Two-dimensional coherent spectroscopy of disordered superconductors in the narrow-band and broad-band limits

We theoretically analyze two-dimensional coherent spectroscopy (2DCS) signals for disordered superconductors in two limits: One is the narrow-band limit with sinusoidal pulse waves, and the other is the broad-band limit with delta-function pulses. While the 2DCS signal in the narrow-band limit is related to the third-order nonlinear susceptibilities $χ^{(3)}(3Ω; Ω, Ω, Ω)$ (third harmonic generation) and $χ^{(3)}(Ω; Ω, Ω, -Ω)$ (ac Kerr effect), we find that in the broad-band limit the signal along the diagonal and horizontal lines in the two-dimensional frequency space is related to another nonlinear susceptibility $χ^{(3)}(Ω; Ω, 0, 0)$ (dc Kerr effect). We numerically evaluate those susceptibilities for a lattice model of superconductors based on the BCS mean-field theory and self-consistent Born approximation for impurities. The 2DCS signals in the narrow-band and broad-band limits show threshold and resonance behaviors at the superconducting-gap frequency, respectively, whose physical origin is discussed in light of quasiparticle and Higgs-mode excitations.