Signatures of superconducting Higgs mode in irradiated Josephson junctions

TL;DR

This work demonstrates that the Higgs mode in superconductors can be unambiguously detected via standard transport measurements in microwave-irradiated, highly asymmetric, transparent Josephson junctions. Using a self-consistent Floquet-Keldysh framework, the authors show two signatures: (i) a resonant enhancement and sign change of the second harmonic of the current-phase relation under phase bias, and (ii) a Higgs-renormalized enhancement of the second harmonic of the AC Josephson current observable through Shapiro steps under DC bias and irradiation. The findings provide a practical route to detect the Higgs mode in conventional s-wave superconductors and clarify how dynamical proximity and retardation effects govern OP dynamics under drive. The results have implications for using transport measurements as probes of collective modes in non-equilibrium superconducting systems and may guide experimental realizations with Al-based junctions.

Abstract

The Higgs mode, originally proposed in the context of superconductivity, corresponds to oscillations of the amplitude of the superconducting order parameter. Recent THz-domain optical studies have found signatures consistent with the Higgs mode, but its unambiguous detection is still challenging. We predict that the existence of the Higgs mode can be unambiguously revealed by standard measurements of the transport characteristics in microwave-irradiated asymmetric and transparent Josephson junctions. One signature of the Higgs mode in a Josephson junction is the microwave-induced enhancement of the second harmonic of the equilibrium current-phase relation (at zero DC bias voltage), whose sign differs from its expected value in the absence of the Higgs mode. As the radiation frequency is varied, this enhancement exhibits resonant behavior when the microwave frequency is tuned across the Higgs mass. The second signature that we propose is the enhancement of the second harmonic of the AC Josephson current at finite DC voltage bias, which can be probed in a customary analysis of the Shapiro steps in a microwave-irradiated junction.

Figures (6)

