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Physical Review A

Semiclassical analysis of quantum localization of the periodically kicked Rydberg atom

S. Yoshida, Frank Großmann, Emil Persson, Joachim Burgdörfer

Published Apr 28, 2004• 20 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.69.043410

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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 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 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 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 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 ${\ensuremath{\tau}}_{L}<t<{t}_{D}$ is characterized by highly irregular, fractal-like fluctuations.
Atom-optics kicked rotor represents an experimentally realizable version of the paradigmatic quantum kicked rotor system. After a short initial diffusive phase the cloud settles down to a stationary state due to the onset of dynamical localization. In this work we realise an enhancement of localization by modification of the kick sequence. We experimentally implement the modification to this system in which the sign of the kick sequence is flipped by allowing for a free evolution of the wavepackets for half the Talbot time after every MMM kicks. Depending on the value of MMM, this modified system displays a combination of enhanced diffusion followed by asymptotic localization. This is explained as resulting from two competing processes -- localization induced by standard kicked rotor type kicks, and diffusion induced by half Talbot time evolution. The evolving states display a localized but non-exponential wave function profiles. This provides another route to quantum control in kicked rotor class of systems. The numerical simulations agree well with the experimental results.
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.
An initial coherent state is propagated exactly by a kicked quantum Hamiltonian and its associated classical stroboscopic map. The classical trajectories within the initial state are regular for low kicking strengths, then bifurcate and become mainly chaotic as the kicking parameter is increased. Time-evolution is tracked using classical, quantum and semiclassical Wigner functions, obtained via the Herman-Kluk propagator. Quantitative comparisons are also included and carried out from probability marginals and autocorrelation functions. Sub-Planckian classical structure such as small stability islands and thin/folded classical filaments do impact semiclassical accuracy, but the approximation is seen to be accurate for multiple Ehrenfest times.