Polarization dependence of multiphoton transitions

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Abstract
In order to determine the polarization dependence of multiphoton transitions to excitonic states in solids, we develop a symmetry analysis of the transition-rate formula that includes the study of its transformation properties under permutation of the photon indices. These properties play an important role when some or all of the photons have equal frequency or polarization. The way permutation invariance affects the transition rate depends on the symmetry of the excited state. We show how the number of dynamical parameters in the polarization dependence is reduced when some of the photons are of equal frequency and how this effect can lead to more stringent selection rules. We apply the theory to the case of two-, three-, and four-photon transitions. We describe a procedure that gives the polarization dependence for any crystal point group. In particular, we point out that the more stringent selection rules that are found in the case of photons of equal frequency are the same as in the case of photons of equal polarization. This property is related to the invariance of the transition rate under permutations of all the photon indices.