Power attenuation in millimeter-wave and terahertz superconducting rectangular waveguides: linear response, TLS loss, and Higgs-mode nonlinearity

Superconducting waveguides are a promising platform for ultralow-loss transmission in the millimeter-wave to terahertz band under cryogenic conditions, with potential applications in astronomical instrumentation and emerging quantum technologies. We develop a framework, based on microscopic superconductivity theory, to evaluate the power-flow attenuation constant α of superconducting rectangular waveguides in the 100 GHz--THz range, applicable to arbitrary electronic mean free paths ell from the dirty limit ellllξ₀ to the clean limit ellggξ₀. We also derive an analytical expression for two-level-system (TLS)-induced attenuation α_{rm TLS} in thin native oxide layers within the standard TLS model. Using this framework, we perform numerical evaluations of α for representative materials over standard waveguide sizes from WR15 to WR1. In the high-frequency regime f gtrsim 0.5 Δ/h, low attenuation favors the clean regime ellgtrsimξ₀, indicating that high-purity materials can achieve very low attenuation below their gap frequency. For the TLS contribution, using parameter values representative of native Nb oxides, we find that α_{rm TLS} can become relevant at sufficiently low temperatures T/T_clesssim 0.1-0.2, where quasiparticle dissipation is exponentially suppressed. Finally, we extend the discussion to the strong-excitation regime using a recently developed nonlinear-response theory within the Keldysh--Usadel framework of nonequilibrium superconductivity and show that nonlinear dissipation produces a Higgs-mode peak in α near fsimeq Δ/h via a Kerr-type nonlinearity of the dissipative conductivity. This peak provides a distinct hallmark of the Higgs mode that has been largely overlooked so far.