Collective modes in pumped unconventional superconductors with competing ground states

Motivated by the recent development of terahertz pump-probe experiments, we investigate the short-time dynamics in superconductors with multiple attractive pairing channels. Studying a single-band square lattice model with a spin-spin interaction as an example, we find the signatures of collective excitations of the pairing symmetries (known as Bardasis-Schrieffer modes) as well as the order parameter amplitude (Higgs mode) in the short-time dynamics of the spectral gap and quasiparticle distribution after an excitation by a pump pulse. We show that the polarization and intensity of the pulse can be used to control the symmetry of the nonequilibrium state as well as frequencies and relative intensities of the contributions of different collective modes. We find particularly strong signatures of the Bardasis-Schrieffer mode in the dynamics of the quasiparticle distribution function. Our work shows the potential of modern ultrafast experiments to address the collective excitations in unconventional superconductors and highlights the importance of subdominant interactions for the nonequilibrium dynamics in these systems.

Figure 1

Phase diagram of $H_{\text{MF}}$ at $T=0$ for the fillings where the $d_{x^2-y^2}$-wave order parameter closely competes with that of the extended $s$-wave symmetry. The solid lines refer to the actual ground-state values for ${\mathrm{\Delta }}_s$ and ${\mathrm{\Delta }}_{d_{x^2-y^2}}$, while the dashed lines show the behavior of a pure $s$-wave or pure $d_{x^2-y^2}$-wave solution, ignoring the possibility of coexistence.

Figure 2

Calculated evolution of the superconducting gap amplitudes for an $s$-wave ground state and an applied light pulse with polarization at an angle $\varphi =0$ (solid line) and an angle $\varphi =\pi /4$ (dashed line) to the $k_x$ axis. (a) and (b) show the short-time dynamics of the $s$- and $d_{x^2-y^2}$-wave order parameter, respectively. To make the existence of a second frequency in (a) clear, arrows illustrate the beating pattern. (c) and (d) refer to the Fourier transform of $|{\mathrm{\Delta }}_s|$ and $|{\mathrm{\Delta }}_{d_{x^2-y^2}}\left(t\right)|$, respectively. As $|{\mathrm{\Delta }}_{d_{x^2-y^2}}\left(t\right)|$ remains 0 for $\varphi =\pi /4$, no Fourier transform of this case is shown in (d).

Figure 3

Fluence dependence of the Bardasis-Schrieffer and the Higgs mode in the $s$-wave ground state for the filling (a) $n=0.38$ and (b) $n=0.411$, away (a) and close (b) to the phase transition to the $s+i{d}_{x^2-y^2}$ ground state, respectively.

Figure 4

Frequency of the Bardasis-Schrieffer mode for different band fillings close to the transition point between extended $s$-wave and $d_{x^2-y^2}$-wave ground states at $n\approx 0.415$. The numerical results (squares) and the results of the linearized equations (solid blue curve) are in good agreement.

Figure 5

Oscillation of the quasiparticle distribution, projected into fully symmetric $s$-wave (a) and $d_{x^2-y^2}$-wave (b) parts. The oscillation of the $k_x$-integrated $d_{x^2-y^2}$-wave part in (c) shows a signature of the Bardasis-Schrieffer mode and of the Higgs mode as shown in its Fourier transform (d).