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Physical Review A

Floquet analysis of the dynamical stabilization of the kicked hydrogen atom

S. Yoshida, Carlos O Reinhold, Peter Kristöfel, Joachim Burgdörfer, Shinichi Watanabe, F. B. Dunning

Published Jun 1, 1999• 17 citations
Publisher

Abstract

We analyze the quantum dynamics of the one-dimensional periodically kicked Rydberg atom. The time-dependent Schr\"odinger equation is solved by employing a nonunitary representation of the period-one evolution operator in a finite basis set that accounts for the outbound probability flux. We find a direct correspondence between stable classical islands in phase space and the quantum Husimi distributions of stable Floquet states. These results explain the pronounced peak recently found experimentally in the frequency-dependent survival probability of Rydberg states subject to a sequence of half-cycle pulses.

DOI
10.1103/physreva.59.r4121

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We analyze the quantum dynamics of the one-dimensional periodically kicked Rydberg atom. The time-dependent Schr"odinger equation is solved by employing a nonunitary representation of the period-one evolution operator in a finite basis set that accounts for the outbound probability flux. We find a direct correspondence between stable classical islands in phase space and the quantum Husimi distributions of stable Floquet states. These results explain the pronounced peak recently found experimentally in the frequency-dependent survival probability of Rydberg states subject to a sequence of half-cycle pulses.
Measurements in which K(np)K(\mathrm{np})K(np) Rydberg atoms with $n\ensuremath{\sim}388$ are subject to a train of up to 50 half-cycle pulses are reported and reveal a peak in the atomic survival probability for pulse repetition frequencies near the classical orbital frequency. Classical calculations show that this behavior is associated with dynamical stabilization and the periodic orbits in phase space around which stable islands form are identified. This work provides the first direct evidence for dynamical stabilization of a periodically kicked Rydberg atom.
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.
We compare the classical and quantum evolution for the impulsively driven hydrogen atom which is subject to a sequence of periodic pulses ({open_quotes}kicks{close_quotes}). The classical system possesses a divided phase space with regions of regular and chaotic motion present. We solve the three-dimensional Schrodinger equation using a basis of {approx_equal}1000 states including pseudostates in the continuum. The dynamics is analyzed in terms of quasi-eigenstates and quasi-eigenenergies of the period-one evolution operator. We find a localization and stabilization in the quantum evolution due to the existence of resonances which are absent in the classical evolution. We show that the classical-quantum correspondence can be restored when quantum resonances are destroyed by a stochastic rather than a periodic sequence of kicks.
The response of quasi-one-dimensional very-high-nnn Rydberg atoms to a periodic train of short unidirectional electric-field pulses, termed half-cycle pulses (HCPs), is studied at high scaled frequencies. Pronounced nonmonotonic structure in the survival probability is observed as NNN, the number of HCPs in the train is increased, the survival probability actually increasing with NNN over certain ranges of NNN. This behavior is very sensitive to the polarization of the Rydberg states and is explained with the aid of classical simulations.
The periodically kicked Rydberg atom displays quantum localization, features of which depend on the orientation and strength of the unidirectional kicks. They include scarring of the wave function, localization by cantori, and exponential localization in the regime of strong perturbation resembling dynamical localization. Using the semiclassical Herman-Kluk propagator we investigate the degree to which semiclassical dynamics can mimic quantum localization. While the semiclassical approximation has difficulties to reproduce the scarred wave functions, the exponential tail which is a typical signature of the dynamical localization is well represented in the case of strong classical diffusion. Also the localization by broken tori is observed in the semiclassical recurrence probability for short times but the deviation from the corresponding quantum dynamics becomes more pronounced for the long-time evolution.