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Optics Communications

Non-perturbative degenerate four-wave mixing in atomic thermal vapors: A spatial harmonic expansion approach

Alisson D. M. de Lima, Thierry Passerat de Silans, Jesus P. Lopez

Published Sep 14, 2026• 0 citations
Publisher

Abstract

We investigate Degenerate Four-Wave Mixing (DFWM) in a thermal cesium vapor using a nearly collinear, co-propagating geometry under deeply non-perturbative conditions (I≈150Is). While standard theoretical treatments that coherently average over the atomic velocity distribution predict a pronounced spectral dip at resonance, our experimental measurements consistently exhibit a single, smooth, Doppler-broadened profile. To resolve this discrepancy, we develop a non-perturbative theoretical model based on a spatial harmonic expansion of the optical Bloch equations, replacing the standard coherent integration with an incoherent summation over velocity classes. By combining this incoherent velocity averaging with a Beer-Lambert attenuation correction, our model accurately reproduces both the experimentally observed smooth spectral line shape and the sublinear power dependence characteristic of highly saturated multi-photon processes. These results suggest that non-homogeneous spatial dephasing processes in dense, highly saturated thermal media play an important role in the dynamics of the interaction.

DOI
10.1016/j.optcom.2026.133756

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We investigate Degenerate Four-Wave Mixing (DFWM) in a thermal cesium vapor using a nearly collinear, co-propagating geometry under deeply non-perturbative conditions (I≈150Is). While standard theoretical treatments that coherently average over the atomic velocity distribution predict a pronounced spectral dip at resonance, our experimental measurements consistently exhibit a single, smooth, Doppler-broadened profile. To resolve this discrepancy, we develop a non-perturbative theoretical model based on a spatial harmonic expansion of the optical Bloch equations, replacing the standard coherent integration with an incoherent summation over velocity classes. By combining this incoherent velocity averaging with a Beer-Lambert attenuation correction, our model accurately reproduces both the experimentally observed smooth spectral line shape and the sublinear power dependence characteristic of highly saturated multi-photon processes. These results suggest that non-homogeneous spatial dephasing processes in dense, highly saturated thermal media play an important role in the dynamics of the interaction.
We report on the observation of enhanced four-wave mixing via crossover resonance in a Doppler-broadened cesium vapor. Using a single laser frequency, a resonant parametric process in a double-Λ level configuration is directly excited for a specific velocity class. We investigate this process in different saturation regimes and demonstrate the possibility of generating intensity correlation and anticorrelation between the probe and conjugate beams. A simple theoretical model is developed that accounts qualitatively well to the observed results.
We develop a model to calculate nonlinear polarization in a nondegenerate four-wave mixing in diamond configuration which includes the effects of hyperfine structure and Doppler broadening. We verify the model against the experiment with 52S1/2,52P3/2,52D3/25{\phantom{\rule{0.16em}{0ex}}}^{2}{S}_{1/2},5{\phantom{\rule{0.16em}{0ex}}}^{2}{P}_{3/2},5{\phantom{\rule{0.16em}{0ex}}}^{2}{D}_{3/2}52S1/2​,52P3/2​,52D3/2​, and 52P1/25{\phantom{\rule{0.16em}{0ex}}}^{2}{P}_{1/2}52P1/2​ levels of rubidium 85. Treating the multilevel atomic system as a combination of many four-level systems we are able to express the nonlinear susceptibility of a thermal ensemble in a low-intensity regime in terms of Voigt-type profiles and obtain an excellent conformity of theory and experiment within this complex system. The agreement is also satisfactory at high intensity and the analytical model correctly predicts the positions and shapes of resonances. Our results elucidate the physics of coherent interaction of light with atoms involving higher excited levels in vapors at room temperature, which is used in an increasing range of applications.
Saturation effects in gas-phase degenerate four-wave mixing (DFWM) are investigated for conditions of interest for diagnostic applications in flames and plasmas. In particular, the case in which Doppler and collisional broadening are comparable, as is often the case for flame species such as NO and OH, is investigated. DFWM line shapes and signal intensities are calculated nonperturbatively and compared with high-resolution laser measurements. In the nonperturbative calculations the time-dependent density-matrix equations for a two-level system interacting with three laser fields are integrated directly on a grid of spatial locations along the phase-matching axis. The electric-field amplitude for the DFWM signal is determined by multiplying the time-varying, laser-induced polarization at each grid point by the appropriate phase factor and then by summing over all grid points. The calculations are in excellent agreement with measurements of DFWM line shapes and signal intensities for NO in a buffer gas of He over a wide range of He pressure. Under saturation conditions the pressure dependence of the DFWM signal is reduced greatly compared with the unsaturated case. The signal level is still dependent on the ratio of pure dephasing to quenching collisions, even for saturation conditions. From the standpoint of minimal dependence on collisional processes it appears that operation with pump-laser intensities approximately equal to the saturation intensity is optimal. The dependence of the DFWM line shape and signal intensity on the ratio of probe to pump-laser intensity is also investigated.
