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Physical Review A

Four-wave mixing in a Λ system

P. R. Berman, Xiaodong Xu

Published Nov 12, 2008• 9 citations

Abstract

The four-wave mixing (4WM) signal is calculated for a three-level quantum system in a $\ensuremath{\Lambda}$ configuration (upper level 2, lower levels 1 and 3) that is driven by three fields. Pump fields $E$ and ${E}^{\ensuremath{'}}$ drive the 1-2 and 3-2 transitions, respectively. The probe field is detuned from field $E$ by an amount $\ensuremath{\Delta}$. If the detuning of field $E$ from the 1-2 transition frequency is equal to that of field ${E}^{\ensuremath{'}}$ from the 3-2 transition frequency (dark state pumping), there is a dark state in the absence of the probe field. Rather surprisingly, even though the probe field destroys the dark state for $\ensuremath{\Delta}\ensuremath{\ne}0$, the 4WM mixing vanishes for all $\ensuremath{\Delta}$ in the limit of dark-state pumping. The 4WM spectra are explained within the context of a semiclassical, dressed-state picture that involves the sequential emission of a probe and signal photon.

DOI: 10.1103/physreva.78.053407

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The four-wave mixing (4WM) signal is calculated for a three-level quantum system in a

\ensuremath{\Lambda}

configuration (upper level 2, lower levels 1 and 3) that is driven by three fields. Pump fields

E

and

{E}^{\ensuremath{'}}

drive the 1-2 and 3-2 transitions, respectively. The probe field is detuned from field

E

by an amount

\ensuremath{\Delta}

. If the detuning of field

E

from the 1-2 transition frequency is equal to that of field

{E}^{\ensuremath{'}}

from the 3-2 transition frequency (dark state pumping), there is a dark state in the absence of the probe field. Rather surprisingly, even though the probe field destroys the dark state for

\ensuremath{\Delta}\ensuremath{\ne}0

, the 4WM mixing vanishes for all

\ensuremath{\Delta}

in the limit of dark-state pumping. The 4WM spectra are explained within the context of a semiclassical, dressed-state picture that involves the sequential emission of a probe and signal photon.

We study laser-induced transparency and dark-line effects in a two-level atomic system where the upper transition level is coupled by a laser field to a dressed system consisting of two lower-energy metastable states mixed by a second laser field (\ensuremath{\Pi} configuration). We show that the emission (absorption) spectrum of the two-level system develops two dynamic dark lines (transparency holes) with tunable central frequencies equal to the Rabi frequency of the second field with positive and negative signs, and widths proportional to that of the first field. The detailed evolution of these effects at low- and high-field intensities and the role of quantum interferences are discussed by presenting a model based on laser-induced continuum structure and adopting a dressed-state picture. We discuss various nonlinear processes in this system and show how the dynamics of emission and absorption spectra are determined by one- and two-photon coupling processes.

We theoretically investigate the four-wave mixing (FWM) characteristics in a Ξ−V type atomic system. The generated FWM is obtained by utilizing the density matrix formalism and solving the master equation. Interesting discrimination in the number of emerged peaks due to the chosen laser scanning schemes was pointed out. The dressed state picture is employed to elucidate the number of peaks emerged as well as the peak positions in the spectra. The dependence of the generated FWM signal accordingly to the change of parameters, such as Rabi frequencies and frequency detunings of the applied fields, are presented and well explained in the simplified dressed state picture. A prominent increase in FWM intensity is obtained when the Rabi frequencies of the coupling and driving field are approximately equal. The frequency detuning of the applied fields causes the shift in the peak positions, peak shape, and even brings about the FWM intensity enhancement.

Four-wave mixing (FWM) in a cycling transition Fg =1 ↔ Fe =0 is investigated. Subject to a linearly polarized pump field with frequency ω c and a magnetic field, and simultaneously saturated and probed by the circularly polarized fields (σ ±) with frequencies ω s and ω p , three kinds of FWM fields with frequencies: 2ω s − ω p, ω c+ω s − ω p and ω c − ω s+ω p, emit. The multi-peaked structure of FWM spectra is obtained and explained as the resonant emission between the ‘virtual states’ in the dressed picture. The generated FMW signals have high quantum efficiency for a wide range of drive Rabi frequency and saturation Rabi frequency. Finally, the emitted fields ω c + ω s − ω p and ω c − ω s+ω p have about the same intensity for the well matching among the Zeeman splitting, drive Rabi frequency and saturation Rabi frequency.