Measuring intrinsic relaxation rates in superconductors using nonlinear response

TL;DR

This work develops a quantitative framework to extract intrinsic relaxation rates $1/T_1$ and $1/T_2$ in clean superconductors from nonlinear terahertz response, using Anderson pseudospin dynamics to link gap evolution and third-harmonic generation to damping. It analyzes both $s$- and $d$-wave pairing, incorporating energy-dependent damping and polarization control to excite/read out modes in different irreducible representations ($A_{1g}$, $B_{1g}$, $B_{2g}$). Key findings include a collisionless Higgs mode with a $t^{-1/2}$ decay in $s$-wave and a faster $1/t$ decay in $d$-wave, with damping introducing exponential factors and channel-dependent recovery governed by $T_1$ and $T_2$, respectively. Polarization provides a powerful experimental knob to selectively probe relaxation of specific irreps, enabling detailed characterization of underlying damping mechanisms relevant for superconducting materials.

Abstract

We discuss intrinsic relaxation rates in superconductors, and how they may be measured using non-linear optical (terahertz) response. We consider both $s$ and $d$-wave superconductors, both with and without a phenomenological (energy dependent) damping. Intrinsic relaxation rates of interest include the Higgs mode decay rate, the quasiparticle redistribution rate ($1/T_1$) and the quasiparticle dephasing rate ($1/T_2$), where the latter two rates are zero in the pure BCS model, but non-zero in the presence of damping. Using the Anderson pseudospin formalism, we illustrate how these intrinsic relaxation rates are related to measurable quantities such as the time-dependent gap function and the non-linear current (a.k.a. third harmonic generation). Hence, we show how intrinsic relaxation rates may be experimentally extracted and discuss what one may thereby learn about the underlying damping. We also discuss the effects of polarization control (viz. non-linear response to light polarized in different directions), which offers a useful experimental knob, especially for $d$-wave superconductors, enabling selective excitation of modes in different irreducible representations (and readout of their corresponding relaxation rates).

Figures (5)

• Figure 1: (a) Gap dynamics in the collisionless limit ($1/T_1=1/T_2=0$) with $x$- and $x'$-polarized pump pulses. Inset: Higgs amplitude decay on a log–log scale, showing a $t^{-1/2}$ power-law behavior. (b) Nonlinear current dynamics in the collisionless limit. Inset: amplitude of $J_{xxxx}=J_{A_{1g}+B_{1g}}$ on a log–log scale, which decays as $1/\sqrt{t}$, analogous to the Higgs amplitude mode. The nonlinear current is shown in arbitrary units.
• Figure 2: (a) Gap dynamics with different pseudospin damping forms under $x$-polarized pump pulse. The damping coefficient is chosen to be $\gamma_0=0.1$. (b) Higgs amplitude oscillation multiplied by $\sqrt{t}$ in a half-log plot. For $\gamma\propto \omega^p$, the exponential decay allows us to extract pseudospin dephasing: $1/T_2(\varepsilon=0)$. (c) Gap recovery for $\gamma\propto |\varepsilon|^p$ shows power-law $t^{-1/p}$. (d) Gap recovery multiplied by $\sqrt{t}$ for $\gamma\propto \omega^p$, allowing extraction of quasiparticle redistribution: $1/T_1(\varepsilon=0)$.
• Figure 3: (a) Gap dynamics of d-wave superconductors under different light polarizations and light intensities $A^2$ with Fermi level $\mu=0$. Inset: Higgs amplitude decay on a log-log scale, showing a much faster power-law decay than in s-wave superconductors. (b) Nonlinear current with different irreps induced by x-polarized ($\alpha=0$) pump pulse. The oscillation amplitude of $J_{B_{1g}}$ decays as $1/t$, as shown in the inset. $J_{A_{1g}}$ has much weaker intensity and $J_{B_{2g}}=0$ because the x-polarized pump cannot excite the $B_{2g}$ irreps.
• Figure 4: Gap dynamics for d-wave superconductors with pseudospin damping forms $\gamma_k = \gamma_0(\omega_k/2)^p$with $\gamma_0=0.1$. In all of these figures, we use x-pump pulse. (b) Nonlinear current with $\gamma_k\propto\omega^3_k$. (c) Gap recovery is dominated by the antinodes and decays as $t^{-4/p}$. (d) Decay of the oscillation amplitude of $J_{xxxx}\approx J_{B_{1g}}\propto e^{-t/T_{2,antinode}}/t$, which is valid for different pseudospin damping forms.
• Figure 5: (a) Nonlinear current induced by an $x'$-polarized pump pulse in the collisionless limit, resolved into different irreducible representations. (b) Oscillation amplitudes in the collisionless limit, showing that the $B_{2g}$ mode decays the fastest. (c)--(d) Nonlinear current with pseudospin damping $\gamma_k \propto \omega_k^3$, where the $B_{2g}$ mode decays more slowly.