Subgap states in dirty superconductors and their effect on dephasing in Josephson qubits

TL;DR

This paper develops a theory for subgap states in dirty superconductors with magnetic impurities, analyzing their impact on dephasing in Josephson qubits, and finds that even weak impurities can significantly limit qubit coherence.

Contribution

The paper introduces a comprehensive theory for subgap tails in diffusive superconductors with magnetic impurities, including mesoscopic and long-wave fluctuation effects, and applies it to qubit dephasing.

Findings

Magnetic impurities at 1 ppm affect large Josephson qubits.

Subgap states contribute to qubit dephasing and limit quality factor.

Dimensional reduction relates different fluctuation contributions.

Abstract

We present a theory of the subgap tails of the density of states in a diffusive superconductor containing magnetic impurities. We show that the subgap tails have two contributions: one arising from mesoscopic gap fluctuations, previously discussed by Lamacraft and Simons, and the other associated to the long-wave fluctuations of the concentration of magnetic impurities. We study the latter both in small superconducting grains and in bulk systems [$d=1,2,3$], and establish the dimensionless parameter that controls which of the two contributions dominates the subgap tails. We observe that these contributions are related to each other by dimensional reduction. We apply the theory to estimate the effects of a weak concentration of magnetic impurities [$\approx 1 {\rm p.p.m}$] on the phase coherence of Josephson qubits. We find that at these typical concentrations, magnetic impurities are relevant for the dephasing in large qubits, designed around a $10 {\rm \mu m}$ scale, where they limit the quality factor to be $Q<10^4-10^5$.