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Journal of Experimental and Theoretical Physics Letters

Effects of two strong fields in resonant four-wave mixing

Sergei A Babin, Evgeniy V. Podivilov, David A. Shapiro

Published Dec 1, 1997• 4 citations
Publisher

Abstract

An explicit solution is obtained for the four-wave mixing ω 4 =ω 1 −ω 2 +ω 3 of two strong fields E 1 , E 3 and two weak fields E 2 , E 4 in a four-level system with large Doppler broadening. Resonance of the intensity dependence of the mixing coefficient is found around equal Rabi frequencies, E 1 · d 1 = E 3 · d 3 , where d 1,3 are the dipole moments of the corresponding transitions. The effect is interpreted as a crossing of quasi-energy levels. Up to 6 peaks appear in the dependence of the conversion coefficient on the detuning of the probe field E 2 . The unexpected additional pair of peaks is a consequence of averaging over velocities. The results permit interpretation of the saturation behavior found in recent experiments on mixing in sodium vapor.

DOI
10.1134/1.567603

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An explicit solution is obtained for the four-wave mixing ω 4 =ω 1 −ω 2 +ω 3 of two strong fields E 1 , E 3 and two weak fields E 2 , E 4 in a four-level system with large Doppler broadening. Resonance of the intensity dependence of the mixing coefficient is found around equal Rabi frequencies, E 1 · d 1 = E 3 · d 3 , where d 1,3 are the dipole moments of the corresponding transitions. The effect is interpreted as a crossing of quasi-energy levels. Up to 6 peaks appear in the dependence of the conversion coefficient on the detuning of the probe field E 2 . The unexpected additional pair of peaks is a consequence of averaging over velocities. The results permit interpretation of the saturation behavior found in recent experiments on mixing in sodium vapor.
The explicit solution is obtained for Raman four-wave mixing ω4=ω1−ω2+ω3\omega_4=\omega_1-\omega_2+\omega_3ω4​=ω1​−ω2​+ω3​ of two strong fields E1,E3E_1, E_3E1​,E3​ and two weak fields E2,E4E_2, E_4E2​,E4​ in a four-level system with the large Doppler broadening. The resonance of the mixing coefficient vs intensity is found near equal Rabi frequencies of strong fields and interpreted as an intersection of quasi-energy levels. Up to 6 peaks appear in the dependence of conversion coefficient on the detuning of the probe field. An unexpected additional pair of peaks is a result of averaging over velocities and disappears at low temperature. The effects allow explaining saturation behavior in recent experiments on the mixing in sodium molecular vapor.
The explicit solution is obtained for four-wave mixing of two strong driving fields at opposite transitions and two weak fields in a four-level system with the large Doppler width. In zero order of the perturbation theory there are two independent two-level systems. A pair of weak fields probe two other allowed transitions. The resonance of the mixing coefficient dependence on intensity is found around equal Rabi splitting in both two level systems. The effect is interpreted as an intersection of quasi-energy levels. Up to 6 peaks appear in the dependence of conversion coefficient on the detuning of the probe field, two of them are a consequence of averaging over velocities and disappear at low temperature. The results allow us to explain the recent experiments on the mixing in sodium vapor.
An explicit solution is obtained for the four-wave frequency mixing ${\ensuremath{\omega}}_{d}={\ensuremath{\omega}}_{a}\ensuremath{-}{\ensuremath{\omega}}_{b}+{\ensuremath{\omega}}_{c}$ of two strong fields a and c and two weak fields b and d in a four-level system with large Doppler broadening in collinear geometry, where the frequencies of weak fields are nearly equal, ${\ensuremath{\omega}}_{b}\ensuremath{\simeq}{\ensuremath{\omega}}_{d},$ and the medium is optically thin. Without weak fields there are two independent two-level systems. A pair of weak fields probes two other allowed transitions. A peak of the mixing coefficient as a function of intensity is found around an equal Rabi splitting of both two-level systems. The effect is based on a resonance between two closed cycles of four-wave mixing via different dressed states. Three, four, or six peaks are predicted in the dependence of the mixing coefficient on the frequency of the weak field; two of them are a consequence of averaging over velocities. The model allows an interpretation of the dependence of the output wave power on the intensity and detuning in recent experiments on frequency mixing in sodium vapor.
Experiments on resonant four-wave mixing demonstrated high- efficiency CW down-conversion of frequency in gas (up to 25%). Meanwhile, the high intensities make the perturbation theory invalid to understand the behavior of output power measured as a function of input powers. The nonlinear susceptibility of optically thin medium is found for difference-sum scheme (omega) 4 equals (omega) 1 - (omega) 2 + (omega) 3, where fields one and three are strong and two and four are weak. In symmetric case (kappa) 2 equals (kappa) 4, the integral over velocity can be calculated analytically for Doppler limits in collinear geometry. The compact explicit formula obtained yields the intensity and frequency dependence. The peak of mixing coefficient as a function of intensity is found around equal Rabi frequencies of fields one and three. The effect is shown to base on resonance between two closed cycles via sublevels of dressed states. The number of peaks in a spectrum varies from three to eight. Explicit formula allows interpreting the measurements and predicting the optimal relation between parameters for high conversion efficiency in gas with large Doppler broadening.
The influence of nonlinear interference processes and of changes in populations in strong electromagnetic fields, and also of the Doppler broadening on resonant four-wave frequency mixing is investigated for a system of Raman-type transitions. The case in which a quantum system interacts with one weak field and two strong fields is considered. Each of the energy levels is assumed to be perturbed by just one strong field. It is shown that the Doppler broadening alters drastically the contributions of the populations of the various levels to the investigated process and that this can alter by several orders of magnitude and nonlinear susceptibilities averaged over the velocities and influence significantly the selection of optimal configurations of the transitions and of the experimental conditions for effective resonant four-wave conversion of the radiation frequencies. A numerical analysis of the derived expressions is used to account for the experimentally observed considerable differences between the spectral dependences of the output power and the conclusions drawn from perturbation theory.
As a rule the thermal motion removes the dark resonance from the spectrum of a three-level system driven by two strong fields. An example has been found where, conversely, the dark resonance manifests itself even after averaging over the velocities. The four-level system is studied with a closed cycle of four resonant fields, when two relatively strong fields interact with opposite transitions. If Rabi frequencies exceed relaxation constants, then the frequency dependence of the nonlinear susceptibility consists of several narrow peaks. Each peak corresponds to a ‘turning point’, the resonant frequency with zero derivative with respect to the velocity. It is shown that two peaks disappear when non-absorbing states coincide with the turning points. The coincidence occurs for equal Rabi frequencies of relatively weak fields.