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Physical Review

Theory of Quantum-Beat and Level-Crossing Experiments Utilizing Electronic Excitation

Robert L. Kelly

Published Jul 8, 1966• 23 citations
Publisher

Abstract

The coherent excitation of several atomic states by inelastic electron scattering and their subsequent radiative decay is considered. General expressions are derived for the photon counting rate in a quantum-beat experiment, and for the total number of photons counted in a level-crossing experiment. The general results are used to calculate the phase of the oscillatory part of the photon counting rate in the Hadeishi-Nierenberg quantum-beat experiment.

DOI
10.1103/physrev.147.376

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The coherent excitation of several atomic states by inelastic electron scattering and their subsequent radiative decay is considered. General expressions are derived for the photon counting rate in a quantum-beat experiment, and for the total number of photons counted in a level-crossing experiment. The general results are used to calculate the phase of the oscillatory part of the photon counting rate in the Hadeishi-Nierenberg quantum-beat experiment.
IMeasurements of space-time correlations in the intensity of a beam of scattereq particles yield information about the scatterer.Goldberger and Watson have developed a theory of such measl~ementG appropriate to the case of ne~rly elastic scattering; in this paper we extend their theory to the case of inelastic scattering.The main result of our work is that under certain experimental conditions the correlated counting rate in an inelastic scattering experiment is of a form equivalent to the correlated counting rate for an incoherent beam of particles emitted by a source.This result applies, in particular, to Raman or Brillouin scattering of light by phonons in an extended target.However, estimates of the signal to noise ratio for this case indicate that it may not be possible to observe intensity correlations in Raman or Brillouin lines using presently available light sources. ABSTRACT OF PART IIThe coherent excitation of several atomic status by inelastic electron' scattering and their subsequent radia ti ve decay is considered.General expressions for the photon counting rate in a quantum beat experiment, and the total number of photons counted. in a le,vel crossing experiment are d.erived.The general results are used to calculate the phase of the oscillatory part of the photon counting rate in the Radeishi-Nierenberg quantum beat experiment.
The results of electron-photon coincidence experiments have given a new insight into the excitation of atoms by electron impact. This article reviews the way in which this new experimental technique has provided an exacting and fundamental test for the theory of the excitation process. The basic experimental method is first described, and a comprehensive treatment of excitation of the n 1 P states of helium is given. The extension of the technique to the study of excitation of the n=2 states of atomic hydrogen is then presented. Coincidence measurements also provide information on the role of spin in the collision, and this subject is treated in detail for excitation of heavy atoms by both unpolarised and spin-polarised beams of electrons. Finally, the electron-photon coincidence method permits observation of the time evolution of excited atomic states, and this review describes the attempts made to observe quantum beats in the radiative decay.
In this paper we present a study of the radiative decay of coherently excited, closely spaced, molecular (or atomic) levels. Interference effects in the decay process cannot be treated in general by displaying the damping matrix in a diagonal form, so that each decay channel is assumed to be characterized by its own lifetime. The problem of the radiative decay of coherently excited indistinguishable levels (characterized by the same symmetry) is treated by a self-consistent extension of the Wigner-Weisskopf method, or alternatively, by the ‘unitarity relations’ method, developed in elementary particles physics. Finally, the Fano configuration interaction method is applied for the decay of a manifold of states characterized by the same polarization. General expressions for photon counting rates are derived and the nature of a general quantum beat experiment is discussed.
A theory of electron-impact excitation of atoms in which fine structure is included as a perturbation is presented in detail. Hyperfine structure is included in the target atom but neglected in the electron-atom interaction. Expressions for electron-photon angular-correlation parameters and scattering cross sections are derived in terms of products of complex scattering amplitudes, with emphasis on electron-sodium excitation. Examples are given in which zero-field quantum beats may be observed in electron-photon angular-correlation and differential-cross-section measurements. Some of the interference terms are only observable in angular-correlation experiments. Examples are also given of partial cross sections for electron-impact excitation of state-to-state hyperfine levels in terms of direct- and exchange-scattering amplitudes plus additional terms due to the inclusion of fine-structure effects. The expressions given allow a comparison between the various possible scattering experiments.
Based on the master equation approach, an expression for the time dependent fluorescence intensity has been derived taking into account both collision effects resulting from the elastic and inelastic scattering from the heat bath and the statistical properties of the incident light. The degree of coherence for the molecular eigenstates is defined. The pure dephasing constant between the excited states attenuates the degree of coherence for the molecular eigenstates during and after the light pulse duration; on the other hand, the pure dephasing between the ground and excited states attenuates it during the pulse duration. A separable nonstationary correlation function for the incident radiation field is introduced to investigate the effects of the field coherence on the time dependent fluorescence. Model calculations of the quantum beats in the fluorescence from molecules with two levels in the excited state in the weak radiation field have been performed to demonstrate the effects of the dephasing constants and the field coherence as well as those of the population decay constants.
A theoretical description is presented for the process in which photons emitted from atoms which have been stepwise excited firstly by electron impact and then by resonant interaction with coherent radiation are detected in coincidence with the inelastically scattered electrons. The optical excitation is modelled using a full quantum electrodynamic theory. Calculations of the Stokes parameters are performed for the Hg (I=0) scheme; electron excitation from the 6 1 S 0 ground state to the 6 1 P 1 state followed by resonant laser excitation to the 6 1 D 2 state, with analysis being performed on the fluorescence from the 6 1 D 2 -6 3 P 1 relaxation channel. It is found that for this case, even though manifestations of optical pumping are evident, the Stokes parameters do not vary by more than +or-4% from the weak intensity limit.