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arXiv:quant-ph/0210025 · Physical Review A

Floquet analysis of atom-optics tunneling experiments

Robert Luter, L. E. Reichl

Published Nov 18, 2002• 25 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreva.66.053615

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arXiv:1203.2792•Physical Review A•May 17, 2012•21 citations
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.
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.
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.
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 study the tunneling dynamics in a time-periodically modulated two-mode Bose-Hubbard model using Floquet theory. We consider situations where the system is in the self-trapping regime and either the tunneling amplitude, the interaction strength, or the energy difference between the modes is modulated. In the former two cases, the tunneling is enhanced in a wide range of modulation frequencies, while in the latter case the resonance is narrow. We explain this difference with the help of Floquet analysis. If the modulation amplitude is weak, the locations of the resonances can be found using the spectrum of the nonmodulated Hamiltonian. Furthermore, we use Floquet analysis to explain the coherent destruction of tunneling (CDT) occurring in a large-amplitude modulated system. Finally, we present two ways to create a NOON state (a superposition of NNN particles in mode 1 with zero particles in mode 2 and vice versa). One is based on a coherent oscillation caused by detuning from a partial CDT. The other makes use of an adiabatic variation of the modulation frequency. This results in a Landau-Zener type of transition between the ground state and a NOON-like state.