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

Journal of Physics B Atomic and Molecular Physics

Effects of finite laser bandwidths in two-photon spectroscopy

Rainer Salomaa

Published Nov 14, 1978• 21 citations
Publisher

Abstract

Discusses within a simple statistical model modifications of two-photon spectra caused by finite bandwidths of the light sources used for excitation. Field fluctuations are shown to affect the two-photon and step-wise contribution differently which provides a way of distinguishing them. The rate equation approximation, which ignores coherent two-photon processes, is shown to be accurate only when the frequency spread of the fields exceeds the detuning parameters and relaxation rates of the three-level system. In an inhomogeneous system satisfying the Doppler-limit criteria the step-wise contribution is efficiently cancelled away demonstrating the poor applicability of a rate description. Only when the field bandwidths are much larger than the Doppler widths are the coherence effects negligible.

DOI
10.1088/0022-3700/11/21/019

Related papers

Two-photon spectroscopy. II. Effects of residual Doppler broadening

Rainer Salomaa, Stig Stenholm

For pt.I see ibid., vol.8, p.1795 (1975) where the theory of Doppler-free two-photon transitions was discussed for the case of one single laser frequency. Here, the effects of two different frequencies are considered. When the residual Doppler width of the two-photon transition is less than the natural linewidth, the previous results are essentially regained. For large residual Doppler shifts, the absorption profile is a power-broadened Lorentzian if the frequency of the probe field is larger than the frequency of the strong field. In the opposite case, the absorption peak splits because of the AC Stark effect. The high-intensity distortions are minimized when equal frequencies can be used. Power-dependent shifts do not ordinarily occur with a resonantly tuned intermediate level. Some modifications of the spectra are introduced by phase-disturbing collisions.

Journal of Physics B Atomic and Molecular Physics•Jun 1, 1976•41 citations

Field-Correlation Effects in Multiphoton Absorption Processes

G. S. Agarwal

The dependence of the multiphoton absorption probability for an atomic system on the statistical properties of the incident light is investigated. It is shown, by means of $n\mathrm{th}$-order time-dependent perturbation theory, that the multiphoton absorption probability depends on the $(n, n)\mathrm{th}$ normally ordered correlation functions of the electromagnetic field. For stationary fields, the transition probability per unit time is constant and is directly related to the reduced cross-spectral correlation function of the electromagnetic field. This expression for the transition probability is then analyzed under various conditions of coherence. Explicit calculations are carried out for thermal light and for laser light, and the transition rates are compared when each has a Lorentzian spectrum. In particular, when the width of the final atomic level is much larger than the bandwidth of the field, the transition rate for thermal light is $n$! times higher than with laser light of the same mean intensity. The other extreme case, i.e., when the width of the final atomic level is much smaller than the bandwidth of the field, is also considered in detail. An equation for the density operator of the field is derived; this equation governs the change in the statistical properties of the photon system in $n$-photon absorption processes. On the basis of this equation, the time-dependent properties of the photon system are then studied and some physical consequences of these equations are discussed. Some numerical solutions relating to the time dependence of the factorial moments in two-photon absorption processes are presented.

Physical Review A•May 1, 1970•216 citations

Occurrence of Split Spectra in Doppler-free Two-photon Spectroscopy

Rainer Salomaa

This paper discusses the origin of splitting of two-photon spectra of three-level cascade configurations having a near resonant intermediate level. A clear distinction between splitting mechanisms in homogeneously and inhomogeneously broadened systems is demonstrated. When one of the laser fields is a weak probe the velocity averaged two-photon response is primarily caused by two resonantly excited velocity groups; the absence of these implies a split spectrum as suggested by Chebotaev and collaborators. A simple derivation of the probe response is presented. When both fields have non-negligible intensities resonant excitation is inhibited and instead all level populations pulsate. Still a split spectrum can appear. A simple graph showing the regions where this occurs is given and analyzed.

Physica Scripta•Apr 1, 1977•10 citations

Rate of resonant two-photon ionization in the presence of a partially coherent radiation field

Jacob L. F. de Meijere, Joseph H. Eberly

A treatment is given of two-photon ionization in which particular attention is paid to the influence of the bandwidth of the ionizing light on the ionization rate. The model adopted for the atom is a common one, including two bound levels and a continuum. The model of the laser is general enough to allow short-term temporal fluctuations of amplitude or phase, and these fluctuations give rise to the laser bandwidth. It is found that the fluctuations have the most interesting effects on the ionization rate when the laser coherence time is shorter than the atomic memory time, as could be expected. In addition to laser bandwidth, we consider the influence on the ionization rate of the detuning of the laser from the intermediate-state resonance, the laser power, and the finite lifetime of the intermediate state. Comparisons with related earlier calculations are made and similarities and differences are pointed out.

