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Physical Review A

Semiclassical theory of Rydberg-wave-packet interferometry

Mark Mallalieu, Carlos R. Stroud

Published Mar 1, 1995• 13 citations
Publisher

Abstract

A semiclassical approximation is derived from the autocorrelation function describing the excitation and detection of Rydberg wave packets by pairs of phase-locked laser pulses. The resulting expression is in terms of a sum over Kepler trajectories, and provides a direct explanation of the periodicity and spreading of the wave packet at short times when the evolution is classical. More surprisingly, this solution also provides an accurate description of the complex revival behavior of the wave packets, which is nonclassical. The phases of the wave packets, which are detected by the pulse-locking scheme, are also accurately given by analytical expressions derived from the semiclassical sum. The phase of the wave packets at short times have a simple interpretation as the phases of Bohr-Sommerfeld, and are related to Berry's phase. The semiclassical expression for the phases of the nonclassical revival wave packets is simpler than the corresponding quantal solution.

DOI
10.1103/physreva.51.1827

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A semiclassical approximation is derived from the autocorrelation function describing the excitation and detection of Rydberg wave packets by pairs of phase-locked laser pulses. The resulting expression is in terms of a sum over Kepler trajectories, and provides a direct explanation of the periodicity and spreading of the wave packet at short times when the evolution is classical. More surprisingly, this solution also provides an accurate description of the complex revival behavior of the wave packets, which is nonclassical. The phases of the wave packets, which are detected by the pulse-locking scheme, are also accurately given by analytical expressions derived from the semiclassical sum. The phase of the wave packets at short times have a simple interpretation as the phases of Bohr-Sommerfeld, and are related to Berry's phase. The semiclassical expression for the phases of the nonclassical revival wave packets is simpler than the corresponding quantal solution.
A semiclassical propagator is applied to describe the dynamics of circular-orbit wave packets which are initially well localized in three dimensions. A sum over classical Kepler trajectories for the wave-packet autocorrelation function is obtained which is extremely accurate well past the classical regime of the evolution regime of the wave packet. The nonclassical nature of the wave-packet evolution at long times is associated with the interference between amplitudes from classical paths at different energies. The time scale of the wave-packet revivals is related to the shearing rate of the corresponding classical ensemble.
We test the ability of semiclassical theory to describe quantitatively the revival of quantum wave packets-a long time phenomena-in the one dimensional quartic oscillator (a Kerr type Hamiltonian). Two semiclassical theories are considered: time-dependent WKB and Van Vleck propagation. We show that both approaches describe with impressive accuracy the autocorrelation function and wave function up to times longer than the revival time. Moreover, in the Van Vleck approach, we can show analytically that the range of agreement extends to arbitrary long times.
The quantum theory of atoms is based on a Hamiltonian which is just that of the classical Kepler theory of planetary orbits. In spite of this common foundation, the predictions of the quantum theory ordinarily do not much resemble those of the classical theory. Even in the limit of large quantum numbers the energy eigenstates of quantum theory do not look at all like the classical elliptical orbits. Recently, Parker and Stroud1 showed that it is possible to excite a Rydberg atomic wave packet localized in the radial coordinate. Also, Yeazell and Stroud2 showed that a wave packet localized in the angular coordinates can be formed. Both wave packets exhibit an evolution describable, at least in part, by classical theory. However, neither wave packet truly corresponds to a wave packet traveling in a classical elliptical orbit. We discuss the manner in which such a wave packet might be formed using a short-pulsed laser and arf dressing field. The evolution of the resulting wave packet is compared with that of its classical counterpart.
A semiclassical method is discussed for propagating wave packets that incorporates nonlinear classical dynamical information through the use of multiple reference trajectories. We apply the theory to obtain an analytic approximation for the autocorrelation function in the Coulomb potential which is extremely accurate in many circumstances in spite of the Coulomb singularity. All the detailed quantum behavior is reproduced including the long time revivals and the spectrum. The evolving phase space structure of the classical analog reveals how well a particular case will be approximated.
Quantum mechanical wave packet states can approach the classical ideal of a spatially localized particle traveling along a well-defined trajectory. These states offer a system in which the classical limit of quantum atomic theory can be studied. In this work, a wave packet that is radially localized and whose radial motion occurs at the classical orbital period is studied. Despite these strong classical features, the long-term evolution displays the quantum nature of the state. As the wave packet evolves, it becomes dispersed, and the radial motion becomes irregular. However, the wave packet does not remain dispersed, and classical motion returns with the return of the spatial localization. Both the collapse and revival of the wave packet occur because of the quantized nature of the atom. In this experiment, the wave packet was excited to the Rydberg slates by a short, ultraviolet pulse from the ground state of potassium. The probe pulse delay was scanned, and the resulting photo-ionization signal displayed the evolution of the wave packet. The collapse and revival of the radial wave packet were observed.1