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Journal of Physics B Atomic and Molecular Physics

Recoil effects in quantum electronics. II. Saturated absorber spectroscopy

C.G. Aminoff, Stig Stenholm

Published May 11, 1976• 13 citations
Publisher

Abstract

For pt.I see ibid., vol.7, p.1235 (1974). Uses the results of an earlier work to consider recoil effects on spectroscopic features observed with a saturable absorber. Systematic approximation schemes are presented, both the straightforward perturbation expansion in the field amplitudes and a non-perturbative approach, which neglects pulsations in the populations. This method generalizes the REA of recoil-free strong-signal theory and is expected to give good results for velocity-averaged observables. Its validity is discussed and the consequences are compared with those of other calculations. The results of numerical evaluations of spectra are shown, in particular the limit of poorly resolved recoil features is explored. Parameter regions where recoil-induced splittings and asymmetries occur are presented.

DOI
10.1088/0022-3700/9/7/005

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Journal of Physics A Mathematical and General•Apr 1, 1977•12 citations
For pt.I see ibid., vol.7, p.1235 (1974). Uses the results of an earlier work to consider recoil effects on spectroscopic features observed with a saturable absorber. Systematic approximation schemes are presented, both the straightforward perturbation expansion in the field amplitudes and a non-perturbative approach, which neglects pulsations in the populations. This method generalizes the REA of recoil-free strong-signal theory and is expected to give good results for velocity-averaged observables. Its validity is discussed and the consequences are compared with those of other calculations. The results of numerical evaluations of spectra are shown, in particular the limit of poorly resolved recoil features is explored. Parameter regions where recoil-induced splittings and asymmetries occur are presented.
Derives the equations of motion for the density matrix of a two-level molecule interacting with two oppositely propagating semiclassical waves. The equations are shown to correctly reproduce earlier results by Kol'chenko et al. (1969) for one travelling wave, and their perturbation treatment for a standing wave. The saturated absorber experiment is treated in the case when one travelling wave is weak enough to allow a perturbation expansion in its amplitude. The velocity dependent absorption function is investigated in detail, and the effect of recoil on the processes: Raman scattering, Rayleigh scattering and dynamic Stark splitting is obtained and discussed. These have been treated without recoil by Haroche and Hartmann (1972) and a detailed comparison with their work is included. Finally there is a brief discussion of the importance of recoil for different situations in quantum electronics.
Previously derived equations describing recoil effects in saturated absorption by a gas of two-level atoms or molecules are solved numerically for the case of standing plane-wave excitation and large Doppler broadening. The numerical techniques and accuracy estimates are discussed briefly. The lineshapes obtained are well approximated by the superposition of two Lorentzian functions whose width, heights, and separation parametrise the lineshape. The dependence of these parameters on saturation, recoil splitting, and relaxation rates is described and compared with corresponding results in the rate equation approximation and in the weak-probe-beam approximation. Saturation broadening depends strongly on these latter approximations, but very little on recoil splitting. When multiphoton processes are retained in the equations, increasing saturation causes a slight decrease in the fitted separation of the peaks. The ratio of the heights of the two Lorentzians is closely related to the ratio of relaxation rates for the level populations.
The atomic-recoil effect leads to large (25%) asymmetries in a nearly ideal form of saturation spectroscopy based on Ca atoms that have been laser-cooled to . Starting with spectra from the more familiar Doppler-broadened domain, we show how the fundamental asymmetry between absorption and stimulated emission of light manifests itself when shorter spectroscopic pulses lead to the Fourier transform regime. These effects occur on frequency scales much larger than the size of the recoil shift itself, and have not been described before in saturation spectroscopy. These results directly impact precision spectroscopic measurements.
When a pump field and probe field simultaneously drive an electronic-state atomic transition, the probe field absorption spectrum can consist of an absorption peak centered near \ensuremath{\Delta}'=0 and amplification peak centered near \ensuremath{\Delta}'=2\ensuremath{\Delta} (\ensuremath{\Delta} and \ensuremath{\Delta}' are the pump and probe field detunings from the atomic transition frequency, respectively). This type of spectrum is seen in the limit \ensuremath{\Vert}\ensuremath{\Delta}\ensuremath{\Vert}\ensuremath{\gg}\ensuremath{\chi}\ensuremath{\gg}\ensuremath{\Gamma}, where \ensuremath{\chi} is the Rabi frequency associated with the pump field and 2\ensuremath{\Gamma} is the homogeneous width associated with the atomic transition. For atoms cooled below the recoil limit of laser cooling, the qualitative nature of the probe absorption spectrum can undergo a dramatic change. Provided that the recoil splitting ${\mathrm{\ensuremath{\omega}}}_{\mathit{k}}$ is larger than the homogeneous decay rate (as might occur in the case of a forbidden transition), the absorption and amplification features each split into an absorption-amplification doublet. In addition, structure is found in the probe absorption spectrum near \ensuremath{\Delta}'=\ensuremath{\Delta}; this structure consists of two absorption-amplification doublets. Both doublets can be resolved if ${\mathrm{\ensuremath{\omega}}}_{\mathit{k}}$>\ensuremath{\Gamma}. If ${\mathrm{\ensuremath{\omega}}}_{\mathit{k}}$, one of the doublets can be resolved provided that ${\mathrm{\ensuremath{\omega}}}_{\mathit{k}}$>${\mathrm{\ensuremath{\Gamma}}}_{\mathit{A}}$, where ${\mathrm{\ensuremath{\Gamma}}}_{\mathit{A}}$ is some effective atomic ground state width in the problem. The positions, widths, and relative weights of all the components are readily predicted using a dressed-atom theory in which quantization of the center-of-mass momentum is included. An analytical expression for the probe field spectrum is obtained for a simple case in which spontaneous decay to the lower level is neglected. Validity criteria for the results are discussed.
The problem of saturated absorption by a gas of two-level molecules, including recoil effects, is formulated in a new quantum representation for the molecular motion. The relationship to other representations is clarified in a simple chart. The new formulation requires the solution of simultaneous linear differential equations with mixed boundary conditions, instead of linear difference equations in two variables followed by an integration over velocity. Analytic solutions are presented for one running wave with Doppler line broadening, and for a weak probe beam in the Doppler limit. The latter lineshape is analysed to show when power broadening obscures the recoil splitting.