• Figure 1: Illustration of the JJ, with two leads of length $L_{L/R}\sim\xi_{sc,L/R}$ comparable to the superconducting coherence length (subscript $L/R$ denotes left/right) forming a bridge. The outer ends of the leads are connected to macroscopic superconducting reservoirs (widening triangles). The JJ has high transparency, and it is highly asymmetric with unequal equilibrium gaps, $\Delta_{0,L}\ll\Delta_{0,R}$ without loss of generality. We find that the OP develops a time-dependent component, denoted $\delta\Delta_{L/R}(t)$, representing the Higgs mode. This Higgs mode, corresponding to radial oscillations of the OP (red balls) in the free-energy landscape of the OP (mexican hat), is excited by radiating tunneling Cooper pairs. An external radiation (blue waves) is used to create SSs, which reveal the presence and the strength of the Higgs-induced Josephson current.
• Figure 2: (a) Numerically obtained first Floquet harmonic of the OP modulation in the left lead at the junction, $\delta\Delta_{1;j=N_L}$ [$n=1$, $j=N_L$ cf. Eq. \ref{['deltafloquet']}], for $N_L=18$, $N_R=5$sizejust, $\mathcal{T}/\zeta=0.4$ (transparency $\approx0.48$Cuevas2002), $\Gamma=0.0125\Delta_{0,R}=0.0025\zeta$, and $\omega_r\approx 2.29\Delta_{0,L}\approx 1.14\omega_{H,L}$, just over the Higgs resonance. As anticipated from Eq. \ref{['deltaDeltazerov']}, the dominant contribution varies as $\sin(\phi_0)$ (shown here, denoted by the subscript). The $\cos(\phi_0)$ component (not shown) is much weaker. The predicted $J_1(\alpha)$ dependence is also confirmed, as indicated by the blue circles. (b) The second harmonic component of the CPR, $\bar{I}^{(2)}$, as a function of $\alpha$. We present the normalized quantity $\bar{I}^{(2)} e R_N / \Delta_{0,R}$, where $R_N$ is the numerically obtained normal-state resistance. For reference, in symmetric ($\Delta_{0,L}=\Delta_{0,R}=\Delta_{0}$) tunnel JJs $I^{(1)}$ satisfies the Ambegaokar--Baratoff result $I^{(1)} e R_N / \Delta_0 = \pi/2$Ambegaokar1963. In the Higgs-free case (red), obtained by retaining only the zeroth Floquet component of the OP in the self-consistency equation Eq. \ref{['gapeq']}, we confirm the expected $J_0(2\alpha)$ dependence [cf. Eq. \ref{['IDCAA']}]. With Higgs renormalization included (blue), $\bar{I}^{(2),\text{Higgs}}$ is much larger. It starts out positive at $\alpha=0$, in contrast to the Higgs-free case. Moreover, inspired by Eq. \ref{['Iup2Higgs']}, the numerical results are well described by a fit of the form $\sum_m p_m [J_m(\alpha)]^2$, with the coefficients $p_m$ obtained from least-squares regression. We find that $p_{m\geq 5}$ are negligible. (c) We show the CPR $I_0$ with varying $\alpha$, in the absence of Higgs renormalization. Since $\bar{I}^{(2),\text{no Higgs}}$ is negligible, the CPR is not noticeably altered from $I_0\sim I^{(1)}J_0(\alpha)\sin(\phi_0)$. (d) Same as (c), but now we include the Higgs renormalization. Since $\bar{I}^{(2),\text{Higgs}}$ is large, it imparts a $\sin(2\phi_0)$ phase dependence, with the $\alpha-$dependence following from Eq. \ref{['Iup2Higgs']} [see also panel (b)].
• Figure 3: (a--d) Numerically obtained CPR ($I_0$) at $\alpha=2$, in a system with $N_L=18$, $N_R=5$sizejust, $\mathcal{T}/\zeta=0.4$ (transparency $\approx 0.48$Cuevas2002), and $\Gamma=0.0125\Delta_{0,R}=0.0025\zeta$. We present the normalized quantity $I_0 e R_N / \Delta_{0,R}$, where $R_N$ is the numerically obtained normal-state resistance. (a--b) CPR as a function of the equilibrium gap asymmetry $\Delta_{0,L}/\Delta_{0,R}$. Each panel considers a different $\omega_r$. The black dashed line marks the point $0.5\omega_r/\Delta_{0,R}=\Delta_{0,L}/\Delta_{0,R}$, corresponding to the resonance condition $\omega_r=2\Delta_{0,L}$. The Higgs renormalization weakens as $2\Delta_{0,L}$ exceeds $\omega_r$, restoring the conventional $\sim\sin(\phi_0)$ behavior. In contrast, when $2\Delta_{0,L}$ moves below $\omega_r$, the CPR develops a negative dip near $\phi_0=0$, originating from the Higgs-induced term $\bar{I}^{(2),\text{Higgs}}\sin(2\phi_0)$ with $\bar{I}^{(2),\text{Higgs}}<0$ [see Fig. \ref{['Fig2']}(b) for the sign at $\alpha=2$]. Panels (c–d) show the corresponding results without Higgs renormalization. We do not find any negative dips in $I_0$. (e) The component of the OP varying as $\sin(\phi_0)$ (the $\cos(\phi_0)$ component is negligible) in the left lead at the junction ($x=N_L$), which is peaked at the Higgs resonance $\omega_r=2\Delta_{0,L}$. This resonance is marked by a dashed line in all remaining panels. (f) Second harmonic $\bar{I}^{(2),\text{Higgs}}$ as a function of $\omega_r$ and $\Delta_{0,L} / \Delta_{0,R}$, showing the Higgs resonance. The Higgs-free counterpart in panel (g) exhibits the opposite sign [see Fig. \ref{['Fig2']}(b)]. In this case, the peak at $\omega_r=2\Delta_{0,L}$ corresponds to the pair-breaking threshold.
