Theory of light-induced resonances with collective Higgs and Leggett modes in multiband superconductors

We theoretically investigate coherent optical excitations of collective modes in two-band BCS superconductors, which accommodate two Higgs modes and one Leggett mode corresponding, respectively, to the amplitude and relative-phase oscillations of the superconducting order parameters associated with the two bands. We find, based on a mean-field analysis, that each collective mode can be resonantly excited through a nonlinear light-matter coupling when the doubled frequency of the driving field coincides with the frequency of the corresponding mode. Among the two Higgs modes, the higher-energy one exhibits a sharp resonance with light, while the lower-energy mode has a broadened resonance width. The Leggett mode is found to be resonantly induced by a homogeneous ac electric field because the leading nonlinear effect generates a potential offset between the two bands that couples to the relative phase of the order parameters. The resonance for the Leggett mode becomes sharper with increasing temperature. All of these light-induced collective modes along with density fluctuations contribute to the third-harmonic generation. We also predict an experimental possibility of optical detection of the Leggett mode.

Figure 1

$\left|L\right(\omega \left)\right|$ at $T=0$ for several values of the dimensionless interband pairing interaction $|{\lambda }_I|$ varied from 0 to 0.071 (a) and from 0.095 to 0.38 (b), for ${\lambda }_{\alpha \alpha }=-0.28,{\lambda }_{\beta \beta }=-0.96,{\lambda }_{\alpha \beta }/{\lambda }_{\beta \alpha }=0.73,{\mathrm{\Delta }}_{\alpha }/{\omega }_c=0.31,{\mathrm{\Delta }}_{\beta }/{\omega }_c=0.96,$ and $c_{\alpha 0}-c_{\beta 0}=1$. Position of the peak defines ${\omega }_{\mathrm{L}}$.

Figure 2

The energy of the Leggett mode (red curve) and half-width (full width at half maximum; FWHM) of $\left|L\right(\omega \left)\right|$ (orange shading) against the interband pairing interaction $|{\lambda }_I|$. The horizontal dashed line indicates $2{\mathrm{\Delta }}_{\alpha }$ (the smaller of the two gaps), while the vertical one the critical value of $|{\lambda }_I|$ at which ${\omega }_{\mathrm{L}}$ reaches $2{\mathrm{\Delta }}_{\alpha }$. Values of the used parameters are the same as in Fig. 1.

Figure 3

$|H_{\alpha }\left(\omega \right)|$ (a) and $|H_{\beta }\left(\omega \right)|$ (b) for several values of $|{\lambda }_I|$ varied from 0 to 0.38, with $c_{\alpha 1}=c_{\beta 1}=-1$ and the same parameters as Fig. 1.

Figure 4

Dependence of the Higgs modes on the interband pairing interaction $|{\lambda }_I|$. Green (blue) line represents the mode energy ${\omega }_{\mathrm{H}\alpha }$ (${\omega }_{\mathrm{H}\beta }$), defined as the peak of $|H_{\alpha }\left(\omega \right)|$ ($|H_{\beta }\left(\omega \right)|$), associated with the smaller gap $2{\mathrm{\Delta }}_{\alpha }$ (larger gap $2{\mathrm{\Delta }}_{\beta }$). The half-widths are indicated by green (blue) regions. Left dashed line indicates $|{\lambda }_I|$ at which the upper half-width of $|H_{\alpha }\left(\omega \right)|$ diverges, while the right one the $|{\lambda }_I|$ at which the peak of $|H_{\alpha }\left(\omega \right)|$ at $\omega =2{\mathrm{\Delta }}_{\alpha }$ disappears. Used parameters are the same as Fig. 3.

Figure 5

(a) Dependence of the gaps (horizontal axis) $2{\mathrm{\Delta }}_{\alpha }\left(T\right)$ and $2{\mathrm{\Delta }}_{\beta }\left(T\right)$ (green and blue dashed lines, respectively) against temperature (vertical axis), for ${\lambda }_{\alpha \alpha }=-0.28,{\lambda }_{\beta \beta }=-0.96,{\lambda }_{\alpha \beta }/{\lambda }_{\beta \alpha }=0.73$ and ${\lambda }_I=-0.19$. Temperatures chosen in (b)–(d) are indicated by horizontal lines: $k_{B}T/{\omega }_c=0$ (blue), 0.15 (green), 0.23 (khaki), 0.35 (orange), and 0.43 (red). $\left|L\right(\omega \left)\right|$ (b), $|H_{\alpha }\left(\omega \right)|$ (c) and $|H_{\beta }\left(\omega \right)|$ (d) for several values of $T$ with $c_{\alpha 0}-c_{\beta 0}=1$ and $c_{\alpha 1}=c_{\beta 1}=-1$.

Figure 6

Temperature dependence of the Leggett mode energy ${\omega }_{\mathrm{L}}$ (red curve) and the Higgs mode energies ${\omega }_{\mathrm{H}\alpha }$ (green) and ${\omega }_{\mathrm{H}\beta }$ (blue). Their resonance widths are indicated by orange, green and blue shadings, respectively. The green line denoting ${\omega }_{\mathrm{H}\alpha }$ overlaps with the red line up to the right edge of the green region, at which the green line is terminated. Used parameters are the same as Fig. 5.

Figure 7

Amplitude of the THG spectra for the incident light polarized along the $[1,0]$ (a) and $[1,1]$ (b) directions of the square lattice. In (a), the red and blue curves show $|j_{\mathrm{L}}^{\left(3\right)x}|$ and $|j_{\mathrm{d}}^{\left(3\right)x}|$, respectively, while in (b), the blue and green curves show $|j_{\mathrm{d}}^{\left(3\right)x}|$ and $|j_{\mathrm{H}}^{\left(3\right)x}|$, respectively. Note that $|j_{\mathrm{L}}^{\left(3\right)x}|$ and $|j_{\mathrm{H}}^{\left(3\right)x}|$ have no polarization dependence. The vertical axis in (b) is normalized by ${({\mathrm{\Delta }}_{\beta }/V_{\beta \beta })}^2$. The positions of $\mathrm{\Omega }={\mathrm{\Delta }}_{\alpha },{\omega }_{\mathrm{L}}/2,{\mathrm{\Delta }}_{\beta }$ are indicated by dashed lines and arrows, respectively. The parameters used here are explained in the text.