Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Physical Review A

Theory of two strongly interacting resonances in multiphoton ionization

Burke Ritchie

Published Feb 1, 1980• 22 citations
Publisher

Abstract

The author presents a theory for $N$-photon ionization in which two intermediate states can be in resonance with the ground state over a narrow range of photon frequencies. The theory is applied to the three-photon $6F$ resonance-state spin-orbit doublet in the four-photon ionization of Cs. The amplitude for the process is the sum of two components, each containing a complex line-shape function. In the limit of zero splitting, the previously studied isolated resonance result, in which the line shape is Lorentzian is recovered. For the splitting in Cs ($6F$), the absorption profile is studied as a function of the splitting parameters of the state, the $6S\ensuremath{\rightarrow}6F$ three-photon Rabi rate, and the field-induced shift and width for the $6S\ensuremath{\rightarrow}6F$ transition. This profile depends in a complicated way on the relative magnitudes of the shift and the spacing between the fine-structure levels.

DOI
10.1103/physreva.21.629

Related papers

Line shape, frequency shift, Rabi splitting, and two-photon resonances in four-level double-resonance spectroscopy with closely spaced intermediate levels

Swapan Kumar Mandal, Pradip Narayan Ghosh

The interaction of a four-level atomic system with two electromagnetic radiations has been studied. The two intermediate levels are closely spaced such that a single radiation frequency can induce transitions from the ground state to both of them. In a cascade-type double resonance, the pump electric-field amplitude is much stronger than the amplitude of the signal field, which can induce transitions from both of the intermediate levels to the upper state. The 15 optical Bloch equations obtained from the Liouville equations are solved analytically to obtain the Doppler-free signal line shape. The computed signals, exhibiting two Lorentzian line profiles, are presented for different pump frequencies. In addition to these one-photon transitions, we obtain a weak two-photon resonance peak. The two-photon resonance becomes weaker when the detuning of the pump frequency from a resonance frequency is increased. When the pump frequency ${\mathrm{\ensuremath{\Omega}}}_{\mathit{p}}$ is intermediate between the pump resonance frequencies ${\mathrm{\ensuremath{\omega}}}_{1}$ and ${\mathrm{\ensuremath{\omega}}}_{2}$, the two-photon resonance is very weak and almost vanishes when ${\mathrm{\ensuremath{\Omega}}}_{\mathit{p}}$=(${\mathrm{\ensuremath{\omega}}}_{1}$+${\mathrm{\ensuremath{\omega}}}_{2}$)/2. When the pump frequency is on resonance with one of the transition frequencies, the Rabi splitting of the signal shows asymmetry because of the presence of a close-lying transition which is pumped off resonance. The intensity-dependent shifts of the transitions are such that they repel each other more and more as the intensity of the pump field is increased. These shifts arise from the nonresonant interaction between the atom and the field.

Physical Review A•Jun 1, 1993•18 citations

Theory of resonant multiphoton processes

C. S. Chang

The present paper applies the method of high-intensity quantum electrodynamics developed earlier to several problems of current experimental interest. In this paper the intensitydependent shifts and widths of atomic levels are calculated and applied to a given transition near resonance. The technique is demonstrated with three examples: the multiphoton ionization of atomic cesium and hydrogen in atomic beams and third-harmonic production in an atomic vapor. It is found that the multiphoton-ionization rate in a relatively weak laser beam (of order ${10}^{8}$ W/${\mathrm{cm}}^{2}$) is determined by the intensity-dependent level width ($6f$ configuration) as well as the interference of the fine-structure states, and that the rates for hydrogen, in a much more intense beam (of order ${10}^{12}$ W/${\mathrm{cm}}^{2}$) are determined by both the intensity-dependent widths and shifts a$\stackrel{\mathrm{`}}{\mathrm{s}}$ well as the wave-function renormalization constants. The agreement of these two examples of calculations with the experiments done recently is satisfactory. For third-harmonic production the role of two-photon resonance is emphasized, and the dependence of the generated third-harmonic intensity on the intensity and frequency of the fundamental is discussed. Near resonance, the ratio of third-harmonic intensity to fundamental intensity saturates.

Physical Review A•May 1, 1974•14 citations

Rate Theory for the Four-Photon Ionization of Cs near the 6F Resonance

Burke Ritchie

A previously presented narrow-bandwidth theory is used to interpret recent data on the four-photon ionization of Cs near the $6F$ resonance. The theory is based on treating the amplitude of the radiation field as a constant relative to its rapid phase oscillations. The ionization rate adiabatically follows a pulsed amplitude leading to maximum ionization for times at which the pulse develops sufficient intensity to shift the $6S\ensuremath{\rightarrow}6F$ transition into resonance.

