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Physical Review A

Three-dimensional kicked hydrogen atom

Matthias Klews, Wolfgang Schweizer

Published Oct 2, 2001• 15 citations
Publisher

Abstract

We present an effective computational method and quantum results for the three-dimensional propagation of wave packets in the hydrogen atom driven by a train of short unidirectional electric-field pulses. We studied the dynamics of wave packets that are initially in a Stark state with different parabolic quantum numbers and observed quantum localization in the high-frequency domain.

DOI
10.1103/physreva.64.053403

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arXiv:0903.1611•Physical Review A•Apr 20, 2009•8 citations
We present an effective computational method and quantum results for the three-dimensional propagation of wave packets in the hydrogen atom driven by a train of short unidirectional electric-field pulses. We studied the dynamics of wave packets that are initially in a Stark state with different parabolic quantum numbers and observed quantum localization in the high-frequency domain.
We study the three-dimensional (3D) unidirectionally kicked Rydberg atom. For parabolic initial states elongated in the direction of the kicks we show that the ionization of the quantum system is suppressed as compared to the classical counterpart and that the quantum wave function is localized along all degrees of freedom, whereas the classical system is globally diffusive. We discuss the connection to the previously studied one-dimensional (1D) model of the kicked Rydberg atom and verify that the 1D model is a good approximation to the 3D quantum case in the limiting case of the most elongated initial states. We further study the quantum phase-space distribution (Husimi distribution) of the eigenstates of the period-one time-evolution (Floquet) operator and show that the eigenstates are localized in phase space. For the most elongated parabolic initial state, we are able to identify the unstable periodic orbits around which Floquet states localize. We discuss the possibility of observing quantum localization in high Rydberg states in n>100.n>100.n>100.
We study the three-dimensional (3D) unidirectionally kicked Rydberg atom. For parabolic initial states elongated in the direction of the kicks we show that the ionization of the quantum system is suppressed as compared to the classical counterpart and that the quantum wave function is localized along all degrees of freedom, whereas the classical system is globally diffusive. We discuss the connection to the previously studied one-dimensional (1D) model of the kicked Rydberg atom and verify that the 1D model is a good approximation to the 3D quantum case in the limiting case of the most elongated initial states. We further study the quantum phase-space distribution (Husimi distribution) of the eigenstates of the period-one time-evolution (Floquet) operator and show that the eigenstates are localized in phase space. For the most elongated parabolic initial state, we are able to identify the unstable periodic orbits around which Floquet states localize. We discuss the possibility of observing quantum localization in high Rydberg states in ng100.ng100.ng100.
It is well known that for the nonrelativistic hydrogen atom it is possible to separate the Schrödinger equation in parabolic as well as spherical coordinates. The eigenfunctions obtained in these coordinate systems are each a suitable basis set in the absence of an electric field, but only the parabolic states, the Stark eigenfunctions, retain their character in the presence of a weak field. The properties of coherent superpositions of these Stark states are investigated and the motion of the resulting wave packet described. It is shown that a properly constituted superposition will mimic classical motion. It is also shown that the constant energy separation between adjacent Stark states of a given principal quantum number leads to periodic motion. In general, they split into distinct “clumps” of probability, but revive to form a single packet after each period. It is, however, possible to construct a packet that maintains its shape for all time.
The characteristics of strongly-polarized quasi-one-dimensional (quasi-1D) Rydberg atoms produced by photoexcitation of selected Stark states in the n = 350 Stark manifold are investigated by studying their ionization when subject to sub-nanosecond unidirectional electric field pulses, termed half-cycle pulses (HCPs), directed both parallel and perpendicular to the atomic axis. The observed differences in the ionization characteristics provide a measure of the anisotropy of the momentum distribution and thus of the quasi-1D nature of the states. The response of such states to a train of HCPs directed parallel and anti-parallel to the axis of polarization of the initial state is also investigated. Pronounced differences in the survival probability are observed confirming theoretical predictions that the dynamical response of the system will be very different for these two conditions. The presence of noise is also observed to play a crucial role in the dynamics. Reasonable agreement between experiment and theory is achieved.
Nondispersive wave packets in a fictitious time variable are calculated analytically for the field-free hydrogen atom. As is well known by means of the Kustaanheimo-Stiefel transformation the Coulomb problem can be converted into that of a four-dimensional harmonic oscillator, subject to a constraint. This regularization makes use of a fictitious time variable, but arbitrary Gaussian wave packets in that time variable in general violate that constraint. The set of ``restricted Gaussian wave packets'' consistent with the constraint is constructed and shown to provide a complete basis for the expansion of states in the original three-dimensional coordinate space. Using that expansion arbitrary localized Gaussian wave packets of the hydrogen atom can be propagated analytically and exhibit a nondispersive periodic behavior as functions of the fictitious time. Restricted wave packets with and without well-defined angular momentum quantum numbers are constructed. They will be used as trial functions in time-dependent variational computations for the hydrogen atom in static external fields in the subsequent paper [T. Fab\ifmmode \check{c}\else \v{c}\fi{}i\ifmmode \check{c}\else \v{c}\fi{}, J. Main, and G. Wunner, Phys. Rev. A 79, 043417 (2009)].