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

Physical Review A

Resonant multiwave-mixing spectra of gas-phase sodium: Nonperturbative calculations

Robert P. Lucht, Rick P. Trebino, Larry A. Rahn

Published Jun 1, 1992• 23 citations
Publisher

Abstract

We perform nonperturbative calculations of resonant multiwave mixing of two high-intensity laser beams of different frequencies both for a general three-level system and for the 3 $^{2}$${\mathit{S}}_{1/2}$--3 $^{2}$${\mathit{P}}_{1/2}$ electronic resonance of the sodium atom. Our calculations proceed by direct numerical integration of the density-matrix equations and subsequent Fourier analysis of the calculated time-dependent density-matrix elements. We examine the case where two nearly degenerate input beams are each detuned many collisional linewidths from electronic resonance, and the frequency difference between the input beams is tuned through a ground-state resonance. Unlike previous high-intensity analyses, each input beam can be resonant with numerous transitions simultaneously; for a three-level system, we refer to this as an ``M-type'' system, in analogy with \ensuremath{\Lambda}- and V-type systems. Calculated four-, six-, and eight-wave-mixing spectra exhibit subharmonic resonances of the type observed experimentally by Trebino and Rahn [Opt. Lett. 12, 912 (1987)]. The three-level system is studied to gain insight into the physics of these subharmonic resonances. For more quantitative comparison with experiment, we use a model that includes the 16 Zeeman and hyperfine states in the 3 $^{2}$${\mathit{S}}_{1/2}$ and the 3 $^{2}$${\mathit{P}}_{1/2}$ levels, and accounts for arbitrary linear polarization of the input laser radiation. The relative intensities of the resonance features in the calculated multiwave-mixing spectra show good agreement with experiment. These resonance features broaden and shift as laser intensity increases, as predicted for four-wave-mixing processes in previous, less general high-intensity analyses. Both theory and experiment also show additional resonant features in the eight-wave-mixing spectra that appear at very high intensity.

DOI
10.1103/physreva.45.8209

Related papers

Nonperturbative Calculation Of Nearly Degenerate Multi-wave-mixing Spectra

Robert P. Lucht, Rick P. Trebino, Larry A. Rahn

Trebino and Rahn observed subharmonic resonances in high-order wave-mixing spectra of the sodium atom in flames. They used two pulse-amplified, single-mode dye lasers to generate two nearly degenerate pump beams at 590 nm. The pump beams were tuned 2 cm-1away from the sodium D1line resonance. At low laser intensity resonances at the hyperfine splitting ωhfs=0.06 cm-1and at zero frequency difference were clearly visible. At higher intensities the hyperfine resonances broadened and a subharmonic resonance at 1/2 ωhfsappeared. Trebino and Rahn also used higher order wave mixing geometries to observe subharmonic resonances at 1/3, 1/4, and 1/5 ωhfs. Trebino and Rahn have used diagrammatic perturbation theory to calculate high-order wave-mixing spectra.

Digest on Nonlinear Optics: Materials, Phenomena and Devices•Jan 1, 1990•0 citations

Thirteen pump-probe resonances of the sodium D1 line

Vincent Wong, Robert W. Boyd, Carlos R. Stroud +2 more

We present the results of a pump-probe laser spectroscopic investigation of the Doppler-broadened sodium $D1$ resonance line. We find 13 resonances in the resulting spectra. These observations are well described by the numerical predictions of a four-level atomic model of the hyperfine structure of the sodium $D1$ line. We also find that many, but not all, of these features can be understood in terms of processes originating in a two-level or three-level subset of the full four-level model. The processes we observed include forward near-degenerate four-wave mixing and saturation in a two-level system, difference-frequency crossing and nondegenerate four-wave mixing in a three-level V system, electromagnetically induced transparency and optical pumping in a three-level lambda system, cross-transition resonance in a four-level double-lambda system, and conventional optical pumping. Most of these processes lead to sub-Doppler or even subnatural linewidths. The dependence of these resonances on the pump intensity and pump detuning from atomic resonance are also studied.

