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Physical Review A

Relation between collective Lamb shifts and the high-pressure odd-photon cancellation effect

M. G. Payne, W. R. Garrett, Roger C. Hart, Irene Datskou

Published Sep 1, 1990• 37 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.42.2756

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Large pressure-dependent cooperative shifts in multicolor odd-photon resonant atomic excitation

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Effects of four-wave mixing on four-photon resonance excitation and ionization in the presence of a three-photon intermediate state resonance enhancement

M. G. Payne, J. C. Miller, Roger C. Hart +1 more

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
When three-photon atomic excitation is produced by two laser beams of angular frequencies \ensuremathωL1{\mathrm{\ensuremath{\omega}}}_{\mathit{L}1}\ensuremathωL1​ and \ensuremathωL2{\mathrm{\ensuremath{\omega}}}_{\mathit{L}2}\ensuremathωL2​, 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 \ensuremathωL2{\mathrm{\ensuremath{\omega}}}_{\mathit{L}2}\ensuremathωL2​/\ensuremathωL1{\mathrm{\ensuremath{\omega}}}_{\mathit{L}1}\ensuremathωL1​ 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 \ensuremathωL2{\mathrm{\ensuremath{\omega}}}_{\mathit{L}2}\ensuremathωL2​/\ensuremathωL1{\mathrm{\ensuremath{\omega}}}_{\mathit{L}1}\ensuremathωL1​, 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.
We show that when three-photon excitation of a dipole-allowed transition is mediated by two laser beams at angular frequencies ${\mathrm{\ensuremath{\omega}}}_{\mathit{L}1}$ and ${\mathrm{\ensuremath{\omega}}}_{\mathit{L}2}$ that are crossed at an angle \ensuremath{\theta}, large cooperative pressure-dependent shifts occur in the line shape for resonant excitation. When excitation of the state is by a 2${\mathrm{\ensuremath{\omega}}}_{\mathit{L}1}$\ifmmode\pm\else\textpm\fi{}${\mathrm{\ensuremath{\omega}}}_{\mathit{L}2}$ process and either \ensuremath{\theta}\ensuremath{\ll}1 or ${\mathrm{\ensuremath{\omega}}}_{\mathit{L}1}$\ensuremath{\gg}${\mathrm{\ensuremath{\omega}}}_{\mathit{L}2}$, the cooperative shift is large compared with the pressure-induced width of the line. For the 2${\mathrm{\ensuremath{\omega}}}_{\mathit{L}1}$+${\mathrm{\ensuremath{\omega}}}_{\mathit{L}2}$ case the shift is toward higher excitation energy, while for 2${\mathrm{\ensuremath{\omega}}}_{\mathit{L}1}$-${\mathrm{\ensuremath{\omega}}}_{\mathit{L}2}$ the shift is to lower excitation energies. In cases where the resonance can be reached by more than one pathway, differing only in the directions of propagation of the photons involved in the excitation, multiple peaks are predicted in the excitation line shape.
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{]}^{1/2}]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{]}^{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 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.
Through multiphoton ionization measurements, the polarization effects in destructive quantum interference under three-photon resonant excitation have been studied. Recent observations [V. Peet, Phys. Rev. A 74, 033406 (2006)] have indicated that contrary to the well-known pattern of a total suppression of resonance excitation, the destructive interference becomes incomplete if three-photon transition is driven by crossed beams with orthogonal polarization planes. These observations have been tested for a more general case of two-color excitation and very similar polarization-dependent anomalies in the interference character have been registered. It has been shown that the destructive interference is modified and the resonance excitation does occur if two crossed laser beams have opposite circular polarizations. The pressure-induced evolution of the uncanceled ionization peaks has the ratio of blue shift to width close to 0.5 exactly as it is known for resonance ionization peaks registered under excitation by counterpropagating laser beams.
Three-photon-resonant excitation and subsequent ionization of xenon at high concentrations has been studied utilizing both Bessel and Gaussian beams. Comparisons of analogous results with the two modes of focusing are made. The excitation-ionization profiles can be analyzed on the basis of the well-studied interference effect involving combined contributions of three- and one-photon transition amplitudes in resonant atomic excitation. We show that the principal features of the excitation problem with Bessel beams can be simulated by a simple picture involving crossed plane-wave beams. Using this approach it has been shown that pressure-broadened and shifted lines that result from the interference-based cooperative Lorentzian line shift constitutes the main pressure-dependent peak of off-resonance excitation profiles produced in Bessel beams.
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.