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Physical Review A

Frequency shift in three-photon resonance

Richard M. Friedberg, Sven R. Hartmann, Jamal T. Manassah

Published Jan 1, 1989• 25 citations
Publisher

Abstract

We calculate the line shift in the three-photon resonance-absorption experiment performed by Ferrel, Payne, and Garrett [Phys. Rev. A 36, 81 (1988)] and find it large, to the violet, and close to experiment. In comparison, the calculated single-photon resonance shift in absorption is a factor of 30 times smaller and to the red. The three-photon resonance shift depends on the relative propagation directions of the excitation beams, and we present a formula for the case where only two excitation beams are present.

DOI
10.1103/physreva.39.93

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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}${\mathit{k}}_{\mathit{r}}$L\ensuremath{\Vert}\ensuremath{\gg}1, where \ensuremath{\Delta}${\mathit{k}}_{\mathit{r}}$ 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}${\mathit{k}}_{\mathit{r}}$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.

Physical Review A•Dec 1, 1991•19 citations
We calculate the line shift in the three-photon resonance-absorption experiment performed by Ferrel, Payne, and Garrett [Phys. Rev. A 36, 81 (1988)] and find it large, to the violet, and close to experiment. In comparison, the calculated single-photon resonance shift in absorption is a factor of 30 times smaller and to the red. The three-photon resonance shift depends on the relative propagation directions of the excitation beams, and we present a formula for the case where only two excitation beams are present.
The authors calculate the three-photon resonance absorption shift obtained with two non-collinear excitation beams of arbitrary orientation incident on an ensemble of collisionally broadened two-level atoms excited close to one-third of the resonance frequency. For counterpropagating beams the absorption resonance is narrow and shifted by the order of the linewidth to the violet. As the beams are angled away from the counterpropagating condition the shift increases, and if the angle approaches copropagation the shift becomes very large. When the shift exceeds the single-photon absorption linewidth by more than L/ lambda , the shifted line weakens and disappears, but a separate unshifted peak persists at copropagation. The third-harmonic generation response is more complicated. Except in the region where the beams are almost copropagating the shift in third-harmonic generation response follows the shift in the absorption response. In contrast to the absorption response the third-harmonic generation response increases as the excitation beams approach copropagation, then peaks, then diminishes, broadens and forms a complicated structure whose character is determined by phase matching.
The physics of the two-photon absorption process is investigated for a three-state system. The density-matrix equations for the two-photon interaction are solved in the steady-state limit assuming that the pump laser radiation is monochromatic. Collisional broadening, saturation, and Stark shifting of the two-photon resonance are investigated in detail by numerical solution of the steady-state density-matrix equations. Analytical expressions for the saturation intensity and the Stark shift are derived for the case where the single-photon transitions between the intermediate state and the initial and final states are far from resonance with the pump laser. For this case, it is found that the direction of the Stark shift is dependent on the relative magnitudes of the dipole-moment matrix elements for the single-photon transitions that couple the intermediate state with the initial and final states. Saturation and Stark shifting are also investigated for the case where the single-photon transitions between the intermediate state and the initial and final states are close to resonance with the pump laser.
In a recent paper [Phys. Rev. A 82, 052519 (2010)], the systematic shifts of a resonance due to quantum-mechanical interference from a distant neighboring resonance were derived. In that paper, the simplest three-level closed system was used to calculate analytic expressions for shifts in the resonant line centers. Here, we extend the analysis to the more relevant four-level system, which consists of two ground states and two excited states and which incorporates the physics of dark states. The shifts are shown to depend on the type of experiment performed and can be much larger than the shifts for the three-level system. The analytic formulas obtained are applied to the 23S2{}^{\phantom{\rule{0.16em}{0ex}}\phantom{\rule{0.16em}{0ex}}3}S23S-to-23P2{}^{\phantom{\rule{0.16em}{0ex}}\phantom{\rule{0.16em}{0ex}}3}P23P transitions in atomic helium, where significant shifts are found.
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 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.