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Journal of the Optical Society of America B

Resonances in the collision-induced spectrum of six-wave mixing susceptibility: generation of subharmonics

Aditi Ray, Ravinder Rupchand Puri

Published Oct 1, 2000• 1 citation
Publisher

Abstract

Resonances of the six-wave mixing susceptibility χ(5)(ω1, ω1, ω1,-ω2,-ω2) at frequency 3ω1-2ω2 of a system of two-level atoms dissipating through the spontaneous emission and collisions and driven by a weak bichromatic external field of frequencies ω1 and ω2 in a finite Q cavity are analyzed. It is shown that the atomic collisions not only induce the usual Bloembergen resonances but that they also play a major role in enhancing the transitions among the higher-order dressed states, thereby bringing out some of the quantum resonances that are suppressed in the absence of collisions. The important contribution of collisions in this higher-order nonlinear process is the generation of the subharmonic resonances at ±1/2 times the hyperfine splitting of the first and the second excited manifolds.

DOI
10.1364/josab.17.001704

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Resonances of the six-wave mixing susceptibility χ(5)(ω1, ω1, ω1,-ω2,-ω2) at frequency 3ω1-2ω2 of a system of two-level atoms dissipating through the spontaneous emission and collisions and driven by a weak bichromatic external field of frequencies ω1 and ω2 in a finite Q cavity are analyzed. It is shown that the atomic collisions not only induce the usual Bloembergen resonances but that they also play a major role in enhancing the transitions among the higher-order dressed states, thereby bringing out some of the quantum resonances that are suppressed in the absence of collisions. The important contribution of collisions in this higher-order nonlinear process is the generation of the subharmonic resonances at ±1/2 times the hyperfine splitting of the first and the second excited manifolds.
The four-wave mixing susceptibility χ(3)(ω1, ω1, -ω2) of a system of two-level atoms driven by weak external fields of frequencies ω1 and ω2 in a high-Q cavity is evaluated in the presence of atomic collisions. That susceptibility exhibits resonances corresponding to dipole-allowed transitions between the first two excited manifolds of the dressed states and the ground state and also the collision-induced Bloembergen resonances. The resonances corresponding to transitions between the first two manifolds of excited states that characterize the quantum and nonlinear nature of the atom–field interaction are, however, flattened in the absence of collisions. The collisions thus induce the quantum effects in four-wave mixing in a high-Q cavity.
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.
Using four-, six-, and eight-wave-mixing geometries, we observe subharmonic resonances in collision-enhanced spectra in a sodium-seeded hydrogen-air flame. Appearing at high intensity, these resonances occur at +/-1/2, +/-1/3, +/-1/4, and +/-1/5 times the frequency of the ground-state hyperfine splitting omega(hfs). We argue that these resonances result from higher-order processes, specifically chi((n)), where n >/= 5. In particular, to describe the resonances at +/-omega(hfs)/5, at least chi((13)) is required. These high-order collision-enhanced effects are true perturbative nonlinearities and not sequential four-wave mixing processes.
A theory of four-wave mixing including effects of velocity-changing collisions is presented. Three fields with frequencies \ensuremath{\omega}, \ensuremath{\omega}, and \ensuremath{\omega}+\ensuremath{\delta} are incident on a vapor of ``two-level'' atoms having upper state b and lower state a. Two of the fields are counterpropagating and the third (of frequency \ensuremath{\omega}+\ensuremath{\delta}) makes a small angle with one of the others. The frequency \ensuremath{\omega} is a nearly resonant (inside the Doppler width) with the a-b transition frequency. The phase-conjugate signal emitted at frequency \ensuremath{\omega}-\ensuremath{\delta} is calculated as a function of \ensuremath{\delta}. Using a simple collision model in which collisions are phase interrupting in their effect on atomic coherence and velocity-changing in their effect on level populations, we discuss the conditions under which resonances characterized by the upper or lower radiative and collision rates can be observed. Assuming that the total (a+b) state population is conserved in the absence of collisions, it is shown that velocity-changing collisions can ``open'' the system and lead to a resonance characterized by the lower-state width (convoluted with the residual Doppler width). With increasing pressure, the width of this induced resonance structure decreases monotonically. For sufficiently high pressure, the collisional redistribution of velocity classes is complete---the system ``recloses'' and the narrow resonance disappears. The interplay of the collision-induced opening, line narrowing, and reclosing of the system is discussed, as is the relationship of these narrow resonances to the so-called pressure-induced extra resonances of Bloembergen and co-workers [Indian J. Pure Appl. Phys. 16, 151 (1978); Phys. Rev. Lett. 46, 111 (1981)].
We report an experimental study of resonant six-wave mixing in coherently prepared Rb atoms. Electromagnetically induced transparency in a four-level atomic system suppresses the linear susceptibility and enhances the nonlinear susceptibilities, which leads to the resonantly enhanced, slow-photon six-wave mixing at low light intensities. The light emission in the six-wave mixing process can be viewed as resulting from diffraction of slow light off a resonant nonlinear grating induced in the four-level system by a standing-wave pump field.
The four-wave-mixing signal at $2{\ensuremath{\omega}}_{1}\ensuremath{-}{\ensuremath{\omega}}_{2}$ from a collective system of NNN two-level atoms dissipating through spontaneous emission and dephasing and driven by weak external fields of frequencies ${\ensuremath{\omega}}_{1}$ and ${\ensuremath{\omega}}_{2}$ in a cavity of arbitrary QQQ is found by evaluating exactly the susceptibility ${\ensuremath{\chi}}^{(3)}({\ensuremath{\omega}}_{1},{\ensuremath{\omega}}_{1},\ensuremath{-}{\ensuremath{\omega}}_{2})$. The exact results show that the customarily employed secular approximation for treating the problem of a strongly coupled atom-cavity system is hopelessly inadequate for describing the process of four-wave mixing.