Higgs modes in proximized superconducting systems

The proximity effect in hybrid superconducting–normal-metal structures is shown to affect strongly the coherent oscillations of the superconducting order parameter $\mathrm{\Delta }$ known as the Higgs modes. The standard Higgs mode at frequency $2\mathrm{\Delta }$ is damped exponentially by the quasiparticle leakage from the primary superconductor. Two new Higgs modes with the frequencies depending on both the primary and induced gaps in the hybrid structure are shown to appear due to the coherent electron transfer between the superconductor and the normal metal. Altogether, these three modes determine the long-time asymptotic behavior of the superconducting order parameter disturbed either by the electromagnetic pulse or the quench of the system parameters and thus are of crucial importance for the dynamical properties and restrictions on the operating frequencies for superconducting devices based on the proximity effect used, e.g., in quantum computing, in particular, with topological low-energy excitations.

Figure 1

Illustration of three Higgs modes in SIN structure, being coherent Cooper pair splitting-recovery processes with three different frequencies $\omega$, decay rates, and amplitudes $A$.

Figure 2

Diagrams for the Green's functions in the superconductor and the normal metal. (a) Exact Dyson equation for the Green's functions (before averaging). (b) Self-consistent Born approximation for the averaged Green's function. The $\times$ symbol denotes tunneling; the dashed line denotes the correlator between the matrix elements of the tunneling operator.

Figure 3

The real and imaginary parts of ${[1-K_{\mathrm{\Delta }}\left(\omega \right)]}^{-1}$ in the case of the finite tunneling rates ${\mathrm{\Gamma }}_s$ and ${\mathrm{\Gamma }}_n$. The inset shows the singularity at frequency $2{\mathrm{\Delta }}_i$ which cannot be observed in the regular scale. For the given parameters ${\mathrm{\Gamma }}_s=0.1{\mathrm{\Delta }}_0$ and ${\mathrm{\Gamma }}_n=0.3{\mathrm{\Delta }}_0$ the induced gap ${\mathrm{\Delta }}_i$ is approximately equal to $0.2{\mathrm{\Delta }}_0$. Here ${\mathrm{\Delta }}_{\mathrm{\Sigma }}={\mathrm{\Delta }}_0+{\mathrm{\Delta }}_i$. The dashed line shows ${[1-K_{\mathrm{\Delta }}\left(\omega \right)]}^{-1}$ for the isolated superconductor ${\mathrm{\Gamma }}_s={\mathrm{\Gamma }}_n=0$.

Figure 4

Area of analyticity of the kernel $K_{\mathrm{\Delta }}$. The four black dots are the branch points $\omega =\pm {\mathrm{\Delta }}_0-i{\mathrm{\Gamma }}_s$ and $\omega =\pm 2{\mathrm{\Delta }}_0-2i{\mathrm{\Gamma }}_s$; the thick black lines are the branch cuts. The thick gray line shows the integration contour in Eq. (41).