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Physical Review A

Transit-time effects in optically pumped coupled three-level systems

John E. Thomas, W. W. Quivers

Published Nov 1, 1980• 50 citations
Publisher

Abstract

A density-matrix calculation of the line shape for an optically pumped coupled three-level system subjected to two traveling wave-laser fields with Gaussian intensity profiles is presented to third-order in perturbation theory. A closed-form expression for the line shape in the extreme transit-time limit is obtained. The line shape of the narrow resonance (Raman-type processes) is found to be an exponential function of the detuning, with half-width at half maximum ${\ensuremath{\Delta}}_{\mathrm{HWHM}}^{(\mathrm{Hz})}\ensuremath{\simeq}0.98(\frac{u}{2\ensuremath{\pi}D})$ ($D$ is the intensity $\frac{1}{e}$ diameter and $u$ is the thermal speed). The saturation intensity is also modified by the transit rate ${\ensuremath{\gamma}}_{t}=\frac{\sqrt{2}u}{D}$ (rad/sec).

DOI
10.1103/physreva.22.2115

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A density-matrix calculation of the line shape for an optically pumped coupled three-level system subjected to two traveling wave-laser fields with Gaussian intensity profiles is presented to third-order in perturbation theory. A closed-form expression for the line shape in the extreme transit-time limit is obtained. The line shape of the narrow resonance (Raman-type processes) is found to be an exponential function of the detuning, with half-width at half maximum \ensuremathΔHWHM(Hz)\ensuremath≃0.98(u2\ensuremathπD){\ensuremath{\Delta}}_{\mathrm{HWHM}}^{(\mathrm{Hz})}\ensuremath{\simeq}0.98(\frac{u}{2\ensuremath{\pi}D})\ensuremathΔHWHM(Hz)​\ensuremath≃0.98(2\ensuremathπDu​) (DDD is the intensity 1e\frac{1}{e}e1​ diameter and uuu is the thermal speed). The saturation intensity is also modified by the transit rate \ensuremathγt=2uD{\ensuremath{\gamma}}_{t}=\frac{\sqrt{2}u}{D}\ensuremathγt​=D2​u​ (rad/sec).
A density-matrix calculation of the line shape for a Doppler-broadened two-level system subjected to a plane-standing-wave laser field with a Gaussian intensity profile is presented to fifth order in perturbation theory. The primary new result is that, for fixed pressure and intensity, the power-broadening contribution to the linewidth is at a minimum when the beam radius is approximately equal to the mean free path. In addition, an analytic solution for the linewidth in the collision-dominated regime is presented for general collision parameters.
This paper describes a new class of techniques, called time-delayed laser saturation spectroscopy, which combine frequency- and time-domain methods of laser spectroscopy to provide a way of studying a molecular system as it evolves from an initially prepared stationary state to a second, final state. The specific example analyzed here is three-level free induction decay, in which the time-dependent gain of a Doppler-broadened molecular transition is probed after the sudden termination of an intense field resonating with a coupled transition. The theoretical calculation is based on the coupled density-matrix equations of motion in the slowly-varying envelope approximation. The time-delayed line shapes, which may be studied in either transmission or side fluorescence, exhibit linewidth asymmetries, line-shape deformations, Ramsey-type fringes, power broadening and dephasing, and dynamic Stark splittings and oscillatory decays. The technique provides a unique way of distinguishing the influence of Raman-type processes from that of population saturation and a means to separately measure the associated decay rates. The relationship of the present work to other studies is also discussed.
Previous theoretical results on the influence of a laser on the line shape of a coupled transition-laser-induced line narrowing-have been restricted to the case where the laser is detuned from the center of its atomic gain profile or is in the form of a traveling wave. This paper extends those results to include the case where the laser is an intense standing-wave field tunable to the center of its atomic gain profile (conditions for Lamb dip). The effect may be observed either in transmission, by probing the coupled transition with a weak traveling-wave field coaxial with the laser field, or else in spontaneous emission from the coupled transition viewed along the axis of the laser field. It is now well known that for a laser detuned from the center of its atomic gain profile, two narrow Lorentzian resonances of different widths appear superimposed upon the broad background signal at frequencies symmetrically located about the corresponding line center. When the laser is tuned to the center of its gain profile, however, additional fine structure develops. This structure, which is particularly significant when the laser field is intense, may have important applications in highresolution spectroscopy and laser-frequency stabilization. In this paper the laser frequency may be smaller or larger than the frequency of the coupled transition. In the latter case an intense laser introduces additional splitting effects, even when the laser is detuned. Splitting effects due to weakly saturating laser fields are also discussed. The problem is formulated by expanding elements of the ensemble-averaged density matrix in an infinite series of spatial Fourier components. A perturbation technique is employed, valid for a weak probe field and a standing-wave field of arbitrary intensity. One obtains an expression for emission induced by the probe field due to atoms moving with one velocity, written in terms of continued fractions in the general case and with Bessel functions in an important special case. This expression is integrated over the atomic velocity distribution by means of a computer to obtain the total emission due to atoms moving with all velocities. In some cases the integrated expressions may be written in closed algebraic form. A detailed discussions of line shapes and of the physical processes involved is included.
The density matrix equations have been solved in conjunction with the time-dependent field amplitude equations for a three level optically pumped gas. The solutions are valid in the limit in which the pumping pulse is longer than the dephasing time of the system and true coherence effects may be neglected. In order to obtain closed form solutions, it is assumed that the FIR field is smaller than the pump field and saturation effects associated with this field are absent. When the pump pulse is resonant with the absorbing FIR transition, particularly simple solutions for the time-dependent FIR field amplification are obtained. These solutions exhibit the specific dependence of the FIR output pulse on the shape of the input pulse and display certain pulse narrowing features.
A theory of saturation spectroscopy is developed which accounts for effects of level degeneracy. The theory can be applied to those situations in which a pump laser field of arbitrary strength and polarization acts between two manifolds of degenerate or nearly degenerate atomic levels, while a probe laser field of arbitrary strength and polarization acts between a coupled manifold of levels. As such, this theory of "three-level" systems could, in principle, include effects arising from level degeneracy, optical pumping, atomic motion, transient laser fields, different polarizations and strengths for pump and probe fields, and atomic transit across the laser fields. The present calculation, however, is limited to a study of optical pumping and field polarization effects for pump fields of arbitrary strength and weak probe fields interacting with a single velocity subclass of atoms. Probe absorption line shapes clearly display the effects of pump-field strength (Rabi splittings), pump-field polarization (positions and number of Rabi-split resonance peaks), pump-field detuning (positions of resonance peaks), and optical pumping (relative strengths of the resonance peaks). The calculation is carried out with the use of an irreducible tensor basis for the atomic density matrix although a standard (mmm-basis) calculation is included in an appendix.