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arXiv:quant-ph/0609031 · Physical Review A

Transient localization in the kicked Rydberg atom

Emil Persson, Sebastian Fürthauer, Sandro Wimberger, Joachim Burgdörfer

Published Nov 22, 2006• 5 citations
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Abstract

We investigate the long-time limit of quantum localization of the kicked Rydberg atom. The kicked Rydberg atom is shown to possess in addition to the quantum localization time ${\ensuremath{\tau}}_{L}$ a second crossover time ${t}_{D}$ where quantum dynamics diverges from classical dynamics towards increased instability. The quantum localization is shown to vanish as either the strength of the kicks at fixed principal quantum number or the quantum number at fixed kick strength increases. The survival probability as a function of frequency in the transient localization regime ${\ensuremath{\tau}}_{L}<t<{t}_{D}$ is characterized by highly irregular, fractal-like fluctuations.

DOI
10.1103/physreva.74.053417

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We investigate the long-time limit of quantum localization of the kicked Rydberg atom. The kicked Rydberg atom is shown to possess in addition to the quantum localization time \ensuremathτL{\ensuremath{\tau}}_{L}\ensuremathτL​ a second crossover time tD{t}_{D}tD​ where quantum dynamics diverges from classical dynamics towards increased instability. The quantum localization is shown to vanish as either the strength of the kicks at fixed principal quantum number or the quantum number at fixed kick strength increases. The survival probability as a function of frequency in the transient localization regime \ensuremathτL<t<tD{\ensuremath{\tau}}_{L}<t<{t}_{D}\ensuremathτL​<t<tD​ is characterized by highly irregular, fractal-like fluctuations.
The quantum localization of the periodically kicked Rydberg atom in the limit of high frequencies $\ensuremath{\nu}$ is studied. We show that the quantum suppression of fast chaotic ionization as predicted by classical dynamics persists as $\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\nu}}\ensuremath{\infty}.$ Properties of localization in the regime of strong coupling to the continuum due to one-photon transitions are discussed. For the unidirectionally kicked atom, the high-frequency limit of localization is determined by the zero-frequency Stark Hamiltonian. The persistence of quantum localization due to interference of classical trajectories can be understood in terms of smearing out of the instabilities at finite $\ensuremath{\Elzxh}.$ Unstable trajectories whose action differs from each other and from Stark orbits in less than $\ensuremath{\Elzxh}$ contribute to quantum localization rather than to chaotic ionization.
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.
We discuss quantum relaxation and fluctuation phenomena in the light of the corresponding classical behaviour. Quantum exponential localization and tunneling effects lead to the universal algebraic decay of the survival probability P ( t ) ∝ 1/ t . This behaviour is observed in particular in the time dependence of the ionization probability of Rydberg atoms driven by a monochromatic field. At a fixed time the quantum survival probability strongly fluctuates as some parameter is varied. These fluctuations have a fractal structure which is quantitatively related to the time dependence of the survival probability.
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.