Nonlinear optical effects and third-harmonic generation in superconductors: Cooper pairs versus Higgs mode contribution

Abstract

In conventional superconductors, the energy gap lies in the frequency range of few tenths of terahertz (THz), making THz spectroscopy the best tool to access its fundamental excitations. Recently it has been shown by R. Matsunaga $e\phantom{\rule{0}{0ex}}t$ $a\phantom{\rule{0}{0ex}}l$. that the use of intense, coherent multicycle THz pulses allows one to measure, in a NbN film, a component of the transmitted pulse oscillating three times faster than the incident light. It is found that this effect, named third-harmonic generation, has its maximum intensity at the temperature below ${T}_{c}$, where the light frequency $\ensuremath{\omega}$ matches the superconducting gap value $\mathrm{\ensuremath{\Delta}}$(T), pointing to a resonant process involving excitations specific of the superconducting state. What is the nature of this resonance? While previous work attributed this resonance to the Higgs mode, i.e., to amplitude fluctuations of the superconducting order parameter, the present paper comes to a different conclusion. By providing a detailed microscopic derivation of the nonlinear optical response, the authors show that the 3$\ensuremath{\omega}$ current response is controlled by the lattice-modulated density fluctuations. As a consequence, the third-harmonic generation turns out to be largely dominated by Cooper-pair excitations, which pile up at 2$\mathrm{\ensuremath{\Delta}}$. At the same time, in analogy with the standard Raman response, the Higgs signal is suppressed by the extremely small coupling to the probing field. The authors discuss also the polarization dependence of the nonlinear 3$\ensuremath{\omega}$ response, which opens the route to a Raman-like symmetry-selective probe of the superconducting excitations. This result offers challenging perspectives for THz spectroscopy of several systems, including cuprate superconductors.