Josephson effect between superconducting nanograins with discrete energy levels

TL;DR

This paper explores the Josephson effect between superconducting nanograins with discrete energy levels, revealing a reentrant behavior of the Josephson energy as a function of level spacing using advanced numerical methods.

Contribution

It introduces a density matrix renormalization group approach to study Josephson effects in regimes where traditional BCS theory fails, highlighting the reentrant behavior of Josephson energy.

Findings

Josephson energy exhibits reentrant behavior with increasing level spacing.

DMRG accurately models ground states of coupled nanograins.

Tight-binding approximation explains the reentrant mechanism.

Abstract

We investigate the Josephson effect between two coupled superconductors, coupled by the tunneling of pairs of electrons, in the regime that their energy level spacing is comparable to the bulk superconducting gap, but neglecting any charging effects. In this regime, BCS theory is not valid, and the notion of a superconducting order parameter with a well-defined phase is inapplicable. Using the density matrix renormalization group, we calculate the ground state of the two coupled superconductors and extract the Josephson energy. The Josephson energy is found to display a reentrant behavior (decrease followed by increase) as a function of increasing level spacing. For weak Josephson coupling, a tight-binding approximation is introduced, which illustrates the physical mechanism underlying this reentrance in a transparent way. The DMRG method is also applied to two strongly coupled superconductors and allows a detailed examination of the limits of validity of the tight-binding model.