A wide variety of optical measurements are key tools for various properties of materials. Especially, nonlinear optics is one of the central fields in physics and has been attracting much attention from fundamental science to technology. For instance, one can apply the second-order nonlinear optical responses to the identification of the phase of exotic matters. It has been demonstrated by measuring the second-harmonic light and direct current generations under the irradiating light . On the other hand, optical responses manifest richer physical consequences such as kinetic inductance in superconductors due to the Cooper pairs’ quantum condensation. This may imply synergy between nonlinear optics and superconducting science, namely, superconducting nonlinear optics.
Superconducting nonlinear optical responses may lead to various applications. Owing to the dissipationless property, superconductors may realize efficient interconversions between current and lights. In addition, the responses are expected to be a useful tool for probing the symmetry violations in complex ordered states such as parity symmetry breaking. In light of these interests, a theory of the second-order nonlinear response in superconductors was recently formulated, and a specific Bogoliubov-de Gennes Hamiltonian with a sublattice degree of freedom was investigated numerically . On the other hand, it has been unclear what material parameter is crucial for the superconducting nonlinear optical response. It is desirable to perform a comprehensive analysis of the superconducting nonlinear optical responses based on a canonical model.
To this end, we investigate the second-order nonlinear responses in superconductors with single-band model Hamiltonians . Focusing on the role of antisymmetric spin-orbit coupling, we present a systematic study of the superconducting nonlinear optical responses. The superconductivity-induced nonlinear optical responses disappear under some conditions on pair potential, whereas the response arises with the coexistence of intraband and interband pairing. Our study identifies a basic ingredient in superconducting nonlinear optics and may cover a broad range of noncentrosymmetric superconductors such as heavy-fermion and two-dimensional superconductors .
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 Non-Centrosymmetric Superconductors: Introduction and Overview, edited by E. Bauer and M. Sigrest, Lecture Notes in Physics Vol. 847 (Springer,Berlin, 2012); Y. Saito et al., Nat. Phys. 12, 144 (2016).
Keywords: nonlinear optical response, nonreciprocal response, noncentrosymmetric superconductors