Casimir Effect in Josephson Junctions

In a Josephson junction, the supercurrent is determined by both the discrete sub-gap part of the spectrum due to Andreev bound states and the continuous part of the spectrum from energy states outside the superconducting gap. We consider the cohesive force exerted on a junction, which is thermodynamically conjugated to the superflow, and comment on its connection to the Casimir effect in quantum electrodynamics. In contrast to the supercurrent, it is shown that in ballistic short junctions, the force is predominantly contributed by the continuum. Its magnitude is universally defined by the energy gap and coherence length of the superconductor per spin-dependent transverse mode. This force scales non-analytically with the junction length and is periodic with the superconducting phase. For long ballistic junctions, the force results from the interplay of oscillatory contributions originating from both bound states and the continuum. The resulting asymptotic limit for the force is established, including the correction terms. Thermal and impurity effects on the force are briefly discussed.