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Physical Review A

Scattering of weak low-frequency light by three-level systems

Michał Berent, R. Parzyński

Published Sep 24, 2009• 4 citations
Publisher

Abstract

We calculate analytically both coherent and incoherent parts of the total spectrum of light scattered by a three-level system in either lambda or ladder configurations. The spectrum is restricted to the cases of the incident light pulse weakly depopulating the initial level and of the carrier frequency much lower than the excitation frequency in the system. We give special attention to the spectra under the conditions of different multiphoton resonances in the system and emphasize the effects resulting from the presence of the third level. The analytic theory is shown to explain the numerically calculated spectra as far as the number of frequency components, their positions in the spectrum, relative intensities, and sensitivity to the pulse shape are concerned.

DOI
10.1103/physreva.80.033834

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We calculate analytically both coherent and incoherent parts of the total spectrum of light scattered by a three-level system in either lambda or ladder configurations. The spectrum is restricted to the cases of the incident light pulse weakly depopulating the initial level and of the carrier frequency much lower than the excitation frequency in the system. We give special attention to the spectra under the conditions of different multiphoton resonances in the system and emphasize the effects resulting from the presence of the third level. The analytic theory is shown to explain the numerically calculated spectra as far as the number of frequency components, their positions in the spectrum, relative intensities, and sensitivity to the pulse shape are concerned.
The power spectrum of the light scattered by a two-level atom in the presence of a coherent continuous-pulse-train driving field is analyzed. Separate expressions for the coherent and incoherent components of the spectrum are obtained. The coherent part of the spectrum consists of a series of spikes (delta functions) which are displaced from the optical carrier by multiples of the pulse repetition frequency. The incoherent spectrum can have fixed peaks at these same frequencies plus sidebands equally spaced on opposite sides of these fixed peaks. That is, the spectrum consists of a series of triplets. The special case of a two-mode driving field is treated numerically and graphs of spectra are presented.
We have performed a numerical calculation for the excitation spectra arising from the interaction of a three-level atom in the lambda configuration with a strong electromagnetic field whose frequency mode is initially populated. The excitation spectra are considered for equal detunings, which refer to the cooperative two-photon resonance process, as well as for different detunings. The computed cooperative two-photon resonance spectra are compared with those derived by analytical methods. In the presence of different detunings, conditions are established under which the laser field induces sidebands at the frequency equal to that for the forbidden transition. Numerical calculations for a wide range of Rabi frequencies and detunings are presented graphically. Detailed discussion of the computed spectral intensities is given and a comparison is made between the spectra of three-level atoms in the lambda and cascade configurations.
Resonant scattering (redistribution) of weak monochromatic light by atoms in the presence of collisions has been considered by Omont et al., who allowed for the broadening of both upper and lower levels of the transition. Mollow (who also has considered this problem for intense radiation) has shown that their results are applicable only if collisional repopulation of the levels (in contrast to collisional depopulation) is ignored. We here consider in detail the effect of repopulation. It leads, in the weak-field limit, to a true-coherent component ($\ensuremath{\delta}$-function spectrum) at the driving frequency, as well as a component at this frequency whose width is essentially the inelastic collisional rate of the lower state. An additional redistributed component due to repopulation of the upper level is also obtained.
An analytic study is presented of the efficient multiphoton excitation and strong harmonic generation in three-level systems specified by a pair of nearly degenerate, strongly dipole-coupled excited states. Such systems are physically formed by the three lowest states in, e.g., the hydrogen atom or evenly charged homonuclear diatomic molecular ions under reasonably chosen laser intensities. As a detailed analytic result, we found that the laser pulse of photon energy 2.05eV2.05\text{eV}2.05eV, duration 0.23ps0.23\text{ps}0.23ps, and intensity $5\ifmmode\times\else\texttimes\fi{}{10}^{13}\frac{\mathrm{W}}{{\mathrm{cm}}^{2}}$ is able to produce complete inversion of the initial population in the hydrogen atom through the five-photon excitation. At the same photon energy, the pulse of duration 0.41ps0.41\text{ps}0.41ps and intensity $3.44\ifmmode\times\else\texttimes\fi{}{10}^{14}\frac{\mathrm{W}}{{\mathrm{cm}}^{2}}$ was found to produce the same effect in the molecular ion but through the nine-photon excitation. We show that the accompanying scattering of light has very rich spectrum differing substantially from that of the two-level system.
We derive and compare the optical fluorescence spectra for all three transitions for three-level systems coherently coupled with two fields in a cascade or lambda configuration. It is shown that the total intensity of fluorescence at the combinational frequency (the sum or difference of the driving field frequencies in the ladder and lambda systems, respectively) may essentially exceed the fluorescence intensity at the driven transitions under the condition of two-photon resonance. The influence of additional homogeneous and inhomogeneous line broadening on the distribution of intensities between different fluorescence channels is studied. In particular, Doppler broadening is found to suppress fluorescence from the driven transitions but in some cases not from the combinational transition.