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Physical Review A

Effect of pressure-dependent quantum interference on the ac Stark shifting of two-photon resonances

M. G. Payne, J.Y. Zhang, W. R. Garrett

Published Sep 1, 1993• 17 citations
Publisher

Abstract

We describe ac Stark shift production and measurement in a two-laser mode where a first laser is tuned through a two-photon resonance between the ground state and an excited state in order to record a line shape for the corresponding resonantly enhanced multiphoton ionization. A second laser is tuned to a fixed wavelength which is close to a resonance between the upper state in the two-photon transition and a third state which has dipole-allowed transitions back to the ground state. At low concentrations the second laser leads to very large ac Stark shifts in the two-photon excitation and in the related resonantly enhanced multiphoton ionization. However, at high concentrations a destructive interference occurs involving the three-photon coupling between the third state and the ground state and the one-photon coupling due to the four-wave-mixing field at frequency 2${\mathrm{\ensuremath{\omega}}}_{\mathit{L}1}$\ifmmode\pm\else\textpm\fi{}${\mathrm{\ensuremath{\omega}}}_{\mathit{L}2}$. This interference leads to a pressure-dependent suppression of the amplitude for the third state. When this destructive interference occurs, the ac Stark shift due to the second laser undergoes a corresponding pressure-dependent suppression. The criteria for the pressure to be high enough to completely suppress the ac Stark shift is that the real part of phase mismatch \ensuremath{\Vert}\ensuremath{\Delta}k\ensuremath{\Vert} at the angular frequency 2${\mathrm{\ensuremath{\omega}}}_{\mathit{L}1}$\ifmmode\pm\else\textpm\fi{}${\mathrm{\ensuremath{\omega}}}_{\mathit{L}2}$ and (or) the absorption coefficient \ensuremath{\beta} at the same frequency should be very large, so that with unfocused focused beams \ensuremath{\Vert}\ensuremath{\Delta}k\ensuremath{\Vert}L>5[(${\mathrm{\ensuremath{\Delta}}}_{\mathit{s}}^{(\mathrm{max})}$\ensuremath{\tau}${]}^{1/2}$, where L is the thickness of gas penetrated before the shift is measured, ${\mathrm{\ensuremath{\Delta}}}_{\mathit{s}}^{(\mathrm{max})}$ is the maximum ac Stark shift at very low pressure, and \ensuremath{\tau} is the pulse length of the lasers.At positions and concentrations such that 1\ensuremath{\Vert}\ensuremath{\Delta}k\ensuremath{\Vert}L\ensuremath{\le}2[(${\mathrm{\ensuremath{\Delta}}}_{\mathit{s}}^{(\mathrm{max})}$\ensuremath{\tau})${]}^{1/2}$ the ac Stark shift is present but the two-photon excitation is suppressed due to a destructive interference between excitation by two photons from the first laser and by two-photon excitation between a four-wave-mixing photon and a photon from the second laser.

