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arXiv:quant-ph/9711073 · Physics of Atomic Nuclei

Revivals of Rydberg Wave Packets

Robert Bluhm, V. Alan Kostelecký, Bogdan Tudose

Published Nov 28, 1997• 2 citations
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Abstract

We examine the revival structure of Rydberg wave packets. These wave packets exhibit initial classical periodic motion followed by a sequence of collapse, fractional/full revivals, and fractional/full superrevivals. The effects of quantum defects on wave packets in alkali-metal atoms and a squeezed-state description of the initial wave packets are also described. We then examine the revival structure of Rydberg wave packets in the presence of an external electric field, i.e., the revival structure of Stark wave packets. These wave packets have energies that depend on two quantum numbers and exhibit new types of interference behavior.

DOI
10.48550/arxiv.quant-ph/9711073

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arXiv:quant-ph/9711061•Nov 25, 1997•1 citation
We examine the revival structure of Rydberg wave packets. These wave packets exhibit initial classical periodic motion followed by a sequence of collapse, fractional/full revivals, and fractional/full superrevivals. The effects of quantum defects on wave packets in alkali-metal atoms and a squeezed-state description of the initial wave packets are also described. We then examine the revival structure of Rydberg wave packets in the presence of an external electric field, i.e., the revival structure of Stark wave packets. These wave packets have energies that depend on two quantum numbers and exhibit new types of interference behavior.
The revival structure and evolution of Rydberg wave packets are studied on a time scale much greater than the revival time trevt_{\rm rev}trev​. We find a new level of revival structure and periodic motion different from that of the known fractional revivals. The new sequence of revivals culminates with the formation of a wave packet that more closely resembles the initial packet than does the full revival at time trevt_{\rm rev}trev​. We refer to such a revival as a superrevival. We also show that an initial radial wave packet may be described as a type of squeezed state known as a radial squeezed state. Our results apply not only for hydrogenic wave packets, but for wave packets in alkali-metal atoms as well in the context of quantum defect theory.
This paper begins with an examination of the revival structure and long-term evolution of Rydberg wave packets for hydrogen. We show that after the initial cycle of collapse and fractional or full revival, which occurs on the time scale trev{\mathit{t}}_{\mathrm{rev}}trev​, a new sequence of revivals begins. We find that the structure of the new revivals is different from that of the fractional revivals. The new revivals are characterized by periodicities in the motion of the wave packet with periods that are fractions of the revival time scale trev{\mathit{t}}_{\mathrm{rev}}trev​. These long-term periodicities result in the autocorrelation function at times greater than trev{\mathit{t}}_{\mathrm{rev}}trev​ having a self-similar resemblance to its structure for times less than trev{\mathit{t}}_{\mathrm{rev}}trev​. The new sequence of revivals culminates with the formation of a single wave packet that more closely resembles the initial wave packet than does the full revival at time trev{\mathit{t}}_{\mathrm{rev}}trev​, i.e., a superrevival forms. Explicit examples of the superrevival structure for both circular and radial wave packets are given. We then study wave packets in alkali-metal atoms, which are typically used in experiments. The behavior of these packets is affected by the presence of quantum defects that modify the hydrogenic revival time scales and periodicities. Their behavior can be treated analytically using supersymmetry-based quantum-defect theory. We illustrate our results for alkali-metal atoms with explicit examples of the revival structure for radial wave packets in rubidium.
After a Rydberg wave packet forms, it is known to undergo a series of collapses and revivals within a time period called the revival time trevt_{\rm rev}trev​, at the end of which it resembles its original shape. We study the behavior of Rydberg wave packets on time scales much greater than trevt_{\rm rev}trev​. We find that after a few revival cycles the wave packet ceases to reform at multiples of the revival time. Instead, a new series of collapses and revivals commences, culminating after a time period tsr≫trevt_{\rm sr} \gg t_{\rm rev}tsr​≫trev​ with the formation of a wave packet that more closely resembles the initial packet than does the full revival at time trevt_{\rm rev}trev​. Furthermore, at times that are rational fractions of tsrt_{\rm sr}tsr​, we show that the motion of the wave packet is periodic with periodicities that can be expressed as fractions of the revival time trevt_{\rm rev}trev​. These periodicities indicate a new type of fractional revival, occurring for times much greater than trevt_{\rm rev}trev​. We also examine the effects of quantum defects and laser detunings on the revival structure of Rydberg wave packets for alkali-metal atoms.
The revival structure of Stark wave packets is considered. These wave packets have energies depending on two quantum numbers and are characterized by two sets of classical periods and revival times. The additional time scales result in revival structures different from those of free Rydberg wave packets. We show that Stark wave packets can exhibit fractional revivals. We also show that these wave packets exhibit particular features unique to the Stark effect. For instance, the wave functions can be separated into distinct sums over even and odd values of the principal quantum number. These even and odd superpositions interfere in different ways, resulting in unexpected periodicities in the interferograms of Stark wave packets.
Experimental and theoretical recurrence spectra of Rydberg electron wave packets are presented. Fractional revivals up to the seventh order are observed. In addition we observe a new interference effect: a forerunner wave packet in the first full revival, due to second-order dispersion. Also it is shown that the quantum defect of rubidium shifts the peaks in the interference pattern by the quantum defect modulo one times the classical orbiting period, while it leaves the envelope of the recurrence spectrum unaffected. All these quantum mechanical effects are explained by a Taylor expansion of the phase factors of the stationary states of which the wave packet is composed.
We present a generic treatment of wave-packet revivals for quantum-mechanical systems. This treatment permits a classification of certain ideal revival types. For example, wave packets for a particle in a one-dimensional box are shown to exhibit perfect revivals. We also examine the revival structure of wave packets for quantum systems with energies that depend on two quantum numbers. Wave packets in these systems exhibit quantum beats in the initial motion as well as new types of long-term revivals. As an example, we consider the revival structure of a particle in a two-dimensional box.