Physical Review A•Apr 1, 1978•87 citations

TWO-PHOTON MULTIWAVE MIXING (DOPPLER-FREE SPECTROSCOPY).

Barbara A. Capron

This dissertation examines aspects of the interaction of multiple coherent light fields for the two-photon two-level model. In this model the interacting energy levels are not connected by an atomic dipole and a two-photon transition between them is necessary. We employ the density matrix formalism allowing easy comparison between the one- and two-photon two-level models. Significant differences are found due to dynamic Stark shifts and conjugate scattering off the pump-induced two-photon coherence. Averages over Doppler broadening are performed and the new upper-level relaxation mechanisms of decay to an intermediate nonresonant level and ionization from the upper state are included. The new relaxation mechanisms, introduced to the theory to better model experiments, are similar except that ionization is intensity dependent. They cause the resulting probe absorption spectra to become more complex and in general asymmetric. Doppler broadening is also important in experiments using gases. We analytically average over a Lorentzian velocity distribution for both co- and counterpropagating pump and probe beams. For copropagating fields the results are similar to those for the one-photon case averaged over inhomogeneous broadening, whereas counterpropagating pump and probe fields yield the so-called Doppler-free configuration that is normally only modelled to third order in the pump amplitude. We consider the pump field amplitude to all orders and find that as long as the width of the Doppler velocity distribution is significantly larger than the two-photon Rabi frequency the results are Doppler-free. The final part of the dissertation treats the question of two-photon squeezed states. This requires quantized sidemodes. Squeezed states are minimum uncertainty states with unequal variances in the two quadratures of the electromagnetic field amplitude. One way to generate these states is via multiwave mixing and we present here the first calculation for nondegenerate two-photon multiwave mixing as it applies to squeezed states. We find that in general two-photon squeezed states require lower intensities and detuning than those predicted by the one-photon model.

•Jan 1, 1986•0 citations

AC Stark effect and fluorescence using modulated laser beams. I. Effect of laser fluctuations and arbitrary relaxation parameters

Ragini Saxena, G. S. Agarwal

The influence of relaxation parameters, as well as temporal fluctuations in the laser field on the modulation rate and phaseshift of the fluorescence from a two-level atom excited by a weakly modulated source is studied in detail. It is shown that, for intense resonant laser fields, the resonance in the modulation rate occurs at the modulation frequency Omega =2 due to the dynamical Stark splitting and has a width equal to 1 / 2 (T 1 -1 +T 2 -1 ). When temporal fluctuations of the exciting laser field are taken into account, the phaseshift of the scattered light is found to be strongly dependent on the bandwidth of the exciting source, as well as the frequency of modulation; hence experimental results on the determination of the lifetime by the phaseshift method have to be interpreted carefully. Numerical results for the modulation rates and phaseshift at different field strengths and various values of T 1 -1 , T 2 -1 , Delta laser bandwidth and Doppler broadening are presented.

