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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

Effects of strong driving field and spontaneous decay in resonant four-wave mixing

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

Published Oct 21, 1998• 0 citations
Publisher

Abstract

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.

DOI
10.1117/12.328245

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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 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.
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.
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.
The purpose of this research is to study the dynamic behavior of a resonant system using four-wave-mixing spectroscopy (FWM). From this study, we have obtained rich information about the FWM spectral response of systems which have different state-specific reservoir couplings. Atomic sodium (Na) is used as our model system. It is shown that the FWM spectral response depends strongly on how each level couples to the vacuum radiation field (through spontaneous emission). This effect is illustrated in the line shape obtained from a nearly degenerate FWM (NDFWM) experiment for a simple two-level system in Na. A narrow spike on top of a wide resonance is observed at zero pump-probe detuning (δ\deltaδ = 0) when the upper-state decay rate $\gamma\sb2$ to the reservoir is larger than the lower-state decay rate $\gamma\sb1$. The width of the spike is 2$\gamma\sb1$ and the width of the wide resonance is determined by the total upper-level decay rate (2$\gamma\sbsp{2}{\rm t}$). From the NDFWM spectrum, all the decay rates characterizing a two-level system may be measured directly. If magnetic substate degeneracy is included, the NDFWM line shape has contributions arising from orientation and alignment gratings, as well as from population gratings. For a two-level system closed with respect to population, the orientation or alignment gratings may contribute a narrow resonance at δ\deltaδ = 0 in the NDFWM response. These features are demonstrated for the first time by using NDFWM. Reservoir coupling can also occur through collisions with externally added gas. Collisions can cause a closed two-level system to become open, resulting in NDFWM spectra that show a narrow resonance at δ\deltaδ = 0 with a width below the residual Doppler width (Dicke narrowing). This is also the first time that Dicke narrowing has been observed in population terms. Finally, we present a calculation of the NDFWM response for strong degenerate pump fields and a weak probe field. We compare the differences between NDFWM spectra of closed two- and three-level systems in the presence of strong pumps.