DOI
10.1103/physreva.48.2334

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We describe ac Stark shift production and measurement in a two-laser mode where a first laser is tuned through a two-photon resonance between the ground state and an excited state in order to record a line shape for the corresponding resonantly enhanced multiphoton ionization. A second laser is tuned to a fixed wavelength which is close to a resonance between the upper state in the two-photon transition and a third state which has dipole-allowed transitions back to the ground state. At low concentrations the second laser leads to very large ac Stark shifts in the two-photon excitation and in the related resonantly enhanced multiphoton ionization. However, at high concentrations a destructive interference occurs involving the three-photon coupling between the third state and the ground state and the one-photon coupling due to the four-wave-mixing field at frequency 2\ensuremathωL1{\mathrm{\ensuremath{\omega}}}_{\mathit{L}1}\ensuremathωL1​\ifmmode\pm\else\textpm\fi{}\ensuremathωL2{\mathrm{\ensuremath{\omega}}}_{\mathit{L}2}\ensuremathωL2​. This interference leads to a pressure-dependent suppression of the amplitude for the third state. When this destructive interference occurs, the ac Stark shift due to the second laser undergoes a corresponding pressure-dependent suppression. The criteria for the pressure to be high enough to completely suppress the ac Stark shift is that the real part of phase mismatch \ensuremath{\Vert}\ensuremath{\Delta}k\ensuremath{\Vert} at the angular frequency 2\ensuremathωL1{\mathrm{\ensuremath{\omega}}}_{\mathit{L}1}\ensuremathωL1​\ifmmode\pm\else\textpm\fi{}\ensuremathωL2{\mathrm{\ensuremath{\omega}}}_{\mathit{L}2}\ensuremathωL2​ and (or) the absorption coefficient \ensuremath{\beta} at the same frequency should be very large, so that with unfocused focused beams \ensuremath{\Vert}\ensuremath{\Delta}k\ensuremath{\Vert}L>5[(\ensuremathΔs(max){\mathrm{\ensuremath{\Delta}}}_{\mathit{s}}^{(\mathrm{max})}\ensuremathΔs(max)​\ensuremath{\tau}]1/2{]}^{1/2}]1/2, where L is the thickness of gas penetrated before the shift is measured, \ensuremathΔs(max){\mathrm{\ensuremath{\Delta}}}_{\mathit{s}}^{(\mathrm{max})}\ensuremathΔs(max)​ is the maximum ac Stark shift at very low pressure, and \ensuremath{\tau} is the pulse length of the lasers.At positions and concentrations such that 1\ensuremath{\Vert}\ensuremath{\Delta}k\ensuremath{\Vert}L\ensuremath{\le}2[(\ensuremathΔs(max){\mathrm{\ensuremath{\Delta}}}_{\mathit{s}}^{(\mathrm{max})}\ensuremathΔs(max)​\ensuremath{\tau})]1/2{]}^{1/2}]1/2 the ac Stark shift is present but the two-photon excitation is suppressed due to a destructive interference between excitation by two photons from the first laser and by two-photon excitation between a four-wave-mixing photon and a photon from the second laser.
We report detailed experimental results in xenon that confirm theoretical predictions [Payne et al., Phys. Rev. A 48, 2334 (1993)] on the interference-based suppression of ac Stark shifting of even-photon resonances under circumstances where the shift is produced through strong coupling to a state that is also coupled back to the ground state. In the copropagating-beam configuration and in the pressure range of 0.5--250 Torr, the multiphoton ionization spectral line shape is unchanged regardless of the presence of the second laser field, i.e., the ac Stark shift introduced by the second laser is totally suppressed due to a complete destructive interference between two excitation pathways. In the counterpropagating-beam configuration, however, the ac Stark shift persists in the range of the pressure studied. By choosing a different set of energy levels where no four-wave-mixing field is permitted, we show that the very same ac Stark shift introduced by the second laser field persists even with copropagating-beam configuration, as predicted by theory.
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 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 report an experiment in xenon which confirms the theoretical predictions (Payne et al, Phys. Rev. A48, 2334 (1993)) on the pressure dependent suppression of the ac Stark shifting of even‐photon resonances. In the co‐propagating configuration the ac Stark due to a second laser is totally suppressed at elevated concentrations due to a destructive interference between two excitation pathways. The ac Stark shift persists in the counter‐propagating configuration because of the absence of destructive interference.
When three-photon atomic excitation is produced by two laser beams of angular frequencies ${\mathrm{\ensuremath{\omega}}}_{\mathit{L}1}$ and ${\mathrm{\ensuremath{\omega}}}_{\mathit{L}2}$, which are crossed at an angle \ensuremath{\theta}, large cooperative pressure-dependent shifts can be produced in the peak position of the excitation line shape. Such shifts occur when the transition between states is dipole allowed in both one and three photons. When \ensuremath{\theta} or ${\mathrm{\ensuremath{\omega}}}_{\mathit{L}2}$/${\mathrm{\ensuremath{\omega}}}_{\mathit{L}1}$ is small, these shifts are very large compared to the pressure broadened width of the line. Our analysis shows that for small \ensuremath{\theta} or small ${\mathrm{\ensuremath{\omega}}}_{\mathit{L}2}$/${\mathrm{\ensuremath{\omega}}}_{\mathit{L}1}$, where the shift is large, the excitation is suppressed near the unperturbed resonance due to a destructive interference between the three-photon excitation by the laser and one-photon excitation due to the four-wave-mixing field, whereas the shifted resonance appears at the position where the atoms interact through ``resonant'' four-wave-mixing photons that are phase matched with the laser field. By studying an experimental situation in xenon we show that the theoretical predictions are in precise agreement with experiment for several angles and a wide range of pressures.