Evanescent-wave Johnson Noise from Superconductors
We compute the evanescent-wave Johnson noise (EWJN) in the vacuum half-space above a superconductor, and the resulting relaxation time ($T_1$) of spin and charge qubits placed at nanometer distances from the surface. The electromagnetic response is described by a single microscopic transverse current-response kernel $Q(q, ω)$ for a BCS superconductor. This is computed for varying densities of both non-magnetic impurities and magnetic impurities, for arbitrary frequency and temperature and for wave vectors $q \ll k_F$ (the Fermi wavevector). When combined with the fluctuation-dissipation theorem and the nonlocal surface impedances of the half-space, this yields the magnetic and electric field noise at any distance $z \gg k_F^{-1}$ from the surface, from which we obtain $T_1$. Just below $T_c$ the magnetic noise is enhanced relative to the normal state by the coherence (Hebel-Slichter-type) peak of the dissipative conductivity and drops exponentially at lower temperatures; the electric noise shows no coherence peak. The theory predicts that there is a zero-temperature noise floor induced by magnetic impurities. In the gapless regime produced by pair breaking, the finite subgap density of states $ν(0)$ yields a temperature-independent noise spectral density and a relaxation rate bounded by $T_1^{-1}(T)\le[ν(0)/ν_F]^{2}\,T_{1,N}^{-1}(T)$ for $T\ll T_c$, with equality in the extreme nonlocal regime. Here $ν(0)$ and $ν_F$ are the superconducting and normal-state densities of states at the Fermi energy, and $T_{1,N}(T)$ is the relaxation time the same electrode would produce in its normal state at the same temperature.