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arXiv:nlin/0307038 · Physical Review A

Experimentally attainable example of chaotic tunneling: The hydrogen atom in parallel static electric and magnetic fields

Dominique Delande, Jakub Zakrzewski

Published Dec 24, 2003• 13 citations
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Abstract

Statistics of tunneling rates in the presence of chaotic classical dynamics is discussed on a realistic example: a hydrogen atom placed in parallel, uniform, static electric, and magnetic fields, where tunneling is followed by ionization along the fields direction. Depending on the magnetic quantum number, one may observe either a standard Porter-Thomas distribution of tunneling rates or, for strong scarring by a periodic orbit parallel to the external fields, strong deviations from it. For the latter case, a simple model based on random matrix theory gives the correct distribution.

DOI
10.1103/physreva.68.062110

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Statistics of tunneling rates in the presence of chaotic classical dynamics is discussed on a realistic example: a hydrogen atom placed in parallel, uniform, static electric, and magnetic fields, where tunneling is followed by ionization along the fields direction. Depending on the magnetic quantum number, one may observe either a standard Porter-Thomas distribution of tunneling rates or, for strong scarring by a periodic orbit parallel to the external fields, strong deviations from it. For the latter case, a simple model based on random matrix theory gives the correct distribution.
A discussion regarding the energy levels spectrum of quantum systems whose classical analogous has states of chaotic motion is presented. The chaotic dynamics of the classical underlying system has its manifestation in the wave functions (in the form of “scars”) and in the energy levels (in the form of “statistical repulsion” of the energy levels). The above mentioned signatures are named “quantum chaos”. A typical study of quantum chaos requires finding accurate energy eigenvalues of highly excited states, to calculate the nearest neighbors spacing between levels, to perform the “unfolding” of the spectrum in order to separate the fluctuations, and finally to find the probability distribution of the unfolded spectrum. This is exemplified by the hydrogen atom under uniform magnetic field and a quadrupole electric field.
We review recent progress in attaining a quantitative understanding of the scarring phenomenon, the non-random behaviour of quantum wavefunctions near unstable periodic orbits of a classically chaotic system. The wavepacket dynamics framework leads to predictions about statistical long-time and stationary properties of quantum systems with chaotic classical analogues. Many long-time quantum properties can be quantitatively understood using only short-time classical dynamics information; these include wavefunction intensity distributions, intensity correlations in phase space and correlations between wavefunctions, and distributions of decay rates and conductance peaks in weakly open systems. Strong deviations from random matrix theory are predicted and observed in the presence of short unstable periodic orbits.
A quantum mechanical study of a simple two-dimensional problem identifies lines in coordinate space about which the probability densities are localised. These lines in general do not correspond to extrema in the system's static potential energy surface. The subject of this study has been the hydrogen atom in externally applied parallel electric and magnetic and magnetic fields. Probability density plots of eigenstates, in which the electron experiences comparable influences from all three fields: Coulomb, electric and magnetic, are each found to display a localisation at a polar angle related to an invariant previously obtained in a classical study of the problem. The line about which localisation occurs is found to vary with energy and is in coincidence with a periodic trajectory found within a classical framework.
New experimental results are presented for short-pulse microwave ionization of highly excited hydrogen atoms. A comparison of these results with quantum numerical computations and analytical predictions provides for the first time experimentally grounded evidence of the localization phenomenon that leads to the suppression of the quantum version of the chaotic diffusion in action space occurring in the classical limit. The onset of chaotic motion in externally driven classical systems is a key problem in nonlinear dynamics that is now well understood in its essential aspects. In many physically interesting cases, as the perturbation strength increases beyond some critical value, the system starts absorbing energy in a diffusive way. The question whether “diffusive” excitation processes can take place also in externally driven quantum systems is then a very interesting one for the physics of atoms and molecules in external electromagnetic fields. This is a deep question involving the nature and the validity of the quasiclassical approximation when the underlying classical dynamics is chaotic. The question of “quantum diffusion” was first addressed in a simple model system, the kicked rotator. It was chosen because the features of classical chaos in it were relatively clean and well understood. Besides that, the numerical simulation of its quantum dynamics could be easily accomplished. The major indication of this model was that quantum mechanics suppresses the classical chaotic diffusion in action via a destructive interference effect similar to that responsible for the Anderson localization well known in condensed matter physics.
The dynamics of Rydberg states of a hydrogen atom subjected simultaneously to uniform static electric field and two microwave fields with commensurate frequencies is considered in the range of small field amplitudes. In the certain range of the parameters of the system the classical secular motion of the electronic ellipse reveals chaotic behavior. Quantum mechanically, when the fine structure of the atom is taken into account, the energy level statistics obey predictions appropriate for the symplectic Gaussian random matrix ensemble.
With a mean-field treatment, the spin tunneling of spin-1 Bose-Einstein condensates in a linearly sweeping magnetic field is investigated. We focus on a typical alkali Bose atoms 87Rb condensate and study its tunneling dynamics under different sweeping rates of external magnetic field. In the adiabatic (i.e., slowly sweeping) and sudden (i.e., fast sweeping) limits, no tunneling is observed. In the case of moderate sweeping rates, the tunneling dynamics is found to be very sensitive to the sweeping rate, showing a chaotic-like tunneling regime. By magnifying the apparently chaotic regime, however, we find interestingly that the plot becomes resolvable at a resolution of 10-8T/s where the tunneling probability with respect to the sweeping rate shows a regular pseudo-periodic pattern. Moreover, a conserved quantity standing for the magnetization in experiments is found which can dramatically affect the above picture of the spin tunneling. Theoretically, we give a reasonable interpretation to the above findings.