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Quantum Electronics

Resonant four-wave frequency mixing in Doppler-broadened transitions

A. K. Popov, S. A. Myslivets

Published Nov 30, 1997• 6 citations
Publisher

Abstract

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.

DOI
10.1070/qe1997v027n11abeh001101

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arXiv:physics/9707003•Jul 4, 1997•1 citation
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.
An investigation is reported of resonant four-wave frequency mixing of the type ω 4 =ω 1 —ω 2 +ω 3 under coherent population trapping conditions. Such mixing may appear when radiations of frequencies ω 1,2 are strong and when ω 1 —ω 2 =ω 20 (ω 20 is the frequency of a dipole-forbidden transition). The density matrix method is used to calculate (taking quantum interference into account) the absorption coefficients, the refractive indices, and the nonlinear susceptibilities exactly for the strong fields and in the first order of perturbation theory for the weak fields. A study is made of the dependence of the power of nonlinear optical generation on various characteristics of the transitions and of the radiation, subject to the propagation effects. The cases of homogeneous and Doppler broadening of the resonant transitions are considered. The characteristics of transitions in the barium atom are used in numerical illustrations. It is shown that the efficiency of resonant nonlinear-optical conversion of radiation can be enhanced by utilising interference processes in quantum transitions, which represent coherent population trapping.
An influence of nonlinear interference processes at quantum transitions in strong resonance electromagnetic fields on absorption, amplification and refractive indices as well as on four-wave mixing processes is investigated. Doppler broadening of the coupled transitions, incoherent excitation, relaxation processes, as well as power saturation processes associated with the coupled levels are taken into account. Both closed (ground state is involved) and open (only excited states are involved) energy level configurations are considered. Common expressions are obtained which allow to analyze the optical characteristics (including gain without inversion and enhanced refractive index at vanishing absorption) for various V, (Lambda) and H configurations of interfering transitions by a simple substitution of parameters. Similar expressions for resonant four-wave mixing (FWM) in Raman configurations are derived too. Crucial role of Doppler broadening is shown. Theory is applied for numerical analysis of some recent and potential experiments.
We study the effect of nonlinear interference processes on quantum transitions in intense resonant electromagnetic fields with allowance for changes in level populations, relaxation processes, incoherent excitation, and Doppler broadening of the transitions, on the absorption, amplification, and refraction of the interacting fields. The theory is generalized to the case of nonlinear interference interaction of two intense fields in open and closed three-level quantum systems. Using the density matrix, we derive general expressions in the case of stationary interaction that make it possible to analyze the optical characteristics for various configurations of the interfering transitions by a simple substitution of parameters. The possibility of amplifying light without saturated population inversion in a resonant transition is discussed. We formulate the conditions for such amplification and use examples to show that under appropriate changes in the initial level populations and the intensity of the auxiliary light, the inversionless amplification coefficient does not decrease with increasing intensity of the amplified radiation. We also show that allowance for these accompanying processes greatly affects the choice of optimal conditions for interference in optical transitions. As an illustration we list the results of a numerical analysis of possible experiments.
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.
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 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.