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Journal of the Optical Society of America B

ac Stark shift of a two-photon transition induced by a model stochastic field

James C. Camparo, P. Lambropoulos

Published Dec 1, 1992• 14 citations
Publisher

Abstract

We have investigated the ac Stark shift of a two-photon transition induced by a model stochastic field by using a Monte Carlo technique. The model field has a stochastic single mode, which can possess both colored amplitude and colored frequency fluctuations. The stochastic field’s modification of the ac Stark shift is quantified by a parameter M, which is the ratio of the Stark shift in the model stochastic field to the Stark shift that would be obtained in a phase-diffusion field of equivalent linewidth (i.e., a field with frequency fluctuations but no amplitude fluctuations). In the regime of weak fields, below saturation of the bound-bound transition, we find that M is greater than unity: the Stark shift is enhanced by the stochastic field. Moreover, in this regime M is an increasing function of the field’s degree of photon bunching. In strong fields, where the bound–bound transition is nearly saturated, the enhancement of the Stark shift is diminished, so much so that in extremely strong fields M is actually less than unity. Our calculations indicate that the ac Stark shift’s modification occurs through two distinct processes, depending on the strength of the field. In weak fields enhancement of a multiphoton transition’s Stark shift is influenced primarily by the intrinsic correlation between ac-Stark-shift fluctuations and Rabi-frequency fluctuations. In strong fields the fluctuating Stark shifts give rise to an asymmetric resonance line shape in a fashion analogous to inhomogeneous broadening. The line shape’s peak position then has a sublinear dependence on the stochastic field’s intensity, and this yields a diminished value of M.

DOI
10.1364/josab.9.002163

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Physical Review A•Oct 1, 1984•8 citations
We have investigated the ac Stark shift of a two-photon transition induced by a model stochastic field by using a Monte Carlo technique. The model field has a stochastic single mode, which can possess both colored amplitude and colored frequency fluctuations. The stochastic field’s modification of the ac Stark shift is quantified by a parameter M, which is the ratio of the Stark shift in the model stochastic field to the Stark shift that would be obtained in a phase-diffusion field of equivalent linewidth (i.e., a field with frequency fluctuations but no amplitude fluctuations). In the regime of weak fields, below saturation of the bound-bound transition, we find that M is greater than unity: the Stark shift is enhanced by the stochastic field. Moreover, in this regime M is an increasing function of the field’s degree of photon bunching. In strong fields, where the bound–bound transition is nearly saturated, the enhancement of the Stark shift is diminished, so much so that in extremely strong fields M is actually less than unity. Our calculations indicate that the ac Stark shift’s modification occurs through two distinct processes, depending on the strength of the field. In weak fields enhancement of a multiphoton transition’s Stark shift is influenced primarily by the intrinsic correlation between ac-Stark-shift fluctuations and Rabi-frequency fluctuations. In strong fields the fluctuating Stark shifts give rise to an asymmetric resonance line shape in a fashion analogous to inhomogeneous broadening. The line shape’s peak position then has a sublinear dependence on the stochastic field’s intensity, and this yields a diminished value of M.
We have developed a computational framework for the study of the effects of field fluctuations on strongly driven nonlinear transitions. Nonlinearity may stem from cither the strong coupling (saturation) of a resonant transition or its multiphoton character, or it may stem from both in the most general case. Our approach is based on a Monte Carlo simulation of stochastic field fluctuations, which allows maximum flexibility in the stochastic nature of the field and in the type of process that is studied. Here, we report on a more general problem than has previously been considered in the interaction of atoms with stochastic fields, namely, the three-photon resonant five- photon ionization of xenon, 3 + 2 photoionization of xenon has been the subject of a fair number of experimental investigations. It involves nonlinearity in with a bound-bound and a bound-free transition as well as an ac-Stark shift. Consequently, there are correlations among three fluctuating parameters: the three-photon Rabi frequency, the two-photon ionization rate, and the intensity-dependent ac-Stark shift, and it is these correlations that make the stochastic-field problem almost intractable to traditional methods of analysis.
The authors investigate the saturation and Stark splitting of an atomic transition in (i) a phase-diffusion field and (ii) a chaotic field of arbitrary bandwidth. The theory takes into account the infinite sequence of field-correlation functions. It is shown that a chaotic field is less effective than a phase-diffusion field in saturating a single- or multiphoton transition. This is contrary to the weak-field case, where the intensity fluctuations and the associated photon bunching make the chaotic field more effective in exciting a multiphoton transition. It is also shown that the Stark splitting of an atomic transition, as observed in double resonance, is influenced dramatically by the intensity fluctuations in the chaotic field.
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.
In free space, where the vacuum mode density varies relatively slowly in the spectral region of an atomic transition, an atom’s real-field and vacuum interactions are effectively decoupled. Consequently, spontaneous decay and the Lamb shift are, for all practical purposes, independent of any real-field atomic perturbation. However, in a colored vacuum (i.e., a low-Q cavity) the vacuum mode density can change significantly in the vicinity of an atomic transition, so that a Stark shift will alter an atom’s vacuum environment and thereby couple real-field and vacuum effects. Since the ac Stark shift is an inherent aspect of multiphoton processes, this coupling is unavoidable for highly nonlinear field–atom interactions that occur in cavities. Here we consider this effect for 3+2 resonance-enhanced multiphoton ionization of xenon.
Abstract We demonstrate how strong-field multiphoton transitions between dynamically shifted atomic levels can be traced in the energy spectra of emitted photoelectrons. Applying an ultrafast and intense laser pulse, two-photon Rabi oscillations are induced between two bound states of an atom. A third photon from the same pulse directly ionizes the atom, thus the emitted photoelectrons coherently probe the underlying dynamics. As the instantaneous energy of photoelectrons follows the pulse intensity envelope, modulated by the ac Stark shifts, electrons emitted with the same energy but at different times—at the rising and falling edge of the pulse—will interfere leading to pronounced dynamic interference pattern in the spectra. We investigate this phenomenon both numerically and analytically by developing a minimal three-state model that incorporates two-photon coupling and dynamically shifted atomic levels. On the example of atomic lithium (2 s → → 4 s → continuum) we show how the individual ac Stark shifts and the two-photon Rabi frequency are reflected through the asymmetry, shifting and splitting of the interference structure of the computed photopeaks.
The influence of ac Stark shifts on the three-photon ionization of cesium in a moderate-intensity laser beam is investigated. The ionization process involves a two-photon resonance with the 8d8d8d fine-structure levels. The work consists of two related parts. In the first part, atomic parameters, such as the two-photon Rabi rate and the ac Stark shift, are calculated in the nonrelativistic Hartree-Fock approximation. In the second part, the atomic parameters thus obtained are used as coefficients in a system of coupled differential equations representing the time evolution of the density matrix. Ionization line shapes are generated for a transform-limited pulsed laser of moderate intensity with a single-mode transverse structure. As the time history of the intensity profile is varied from square to smooth, the ionization line shape becomes asymmetric due to the time dependence of the ac Stark shift. The effects of varying the spatial intensity distribution of the laser light to include transverse centering and longitudinal focusing are also studied.