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Physical Review A

Frequency shift in three-photon resonant four-wave mixing by internal atom-field interaction

Natalia R. de Melo, S. S. Vianna

Published Nov 11, 2015• 4 citations
Publisher

Abstract

We report on experimental results of four-wave mixing processes in rubidium vapor where coherence is induced on the three-photon resonant transition from $5s$ to $6p$ states via intermediate Rydberg levels. It is shown that the use of two beams in a noncollinear configuration, i.e., $\ensuremath{\theta}\ensuremath{\ne}0$, and high atomic density unveil new features. First, the $\ensuremath{\theta}=0$ (collinear configuration) odd-photon destructive interference between the incident and generated fields is strongly inhibited for $\ensuremath{\theta}\ensuremath{\ne}0$. Second, most importantly, the observed cooperative frequency shift of the three-photon transition is strongly enhanced for small, but nonzero, values of $\ensuremath{\theta}$ due to the factor ${(1\ensuremath{-}cos\ensuremath{\theta})}^{\ensuremath{-}1}$, which is not present if the generated radiation field is not considered self-consistently in the Maxwell-Bloch equations.

DOI
10.1103/physreva.92.053830

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We report on observations of four-wave mixing in rubidium vapor via two-photon resonant excitation with nd and (n+2)s states, for n=14–20. The intensity of the generated beam is studied in a collinear configuration, and the results are consistent with the predictions of a theoretical model based on the semiclassical Maxwell–Bloch equations, considering interference effects associated with multiple excitation pathways and with internally generated fields. We have also observed spectral broadening of the two-photon resonant lines, which is well described by a linear dependence on atomic density, in accordance with collision-induced broadening.
We present a theory for two- and three-photon excitation, optical shifting, and four-wave mixing when a first laser is tuned onto, or near, a two-photon resonance and a second much more intense laser is tuned near or on resonance between the two-photon resonance and a second excited state. When the second excited state has a dipole-allowed transition back to the ground state and the concentration is sufficiently high, a destructive interference is produced between three-photon coupling of the ground state and the second excited state and one-photon coupling between the same states by the internally generated four-wave mixing field. This interference leads to several striking effects. For instance, as the onset of the interference occurs, the optical shifts in the two-photon resonance excitation line shape become smaller in copropagating geometry so that the line shapes for multiphoton ionization enhanced by the two-photon resonance eventually become unaffected by the second laser. In the same range of concentrations the four-wave mixing field evolves to a concentration-independent intensity. With counterpropagating laser beams the line shape exhibits normal optical shifts like those observed for both copropagating and counterpropagating laser beams at very low concentrations. The theoretical work presented here extends our earlier works by including the effect of laser bandwidth and by removing the restriction of having the second laser be tuned far from three-photon resonance. In this way we have now included, as a special case, the effect of both laser bandwidth and interference on laser-induced transparency. Unlike other effects related to odd-photon destructive interference, the effect of a broad bandwidth is to bring about the predicted effects at much lower concentrations. Studies in rubidium show good agreement between theory and experiment for both ionization line shapes and four-wave mixing intensity as a function of concentration.
We describe several novel experimental effects resulting from optical coherences in multilevel atoms driven with coherent laser fields. Four configurations were explored using Rb vapor cells: lambda dark-line resonances, single beam double-lambda oscillations, and cascade two-photon transitions in co- and counter-propagating geometries. Experiments were performed with high spectral-resolution using low-power diode lasers sources and optically thick cells. Two of the most striking effects we observed are: (1) a self-oscillation that occurs at 3.0 GHz on the 85Rb hyperfine frequency when a single laser field pumps the atoms, and (2) a coherent blue-beam emission that we observe in the co-propagating cascade two-photon case. These phenomena lead to complex lineshapes, and the effects sometimes dominate the well-known 3-level coherence effects. Additional higher-order mixing or cascaded interferences are apparent in all four experimental configurations. Many of these can be viewed as multi-wave mixing enhanced by optical coherences around closed-loop paths of atomic energy levels. The effects are particularly strong when four-photon closed-loop paths with resonant energy levels are possible, as in, the double-lambda system or as a four-wave mixing “box.”
We consider effects which occur when four-wave sum frequency generation and multiphoton ionization are induced by lasers tuned near a three-photon resonance and simultaneously near or at a dipole allowed four-photon resonance. In studies with unfocused laser beams, if the phase mismatch of the generated four-wave-mixing field is large and the related two-photon resonance for the absorption of a four-wave-mixing photon and a laser photon results in strong absorption of the four-wave-mixing field, a coherent cancellation occurs between the pumping of the resonance by two- and four-photon processes. This interference effect occurs when the first laser is tuned on either side of the three-photon resonance and \ensuremath{\Vert}\ensuremath{\Delta}kr{\mathit{k}}_{\mathit{r}}kr​L\ensuremath{\Vert}\ensuremath{\gg}1, where \ensuremath{\Delta}kr{\mathit{k}}_{\mathit{r}}kr​ is the mismatch and L is the length of the path of the laser beams in the gas. With focused laser beams large differences occur between ionization with unidirectional beams and with counterpropagating laser beams when \ensuremath{\Vert}\ensuremath{\Delta}kr{\mathit{k}}_{\mathit{r}}kr​b\ensuremath{\Vert}\ensuremath{\gg}1, where b is the confocal parameter of the focused laser beams. Strong absorption of the four-wave-mixing field is shown not to be necessary for strong destructive interference with focused laser beams when the phase mismatch is large. This work also suggests an explanation for earlier experiments where the presence of a four-photon resonance enabled the generation of third-harmonic light in a positively dispersive wavelength region. We argue that this process can occur when the laser used to achieve the four-photon resonance is focused on the small z (z is the coordinate in the direction of propagation) side of the focal point of the laser responsible for the third-harmonic generation. In this way the absorption of the third-harmonic light on one side of the focal point is made much stronger than on the other side. Correspondingly, contributions to the third-harmonic field from the two sides of the focal point are unequal and cannot undergo complete destructive interference as would otherwise occur in a positively dispersive region.
We have observed an interference in four-wave mixing involving two Rydberg states that play the key role of near two-photon resonant intermediate levels. This interference, due to the coherence of the nonlinear process, can be explained either classically or quantum mechanically and it depends on the lasers' polarizations and detunings. We have performed calculations that are in very good agreement with the experimental data. Our results provide new information about the system, such as ionization rates of the Rydberg levels and the ratio of reduced matrix elements involved.