AC Josephson Signatures of the Superconducting Higgs Mode

TL;DR

This work proposes a transport-based route to excite and detect the superconducting Higgs mode without external irradiation by leveraging the AC Josephson effect in a voltage-biased junction. Using a microscopic Floquet-Keldysh formalism, the authors show that Higgs oscillations are resonantly driven when the Josephson frequency matches the Higgs energy, $\omega_J=\omega_H=2\Delta_0$, and that this coupling generates a strong second-harmonic current at $2\omega_J$, which can dominate the conventional $\omega_J$ response in highly asymmetric-gap junctions with high transparency. The analysis connects a Higgs susceptibility $\chi_{\Delta\Delta}$ to source terms from radiating Cooper pairs, providing a concrete mechanism for how Higgs dynamics imprint on transport observables. The findings hold across model geometries (2D/3D), are robust to Dynes broadening, and offer a practical route to spectroscopically probe the Higgs mode via Josephson radiation, advancing irradiation-free detection of collective modes in superconductors.

Abstract

The Higgs mode in superconductors corresponds to oscillations of the amplitude of the order parameter. While its detection typically entails resonant optical excitation, we present a purely transport-based setup wherein it is excited in a voltage biased Josephson junction. Demonstrating the importance of order parameter dynamics, the interplay of Higgs resonance and Josephson physics enhances the second harmonic Josephson current oscillating at twice the usual Josephson frequency in transparent junctions featuring single-band s-wave superconductors. If the leads have unequal equilibrium superconducting gaps, this second harmonic component may even eclipse its first harmonic counterpart, thus furnishing a unique hallmark of the Higgs oscillations.

Figures (8)

• Figure 1: Schematic of the Higgs mode, corresponding to radial oscillations of the OP (red balls) in the free-energy landscape (mexican hat). It is excited by radiating tunneling Cooper pairs (green balls) in a voltage-biased Josephson junction. We show the leads of the junction, which have lengths comparable to the superconducting coherence length to allow the Higgs oscillations to fully develop. The leads terminate in macroscopic superconducting reservoirs (not shown).
• Figure 2: Numerical results, considering a two-dimensional junction with $N_{L}=36$, $N_{R}=6$, $N_w=12$, $\mu=0$, $\mathcal{T}=0.4\zeta$ (normal state transparency$\ \approx 0.48$Cuevas1996), $\Gamma=0.02\zeta$, and $\zeta=5$ (bandwidth$=40$). We define the SC coherence length of the right lead (remains unchanged) $\xi_{sc,R}=4\zeta/(\pi\Delta_{0,R})$. We show the OP at the $y=6^{\text{th}}$ site along the transverse direction in (a-b). The OP shows minor transverse variations sm. (a) $|\Delta_{2}|$ (frequency $\omega_J=2(eV)$) at the first site on the left SC immediately neighbouring the junction, showing the Higgs resonance at $eV=\Delta_{0,L}$ (orange dashed line). (b) Same as (a), but we show the argument $\angle\Delta_{2}$, such that the OP modulation is $2|\Delta_2|\cos(\omega_J t-\angle\Delta_{2})$, revealing a jump across the resonance (orange dashed line). In (c-f), we define $R_N$ as the numerically obtained normal state resistance. (c) The current components $I_4$ (frequency $2\omega_J=4(eV)$) and $I_2$ (frequency $\omega_J=2(eV)$), at $eV/\zeta=0.02$, on a logarithmic scale. With decreasing $\Delta_{0,L}/\Delta_{0,R}$, $I_4$ strengthens and peaks near the Higgs resonance ($eV/\Delta_{0,R}=\Delta_{0,L}/\Delta_{0,R}$ blue dashed line), eventually dominating $I_2$. (d) Amplitude of $I_4$ with varying voltage and $\Delta_{0,L}$ in the absence of Higgs oscillations, forcing $\delta\Delta=0$. (e) Same as (d), but with Higgs oscillations included. $I_4$ exhibits a peak at the Higgs resonance (orange dashed line) for highly asymmetric junctions (bottom left corner of the plot)---a feature absent in panel (d). The slight deviation from $eV=\Delta_{0,L}$ likely stems from the enhancement of the equilibrium gap near the junction from its bulk value $\Delta_{0,L}$ via the inverse proximity effect sm. (f) $I_4$ (black diamonds) and $I_2$ (red circles) are shown as functions of $\mathcal{T}$, both with (solid markers) and without (empty markers and dashed lines) Higgs oscillations, for $\Delta_{0,L}/\Delta_{0,R} \approx 0.07$ and $eV/\zeta = 0.02 \approx \Delta_{0,L}/\zeta$. Notably, $I_4$ exceeds $I_2$ at large $\mathcal{T}$ only in the presence of the Higgs enhancement.
• Figure 3: (a) $\zeta\Re \chi_{\Delta\Delta}^{-1}(\omega,q) = \zeta(\frac{1}{g}+\frac{1}{2}\Im\chi_{\Delta\Delta,0}(\omega,q))$ for a three-dimensional s-wave SC, showing a dip at the Higgs frequency $\omega_H(q)$. We use $\zeta=10\Delta_0$, $\mu=0$, $\Gamma=0.1\Delta_0$, and $\xi_{\text{sc}}=4\zeta/(\pi\Delta_0)$ is the coherence length. (b) $s_\phi/(\mathcal{T}/\Delta_{0,L}^2)$ for $\zeta=10\Delta_{0,L}$, $\mu=0$, $\Gamma=0.25\Delta_{0,L}$ and $\Delta_{0,R}=5\Delta_{0,L}$. It oscillates at $\omega_J$, and decays into the lead $(x>0)$ over a few $\xi_{\text{sc},L}$.
• Figure S1: Pair susceptibility in a system with $\zeta=20\Delta_0$, $\mu=0$ (half-filled). (a,b) show $\chi_{\Delta\Delta,0}(\omega,q)$ as a function of frequency $\omega$ for $q=0$, for varying $\Gamma$. The $q=0$ Higgs mode occurs at $\omega/(2\Delta_0)=\pm 1$. (c,d) show the full susceptibility $\chi_{\Delta\Delta}(\omega,q=0)$, for varying $\Gamma$. The singularity in $\chi_{\Delta\Delta}$ at $\omega=\pm 2\Delta_0$ gets stronger with decreasing $\Gamma$. (e-f) show the full susceptibility $\chi_{\Delta\Delta}(\omega,q=\xi/v_F)$ for $\Gamma/\Delta_0=0.1$, revealing the Higgs dispersion. We state $q$ in terms of the single-particle dispersion $\xi=v_F q$, linearised near the Fermi energy with the Fermi velocity $v_F$.
• Figure S2: Numerically obtained equilibrium gap at the $y=6^{\text{th}}$ site. We use the same parameters as in Fig. \ref{['Fig2']}. The gap in the left lead is slightly larger near the junction due to the inverse proximity effect, and stabilises to its bulk value over a distance of the order of the coherence length $\xi_L\sim 1/\Delta_{0,L}$ ($\xi_L\neq\xi_R$).