Josephson effect through a quantum dot array

TL;DR

This paper investigates the ground state and Josephson effect in a quantum dot array connected to superconductors, revealing phase diagrams, even-odd effects, and the influence of superconductivity on the system's properties.

Contribution

It introduces a finite Hubbard chain model with effective local pairing to analyze Josephson coupling sign inversion and system size effects in quantum dot arrays.

Findings

Identification of conditions for $$-junction behavior.

Phase diagrams showing even-odd effects as system size increases.

Scaling relations of Josephson current with system size.

Abstract

We analyze the ground state properties of an array of quantum dots connected in series between superconducting electrodes. This system is represented by a finite Hubbard chain coupled at both ends to BCS superconductors. The ground state is obtained using the Lanczos algorithm within a low energy theory in which the bulk superconductors are replaced by effective local pairing potentials. We study the conditions for the inversion of the sign of the Josephson coupling ($\pi$-junction behavior) as a function of the model parameters. Results are presented in the form of phase diagrams which provide a direct overall view of the general trends as the size of the system is increased, exhibiting a strong even-odd effect. The analysis of the spin-spin correlation functions and local charges give further insight into the nature of the ground state and how it is transformed by the presence of superconductivity in the leads. Finally we study the scaling of the Josephson current with the system size and relate these results with previous calculations of Josephson transport through a Luttinger liquid.