• Figure 4: Numerically obtained current harmonics without radiation, in a system with $N_L=18$, $N_R=5$sizejust, $\mathcal{T}/\zeta=0.4$ (transparency $\approx 0.48$Cuevas2002), and $\Gamma=0.0125\Delta_{0,R}=0.0025\zeta$. Panels (a) and (b) show $I_{\omega_J}$ without ($I^{\text{no Higgs}}_{2}$) and with ($I_{2}^{\text{Higgs}}$) Higgs oscillations, respectively; they exhibit no significant qualitative differences. The orange diagonal dashed line marks the Higgs resonance, $2eV=2\Delta_{0,L}$, in all the heat maps. (c) $\Delta_2$ (OP component oscillating at frequency $\omega_J$), normalized by $\Delta_{0,R}$, which exhibits a peak at the Higgs resonance (orange dashed line). The small offset from the resonance likely originates from a local enhancement of $\Delta_{0,L}$ near the junction barrier due to the proximity effect. (d) Cuts of $\Delta_2$, normalized by $\Delta_{0,L}$, as a function of $eV$ for various values of $\Delta_{0,L}$. It is peaked at the Higgs resonance (vertical dashed line), with the peak achieving its maximum value for $\Delta_{0,L}/\Delta_{0,R}\approx 0.07$. Panels (e) and (f) show $|I_{2\omega_J}|$, in the absence ($I^{\text{no Higgs}}_{4}$) and presence ($I_{4}^{\text{Higgs}}$) of Higgs renormalization, respectively. In the absence of Higgs renormalization (e), only a small $2\omega_J$ current appears, attributable to higher-order Josephson effect. In contrast, with Higgs renormalization (f), a pronounced peak emerges at the Higgs resonance (orange dashed line), particularly in the highly asymmetric regime where $\Delta_{0,L} \ll \Delta_{0,R}$ (bottom-left corner, highlighted by the red box). In the same regime (once again, highlighted by a red box), panel (e) shows that $I^{\text{no Higgs}}_{4}$ remains much smaller.
• Figure 5: Numerically calculated Shapiro step height $SS^1_1$ [Eq. \ref{['SSab']}], in a system with $N_L=18$, $N_R=5$sizejust, $\mathcal{T}/\zeta = 0.4$ (corresponding to transparency$\approx 0.48$Cuevas2002), $\Delta_{0,L} = 0.045\Delta_{0,R} = 0.0088\zeta$, and $\Gamma = 0.0125\Delta_{0,R} = 0.0025\zeta$. We plot the normalized quantity $SS^1_1 R_N e / \Delta_{0,R}$ as a function of the radiation strength $\alpha$, for various values of the DC voltage $eV$ sweeping across and beyond the Higgs resonance $eV=\Delta_{0,L}$. We show it in (a) the presence and (b) the absence of Higgs renormalization. In its presence, $SS^1_{1,\text{Higgs}}$ noticeably deviates from the $J_{-1}(\alpha)$ profile, evident most immediately from the changes in the location and magnitudes of the nodes/dips as a function of $\alpha$. On the other hand, in the absence of Higgs renormalization [panel (b)], while $SS^1_{1,\text{no Higgs}}$ is still altered by the presence of $I_{2\omega_J}$ arising solely from higher-order Josephson processes [cf. Eq. \ref{['SS11n']}], for the chosen transparency this contribution is insufficient to generate noticeable deviations from the expected $\sim J_{-1}(\alpha)$ dependence. Panel (c) compares the exact numerical results from (a) (solid lines with markers) with the AA of Eq. \ref{['HiggsSS11']} (dashed lines). The AA is evaluated using the numerically obtained currents [Eq. \ref{['If']}] with $I_{\omega_J} = \Im(I_{20} - I_{-20})$ and $I_{2\omega_J} = \Im(I_{40} - I_{-40})$. Our goal is to test whether the $\alpha$-dependence of $S^1_1$ (but not that on the DC voltage) can be captured by the AA [Eq. \ref{['SS11n']}]. For small $\omega_J$ and $\omega_r$, the AA agrees well with the numerical results, reproducing the Higgs-induced deviations from the $J_{-1}(\alpha)$ profile shown in (b). Finally, panel (d) presents the amplitudes of the $\omega_J$ and $2\omega_J$ currents, with $I_{\omega_J} = I_2$ (red) and $I_{2\omega_J} = I_4$ (blue), both with and without Higgs renormalization, as functions of the bias voltage. $I_4$ decreases away from the resonance.