Physical Review Letters•Nov 5, 1979•5 citations

"Multiple-" versus "multi-" photon absorption in the theory of resonant ionization

Burke Ritchie

The author has identified his result for resonant ionization by the simultaneous absorption of two photons by a ground-state electron [Phys. Rev. A 17, 659 (1978)] with the high-laser-power large-intermediate-state ionization rate result derived by Eberly and co-workers using a steady-state rate theory. This result gives $R{\ensuremath{\sigma}}_{22}$ for all power levels considered, where $R$ is the one-photon ionization rate for the intermediate state and ${\ensuremath{\sigma}}_{22}$ is its steady-state population factor. The result can be cast in this form, but it is found that ${\ensuremath{\sigma}}_{22}$ must be much less than ${\ensuremath{\sigma}}_{11}$, the ground-state population factor. Thus the resonant process can be described by a rate limit for conserving energy by simultaneous absorption only when the ground-state population factor remains unity to order $\frac{{\ensuremath{\Omega}}^{2}}{{R}^{2}}$, where $\ensuremath{\Omega}$ is the Rabi rate for the bound-bound transition. In other words, the simultaneous and sequential processes are physically equivalent in this limit.

Physical Review A•Oct 1, 1979•3 citations

Theory of multiphoton ionization: Near-resonance effects in two-photon ionization

P. Lambropoulos

I present a perturbation theory of multiphoton ionization based on the resolvent operator. The theory is applied to a detailed study of two-photon ionization under near-resonance conditions. It is shown that a distinction must be made between linear level shifts and widths, and resonance shifts and widths which appear in the transition probability. The resonance shifts and widths exhibit saturation effects and their values tend to be much smaller than those of the linear level shifts and widths. Using numerical examples corresponding to tunable dye lasers and alkali-metal atoms, we show that the above shifts can have a dramatic effect on the ionization rate even when the laser is not in exact resonance with the intermediate level. The question of "one-step" vs "two-step" two-photon ionization is also discussed. It is shown that exact resonance does not necessarily imply a two-step process. In fact, in most cases it is quite the opposite. A quantitative criterion for this distinction is established. Finally, a distinction is made between instrumental saturation and saturation of the transition probability. The two are shown to be of different nature. Further extensions of the work are also discussed.

Physical Review A•May 1, 1974•104 citations

Resonant multiphoton ionization of atomic hydrogen

Yves Gontier, N. K. Rahman, M. Trahin

The resonantly enhanced multiphoton ionization of atomic hydrogen is studied for the case of four-photon ionization (three-photon resonant). The time-dependent probabilities, the shape of the resonance profile at different intensities, and the nonlinear index are computed for coherent light at frequencies around the resonance (365 nm). The systematic dependence of the cross section as a function of the principal quantum number of the resonant state (as this tends towards the Rydberg states) is investigated at zero detuning. The probabilities of absorption of one photon more than necessary for the four-photon ionization process (resonantly enhanced above-threshold ionization) is computed and the relevant branching ratios are examined. Then it is shown that the effect of the spatio-temporal structures of the laser pulse is to red shift the maximum of the resonance profile, whereas that of an ideal chaotic light is to blue shift and broaden the resonance peak. The shape, shift, and width of the profile resulting from both these effects are in very fair agreement with the experimental ones.

