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Physical Review A

Analytic calculation of linear susceptibility in velocity-dependent pump-probe spectroscopy

Geol Moon, Heung‐Ryoul Noh

Published Sep 4, 2008• 21 citations
Publisher

Abstract

We present an analytic calculation for linear susceptibility in pump-probe spectroscopy for a Doppler broadened atomic vapor. The analytic form of the population of each magnetic sublevel of a $^{87}\mathrm{Rb}$ atom, illuminated by a weak circularly polarized pump beam, was calculated for the lowest order of laser intensity and used in the calculation for the susceptibility of a counterpropagating (copropagating) probe beam. The obtained analytic results for the susceptibility can be used for calculating the absorption or the refractive index of the probe beam as functions of pump beam diameter and intensity.

DOI
10.1103/physreva.78.032506

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Physical Review A•Dec 27, 2010•14 citations
We present an analytic calculation for linear susceptibility in pump-probe spectroscopy for a Doppler broadened atomic vapor. The analytic form of the population of each magnetic sublevel of a 87Rb^{87}\mathrm{Rb}87Rb atom, illuminated by a weak circularly polarized pump beam, was calculated for the lowest order of laser intensity and used in the calculation for the susceptibility of a counterpropagating (copropagating) probe beam. The obtained analytic results for the susceptibility can be used for calculating the absorption or the refractive index of the probe beam as functions of pump beam diameter and intensity.
We present an analytical study of susceptibility in pump–probe laser spectroscopy for an atomic medium composed of two-level atoms in the Doppler-broadened limit. We derive an accurate analytical formula for susceptibility up to the first (arbitrary) order at the Rabi frequency of the probe (pump) beam. This is expressed in a succinct single term rather than as a complicated summation, as presented in previous papers. We also derive analytical solutions for the linewidth and peak value of the absorption spectrum of the probe beam.
Sub-Doppler resonances observed in miniature cells (V≪1cm3V \ll 1 cm^3V≪1cm3) with Rb or Cs atoms in the field of counterpropagating bichromatic laser beams are of interest for the creation of a miniature quantum frequency standard (QFS) of the optical range. However, a theory of these resonances that takes into full account the real structure of energy levels in an atom has not yet been constructed, which would make it possible to obtain explicit analytical expressions for the absorption coefficient of the medium. This paper addresses this gap using the density matrix formalism and the algebra of irreducible tensor operators. Namely, we obtained an analytical expression for the absorption coefficient of the probe beam in the presence of a more intense counterpropagating beam (pump-probe regime). Moreover, the calculations were carried out for arbitrary parameters of laser beam polarization, arbitrary angular momenta of optical dipole transitions in the atom and various relaxation processes. The constructed theory allows us to correctly take into account the simultaneous action of various nonlinear optical effects, in contrast to the simplified three-level model of the atom, and also to investigate the partial contribution of each of them to the overall observed resonant circuit. The analysis performed, in particular, made it possible to predict an increase in the amplitude of the sub-Doppler resonance when using counterpropagating light beams with opposite circular polarizations in relation to the configuration of counterpropagating beams with orthogonal linear polarizations, which is often used in practice.
We discuss the effect of Doppler broadening on the gain and refractive index experienced by an optical probe when passing through a collection of atoms in the absence of a population inversion. We calculate the gain-absorption spectrum of the medium when the atoms have a Maxwell velocity distribution and compare the atomic response when the pump and probe beams are arranged in copropagating and counterpropagating configurations. The results in these two cases can be qualitatively very different from each other, as we show with the help of numerical calculations. A physical interpretation and the conditions for the validity of the results are provided.
We present analytic solutions for saturated absorption spectra. The analytic forms of all the ground- and excited-state populations were obtained, and the absorption of a counterpropagating probe beam was calculated by solving the rate equations in the presence of a pump laser beam. The analytic solutions were compared with the numerical, experimental results, and Nakayama's model and good agreement was found between them. We found that the analytic theory could provide accurate spectra at an arbitrary pump beam intensity and diameter.
We present a detailed study of the effect of a strong and circularly polarized coupling beam on a probe beam locked to a hyperfine transition of the rubidium D2{D}_{2}D2​ manifold. Experiments were carried out in the co- and counterpropagating configurations. Counterpropagating beams are used to study the effect of the probe beam intensity on absorption and polarization spectra. These data are compared with results of a rate-equation calculation. The absorption spectra show very clear indication of saturation, with different saturation intensities for each atomic line. The slope of the dispersionlike profile of the polarization signal increases with the intensity of the probe beam. These results are in very good agreement with the calculation. Experimental absorption and polarization spectra in the copropagating configuration are also well reproduced by the rate-equation calculation. The dispersion profiles in the polarization signal can be used to lock the coupling laser frequency at well defined frequencies.
This paper presents a simple analytical theory for the velocity-dependent pump-probe laser spectroscopy of 87Rb^{87}\mathrm{Rb}87Rb and 85Rb^{85}\mathrm{Rb}85Rb atoms where the pump and the probe beams are circularly or linearly polarized. The analytical solutions of the line shapes of the velocity-selective optical pumping spectroscopy [G. Moon and H. R. Noh, Phys. Rev. A 78, 032506 (2008)] and saturated absorption spectroscopy [G. Moon and H. R. Noh, J. Opt. Soc. Am. B 25, 701 (2008); 27, 1741 (2010)] obtained in the previous reports, expressed as a sum of several Lorentzian functions, could be approximated as one (or in some cases, two) Lorentzian function(s). In particular, the contributions of the saturation and optical pumping effects could be discriminated explicitly in these simple analytical solutions, which is not possible in existing theories such as Nakayama's model. The simple analytical results for the saturation spectroscopy were compared with experimental results, and good agreement between them was observed.