Fractionalization of a flux quantum in a one-dimensional parallel Josephson junction array with alternating $\pi$ junctions

We study numerically and analytically the properties of a one-dimensional array of parallel Josephson junctions in which every {\em alternate} junction is a $\pi$ junction. In the ground state of the array, each cell contains spontaneous magnetic flux $\Phi\leq\Phi_{0}/2$ which shows {\em antiferromagnetic} ordering along the array. We find that an externally introduced $2\pi$-fluxon $\Phi_{0}$ in such an array is unstable and fractionalizes into two $\pi$ fluxons of magnitude ${1/2}\Phi_{0}$. We attribute this fractionalization to the degeneracy of the ground state of the array. The magnitude of the flux in the fractional fluxons can be controlled by changing the critical current of the $\pi$ junctions relative to the 0 junctions. In the presence of an external current, the fluxon lattice in the antiferromagnetic ground state can be depinned. We also observe a novel resonant structure in the $V$-$I$ characteristics above the depinning current due to the interaction between the fluxon lattice and the array.