intensity of the standing wave increases, the resonances overlap, and the resonances appear as islands of stability surrounded by bands of chaos. We have used our method of state preparation to center the atomic distribution at various positions and at the same momentum as one of the "side" resonances. We observe trapping of the atoms in the resonance when the atomic distribution is centered on the island, but not when the atoms are centered away from the resonance, since they instead diffuse through the chaotic region.

DOI
10.1109/qels.2001.962100

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Summary form only given. Atom optics has proven to be a fruitful testbed for the experimental study of quantum dynamics in classically chaotic systems. However, the quantum behavior in a mixed classical phase space, where both stable and chaotic regions are present, remains an interesting and challenging subject, with much left to understand. To probe local dynamics in mixed phase space, the atomic sample must be prepared such that it is localized in phase space. To accomplish this, cesium atoms are first prepared in a standard magneto-optic trap. They are then further cooled in a 3-D far-detuned optical lattice. A model system for the study of mixed phase space is realized when the atoms are subjected to an amplitude-modulated standing wave, where the intensity varies as sin/sup 2/ (/spl pi/t/T) (T is the pulse duration). There are three primary resonances in this system; one is at zero momentum, and the other two are located symmetrically about p = 0. As the intensity of the standing wave increases, the resonances overlap, and the resonances appear as islands of stability surrounded by bands of chaos. We have used our method of state preparation to center the atomic distribution at various positions and at the same momentum as one of the "side" resonances. We observe trapping of the atoms in the resonance when the atomic distribution is centered on the island, but not when the atoms are centered away from the resonance, since they instead diffuse through the chaotic region.
Summary form only given. In our experiments ultracold rubidium atoms are loaded into an optical standing wave. The amplitude of the standing wave is then modulated, creating a time dependent potential given by V(x,t)=/spl kappa/(1-2/spl epsiv/sin(2/spl pi/f/sub m/t))cos(2kx) where /spl kappa/ is the strength of the potential, /spl epsiv/ is the amplitude of modulation and f/sub m/ is the frequency of modulation. The dynamics of atoms in such a potential is chaotic except for small islands of stability which exist for certain values of /spl kappa/, /spl epsiv/ and f/sub m/. These islands are phase space resonances and rotate in phase space at the modulation frequency. By viewing the dynamics of the interaction stroboscopically, at intervals of the modulation frequency it is possible to observe these resonances in the final atomic velocity distribution.
We present experimental results for the dynamics of cold atoms in a far detuned amplitude-modulated optical standing wave. Phase-space resonances constitute distinct peaks in the atomic momentum distribution containing up to 65% of all atoms resulting from a mixed quantum chaotic phase space. We characterize the atomic behavior in classical and quantum regimes and we present the applicable quantum and classical theory, which we have developed and refined. We show experimental proof that the size and the position of the resonances in phase space can be controlled by varying several parameters, such as the modulation frequency, the scaled well depth, the modulation amplitude, and the scaled Planck's constant of the system. We have found a surprising stability against amplitude noise. We present methods to accurately control the momentum of an ensemble of atoms using these phase-space resonances which could be used for efficient phase-space state preparation.
Cold atoms in optical potentials provide an ideal test bed to explore quantum nonlinear dynamics. Atoms are prepared in a magneto-optic trap or as a dilute Bose–Einstein condensate and subjected to a far detuned optical standing wave that is modulated. They exhibit a wide range of dynamics, some of which can be explained by classical theory while other aspects show the underlying quantum nature of the system. The atoms have a mixed phase space containing regions of regular motion which appear as distinct peaks in the atomic momentum distribution embedded in a sea of chaos. The action of the atoms is of the order of Planck’s constant, making quantum effects significant. This tutorial presents a detailed description of experiments measuring the evolution of atoms in time-dependent optical potentials. Experimental methods are developed providing means for the observation and selective loading of regions of regular motion. The dependence of the atomic dynamics on the system parameters is explored and distinct changes in the atomic momentum distribution are observed which are explained by the applicable quantum and classical theory. The observation of a bifurcation sequence is reported and explained using classical perturbation theory. Experimental methods for the accurate control of the momentum of an ensemble of atoms are developed. They use phase space resonances and chaotic transients providing novel ensemble atomic beamsplitters. The divergence between quantum and classical nonlinear dynamics is manifest in the experimental observation of dynamical tunnelling. It involves no potential barrier. However a constant of motion other than energy still forbids classically this quantum allowed motion. Atoms coherently tunnel back and forth between their initial state of oscillatory motion and the state 180 ° out of phase with the initial state.
Summary form only given. We represent the first experiment evidence of dynamic tunneling of ultra-cold rubidium atoms. Rather then tunneling through a potential well, an atom tunnels from its initial motion into a different and oppositely directed motion. This process is classically forbidden. To observe this effect we have implemented an experiment that probes the dynamics of cold rubidium atoms in a far detuned optical standing wave. Rubidium atoms are prepared in a standard magneto-optic trap. The light intensity of the standing wave is modulated leading to a quantum chaotic mixed phase space for the atomic dynamics, consisting of regions of regular motion encircled by a sea of chaos. Phase space resonances can be observed as distinct peaks (with non-zero mean momentum) in the atomic momentum distribution.
Cold rubidium atoms are subjected to an amplitude-modulated far-detuned standing wave of light to form a quantum-driven pendulum. Here we discuss the dynamics of these atoms. Phase space resonances and chaotic transients of the system exhibit dynamics which can be useful in many atom optics applications as they can be utilized as means for phase space state preparation. We explain the occurrence of distinct peaks in the atomic momentum distribution, analyse them in detail and give evidence for the importance of the system for quantum chaos and decoherence studies.
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.