Physical Review A•Jul 8, 2003•33 citations

Calculation of higher-order wave-mixing spectra

Rick P. Trebino, Larry A. Rahn

Using diagrammatic perturbation theory, we calculate the higher-order susceptibility for the n-wave process, omega(0) = (n/2 - 1)(omega(1) - omega(2)) + omega(1), in a phase-matched n-wave-mixing geometry. We include the 16 Zeeman and hyperfine levels of the sodium ground (3S((1/2))) and excited (3P((1/2))) states, finding resonances at subharmonics [+/-(1/2), +/-?, ..., +/-1(n/2 - 1)] of the ground-level transition frequencies. The computed spectrum for eight-wave mixing is in satisfactory agreement with experiment. In addition, a theoretical twelve-wave-mixing spectrum predicts a new higher-order selection rule.

Optics Letters•Apr 1, 1990•13 citations

Resonant degenerate four-wave mixing spectroscopy of transitions with degenerate energy levels: Saturation and polarization effects

Thomas A. Reichardt, Robert P. Lucht

The physics of the degenerate four-wave mixing process for resonant transitions between two degenerate energy levels is investigated by direct numerical integration of the time-dependent density matrix equations. The Zeeman structure of the upper and lower energy levels is included in a multistate formulation of the density matrix equations. The inclusion of the Zeeman structure enables the investigation of the degenerate four-wave mixing process for different polarization configurations of the forward pump, backward pump, and probe beams. Saturation curves and lineshapes are calculated for different polarization configurations and for numerous low-J transitions. At low laser intensity, the results of our calculations are in excellent agreement with perturbation theory in terms of the relative intensities of the degenerate four-wave mixing signal for linear polarization configurations. As the laser intensity increases and the resonance starts to saturate, we find in general that the relative degenerate four-wave mixing reflectivity increases for the crossed polarization configurations compared to the parallel polarization configuration because the saturation intensity is higher. However, for some resonance transitions, some of the crossed polarization configurations saturate at lower laser intensities than the parallel polarization configuration, even though the reflectivity for these crossed polarization configurations is much lower than for the parallel polarization configuration in the perturbative intensity limit. This result is explained in terms of the coupling of the various Zeeman states during the degenerate four-wave mixing interaction for specific polarization configurations. The effect of saturation on the resonance line shapes for the different polarization configurations is also investigated. Finally, a limited number of calculations are performed for resonances that are Doppler broadened as well as collision broadened. The effect of saturation on the reflectivity of the crossed polarization configurations compared to the parallel polarization configuration is even more significant for resonances with comparable Doppler and collisional broadening.

The Journal of Chemical Physics•Dec 8, 1999•22 citations

On the non-linear spectroscopy including saturated absorption and four-wave mixing in two and multi-level atoms: a computational study

M Patel, Gerhard De Jager, Z Nkosi +2 more

In this paper we report on the study of two and multi-level atoms interacting with multiple laser beams. The semi-classical approach is used to describe the system in which the atoms are treated quantum mechanically via the density matrix operator, while the laser beams are treated classically using Maxwells equations. We present results of a two level atom interacting with single and multiple laser beams and demonstrate Rabi oscillations between the levels. The effects of laser modulation on the dynamics of the atom (atomic populations and coherences) are examined by solving the optical Bloch equations. Plots of the density matrix elements as a function of time are presented for various parameters such as laser intensity, detuning, modulation etc. In addition, phase-space plots and Fourier analysis of the density matrix elements are provided. The atomic polarization, estimated from the coherence terms of the density matrix elements, is used in the numerical solution of Maxwells equations to determine the behaviour of the laser beams as they propagate through the atomic ensemble. The effects of saturation and hole-burning are demonstrated in the case of two counter propagating beams with one being a strong beam and the other being very weak. The above work is extended to include four-wave mixing in four level atoms in a diamond configuration. Two co-propagating beams of different wavelengths drive the atoms from a ground state |1〉 to an excited state |3〉 via an intermediate state |2〉. The atoms then move back to the ground state via another intermediate state |4〉, resulting in the generation of two additional correlated photon beams. The characteristics of these additional photons are studied.

