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Physical Review A

Experimental study of the quantum driven pendulum and its classical analog in atom optics

W. K. Hensinger, Andrew G. Truscott, Ben Upcroft, M. Hug, Howard M. Wiseman, N. R. Heckenberg, Halina H. Rubinsztein-Dunlop

Published Aug 6, 2001• 23 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.64.033407

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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.

Quantum Electronics and Laser Science Conference•Jan 20, 2003•0 citations
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.
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.
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. The quantum driven pendulum can be implemented using laser cooled rubidium atoms which are exposed to a far detuned amplitude modulated standing wave. This system allows controlled variation of the effective Planck's constant and the amount of coupling to the environment (decoherence). Therefore it is possible to undertake experiments in both classical and quantum regimes of the dynamics of these atoms. The potential can be extremely well approximated as one-dimensional, making the system ideal for simulation using a master equation or quantum trajectories. The dynamics of the system feature distinct regions of regular motion (second order resonances) bounded by a sea of chaos. The resonances are observed as side peaks in the atomic momentum distribution, which each contain up to 30% of the atoms initially in the trap. We report on the results of our experiments to achieve quantum phase space tunneling. After adiabatic elimination of the excited state, the Hamiltonian describing the system is given. The evolution of the quantum driven pendulum can be described with the Floquet operator. Eigenstates of the Floquet operator may be identified with fixed points in phase space. Starting the evolution in a superposition of two Floquet states which is localised in one resonance, time evolution leads to localisation in the other fixed point. This is a coherent tunneling process forbidden by classical dynamics. To observe this tunneling process it is important to localise the superposition of Floquet states in one fixed point. We discuss the effectiveness of alternative methods of loading most of the atoms into a single resonance in order to observe quantum phase space tunneling.
We report an experimental and theoretical study of momentum transfer from a modulated standing wave of light to a sample of ultracold atoms. This system is a quantum realization of the periodically driven rotor where the underlying classical phase space goes from stable to chaotic as a control parameter is varied. Our experimental results are in good absolute agreement with a quantum Floquet analysis and with a quantum simulation. We relate the quantum evolution to the underlying classical dynamics in this mixed phase space regime.
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.