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International Journal of Modern Physics B

Tunneling Between Floquet States in Atom Optics Experiments

Robert Luter, L. E. Reichl

Published Sep 30, 2003• 1 citation
Publisher

Abstract

Dynamical tunneling has been observed in atom optics experiments by two groups, one in Texas and the other at NIST. The oscillations observed in these experiments are due to a few Floquet states that were excited by the initial conditions in the experiment. In this paper we focus on the Texas experiment and discuss the properties of the dominant Floquet states for a range of experimental parameters.

DOI
10.1142/s0217979203022076

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Dynamical tunneling has been observed in atom optics experiments by two groups, one in Texas and the other at NIST. The oscillations observed in these experiments are due to a few Floquet states that were excited by the initial conditions in the experiment. In this paper we focus on the Texas experiment and discuss the properties of the dominant Floquet states for a range of experimental parameters.
Dynamical tunneling has been observed in atom optics experiments by two groups, one in Texas using Cesium atoms and the other at NIST using a Bose–Einstein condensate. This tunneling is classically forbidden due to the isolation of the atoms in momentum space. The experimental results are extremely well described by time-periodic Hamiltonians with momentum quantized in units of the atomic recoil. The oscillations observed in these experiments are due to a few Floquet states that were excited by the initial conditions in the experiment. The observed tunneling has a well defined period when only two Floquet states dominate the dynamics. Beat frequencies are observed when three Floquet states dominate. Frequencies which are calculted theoretically match those observed by both experiments. The dynamical origin of the dominant Floquet states is identified as resulting from states which are excited by the initial conditions of the experiment.
Dynamical tunneling has been observed in atom-optics experiments by two groups. We show that the experimental results are extremely well described by time-periodic Hamiltonians with momentum quantized in units of the atomic recoil. The observed tunneling has a well-defined period when only two Floquet states dominate the dynamics. Beat frequencies are observed when three Floquet states dominate. We find frequencies that match those observed in both experiments. The dynamical origin of the dominant Floquet states is identified.
The recent pioneering experiments of the [Nature 412, 52 (2001)] and [Science, 293, 274 (2001)] groups have demonstrated the dynamical tunneling of cold atoms interacting with standing electromagnetic waves. It has been shown [Phys. Rev. Lett. 89, 253201 (2002)], that the tunneling oscillations observed in these experiments correspondingly stems from two- and three-Floquet quantum state mechanism and can be controlled by varying the experimental parameters. The question where are the fingerprints of the classical chaotic dynamics in a quantum dynamical process which is controlled by 2 or 3 quantum states remains open. Our calculations show that although the effective $\ensuremath{\hbar}$ associated with the two experiments is large, and the quantum system is far from its semiclassical limit, the quantum Floquet-Bloch quasienergy states still can be classified as regular and chaotic states. In both experiments the quantum and the classical phase-space entropies are quite similar, although the classical phase space is a mixed regular-chaotic space. It is also shown that as the wave packet which is initially localized at one of the two inner regular islands oscillates between them through the chaotic sea, it accumulates a random phase which causes the decay of the amplitude of the oscillating mean momentum, ⟨P(t)⟩⟨P(t)⟩⟨P(t)⟩, as measured in both experiments. The extremely high sensitivity of the rate of decay of the oscillations of ⟨P(t)⟩⟨P(t)⟩⟨P(t)⟩ to the very small changes in the population of different Floquet-Bloch states, is another type of fingerprint of chaos in quantum dynamics that presumably was measured in the NIST and AUSTIN experiments for the first time.
A Bose-Einstein condensate in a modulated, one-dimensional, anharmonic potential can exhibit dynamical tunneling between islands of regular motion in phase space. With increasingly repulsive atomic interactions, dynamical tunneling is predicted to cease due to self-trapping [S. Wüster ]. This suppression of tunneling oscillations is related to the same phenomenon that occurs in the two-mode dynamics of a repulsively interacting Bose-Einstein condensate in a double-well potential. Here we present a two-mode model for dynamical tunneling based on nonlinear Floquet states and examine the range of validity of the approximation. We characterize nonlinear dynamical tunneling for different trap strengths, modulation amplitudes, and effective Planck constants. Using the linear Floquet states we derive an expression for the critical nonlinearity beyond which tunneling ceases. Finally we demonstrate the dynamical instability of selected nonlinear Floquet states and show how to initialize some Floquet states in experiments. Our detailed survey will enable experiments to target accessible parameter regimes for the study of nonlinear dynamical tunneling.
Dynamical tunneling is a quantum phenomenon where a classically forbidden process occurs that is prohibited not by energy but by another constant of motion. The phenomenon of dynamical tunneling has been recently observed in a sodium Bose-Einstein condensate. We present a detailed analysis of these experiments using numerical solutions of the three-dimensional Gross-Pitaevskii equation and the corresponding Floquet theory. We explore the parameter dependency of the tunneling oscillations and we move the quantum system towards the classical limit in the experimentally accessible regime