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We study the contribution of the Higgs amplitude mode on the nonlinearoptical response of superconductors beyond the BCS approximation by taking intoaccount the retardation effect in the phonon-mediated attractive interaction.To evaluate the vertex correction in nonlinear optical susceptibilities thatcontains the effect of collective modes, we propose an efficient scheme whichwe call the "dotted DMFT" based on the nonequilibrium dynamical mean-fieldtheory (nonequilibrium DMFT) to go around the difficulty of solving theBethe-Salpeter equation and analytical continuation. The vertex correction isrepresented by the derivative of the self-energy with respect to the externaldriving field, which is self-consistently determined by the differentiated("dotted") DMFT equations. We apply the method to the Holstein model, aprototypical electron-phonon-coupled system, to calculate the susceptibilityfor the third-harmonic generation including the vertex correction. The resultsshow that, in sharp contrast to the BCS theory, the Higgs mode can contributeto the third-harmonic generation for general polarization of the laser fieldwith an order of magnitude comparable to the contribution from the pairbreaking or charge density fluctuations. The physical origin is traced back tothe nonlinear resonant light-Higgs coupling, which has been absent in the BCSapproximation.

Nonlinear light-Higgs coupling in superconductors beyond BCS: Effects of the retarded phonon-mediated interaction