We report on the observation of enhanced four-wave mixing via crossover resonance in a Doppler broadened cesium vapor. Using a single laser frequency, a resonant parametric process in a double-ΛΛΛ level configuration is directly excited for a specific velocity class. We investigate this process in different saturation regimes and demonstrate the possibility of generating intensity correlation and anti-correlation between the probe and conjugate beams. A simple theoretical model is developed that accounts qualitatively well to the observed results.
In this work, we present a set of experiments involving four-wave mixing processes in rubidium atoms. We employ several experimental configurations exploring two- and three-level systems in cascade or Λ configurations, all of which are in a copropagating geometry, to investigate different physical phenomena present in each scenario. In the first experiment, we combine a cw diode laser and a 1 GHz femtosecond pulse train to generate coherent blue light through the four-wave mixing process in hot rubidium vapor. The generated coherent signal is analyzed by scanning either the diode frequency or the repetition rate of the frequency comb. In both cases, the cw laser induces an Autler- Townes splitting with a wide separation in the doublet structure due to the configuration of copropagating fields. To model these results, we employ the density matrix formalism to write the Bloch equations and solve them numerically. The solutions allow us to explore other cases beyond those involving our experimental setup, comparing the response of the upper-level population and the generated signal coherence for homogeneously and non-homogeneously broadened systems. Moreover, in this same experiment, we also observed an unexpected interference effect between excitation routes of the four-wave mixing process when the repetition rate of the pulse train is scanned. The theoretical model demonstrates that the interfering effect appears only when the repetition rate is carefully chosen to produce a pair of modes of the frequency comb resonant with both one- and two-photon transitions for the same atomic velocity group. In the second experiment, still with a hot rubidium vapor, we explore two symmetrical degenerate four- wave mixing signals induced by two almost copropagating laser beams, with~ka and~kb wave-vectors, and detected simultaneously in the 2~ka −~kb and 2~kb −~ka directions. Each spectrum has a single peak with a short frequency separation between them, which our model indicates is due to the nonlinear behavior of the refractive indexes, a consequence of the high intensity of both input fields. Finally, we use the same experimental setup for the two symmetrical signals to study the field intensity fluctuations of the degenerate four-wave mixing signals and the transmission signals. In this case, the sample is an ensemble of cold rubidium atoms in a magneto-optical trap. There is an intensity- intensity cross-correlation between the signals due to the resonant phase-noise to amplitude-noise conversion. In this type of cold system, we can neglect the Doppler- broadening, allowing a straightforward study of the dependency of the correlations on the laser detuning. In these measurements, we observe a type of oscillation in the correlation curves that the theoretical model indicates to be connected with Rabi oscillations, revealed by the fluctuations long after the transient period.
We study phase-conjugate degenerate four-wave mixing in an optically pumped, two-level atomic system (the 3s2S1/2,{3s}^{2}{S}_{1/2},3s2S1/2​, F=2,F=2,F=2, ${m}_{\mathrm{F}}=\stackrel{\ensuremath{\rightarrow}}{2}{3p}^{2}{P}_{3/2},$ F=3,F=3,F=3, mF=3{m}_{\mathrm{F}}=3mF​=3 transition in a diffuse, collision-free, thermal beam of sodium). In this investigation, we study the detailed line shape and strength of the four-wave mixing signal as a function of the intensity of the cw pump waves, and of the velocity of the atoms. Our technique allows us to examine the four-wave mixing interaction under conditions in which the Doppler shift, the natural linewidth of the transition, and the Rabi frequency for the interaction are each comparable with one another. From our experimental measurements and computational results, we show that the line shapes are very complex in this regime, and that they depend sensitively upon the atomic velocity, especially in the case when the intensity of the forward and backward pump beams exceeds the saturation intensity for the transition.