Theory of theπState inH3eJosephson Junctions

Abstract

The flow of superfluid ${}^{3}\mathrm{He}\ensuremath{-}B$ through a $65\ifmmode\times\else\texttimes\fi{}65$ array of nanometer-size apertures has been measured recently by Backhaus et al. They find in the current-phase relation a new branch, the so-called $\ensuremath{\pi}$ state. We study two limiting cases which show that the $\ensuremath{\pi}$ state arises from the coupling of the phase degree of freedom to the spin-orbit rotation. The $\ensuremath{\pi}$ state exists in a single large aperture, but is difficult to observe because of hysteresis. A better correspondence with experiments is obtained by assuming a thin wall, where the Josephson coupling between the two sides arises from a dense array of pinholes.