Physical Review A•Jun 1, 1988•15 citations
The author presents a theory for NNN-photon ionization in which two intermediate states can be in resonance with the ground state over a narrow range of photon frequencies. The theory is applied to the three-photon 6F6F6F resonance-state spin-orbit doublet in the four-photon ionization of Cs. The amplitude for the process is the sum of two components, each containing a complex line-shape function. In the limit of zero splitting, the previously studied isolated resonance result, in which the line shape is Lorentzian is recovered. For the splitting in Cs (6F6F6F), the absorption profile is studied as a function of the splitting parameters of the state, the 6S\ensuremath→6F6S\ensuremath{\rightarrow}6F6S\ensuremath→6F three-photon Rabi rate, and the field-induced shift and width for the 6S\ensuremath→6F6S\ensuremath{\rightarrow}6F6S\ensuremath→6F transition. This profile depends in a complicated way on the relative magnitudes of the shift and the spacing between the fine-structure levels.
The interaction of a four-level atomic system with two electromagnetic radiations has been studied. The two intermediate levels are closely spaced such that a single radiation frequency can induce transitions from the ground state to both of them. In a cascade-type double resonance, the pump electric-field amplitude is much stronger than the amplitude of the signal field, which can induce transitions from both of the intermediate levels to the upper state. The 15 optical Bloch equations obtained from the Liouville equations are solved analytically to obtain the Doppler-free signal line shape. The computed signals, exhibiting two Lorentzian line profiles, are presented for different pump frequencies. In addition to these one-photon transitions, we obtain a weak two-photon resonance peak. The two-photon resonance becomes weaker when the detuning of the pump frequency from a resonance frequency is increased. When the pump frequency ${\mathrm{\ensuremath{\Omega}}}_{\mathit{p}}$ is intermediate between the pump resonance frequencies ${\mathrm{\ensuremath{\omega}}}_{1}$ and ${\mathrm{\ensuremath{\omega}}}_{2}$, the two-photon resonance is very weak and almost vanishes when ${\mathrm{\ensuremath{\Omega}}}_{\mathit{p}}$=(${\mathrm{\ensuremath{\omega}}}_{1}$+${\mathrm{\ensuremath{\omega}}}_{2}$)/2. When the pump frequency is on resonance with one of the transition frequencies, the Rabi splitting of the signal shows asymmetry because of the presence of a close-lying transition which is pumped off resonance. The intensity-dependent shifts of the transitions are such that they repel each other more and more as the intensity of the pump field is increased. These shifts arise from the nonresonant interaction between the atom and the field.
The present paper applies the method of high-intensity quantum electrodynamics developed earlier to several problems of current experimental interest. In this paper the intensitydependent shifts and widths of atomic levels are calculated and applied to a given transition near resonance. The technique is demonstrated with three examples: the multiphoton ionization of atomic cesium and hydrogen in atomic beams and third-harmonic production in an atomic vapor. It is found that the multiphoton-ionization rate in a relatively weak laser beam (of order 108{10}^{8}108 W/cm2{\mathrm{cm}}^{2}cm2) is determined by the intensity-dependent level width (6f6f6f configuration) as well as the interference of the fine-structure states, and that the rates for hydrogen, in a much more intense beam (of order 1012{10}^{12}1012 W/cm2{\mathrm{cm}}^{2}cm2) are determined by both the intensity-dependent widths and shifts as‘\stackrel{\mathrm{`}}{\mathrm{s}}s‘ well as the wave-function renormalization constants. The agreement of these two examples of calculations with the experiments done recently is satisfactory. For third-harmonic production the role of two-photon resonance is emphasized, and the dependence of the generated third-harmonic intensity on the intensity and frequency of the fundamental is discussed. Near resonance, the ratio of third-harmonic intensity to fundamental intensity saturates.
A previously presented narrow-bandwidth theory is used to interpret recent data on the four-photon ionization of Cs near the 6F6F6F resonance. The theory is based on treating the amplitude of the radiation field as a constant relative to its rapid phase oscillations. The ionization rate adiabatically follows a pulsed amplitude leading to maximum ionization for times at which the pulse develops sufficient intensity to shift the $6S\ensuremath{\rightarrow}6F$ transition into resonance.
The author has identified his result for resonant ionization by the simultaneous absorption of two photons by a ground-state electron [Phys. Rev. A 17, 659 (1978)] with the high-laser-power large-intermediate-state ionization rate result derived by Eberly and co-workers using a steady-state rate theory. This result gives $R{\ensuremath{\sigma}}_{22}$ for all power levels considered, where RRR is the one-photon ionization rate for the intermediate state and ${\ensuremath{\sigma}}_{22}$ is its steady-state population factor. The result can be cast in this form, but it is found that ${\ensuremath{\sigma}}_{22}$ must be much less than ${\ensuremath{\sigma}}_{11}$, the ground-state population factor. Thus the resonant process can be described by a rate limit for conserving energy by simultaneous absorption only when the ground-state population factor remains unity to order $\frac{{\ensuremath{\Omega}}^{2}}{{R}^{2}}$, where $\ensuremath{\Omega}$ is the Rabi rate for the bound-bound transition. In other words, the simultaneous and sequential processes are physically equivalent in this limit.
I present a perturbation theory of multiphoton ionization based on the resolvent operator. The theory is applied to a detailed study of two-photon ionization under near-resonance conditions. It is shown that a distinction must be made between linear level shifts and widths, and resonance shifts and widths which appear in the transition probability. The resonance shifts and widths exhibit saturation effects and their values tend to be much smaller than those of the linear level shifts and widths. Using numerical examples corresponding to tunable dye lasers and alkali-metal atoms, we show that the above shifts can have a dramatic effect on the ionization rate even when the laser is not in exact resonance with the intermediate level. The question of "one-step" vs "two-step" two-photon ionization is also discussed. It is shown that exact resonance does not necessarily imply a two-step process. In fact, in most cases it is quite the opposite. A quantitative criterion for this distinction is established. Finally, a distinction is made between instrumental saturation and saturation of the transition probability. The two are shown to be of different nature. Further extensions of the work are also discussed.
The resonantly enhanced multiphoton ionization of atomic hydrogen is studied for the case of four-photon ionization (three-photon resonant). The time-dependent probabilities, the shape of the resonance profile at different intensities, and the nonlinear index are computed for coherent light at frequencies around the resonance (365 nm). The systematic dependence of the cross section as a function of the principal quantum number of the resonant state (as this tends towards the Rydberg states) is investigated at zero detuning. The probabilities of absorption of one photon more than necessary for the four-photon ionization process (resonantly enhanced above-threshold ionization) is computed and the relevant branching ratios are examined. Then it is shown that the effect of the spatio-temporal structures of the laser pulse is to red shift the maximum of the resonance profile, whereas that of an ideal chaotic light is to blue shift and broaden the resonance peak. The shape, shift, and width of the profile resulting from both these effects are in very fair agreement with the experimental ones.