Journal of Physics Conference Series•Oct 1, 2017•0 citations

Quantitative analysis of the deflection of a sodium beam by laser radiation

Bernardo Jaduszliwer, GF Shen, Benjamin B. Bederson

We have performed experimental and theoretical studies of the deflection of a sodium atomic beam by a laser tuned to the 3 ${\mathrm{}}^{2}$${S}_{1/2}$,F=2--3 ${\mathrm{}}^{2}$${P}_{3/2}$,F=3 transition. The intensity profile of the deflected beam was measured at different laser power levels and for different laser polarizations. Calculations using a constant force derived from a two-level atom yield deflections which are too large. The results of more realistic simulation calculations agree very well with the measured deflection. Those calculations incorporate the atomic-beam spatial and velocity distributions as well as the laser frequency distribution and polarization state. Optical pumping, loss of atoms to the nonresonant F=1 ground state, and Doppler shifting of the atoms out of resonance are all considered, as also are the statistics and angular distributions of the spontaneous photons.

Physical Review A•Jun 1, 1986•7 citations
We perform nonperturbative calculations of resonant multiwave mixing of two high-intensity laser beams of different frequencies both for a general three-level system and for the 3 ^{2}$${\mathit{S}}_{1/2}--3 ^{2}$${\mathit{P}}_{1/2} electronic resonance of the sodium atom. Our calculations proceed by direct numerical integration of the density-matrix equations and subsequent Fourier analysis of the calculated time-dependent density-matrix elements. We examine the case where two nearly degenerate input beams are each detuned many collisional linewidths from electronic resonance, and the frequency difference between the input beams is tuned through a ground-state resonance. Unlike previous high-intensity analyses, each input beam can be resonant with numerous transitions simultaneously; for a three-level system, we refer to this as an ``M-type'' system, in analogy with \ensuremath{\Lambda}- and V-type systems. Calculated four-, six-, and eight-wave-mixing spectra exhibit subharmonic resonances of the type observed experimentally by Trebino and Rahn [Opt. Lett. 12, 912 (1987)]. The three-level system is studied to gain insight into the physics of these subharmonic resonances. For more quantitative comparison with experiment, we use a model that includes the 16 Zeeman and hyperfine states in the 3 ^{2}$${\mathit{S}}_{1/2} and the 3 ^{2}$${\mathit{P}}_{1/2} levels, and accounts for arbitrary linear polarization of the input laser radiation. The relative intensities of the resonance features in the calculated multiwave-mixing spectra show good agreement with experiment. These resonance features broaden and shift as laser intensity increases, as predicted for four-wave-mixing processes in previous, less general high-intensity analyses. Both theory and experiment also show additional resonant features in the eight-wave-mixing spectra that appear at very high intensity.
Trebino and Rahn observed subharmonic resonances in high-order wave-mixing spectra of the sodium atom in flames. They used two pulse-amplified, single-mode dye lasers to generate two nearly degenerate pump beams at 590 nm. The pump beams were tuned 2 cm-1away from the sodium D1line resonance. At low laser intensity resonances at the hyperfine splitting ωhfs=0.06 cm-1and at zero frequency difference were clearly visible. At higher intensities the hyperfine resonances broadened and a subharmonic resonance at 1/2 ωhfsappeared. Trebino and Rahn also used higher order wave mixing geometries to observe subharmonic resonances at 1/3, 1/4, and 1/5 ωhfs. Trebino and Rahn have used diagrammatic perturbation theory to calculate high-order wave-mixing spectra.
We present the results of a pump-probe laser spectroscopic investigation of the Doppler-broadened sodium D1D1D1 resonance line. We find 13 resonances in the resulting spectra. These observations are well described by the numerical predictions of a four-level atomic model of the hyperfine structure of the sodium D1D1D1 line. We also find that many, but not all, of these features can be understood in terms of processes originating in a two-level or three-level subset of the full four-level model. The processes we observed include forward near-degenerate four-wave mixing and saturation in a two-level system, difference-frequency crossing and nondegenerate four-wave mixing in a three-level V system, electromagnetically induced transparency and optical pumping in a three-level lambda system, cross-transition resonance in a four-level double-lambda system, and conventional optical pumping. Most of these processes lead to sub-Doppler or even subnatural linewidths. The dependence of these resonances on the pump intensity and pump detuning from atomic resonance are also studied.
