Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Journal of Physics B Atomic and Molecular Physics

Recoil effects on Doppler-free lineshapes in non-linear spectroscopy

J.H. Shirley

Published Apr 28, 1980• 8 citations
Publisher

Abstract

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.

DOI
10.1088/0022-3700/13/8/010

Related papers

New quantum representation for treating recoil effects in non-linear spectroscopy

J H Shirely, Stig Stenholm

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.

Journal of Physics A Mathematical and General•Apr 1, 1977•12 citations

Lineshapes in Three-Level Spectroscopy with an Intense Laser Field

Motowo Tsukakoshi, Kōichi Shimoda

Effects of an intense saturating radiation on a probe lineshape under Doppler broadening are studied in a three-level system when the frequency of the probe light is higher than that of the saturating light. The problem is analysed in the approximation of weak probe field using an ensemble averaged density matrix. An analytical form of the lineshape is obtained for the resonant saturating field of high intensity at which the Rabi nutation frequency is larger than the Doppler width of the probe transition. The linewidth of probe absorption depends on the Doppler width and the frequency ratio of the probe light to the saturating light. The computer calculation is carried out to find the lineshape in detail at a light intensity where the Rabi nutation frequency is comparable to the Doppler width.

Japanese Journal of Applied Physics•Apr 1, 1980•6 citations

Theory of lineshapes in optical–optical double resonance spectroscopy

Kevin K. Lehmann

This paper presents a theory of lineshapes for molecular Optical-Optical Double Resonance (DR) Spectroscopy with arbitrary strength for both pump and probe fields using the steady-state solutions for the 3-level density matrix. When Doppler broadening can be neglected, the results are analytical, and the probe spectrum is a pair of Lorentzian lines that display Autler-Townes splitting, each with an angular frequency half-width half maximum equal to the relaxation rates, which are assumed to be equal. When Doppler broadening is introduced, one must resort to numerical integration except for the limit of weak pump and probe fields. When the Doppler width is assumed to be much larger than the pump and probe Rabi frequencies and the probe frequency is higher than that of the pump, the calculated DR lineshapes are found to be Lorentzian with a strong pump field limit width that is proportional to the pump Rabi frequency, which is commonly known as power broadening. However, the width does not equal that Rabi frequency and is different for co- and counter-propagating pump and probe fields. Furthermore, that broadening is largely inhomogeneous, despite the Lorentzian shape. The saturation power is found to be about 4 times higher than for the bare probe transition with the same relaxation rate, dramatically lower than that expected if the width is interpreted as homogeneous.

arXiv:2604.02262•The Journal of Chemical Physics•Aug 14, 2026•0 citations

Effect of optical pumping in saturated absorption spectroscopy: an analytic study for two-level atoms

Heung‐Ryoul Noh

We present the analytic solutions of the saturated absorption spectrum for an atomic medium composed of two-level atoms and describe the effects of saturation and optical pumping on the spectrum. The background-subtracted Doppler-free spectrum is Lorentzian-shaped with an amplitude and width given by 1 − {1 + s 0 [1 + (γ 2 t /2)]} −1/2 and γ t [1 + {1 + s 0 [1 + (γ 2 t /2)]} 1/2 ], respectively, where γ t is the transverse decay rate, γ 2 is the decay rate to energy levels other than the ground state, s 0 is the saturation parameter and t is the interaction time of the atoms with laser beams. Analytic results are compared with numerical results and show good agreement with one another. This paper is intended for graduate students who have started to study basic atomic physics.

European Journal of Physics•Aug 14, 2009•13 citations

Theory of Optical-Microwave Double Resonance I. Fundamentals and Double Resonance of Microwave Detection

Michio Takami

The effect of optical pumping on the lineshapes and the peak values of microwave absorption line is calculated by a density matrix method in the Doppler limit approximation. The calculated double resonance signal consists of two terms. One is proportional to the unperturbed population difference of the optical transition and has a Lorentzian lineshape at weak microwave saturation. In this term the two-photon processes cause broadening of the line. The other is proportional to the population difference of the microwave transition and gives the higher-order dynamic Stark effect. At moderate optical saturation the former term is much greater than the latter. At strong optical saturation the latter term becomes noticeable and lineshape anomaly occurs. The peak values of the double resonance signals are also calculated and compared with an experimental result.

