Two-photon absorption and amplitude mode in conventional superconductors with paramagnetic impurities

The two-photon absorption (TPA) spectrum of BCS superconductors is theoretically investigated. We discuss a possibility of observing the amplitude mode (one of collective modes in superconductors) in this spectrum. An absorption edge by this mode appears at the same frequency as that by quasiparticle excitations in the TPA spectrum of superconductors with time-reversal symmetry, and it will be difficult to distinguish the amplitude mode from quasiparticle excitations. This property originates from the fact that the magnitude of superconducting order parameter is one half of the energy gap in the linear absorption spectrum. In contrast to this one half of the energy gap is smaller than the magnitude of order parameter in superconductors with paramagnetic impurities. Though the amplitude mode is not well-defined owing to the broken time-reversal symmetry, a numerical calculation shows that the peak by this mode persists in the TPA spectrum. This peak appears at the frequency different from that of the absorption edge by quasiparticle excitations, and it is possible to specify the amplitude mode in the TPA spectrum.