Using diagrammatic perturbation theory, we calculate the higher-order susceptibility for the n-wave process, omega(0) = (n/2 - 1)(omega(1) - omega(2)) + omega(1), in a phase-matched n-wave-mixing geometry. We include the 16 Zeeman and hyperfine levels of the sodium ground (3S((1/2))) and excited (3P((1/2))) states, finding resonances at subharmonics [+/-(1/2), +/-?, ..., +/-1(n/2 - 1)] of the ground-level transition frequencies. The computed spectrum for eight-wave mixing is in satisfactory agreement with experiment. In addition, a theoretical twelve-wave-mixing spectrum predicts a new higher-order selection rule.
The physics of the degenerate four-wave mixing process for resonant transitions between two degenerate energy levels is investigated by direct numerical integration of the time-dependent density matrix equations. The Zeeman structure of the upper and lower energy levels is included in a multistate formulation of the density matrix equations. The inclusion of the Zeeman structure enables the investigation of the degenerate four-wave mixing process for different polarization configurations of the forward pump, backward pump, and probe beams. Saturation curves and lineshapes are calculated for different polarization configurations and for numerous low-J transitions. At low laser intensity, the results of our calculations are in excellent agreement with perturbation theory in terms of the relative intensities of the degenerate four-wave mixing signal for linear polarization configurations. As the laser intensity increases and the resonance starts to saturate, we find in general that the relative degenerate four-wave mixing reflectivity increases for the crossed polarization configurations compared to the parallel polarization configuration because the saturation intensity is higher. However, for some resonance transitions, some of the crossed polarization configurations saturate at lower laser intensities than the parallel polarization configuration, even though the reflectivity for these crossed polarization configurations is much lower than for the parallel polarization configuration in the perturbative intensity limit. This result is explained in terms of the coupling of the various Zeeman states during the degenerate four-wave mixing interaction for specific polarization configurations. The effect of saturation on the resonance line shapes for the different polarization configurations is also investigated. Finally, a limited number of calculations are performed for resonances that are Doppler broadened as well as collision broadened. The effect of saturation on the reflectivity of the crossed polarization configurations compared to the parallel polarization configuration is even more significant for resonances with comparable Doppler and collisional broadening.
In this paper we report on the study of two and multi-level atoms interacting with multiple laser beams. The semi-classical approach is used to describe the system in which the atoms are treated quantum mechanically via the density matrix operator, while the laser beams are treated classically using Maxwells equations. We present results of a two level atom interacting with single and multiple laser beams and demonstrate Rabi oscillations between the levels. The effects of laser modulation on the dynamics of the atom (atomic populations and coherences) are examined by solving the optical Bloch equations. Plots of the density matrix elements as a function of time are presented for various parameters such as laser intensity, detuning, modulation etc. In addition, phase-space plots and Fourier analysis of the density matrix elements are provided. The atomic polarization, estimated from the coherence terms of the density matrix elements, is used in the numerical solution of Maxwells equations to determine the behaviour of the laser beams as they propagate through the atomic ensemble. The effects of saturation and hole-burning are demonstrated in the case of two counter propagating beams with one being a strong beam and the other being very weak. The above work is extended to include four-wave mixing in four level atoms in a diamond configuration. Two co-propagating beams of different wavelengths drive the atoms from a ground state |1〉 to an excited state |3〉 via an intermediate state |2〉. The atoms then move back to the ground state via another intermediate state |4〉, resulting in the generation of two additional correlated photon beams. The characteristics of these additional photons are studied.
We have performed experimental and theoretical studies of the deflection of a sodium atomic beam by a laser tuned to the 3 2{\mathrm{}}^{2}2S1/2{S}_{1/2}S1/2​,F=2--3 2{\mathrm{}}^{2}2P3/2{P}_{3/2}P3/2​,F=3 transition. The intensity profile of the deflected beam was measured at different laser power levels and for different laser polarizations. Calculations using a constant force derived from a two-level atom yield deflections which are too large. The results of more realistic simulation calculations agree very well with the measured deflection. Those calculations incorporate the atomic-beam spatial and velocity distributions as well as the laser frequency distribution and polarization state. Optical pumping, loss of atoms to the nonresonant F=1 ground state, and Doppler shifting of the atoms out of resonance are all considered, as also are the statistics and angular distributions of the spontaneous photons.