Japanese Journal of Applied Physics•Jun 1, 1976•41 citations

Two-photon spectroscopy. II. Effects of residual Doppler broadening

Rainer Salomaa, Stig Stenholm

For pt.I see ibid., vol.8, p.1795 (1975) where the theory of Doppler-free two-photon transitions was discussed for the case of one single laser frequency. Here, the effects of two different frequencies are considered. When the residual Doppler width of the two-photon transition is less than the natural linewidth, the previous results are essentially regained. For large residual Doppler shifts, the absorption profile is a power-broadened Lorentzian if the frequency of the probe field is larger than the frequency of the strong field. In the opposite case, the absorption peak splits because of the AC Stark effect. The high-intensity distortions are minimized when equal frequencies can be used. Power-dependent shifts do not ordinarily occur with a resonantly tuned intermediate level. Some modifications of the spectra are introduced by phase-disturbing collisions.

Journal of Physics B Atomic and Molecular Physics•Jun 1, 1976•41 citations
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 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.
Effects of an intense saturating radiation on a probe lineshape under Doppler broadening are studied in a three-level system when the frequency of the probe light is higher than that of the saturating light. The problem is analysed in the approximation of weak probe field using an ensemble averaged density matrix. An analytical form of the lineshape is obtained for the resonant saturating field of high intensity at which the Rabi nutation frequency is larger than the Doppler width of the probe transition. The linewidth of probe absorption depends on the Doppler width and the frequency ratio of the probe light to the saturating light. The computer calculation is carried out to find the lineshape in detail at a light intensity where the Rabi nutation frequency is comparable to the Doppler width.
This paper presents a theory of lineshapes for molecular Optical-Optical Double Resonance (DR) Spectroscopy with arbitrary strength for both pump and probe fields using the steady-state solutions for the 3-level density matrix. When Doppler broadening can be neglected, the results are analytical, and the probe spectrum is a pair of Lorentzian lines that display Autler-Townes splitting, each with an angular frequency half-width half maximum equal to the relaxation rates, which are assumed to be equal. When Doppler broadening is introduced, one must resort to numerical integration except for the limit of weak pump and probe fields. When the Doppler width is assumed to be much larger than the pump and probe Rabi frequencies and the probe frequency is higher than that of the pump, the calculated DR lineshapes are found to be Lorentzian with a strong pump field limit width that is proportional to the pump Rabi frequency, which is commonly known as power broadening. However, the width does not equal that Rabi frequency and is different for co- and counter-propagating pump and probe fields. Furthermore, that broadening is largely inhomogeneous, despite the Lorentzian shape. The saturation power is found to be about 4 times higher than for the bare probe transition with the same relaxation rate, dramatically lower than that expected if the width is interpreted as homogeneous.
We present the analytic solutions of the saturated absorption spectrum for an atomic medium composed of two-level atoms and describe the effects of saturation and optical pumping on the spectrum. The background-subtracted Doppler-free spectrum is Lorentzian-shaped with an amplitude and width given by 1 − {1 + s 0 [1 + (γ 2 t /2)]} −1/2 and γ t [1 + {1 + s 0 [1 + (γ 2 t /2)]} 1/2 ], respectively, where γ t is the transverse decay rate, γ 2 is the decay rate to energy levels other than the ground state, s 0 is the saturation parameter and t is the interaction time of the atoms with laser beams. Analytic results are compared with numerical results and show good agreement with one another. This paper is intended for graduate students who have started to study basic atomic physics.
The effect of optical pumping on the lineshapes and the peak values of microwave absorption line is calculated by a density matrix method in the Doppler limit approximation. The calculated double resonance signal consists of two terms. One is proportional to the unperturbed population difference of the optical transition and has a Lorentzian lineshape at weak microwave saturation. In this term the two-photon processes cause broadening of the line. The other is proportional to the population difference of the microwave transition and gives the higher-order dynamic Stark effect. At moderate optical saturation the former term is much greater than the latter. At strong optical saturation the latter term becomes noticeable and lineshape anomaly occurs. The peak values of the double resonance signals are also calculated and compared with an experimental result.
For pt.I see ibid., vol.8, p.1795 (1975) where the theory of Doppler-free two-photon transitions was discussed for the case of one single laser frequency. Here, the effects of two different frequencies are considered. When the residual Doppler width of the two-photon transition is less than the natural linewidth, the previous results are essentially regained. For large residual Doppler shifts, the absorption profile is a power-broadened Lorentzian if the frequency of the probe field is larger than the frequency of the strong field. In the opposite case, the absorption peak splits because of the AC Stark effect. The high-intensity distortions are minimized when equal frequencies can be used. Power-dependent shifts do not ordinarily occur with a resonantly tuned intermediate level. Some modifications of the spectra are introduced by phase-disturbing collisions.