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Physical Review A

Semiclassical theory of two-photon absorption: Absorption and level shifts in a model system

Michael D. Burrows, W. R. Salzman

Published Apr 1, 1977• 20 citations
Publisher

Abstract

Numerical calculations of level shifts and power-broadening linewidths for double-photon absorption in truncated rigid-rotor models are presented. This work demonstrates the feasibility of a previously suggested calculational method. It is shown that level shifts in double-photon absorption deduced from three-state models are misleading since other states dipole-coupled to states directly involved in the transition make important contributions to the shifts.

DOI
10.1103/physreva.15.1636

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Numerical calculations of level shifts and power-broadening linewidths for double-photon absorption in truncated rigid-rotor models are presented. This work demonstrates the feasibility of a previously suggested calculational method. It is shown that level shifts in double-photon absorption deduced from three-state models are misleading since other states dipole-coupled to states directly involved in the transition make important contributions to the shifts.
The results of semiclassical calculations on one-, two-, three-, and four-photon absorption in truncated models of the hydrogen atom are presented. The temporal behavior near "resonance" is confirmed to be Rabi-like, with a transition probability given by $P(t)={\frac{{\ensuremath{\gamma}}^{2}}{[{(\frac{{\ensuremath{\omega}}_{0}}{N}+\ensuremath{\delta}\ensuremath{-}\ensuremath{\omega})}^{2}+{\ensuremath{\gamma}}^{2}]}}{sin}^{2}{(\frac{\mathrm{Nt}}{2}){[{(\frac{{\ensuremath{\omega}}_{0}}{N}+\ensuremath{\delta}\ensuremath{-}\ensuremath{\omega})}^{2}+{\ensuremath{\gamma}}^{2}]}^{\frac{1}{2}}},$ where the resonance shift $\ensuremath{\delta}$ is proportional to the intensity of the monochromatic driving field with frequency $\ensuremath{\omega}$ and the power broadening $\ensuremath{\gamma}$ is proportional to the N2\frac{N}{2}2N​ power of intensity. NNN is the number of photons involved and $\mathrm{h}{\ensuremath{\omega}}_{0}$ is the energy separation between initial and final states. Calculated values of $\ensuremath{\delta}$ and $\ensuremath{\gamma}$ are given for several models; also, for the two-photon $1s\ensuremath{\rightarrow}2s$ absorption, $\ensuremath{\delta}$ and $\ensuremath{\gamma}$ are given which include the contributions of all bound ppp states.
An unconventional perturbation treatment of the theory of two-photon absorption in a four-level atomic system close to two-photon resonance is developed on the basis of the optical Bloch equations. The results for the longtime populations of the intermediate and upper states are valid in low- and medium-intensity regimes. In both regimes the upper state population shows a lorentzian resonance centred on zero two-photon detuning with a width equal to the natural linewidth of the upper state. The intermediate state population shows a similar behaviour in the intermediate regime only, with no significant dependence on two-photon detuning in the low-intensity regime. However, in both regimes the population of the virtual intermediate state is at least comparable to that of the resonant upper state. The perturbation theory results have been checked by direct numerical calculations based on the Bloch equations. Numerical calculations confirm the presence of a high-intensity regime characterized by power shifts, broadening and saturation effects.
Two-photon transitions have provided the basis for numerous useful nonlinear interactions with radiation. In this paper we calculate the absorption and coupling coefficients for one-, two-, and three-wave mixing in two-photon homogeneously broadened media. The resulting field equations form the basis for understanding saturation spectroscopy, phase conjugation,1 and single-mode instabilities2 in lasers and optical bistability in two-photon media. We assume that the wave frequencies are not resonant with any intermediate state j and that the probe and conjugate waves cannot saturate the two-photon transition. This allows us to use the two-photon two-level model and to transfer much of our knowledge of the one-photon two-level systems to the two-photon case. Two major differences occur between the two models. First, dynamic Stark shifts of the active levels play an important role in generating asymmetric spectra and nonzero dispersion (≃ω s T1) at line center. Second, high pump intensities do not bleach the absorption coefficient at the Rabi sidebands.
The physics of the two-photon absorption process is investigated for a three-state system. The density-matrix equations for the two-photon interaction are solved in the steady-state limit assuming that the pump laser radiation is monochromatic. Collisional broadening, saturation, and Stark shifting of the two-photon resonance are investigated in detail by numerical solution of the steady-state density-matrix equations. Analytical expressions for the saturation intensity and the Stark shift are derived for the case where the single-photon transitions between the intermediate state and the initial and final states are far from resonance with the pump laser. For this case, it is found that the direction of the Stark shift is dependent on the relative magnitudes of the dipole-moment matrix elements for the single-photon transitions that couple the intermediate state with the initial and final states. Saturation and Stark shifting are also investigated for the case where the single-photon transitions between the intermediate state and the initial and final states are close to resonance with the pump laser.
Two-photon absorption is theoretically analyzed within the semiclassical formalism of radiation-matter interaction. We consider an ensemble of inhomogeneously broadened three-level atoms subjected to the action of two counterpropagating fields of the same frequency. By concentrating in the limit of large detuning in one-photon transitions, we solve perturbatively the Bloch equations in a non-usual way. In this way we derive an analytical expression for the width of the two-photon resonance that makes evident sub-Doppler two--photon spectroscopy. We also derive an analytical expression for the Stark shift of the two-photon resonance.
The expression for the absorption coefficient in two-photon resonance transitions is deduced under general conditions. No particular restrictions are imposed on the level system or on the frequencies and intensities of the fields. The following cases are considered: (i) a two-level system (with and without singularities) irradiated by a single wave and (ii) a three-level system irradiated by two waves. Some theoretical and experimental disagreements reported by authors of previous papers are examined and clarified by the present theory.