Nonequilibrium nonlinear response theory of amplitude-dependent dissipative conductivity in disordered superconductors

TL;DR

This paper develops a rigorous nonlinear theory of amplitude-dependent dissipative conductivity in disordered superconductors using the Keldysh–Usadel framework. By perturbatively expanding to third order in the ac-field amplitude, it simultaneously accounts for direct photon action, Eliashberg-driven pair-potential variations, and Higgs-mode dynamics, revealing a Higgs-related resonance near the superconducting gap $\Delta$ in both the second-harmonic response and the nonlinear correction to the first-harmonic current. The study provides explicit microscopic formulas for the nonlinear conductivity and dissipation, showing that Higgs-mode contributions yield a distinct resonance in $\delta\sigma_1$ at $\hbar\omega_{ac}\approx\Delta$ and clarifying the frequency dependence of the quality factor in practical devices, especially at low frequencies where the direct photon term dominates. While powerful for dirty-limit superconductors and subgap operation, the approach is perturbative and omits phonons and nonperturbative field effects, suggesting clear directions for future extensions to capture the full nonlinear Q-theory up to the depairing limit and to incorporate electron-phonon dynamics.

Abstract

This work investigates amplitude-dependent nonlinear corrections to the dissipative conductivity in superconductors, using the Keldysh--Usadel theory of nonequilibrium superconductivity, which captures the nonequilibrium dynamics of both quasiparticles and the pair potential. Our rigorous formulation naturally incorporates both the direct nonlinear action of the photon field and indirect contributions mediated by nonequilibrium variations in the pair potential, namely the Eliashberg effect and the Higgs mode. The third-harmonic current, often regarded as a hallmark of the Higgs mode in disordered superconductors, arises from both the direct photon action and the Higgs mode. Our numerical results are in excellent agreement with previous studies. In contrast, the first-harmonic current, and consequently the dissipative conductivity, receives contributions from all three mechanisms: the direct photon action, the Higgs mode, and the Eliashberg effect. It is shown that that the nonlinear correction to dissipative conductivity in dirty-limit superconductors can serve as a fingerprint of the Higgs mode, appearing as a resonance peak at a frequency near the superconducting gap \( Δ\). In addition, our results provide microscopic insight into amplitude-dependent dissipation at frequencies well below \( Δ\), which is particularly relevant for applied superconducting devices. In particular, the long-standing issue concerning the frequency dependence of the amplitude-dependent quality factor is explained as originating from the direct nonlinear action of the photon field, rather than from contributions by the Higgs mode and the Eliashberg effect. Our practical and explicit expression for the nonlinear conductivity formula makes our results accessible to a broad range of researchers.

Figures (9)

• Figure 1: (a) Zero-harmonic (time-averaged) pair-potential variation $\Psi_{\rm 0H}$ as a function of $\omega_{\rm ac}$, illustrating the positive ($\Psi_{\rm 0H} > 0$) and negative ($\Psi_{\rm 0H} < 0$) Eliashberg effects. (b) Second-harmonic pair-potential variation $|\Psi_{\rm 2H} |$ as a function of $\omega_{\rm ac}$, showing the Higgs-mode resonance near $\omega_{\rm ac} \simeq \Delta(T)$. See also Appendix \ref{['appendix_check_deltaDelta']} for comparison with previous studies.
• Figure 2: Nonlinear corrections to the first-harmonic current. (a) Contribution from the direct nonlinear action of the electromagnetic field. (b) Sum of the contributions mediated by nonequilibrium pair-potential variations, $\delta\Delta_{\mathrm{0H}}$ and $\delta\Delta_{\mathrm{2H}}$. The dashed curve represents $|I_{\rm 1H}^{\rm Eliash}|$. (c) Real and (d) imaginary parts of the total nonlinear correction.
• Figure 3: Third-harmonic current as a manifestation of the nonlinear current response. (a) Contribution from the direct nonlinear action of the electromagnetic field. (b) Contribution mediated by the Higgs mode, $\delta\Delta_{\mathrm{2H}}$. (c) Real and (d) imaginary parts of the total third-harmonic current. The dashed curves represent the direct contribution $I_{\mathrm{3H}}^{qqq}$.
• Figure 4: Nonlinear corrections to the real part of the complex conductivity. (a, b) Total nonlinear correction $\delta \sigma_1$ calculated at $T/T_c = 0.1$ and $0.2$, respectively. (c, d) Decomposition of $\delta \sigma_1$ into individual contributions $\delta \sigma_1^{qqq}$, $\delta \sigma_1^{\rm Higgs}$ and $\delta \sigma_1^{\rm Eliash}$.
• Figure 5: (a, b) Amplitude-dependent dissipative conductivity at various drive frequencies [cf. Fig. 4(b)]. (c) Slope of the nonlinear correction, $d\delta \sigma_1/ds$, as a function of frequency. (d) Switching frequency $\omega_{\rm ac}^*$ as a function of temperature.