[Paper Review] Microwave response of an NS ring coupled to a superconducting resonator

F. Chiodi, M. Ferrier+6 more|arXiv (Cornell University)|May 3, 2010

Physics of Superconductivity and Magnetism11 citations

TL;DR

This study investigates the microwave response of a superconducting ring with a normal-metal (NS) junction coupled to a multimode superconducting resonator. Using linear response measurements at GHz frequencies, it reveals that the dissipative and non-dissipative current responses follow a Debye relaxation law with a characteristic time of ~0.6 ns, attributed to the diffusion time through the normal region. The key finding is that the flux-dependent response exhibits strong harmonic content due to frozen Andreev level populations, deviating significantly from the flux derivative of the Josephson current.

ABSTRACT

A long phase coherent normal (N) wire between superconductors (S) is characterized by a dense phase dependent Andreev spectrum . We probe this spectrum in a high frequency phase biased configuration, by coupling an NS ring to a multimode superconducting resonator. We detect a dc flux and frequency dependent response whose dissipative and non dissipative components are related by a simple Debye relaxation law with a characteristic time of the order of the diffusion time through the N part of the ring. The flux dependence exhibits $h/2e$ periodic oscillations with a large harmonics content at temperatures where the Josephson current is purely sinusoidal. This is explained considering that the populations of the Andreev levels are frozen on the time-scale of the experiments.

Motivation & Objective

• To probe the high-frequency dynamics of Andreev states in a long diffusive normal-superconductor (NS) ring under phase bias.
• To understand the origin of dissipative and non-dissipative current responses at GHz frequencies when the system is out of equilibrium.
• To determine whether the relaxation time in the response is governed by electron-electron, electron-phonon, or diffusion processes.
• To test the hypothesis that Andreev level populations are frozen on the experimental timescale, leading to harmonic content in the flux response.

Proposed method

• The NS ring is inductively coupled to a high-quality (Q ~ 80,000) multimode superconducting resonator operating between 300 MHz and 6 GHz.
• The resonator's frequency and quality factor are measured as functions of the dc flux through the ring, enabling extraction of the real (χ′) and imaginary (χ′′) parts of the current response.
• The experimental data are fitted using a Debye relaxation model: χ(ω,φ,T) = χ_d(φ,T)/(1 + iωτ), with τ ≈ 0.6 ns.
• χ_d(φ,T) is calculated from the flux derivative of the Josephson current, accounting for phase-dependent Andreev level energies and frozen populations.
• Theoretical modeling uses the Usadel equations to compute the temperature- and flux-dependent background contribution F(φ,T), which is subtracted from the data.
• The system is cooled to millikelvin temperatures (20 mK) to suppress thermal fluctuations and enhance sensitivity.

Experimental results

Research questions

• RQ1 What is the origin of the dissipative response in the microwave regime for an NS ring with a long normal region?
• RQ2 Why does the flux-dependent response contain multiple harmonics despite the dc Josephson current being sinusoidal at high temperature?
• RQ3 Is the characteristic relaxation time τ ≈ 0.6 ns related to the diffusion time τ_D through the normal wire?
• RQ4 Can the observed response be explained by frozen populations of Andreev levels that cannot follow rapid phase oscillations?
• RQ5 Does the relaxation dynamics reflect electron-electron scattering, electron-phonon coupling, or diffusion processes?

Key findings

• The dissipative (χ′′) and non-dissipative (χ′) current responses follow a Debye relaxation law with a characteristic time constant τ ≈ 0.6 ns.
• The relaxation time τ is consistent with the diffusion time τ_D through the normal region, indicating that the dynamics are governed by electron diffusion.
• The flux dependence of both χ′ and χ′′ exhibits strong harmonic content, even though the dc Josephson current is sinusoidal at T = 1 K.
• The harmonic content arises because Andreev level populations are frozen on the GHz timescale, preventing them from adiabatically following the phase oscillations.
• The ratio δχ′′/δχ′ shows linear dependence on frequency (2πf), confirming the validity of the Debye relaxation model.
• The observed response cannot be explained by equilibrium population relaxation (τ_in ~ 100 ns), indicating a faster, non-equilibrium relaxation mechanism related to diffusion.