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Physical Review A

Discrepancies in the resonance-fluorescence spectrum calculated with two methods

Qing Yang Xu, Xiangming Hu, Jian-wu Yin

Published Jul 10, 2008• 3 citations
Publisher

Abstract

The discrepancies in the power spectrum of the light scattered by a coherently driven three-level atom are identified with the help of two methods [see, A. S. Manka et al., Phys. Rev. A 43, 3748 (1991); M. R. Ferguson et al., Phys. Rev. A 54, 2379 (1996)]. Moreover, the most prominent differences are the negative values as well as the greatly distorted profile in the spectrum. It is shown that the inhomogeneity or homogeneity of the two-time correlation Bloch vector equation is responsible for such discrepancies.

DOI
10.1103/physreva.78.013407

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The discrepancies in the power spectrum of the light scattered by a coherently driven three-level atom are identified with the help of two methods [see, A. S. Manka et al., Phys. Rev. A 43, 3748 (1991); M. R. Ferguson et al., Phys. Rev. A 54, 2379 (1996)]. Moreover, the most prominent differences are the negative values as well as the greatly distorted profile in the spectrum. It is shown that the inhomogeneity or homogeneity of the two-time correlation Bloch vector equation is responsible for such discrepancies.
There are two alternative methods used in the literature to calculate the incoherent part of the spectrum of light scattered by an atomic system. In the first, one calculates the spectrum of the total light scattered by the system and obtains the incoherent part by subtracting the coherent part. In the second method, one introduces the fluctuation operators and obtains the incoherent part of the spectrum by taking the Fourier transform of the two-time correlation function of the fluctuation operators. These two methods have been recognized for years as completely equivalent for evaluating the incoherent part of the spectrum. In a recent paper, Xu et al. [Phys. Rev. A 78, 013407 (2008)] showed that there are discrepancies between the incoherent parts of the stationary spectrum of a three-level $\ensuremath{\Lambda}$-type system calculated with these two methods. The predicted discrepancies can be severe that over a wide range of the Rabi frequencies and atomic decay rates, the spectrum calculated with the variance method can have negative values. In this Comment, we show that there are no discrepancies between these two methods. We show the equivalence of these two methods that leads to the same incoherent spectra which are positive for all frequencies independent of values of the parameters involved. We also identify the source of the discrepancy, that is, in an incorrect treatment of the incoherent part of the spectrum calculated with the two-time correlation function of the fluctuation operators.
We identify discrepancies between two published predictions of the resonance fluorescence from a two-level atom driven by a strong bichromatic field [J. Opt. Soc. Am. B 8,8,8, 1163 (1991) and Phys. Rev. A 48,48,48, 3092 (1993)]. These discrepancies are shown to arise from subtleties in extracting the time-averaged resonance fluorescence from the expansions used to solve the optical Bloch equations. We construct a solution to these equations that corrects the results of the latter reference and reconciles the two predictions.
Calculations are presented for the degrees of first- and second-order coherence and the spectral distribution of the light emitted in resonance fluorescence by a two-level atom driven by monochromatic incident light. The theory includes the effects of both homogeneous and inhomogeneous broadening, but is restricted to ranges of the parameters for which analytic solutions can be obtained. There is a further restriction to forward scattering in the case where the inhomogeneous broadening is caused by the Doppler effect. The results show marked differences between the effects of the homogeneous and inhomogeneous broadening mechanisms.
The spectral distribution of light scattered by a two-level atom in a laser beam with phase modulation has been calculated. The spectrum has been obtained via numerical solution of the optical Bloch equations using the algorithm of Wilkens and Rza\ifmmode \mbox{\c{}}\else \c{}\fi{}\ifmmode \dot{z}\else \.{z}\fi{}ewski [Phys. Rev. A 40, 3164 (1989)]. The modulation produces additional sidebands in both coherent and incoherent parts of the spectrum; the location of the added components depends on the modulation depth. In the case of two-frequency phase modulation with incommensurate frequencies, wild, ``chaotic'' behavior of the spectrum has been observed, similar to the case of bichromatic amplitude modulation. The intensity distribution is practically insensitive to the chosen constant value of the phase of modulation.
Although the resonance fluorescence spectrum of coherent light scattering from a two-level atom has been discussed in many papers, the effects of longitudinal, transverse relaxation and optical pumping have not yet been considered simultaneously in the motion equations. By solving the optical Bloch equations including the longitudinal relaxation time T 1 , transverse relaxation time T 2 and optical pumping parameter Δ 0 , in the case T 2 /T 1 <<1, we obtain the spectrum with dip in the central peak besides the familiar spectrum of three peaks. In addition, we find that the parameter Δ 0 and detuning Δω have influence on the symmetry, height and width of the spectrum of the three peaks.
The power spectrum of the light scattered by a two-level atom driven near resonance by a monochromatic classical electric field is evaluated. The atom is assumed to relax to equilibrium with the driving field via radiation damping, which is treated by explicitly coupling the atom to the quantized electromagnetic field modes. The power spectrum of the scattered field is directly obtainable from the two-time atomic dipole moment correlation function, which is evaluated by a method based on a Markoff-type assumption analogous to that used to evaluate the time evolution of single-time atomic expectation values.