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Physical Review A

Scattering of Intense Light by a Two-Level System

Rodney J. Gush, Herbert P. Gush

Published Jul 1, 1972• 46 citations
Publisher

Abstract

A general expression is found for the Green's function of a two-level system interacting with a classical monochromatic field of arbitrary frequency and intensity. The Green's function is first used to calculate the transition probability from the lower to the upper state as a function of the frequency and intensity of the field and a comparison of the results is made with experiment. It is then used to study the problem of scattering. It is found that the spectrum of the scattered radiation consists of a line at the frequency of the field (Rayleigh line) and two satellites symmetrically displaced from the Rayleigh line by an amount depending both on the intensity and frequency of the field. In addition there are emissions at three, five, etc., times the frequency of the field, and accompanying satellites. The cross sections for the production of the Rayleigh line and its satellites, and for the third-harmonic line and its satellites have been calculated for a range of values of the frequency and intensity of the incident field.

DOI
10.1103/physreva.6.129

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Journal of Physics B Atomic and Molecular Physics•Feb 21, 1977•148 citations
A general expression is found for the Green's function of a two-level system interacting with a classical monochromatic field of arbitrary frequency and intensity. The Green's function is first used to calculate the transition probability from the lower to the upper state as a function of the frequency and intensity of the field and a comparison of the results is made with experiment. It is then used to study the problem of scattering. It is found that the spectrum of the scattered radiation consists of a line at the frequency of the field (Rayleigh line) and two satellites symmetrically displaced from the Rayleigh line by an amount depending both on the intensity and frequency of the field. In addition there are emissions at three, five, etc., times the frequency of the field, and accompanying satellites. The cross sections for the production of the Rayleigh line and its satellites, and for the third-harmonic line and its satellites have been calculated for a range of values of the frequency and intensity of the incident field.
The behavior of a two-level system interacting with a bichromatic classical field is studied. Analytic expressions are found for the Green's-function operator for the system if the field is weak and near resonance, and a systematic procedure is developed to obtain the Green's-function operator in a more general case. The results are used to calculate the probability of a transition from the lower to the upper state; resonance conditions occur when the level spacing of the system, ${\ensuremath{\omega}}_{ba}$, is near $n{\ensuremath{\omega}}_{\ensuremath{\lambda}}\ensuremath{-}(n\ensuremath{-}1){\ensuremath{\omega}}_{\ensuremath{\mu}}$, where nnn is a positive or negative integer, and ${\ensuremath{\omega}}_{\ensuremath{\lambda}}$ and ${\ensuremath{\omega}}_{\ensuremath{\mu}}$ are the frequencies of the components of the field. The resonance frequency depends on the intensity of the field components. In addition, the cross section for scattering photons out of the field has been calculated. It is found that the spectrum of the scattered radiation consists of lines at frequency ${\ensuremath{\omega}}_{k}={\ensuremath{\omega}}_{\ensuremath{\mu}},{\ensuremath{\omega}}_{\ensuremath{\lambda}}$, and odd harmonics of these, combinations of the fundamental and the harmonics with $\ensuremath{\Delta}={\ensuremath{\omega}}_{\ensuremath{\lambda}}\ensuremath{-}{\ensuremath{\omega}}_{\ensuremath{\mu}}$, and satellites shifted from all of these features by an amount which depends on the intensity of the field and ${\ensuremath{\omega}}_{ba}$. The cross section is intensity dependent particularly insofar as the location of resonance peaks is concerned. It is suggested that this effect may be exploited in transferring intensity modulation of one component of the field to phase modulation in the other component.
The interaction of a two-level atom with two monochromatic electromagnetic fields having arbitrary amplitudes and resonance frequencies is considered. It is shown that the time averaged upper-level population can exceed 0.5 and even reach 0.7. Its dependence on the frequency difference reveals resonance satellite peaks, which are concerned with three- and five-photon combination scattering. The widths and the resonance spectrum shifts of these peaks are found. The frequency spectrum of the spontaneous emission of an atom contains fine structure under some conditions. This structure does not exist in the case of one monochromatic field. An asymmetry in the intensity distribution for the Rabi satellites due to a field non-monochromaticity is predicted.
The Green's-function method is used to study the properties of the spectrum of a three-level system driven by two resonant monochromatic fields. In the case of strong fields, we find that the correlation between two transitions is too small to consider. The Green's functions describing the two transitions consist of five Lorentzian lines rather than seven lines; furthermore, the positions at which sidebands are peaked depend on both Rabi frequencies, and only when two Rabi frequencies reach certain values can five Lorentzian lines appear. The heights of the sidebands become larger with increasing Rabi frequencies. The decay rates also affect the spectrum strongly.
The time evolution of a two-level system interacting with a classical monochromatic radiation field is calculated by Fourier transforming the equations of motion. This can be done exactly if one uses the convolution of the product of two Fourier transformations. The method produces recurrence relationships for the transforms. The time-dependent amplitudes of the states and the transition probability are evaluated by using two such recurrence relationships. The time-averaged probability shows characteristic resonances at odd multiples of the ratio of the natural frequency of the system to the frequency of the radiation field. The peaks of the resonances are displaced by the Bloch-Siegert shift. At these resonances the system is completely saturated with the flipping frequency dependent on the strength of the radiation field. The time dependence and the frequency spectrum of the dipole moment is determined at the main resonance and at the three-quantum resonance. The spectrum contains the zero-frequency terms which are responsible for the coherence resonances.
In order to investigate the effect of perturbation caused by the interaction of a two-level atom with its environment on the coherency of light scattered by the atom when it is irradiated by an intense radiation, Mollow's theory is generalized to include a random modulation of the resonance frequency as well as fluctuations of the incident field. Expressions for the power spectrum, the total intensity, the coherent scattering intensity and the broad Rayleigh scattering intensity are derived. Some general aspects of the problem are discussed.
The general method developed in a previous paper (see ibid., vol.10, p.345 (1977)) is applied to the problem of resonance Raman scattering in very intense laser fields. At very low intensities the spectrum consists of three monochromatic lines (Rayleigh, Raman Stokes and anti-Stokes) which split into a triplet and two doublets when the laser light, tuned on one of the two transitions connecting the upper state to the two lower sublevels, has a sufficiently high intensity to saturate this transition but not the second one. The Raman and Rayleigh lines are completely mixed when the intensity becomes so high that both transitions are saturated. The corresponding evolution of the spectrum is determined. Optical-pumping effects leading to asymmetries in the fluorescence spectrum are taken into account.