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Physical Review

Double-Photon Contributions to Multiple-Photon Processes

Norah V. Cohan, Hendrik F. Hameka

Published Nov 25, 1966• 17 citations
Publisher

Abstract

It is shown that in evaluating the transition probability of a multiple-photon process involving more than two photons, the contributions of the part of the interaction Hamiltonian that is quadratic in the vector potential of the radiation field may be neglected if the vector potential of the radiation field is assumed to be constant in the region of space that we consider.

DOI
10.1103/physrev.151.1076

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It is shown that in evaluating the transition probability of a multiple-photon process involving more than two photons, the contributions of the part of the interaction Hamiltonian that is quadratic in the vector potential of the radiation field may be neglected if the vector potential of the radiation field is assumed to be constant in the region of space that we consider.
We study the contribution to the transition probability of a multiple-photon process for which resonance occurs of the part of the interaction operator quadratic in the vector potential. We consider only the case in which this vector potential is constant within the region of space to which the system is confined. Under this assumption we show that the ratio of the contribution from the quadratic term in the vector potential to that from the linear term is proportional to the square of the energy difference between the initial state of the whole system (material system plus radiation field) and the intermediate state for which resonance occurs. As this energy difference is negligible near resonance, the quadratic term in the vector potential (or double-photon contributions) can be neglected when calculating transition probabilities for multiple-photon processes near resonances in the dipole approximation.
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 interaction is considered between NNN stationary two-level atoms and a radiation field described by a single cavity mode. The state vector for the complete system of atoms plus radiation is expressed as a linear superposition of states constructed from a product of photon states in the nnn representation and products of Pauli spin eigenstates describing all combinations of atoms in the lower and upper energy levels. Equations for the corresponding probability amplitudes are derived by substituting this superposition into the Schr\"odinger equation. The resulting equations are combined into bilinear form and phenomenological damping contributions are added. After the neglect of certain of the bilinear quantities, a master equation is derived which governs the probability Pmn{{P}_{m}}^{n}Pm​n of having mmm atoms in the lower level and nnn photons in the mode. This master equation takes account of multiple single-quantum absorption and emission processes but not of simultaneous multiple processes involving two or more atoms at a time. The equation which governs the expected number of photons 〈n〉〈n〉〈n〉 derived from the master equation bears a close resemblance to a rate equation. The effect of radiation loss from the cavity is incorporated into the master equation. Numerical calculations for a QQQ-spoiled laser show that the statistics of the number of photons in the mode bear a qualitative resemblance to Poisson statistics.
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.