Gauge-invariant microscopic kinetic theory of superconductivity in response to electromagnetic fields II: Application to collective modes

We show that the gauge-invariant kinetic equation of superconductivity provides an efficient approach to study the electromagnetic response of the gapless Nambu-Goldstone and gapful Higgs modes on an equal footing. We prove that the Fock energy in the kinetic equation is equivalent to the generalized Ward's identity. Hence, the gauge invariance directly leads to the charge conservation. Both linear and second-order responses are investigated. The linear response of the Higgs mode vanishes in the long-wave limit. The linear response of the Nambu-Goldstone mode is in consistency with the previous works. Whereas the second-order response exhibits interesting physics. On one hand, a finite second-order response of the Higgs mode is obtained in the long-wave limit. We reveal that this response, which has been experimentally observed, is attributed solely to the drive effect rather than the widely considered Anderson-pump effect. On the other hand, the second-order response of the Nambu-Goldstone mode, free from the influence of the long-range Coulomb interaction and hence the Anderson-Higgs mechanism, is predicted. We find that both Anderson-pump and drive effects play important role in this response. A tentative scheme to detect this second-order response is proposed.