Nonlinear optical response of collective modes in multiband superconductors assisted by nonmagnetic impurities

In multiband superconductors, multiple collective modes exist associated with the multiple order parameters. Oscillations of the amplitude and the relative phase of the order parameters are called Higgs and Leggett modes, respectively. Recently, it has been suggested that nonmagnetic impurity scattering would enhance nonlinear coupling between the Higgs mode and an electromagnetic wave with a frequency located in the superconducting gap region, while its effect on the Leggett mode is still unresolved. Here, we theoretically investigated the nonlinear optical response of multiband Bardeen-Cooper-Schrieffer-type superconductors in the presence of nonmagnetic impurities with a density matrix approach extending the Mattis-Bardeen model of linear response. We found that the drastic enhancement of nonlinear optical response due to the nonmagnetic impurity scattering occurs only for the Higgs modes and not for the Leggett mode. As a result, both the light-induced dynamics of the superconducting gaps and the resulting third-harmonic generation are dominated by the Higgs modes. We also examined the role of quasiparticle excitations to find that they give the subdominant contribution to the third-harmonic generation.

Figure 1

Linear response of a two-band BCS superconductor. (a) Temperature dependence of the superconducting gaps considered in this work. (b) Electric field and (c) the induced electric current above $T_{\mathrm{c}}$ (dotted) and below $T_{\mathrm{c}}$ (solid). (d) Optical conductivity above $T_{\mathrm{c}}$. The solid and open circles correspond to the simulated real and imaginary parts, respectively. Solid lines are Drude model. (e) Optical conductivity below $T_{\mathrm{c}}$. Solid lines are MB model. Dashed line shows the contribution to the real part from the band with the lower gap.

Figure 2

Optical response of a single-band BCS superconductor at $T=0$, dependent on the scattering rate $\gamma$. (a) The real part of the optical conductivity ${\sigma }_1\left(\omega \right)$ for $\gamma /2\mathrm{\Delta }=0.1$ (blue), 0.2 (green), 1 (yellow), 3 (orange), and 5 (red). Pump power spectrum is also shown as the dotted line, whose electric field waveform is plotted in (b). (c) Forced oscillation of the superconducting gap $\delta {\mathrm{\Delta }}^{\prime }\left(t\right)$. Colors are the same as in (a). (d) $\gamma$ dependence of the maximum of $|\delta {\mathrm{\Delta }}^{\prime }\left(t\right)|$ (circles) and ${\sigma }_1\left(\omega \right)$ (squares), normalized to 1 at $\gamma /2\mathrm{\Delta }=1$.

Figure 3

Nonadiabatic excitation of a two-band BCS superconductor. (a) Electric field waveform. (b) Induced Higgs mode. The upper and lower lines correspond to the smaller and larger gaps, respectively. a.u. stands for arbitrary units. (c) Induced Leggett mode, shown in a form of the phase difference $\delta \phi \left(t\right)=\delta {\mathrm{\Delta }}_1^{″}\left(t\right)/{\mathrm{\Delta }}_1-\delta {\mathrm{\Delta }}_2^{″}\left(t\right)/{\mathrm{\Delta }}_2$. (d)–(f) Power spectrum of the electric field, the Higgs mode, and the Leggett mode, respectively. The position of superconducting gaps $2{\mathrm{\Delta }}_{1,2}$ are shown by dotted lines.

Figure 4

Multicycle excitation of a two-band BCS superconductor. (a) Power spectra of electric field with $\mathrm{\Omega }=0.25$, 0.5, 0.75, 1, 1.5, and 3 from left to right, each corresponding to (b)–(g), respectively. Pulse width is $\tau =2\pi \times 1$ for (b)–(f) and $\tau =2\pi \times 0.3$ for (g). ${\sigma }_1\left(\omega \right)$ is also drawn. (b1)–(g1) Electric field waveform for $\mathrm{\Omega }=0.25$, 0.5, 0.75, 1, 1.5, and 3, respectively. (b2)–(g2) Dynamics of $\delta {\mathrm{\Delta }}_{1,2}^{\prime }\left(t\right)$ for each excitation pulse. a.u. stands for arbitrary units. (b3)–(g3) Dynamics of the phase difference $\delta \phi \left(t\right)$.

Figure 5

Schematic level diagrams concerning THG in BCS superconductors. (a) Contribution from quasiparticle excitations. Horizontal lines denote quasiparticle levels, either real or virtual. Wavy arrows indicate photons. (b) Propagation of collective modes. Successive absorption and emission of $\delta \mathrm{\Delta }$ (single-line arrows) induce the self-consistent motion of collective modes (double-line arrows), either Higgs or Leggett. (c) Contribution from collective modes.

Figure 6

Third-order nonlinear response of a two-band BCS superconductor. (a) Top: electric field waveform. Second: total third-order current. Third, fourth, bottom: contribution from the Higgs mode, quasiparticles, and the Leggett mode, respectively, with the last one multiplied by 5000. (b) Intensity of the third harmonic. The topmost line gives the total emission. The other three lines show contributions from the Higgs mode, quasiparticles, and the Leggett mode, from top to bottom. The last one is multiplied by ${10}^5$. Dashed line is intensity of the incident electric field. a.u. stands for arbitrary units. (c) Nonlinear absorption spectrum. The shaded region is outside the detection range of the incident electric field, whose intensity is shown as the dashed line. The topmost line gives the total spectrum. The other three lines are contributions from the Higgs mode, quasiparticles, and the Leggett mode, from top to bottom (seen at the edge of the right shaded region). The last one is multiplied by 5000.