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

TL;DR

The paper develops a microscopic framework to quantify power attenuation in superconducting rectangular waveguides from millimeter-wave to terahertz frequencies, applicable across the full range of mean free paths using the Eilenberger/Keldysh formalism. It provides a compact TLS-induced attenuation expression and evaluates $\alpha$ for NbN, Nb$_3$Sn, and Nb across standard WR geometries, revealing that in the high-frequency, clean regime attenuation is minimized and TLS losses are negligible except at very low $T$. In the nonlinear regime, a Higgs-mode–driven peak in attenuation appears near $f=\Delta/h$ due to Kerr-type nonlinearity, presenting a distinct experimental signature. These results offer material- and geometry-specific guidelines for ultra-low-loss cryogenic interconnects in astronomy and quantum technologies and identify regimes where TLS and nonlinear effects govern performance.

Abstract

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 $100~\mathrm{GHz}$--THz range, applicable to arbitrary electronic mean free paths $\ell$ from the dirty limit $\ell\llξ_0$ to the clean limit $\ell\ggξ_0$. 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 $\ell\gtrsimξ_0$, 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 $T/T_c\lesssim 0.1$-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 $f\simeq Δ/h$ 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.

Figures (12)

• Figure 1: Schematic of a superconducting rectangular waveguide. The wall thickness is assumed to be much larger than the electromagnetic-field penetration depth of the superconducting walls.
• Figure 2: Complex conductivity as a function of frequency for different temperatures, calculated for (a, b) the clean case and (c, d) the moderately dirty case.
• Figure 3: Surface resistance as a function of frequency for different temperatures, calculated for (a) clean and (b) moderately dirty cases.
• Figure 4: Surface resistance as a function of frequency for different mean free paths, ranging from relatively dirty to relatively clean regimes, calculated for (a,b) $T/T_c = 0.2$, (c) $0.5$, and (d) $0.7$.
• Figure 5: Attenuation constant $\alpha$ in (a, b) NbN and (c, d) Nb$_3$Sn rectangular waveguides as a function of frequency $f=2\pi \omega$, calculated for $T/T_c=0.25$ and different mean free paths ranging from clean to dirty cases ($\ell/\xi_0 = 10, 3, 1, 0.3$). Each curve is plotted from 5% above the cutoff frequency of the fundamental ${\rm TE}_{10}$ mode to 5% below the cutoff frequency of the next higher-order mode. The corresponding waveguide designations are given in Table \ref{['Table_waveguide']}.