Dynamics of the Schmid-Higgs Mode in $d$-wave superconductors

TL;DR

This paper investigates the time evolution of the Higgs mode in a $d$-wave superconductor after a sudden perturbation, revealing oscillations at twice the anti-nodal pairing amplitude with a power-law decay.

Contribution

It provides a detailed analysis of the Higgs mode dynamics in $d$-wave superconductors using a quasi-classical formalism and numerical simulations, highlighting the oscillation frequency and decay behavior.

Findings

Oscillation frequency is twice the anti-nodal pairing amplitude.

Amplitude decays as 1/t^2 over time.

Results confirmed by numerical solutions of pseudospin equations.

Abstract

We study the dynamics of the longitudinal collective mode in an unconventional superconductor. For concreteness, we assume that the superconductor is described by a $d$-wave order parameter with $d_{x^2-y^2}$ symmetry. After the superconductor has been suddenly subjected to a perturbation at time $t=0$, the order parameter exhibits a peculiar oscillatory behavior, with the amplitude of the oscillations slowly decaying with time in a power-law fashion. Assuming that the initial perturbation is weak, we use a formalism based on quasi-classical approach to superconductivity to determine both the frequency of the oscillations as well as how fast these oscillations decay with time by evaluating the time dependence of the pairing susceptibility. We find that the frequency of the oscillations is given by twice the value of the pairing amplitude in the anti-nodal direction and its amplitude decays as $1/t^2$. The results are also verified by a direct calculation of the order parameter dynamics by numerically solving the equations of motion for the Anderson pseudospins.