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arXiv:quant-ph/0402155

Two-Photon Absorption

Isabel Pérez–Arjona, Germán J. de Valcárcel, Eugenio Roldán

Published Feb 20, 2004• 0 citations
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Abstract

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.

DOI
10.48550/arxiv.quant-ph/0402155

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Two-photon absorption

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SE ANALIZA TEORICAMENTE LA ABSORCION DE DOS FOTONES DENTRO DEL FORMALISMO SEMICLASICO DE LA INTERACCION ENTRE LA RADIACION Y LA MATERIA. CONSIDERAMOS UN CONJUNTO, CON ENSANCHAMIENTO INHOMOGENEO, DE ATOMOS DE TRES NIVELES SOMETIDO A LA ACCION DE DOS CAMPOS CONTRAPROPAGANTES DE IGUAL FREQUENCIA. RESOLVEMOS PERTURBATIVAMENTE LAS ECUACIONES DE BLOCH DEL SISTEMA DE UNA FORMA NO USUAL CONCENTRANDONOS EN EL LIMITE DE ALTA DESINTONIA DE LAS TRANSICIONES A UN FOTON. DE ESTA FORMA OBTENEMOS UNA EXPRESION ANALITICA PARA LA ANCHURA DE LA RESONANCIA A DOS FOTONES EN LA QUE SE PONE DE MANIFIESTO LA POSIBILIDAD DE ESPECTROSCOPIA SUB”DOPPLER A DOS FOTONES. TAMBIEN OBTENEMOS UNA EXPRESION ANALITICA PARA EL DESPLAZAMIENTO STARK DE LA RESONANCIA A DOS FOTONES. Abstract 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.

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Reference 1 derives the two-photon probe-absorption coefficient for a homogeneously broadened medium subjected to an arbitrarily intense saturator wave. We have generalized this theory to allow for Doppler broadening with a counterrunning probe and saturator waves. We have carried out integrals over a Lorentzian Doppler distribution analytically and have specialized to a number of limits, such as extreme Doppler broadening. In this limit, we find the simple probe absorption coefficient α 1 ≃ 2 α 0 ( γ D 1 I 2 − i ω s T 1 ) , where α0 is the unsaturated two-photon absorption coefficient, I2 is the saturator intensity, γ is the two-photon coherence decay rate constant, ω s is the Stark shift parameter, T1 is the population difference decay time, and D 1 = 1 / [ γ+ i ( ω+ω s I 2 − ν 2 − ν 1 ) ] . For vanishing Stark shifts (ω s = 0), this is the standard two-photon probe absorption coefficient, but for large ω s , the complex Lorentzian D 1 becomes detuned. In the homogeneously broadened limit, the absorption spectrum resembles the one-photon case. In this paper we show the transition between these two limits.

Annual Meeting Optical Society of America•Jan 1, 1985•0 citations

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

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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
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 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.
SE ANALIZA TEORICAMENTE LA ABSORCION DE DOS FOTONES DENTRO DEL FORMALISMO SEMICLASICO DE LA INTERACCION ENTRE LA RADIACION Y LA MATERIA. CONSIDERAMOS UN CONJUNTO, CON ENSANCHAMIENTO INHOMOGENEO, DE ATOMOS DE TRES NIVELES SOMETIDO A LA ACCION DE DOS CAMPOS CONTRAPROPAGANTES DE IGUAL FREQUENCIA. RESOLVEMOS PERTURBATIVAMENTE LAS ECUACIONES DE BLOCH DEL SISTEMA DE UNA FORMA NO USUAL CONCENTRANDONOS EN EL LIMITE DE ALTA DESINTONIA DE LAS TRANSICIONES A UN FOTON. DE ESTA FORMA OBTENEMOS UNA EXPRESION ANALITICA PARA LA ANCHURA DE LA RESONANCIA A DOS FOTONES EN LA QUE SE PONE DE MANIFIESTO LA POSIBILIDAD DE ESPECTROSCOPIA SUB”DOPPLER A DOS FOTONES. TAMBIEN OBTENEMOS UNA EXPRESION ANALITICA PARA EL DESPLAZAMIENTO STARK DE LA RESONANCIA A DOS FOTONES. Abstract 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.
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.
This paper presents new results on aspects of the two-photon laser. A semiclassical investigation of threshold behaviour reveals that the atom-field coupling is already so strong at threshold that a non-perturbative treatment of the dynamics is necessary. A master equation is derived on the basis of an effective Hamiltonian generalized in that it includes the effect of an overall two-photon detuning, the effect of ionization of the upper atomic level and the effect of dynamic Stark shifts. From the diagonal elements of the master equation, a Fokker-Planck equation is obtained and shown to yield a general analytical expression for the photon distribution which is found to simplify considerably in special cases. The agreement with the result obtained through numerical solution of the master equation itself is very good. It is further shown that a detuning from two-photon resonance and a significant Stark shift lead to sub-Poissonian statistics. Finally, time-dependent solutions for the photon distribution under an injected coherent state are presented.
Reference 1 derives the two-photon probe-absorption coefficient for a homogeneously broadened medium subjected to an arbitrarily intense saturator wave. We have generalized this theory to allow for Doppler broadening with a counterrunning probe and saturator waves. We have carried out integrals over a Lorentzian Doppler distribution analytically and have specialized to a number of limits, such as extreme Doppler broadening. In this limit, we find the simple probe absorption coefficient α 1 ≃ 2 α 0 ( γ D 1 I 2 − i ω s T 1 ) , where α0 is the unsaturated two-photon absorption coefficient, I2 is the saturator intensity, γ is the two-photon coherence decay rate constant, ω s is the Stark shift parameter, T1 is the population difference decay time, and D 1 = 1 / [ γ+ i ( ω+ω s I 2 − ν 2 − ν 1 ) ] . For vanishing Stark shifts (ω s = 0), this is the standard two-photon probe absorption coefficient, but for large ω s , the complex Lorentzian D 1 becomes detuned. In the homogeneously broadened limit, the absorption spectrum resembles the one-photon case. In this paper we show the transition between these two limits.
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.