Journal of Physics B Atomic and Molecular Physics•Jun 28, 1979•16 citations
Discusses within a simple statistical model modifications of two-photon spectra caused by finite bandwidths of the light sources used for excitation. Field fluctuations are shown to affect the two-photon and step-wise contribution differently which provides a way of distinguishing them. The rate equation approximation, which ignores coherent two-photon processes, is shown to be accurate only when the frequency spread of the fields exceeds the detuning parameters and relaxation rates of the three-level system. In an inhomogeneous system satisfying the Doppler-limit criteria the step-wise contribution is efficiently cancelled away demonstrating the poor applicability of a rate description. Only when the field bandwidths are much larger than the Doppler widths are the coherence effects negligible.
For pt.I see ibid., vol.8, p.1795 (1975) where the theory of Doppler-free two-photon transitions was discussed for the case of one single laser frequency. Here, the effects of two different frequencies are considered. When the residual Doppler width of the two-photon transition is less than the natural linewidth, the previous results are essentially regained. For large residual Doppler shifts, the absorption profile is a power-broadened Lorentzian if the frequency of the probe field is larger than the frequency of the strong field. In the opposite case, the absorption peak splits because of the AC Stark effect. The high-intensity distortions are minimized when equal frequencies can be used. Power-dependent shifts do not ordinarily occur with a resonantly tuned intermediate level. Some modifications of the spectra are introduced by phase-disturbing collisions.
The dependence of the multiphoton absorption probability for an atomic system on the statistical properties of the incident light is investigated. It is shown, by means of nthn\mathrm{th}nth-order time-dependent perturbation theory, that the multiphoton absorption probability depends on the (n,n)th(n, n)\mathrm{th}(n,n)th normally ordered correlation functions of the electromagnetic field. For stationary fields, the transition probability per unit time is constant and is directly related to the reduced cross-spectral correlation function of the electromagnetic field. This expression for the transition probability is then analyzed under various conditions of coherence. Explicit calculations are carried out for thermal light and for laser light, and the transition rates are compared when each has a Lorentzian spectrum. In particular, when the width of the final atomic level is much larger than the bandwidth of the field, the transition rate for thermal light is nnn! times higher than with laser light of the same mean intensity. The other extreme case, i.e., when the width of the final atomic level is much smaller than the bandwidth of the field, is also considered in detail. An equation for the density operator of the field is derived; this equation governs the change in the statistical properties of the photon system in nnn-photon absorption processes. On the basis of this equation, the time-dependent properties of the photon system are then studied and some physical consequences of these equations are discussed. Some numerical solutions relating to the time dependence of the factorial moments in two-photon absorption processes are presented.
This paper discusses the origin of splitting of two-photon spectra of three-level cascade configurations having a near resonant intermediate level. A clear distinction between splitting mechanisms in homogeneously and inhomogeneously broadened systems is demonstrated. When one of the laser fields is a weak probe the velocity averaged two-photon response is primarily caused by two resonantly excited velocity groups; the absence of these implies a split spectrum as suggested by Chebotaev and collaborators. A simple derivation of the probe response is presented. When both fields have non-negligible intensities resonant excitation is inhibited and instead all level populations pulsate. Still a split spectrum can appear. A simple graph showing the regions where this occurs is given and analyzed.
A treatment is given of two-photon ionization in which particular attention is paid to the influence of the bandwidth of the ionizing light on the ionization rate. The model adopted for the atom is a common one, including two bound levels and a continuum. The model of the laser is general enough to allow short-term temporal fluctuations of amplitude or phase, and these fluctuations give rise to the laser bandwidth. It is found that the fluctuations have the most interesting effects on the ionization rate when the laser coherence time is shorter than the atomic memory time, as could be expected. In addition to laser bandwidth, we consider the influence on the ionization rate of the detuning of the laser from the intermediate-state resonance, the laser power, and the finite lifetime of the intermediate state. Comparisons with related earlier calculations are made and similarities and differences are pointed out.
This dissertation examines aspects of the interaction of multiple coherent light fields for the two-photon two-level model. In this model the interacting energy levels are not connected by an atomic dipole and a two-photon transition between them is necessary. We employ the density matrix formalism allowing easy comparison between the one- and two-photon two-level models. Significant differences are found due to dynamic Stark shifts and conjugate scattering off the pump-induced two-photon coherence. Averages over Doppler broadening are performed and the new upper-level relaxation mechanisms of decay to an intermediate nonresonant level and ionization from the upper state are included. The new relaxation mechanisms, introduced to the theory to better model experiments, are similar except that ionization is intensity dependent. They cause the resulting probe absorption spectra to become more complex and in general asymmetric. Doppler broadening is also important in experiments using gases. We analytically average over a Lorentzian velocity distribution for both co- and counterpropagating pump and probe beams. For copropagating fields the results are similar to those for the one-photon case averaged over inhomogeneous broadening, whereas counterpropagating pump and probe fields yield the so-called Doppler-free configuration that is normally only modelled to third order in the pump amplitude. We consider the pump field amplitude to all orders and find that as long as the width of the Doppler velocity distribution is significantly larger than the two-photon Rabi frequency the results are Doppler-free. The final part of the dissertation treats the question of two-photon squeezed states. This requires quantized sidemodes. Squeezed states are minimum uncertainty states with unequal variances in the two quadratures of the electromagnetic field amplitude. One way to generate these states is via multiwave mixing and we present here the first calculation for nondegenerate two-photon multiwave mixing as it applies to squeezed states. We find that in general two-photon squeezed states require lower intensities and detuning than those predicted by the one-photon model.
The influence of relaxation parameters, as well as temporal fluctuations in the laser field on the modulation rate and phaseshift of the fluorescence from a two-level atom excited by a weakly modulated source is studied in detail. It is shown that, for intense resonant laser fields, the resonance in the modulation rate occurs at the modulation frequency Omega =2 due to the dynamical Stark splitting and has a width equal to 1 / 2 (T 1 -1 +T 2 -1 ). When temporal fluctuations of the exciting laser field are taken into account, the phaseshift of the scattered light is found to be strongly dependent on the bandwidth of the exciting source, as well as the frequency of modulation; hence experimental results on the determination of the lifetime by the phaseshift method have to be interpreted carefully. Numerical results for the modulation rates and phaseshift at different field strengths and various values of T 1 -1 , T 2 -1 , Delta laser bandwidth and Doppler broadening are presented.