Quantum dot attached to superconducting leads: Relation between symmetric and asymmetric coupling

TL;DR

This paper demonstrates that asymmetric coupling in a quantum dot-superconductor system can be related to a symmetric case, simplifying analysis and revealing limitations of Kondo universality in asymmetric setups.

Contribution

It provides conversion formulas linking asymmetric and symmetric couplings, and applies these to experimental data to analyze the 0-π transition in quantum dots.

Findings

Symmetric and asymmetric couplings are related through simple formulas.

Conversion formulas accurately predict the 0-π transition boundary.

Kondo universality does not hold in asymmetric junctions.

Abstract

We study the Anderson single-level quantum dot attached to two BCS superconducting leads with the same gap size. We reveal that a system with asymmetric tunnel coupling to the leads ($\Gamma_{L}\neq\Gamma_{R}$) can be related to the symmetric system with the same net coupling strength $\Gamma=\Gamma_{L}+\Gamma_{R}$. Surprisingly, it is the symmetric case which is the most general, meaning that all physical quantities in case of asymmetric coupling are fully determined by the symmetric ones. We give ready-to-use conversion formulas for the $0-\pi$ phase transition boundary, on-dot quantities, and the Josephson current, and illustrate them on the NRG results of Oguri, Tanaka and Bauer [Phys. Rev. B 87, 075432 (2013)] for the three-terminal setup. We apply our theory to the recent $0-\pi$ transition measurement of Delagrange et al. [Phys. Rev. B 93, 196437 (2016)] and determine the asymmetry of the experimental setup from the measured transition width. Finally, we establish that the widely assumed Kondo "universality" of physical quantities depending only on the ratio of the Kondo temperature and the superconducting gap $T_{K}/\Delta$ cannot hold for asymmetric junctions.