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Canadian Journal of Physics

THEORY OF TWO-PHOTON ABSORPTION

R. Guccione-Gush, Herbert P. Gush, Jan Van Kranendonk

Published Aug 1, 1967• 23 citations
Publisher

Abstract

The two-photon absorption process is discussed using a density matrix formalism. The electromagnetic field is described by means of coherent states, a technique that permits a simple evaluation of the effect of coherence on the transition probability. It is concluded that the latter is the same for a coherent and an incoherent pulse of radiation. In the case of a stationary field containing many modes, the transition probability is nearly the same in the coherent and incoherent cases, whereas for a single mode the transition probability is twice as high for the incoherent field. These conclusions are independent of the multipole expansion of the interaction Hamiltonian and are valid for electric dipole and higher multipole transitions.

DOI
10.1139/p67-204

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The two-photon absorption process is discussed using a density matrix formalism. The electromagnetic field is described by means of coherent states, a technique that permits a simple evaluation of the effect of coherence on the transition probability. It is concluded that the latter is the same for a coherent and an incoherent pulse of radiation. In the case of a stationary field containing many modes, the transition probability is nearly the same in the coherent and incoherent cases, whereas for a single mode the transition probability is twice as high for the incoherent field. These conclusions are independent of the multipole expansion of the interaction Hamiltonian and are valid for electric dipole and higher multipole transitions.
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.
Coherent excitation of an atomic excited state occurs during the propagation of near-resonant light pulses and is responsible for the induced polarization. Simultaneously, incoherent excitation occurs due to the relaxation processes described by the absorption coefficient. Here, the theory for the coherent and incoherent excitation is initially presented in terms of the traditional vector model. While a complete understanding of the two-level system is provided by the vector model, it is shown to be incomplete when the problem of directly monitoring the coherent and incoherent excitation is considered. This is because this latter problem involves more than two levels. For this more complicated multilevel problem, adiabatic states are introduced to gain further understanding. The adiabatic states are the stationary states of the atom in the presence of the near-resonant laser field; they help to explain the intimate connection between the coherent excitation and the two-photon resonance. Experimental measurements of the coherent and incoherent excitation associated with near-resonant pulse propagation in Rb vapor are presented. The double-resonance technique used a relatively strong pulsed dye laser tuned near the $5{S}_{\frac{1}{2}}\ensuremath{\leftrightarrow}5{P}_{\frac{1}{2}}$ transition (7948 \AA{}A) of Rb to produce the coherent and incoherent excitation, and a weak, tunable cw dye laser tuned in the region of the $5{P}_{\frac{1}{2}}\ensuremath{\leftrightarrow}6{D}_{\frac{3}{2}}$ transition (6206 \AA{}A) to monitor this excitation. In agreement with theory, the experimental results demonstrate that coherent excitation is responsible for two-photon absorption, while the incoherent excitation corresponds to one-photon absorption to the 5P125{P}_{\frac{1}{2}}5P21​​ state.
This paper deals with the equations of motion of the density matrix of a two-level atom in the presence of an intense multimode radiation field characterizing a multimode laser. Using a simplified treatment of line broadening which does not include correlations between momentum-changing collisions and pressure-broadening collisions, we obtain expressions which show the effect of pressure, laser-pulse length, and intensity on the excitation. We obtain analytic expressions in limiting cases of the pressure which allow prediction of the degree of excitation from a given laser pulse. In our treatment the phase relaxation time of the atoms is assumed to be fast enough so that coherence effects among the atoms can be ignored. Treating the velocity of the atoms statistically, we solve for three cases: constant velocity, velocity changes that are fast compared to the excitation time but slow compared to the dephasing time, and velocity changes that are fast compared to all other processes. The three cases yield the same formal equation for the two-body process of absorption of photons by incoherent atoms; however, the cross sections differ and are calculated explicitly for each case. Also discussed are the range of laboratory conditions and the relationship between these laboratory conditions and the various cases considered to demonstrate that under most circumstances this simple photon-absorption picture is applicable, provided the cross section is calculated correctly.
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A theory of a quantum amplifier based on two-photon transitions is presented. The model consists of an assemblage of atoms interacting with a single mode of the light field via two-photon absorption and emission. The treatment is quantum mechanical, and the emphasis is on the photon-statistical aspects of the problem. Difference-differential equations for the density matrix of the field mode are derived. The equations are solved for a special case. Differential equations for the expectation vaues of the number of photons and of the field amplitudes, and for their higher statistical moments, are also obtained. Approximate solutions for the average number of photons are found, and the effect of the amplification process on the statistical fluctuations of the number of photons and the field amplitudes is discussed. The results of this study suggest that such an amplifier does not have the coherence properties of a laser amplifier because it tends to increase the statistical fluctuations of the field.
The rate of two-photon absorption from an arbitrary state of the radiation field, and its dependence on the statistical properties of the field, are studied. It is found that when the density operator of the field factors into a product of single-mode density operators (in which case the modes are statistically independent), the rate depends on the spectral composition of the field. When the density operator is nonfactorable, the process depends on a photon doublet density. Therefore, the information obtained from two-photon experiments in the first case can, in principle, also be obtained from single-photon counting, while in the second case it would require coincidence counting.