Casimir-Josephson force on a point contact between two superconductors

We calculate the elongation or contraction force $F$ on a point contact (length $L$) connecting two superconductors with a phase difference $\phi$. When $L$ is small compared to the superconducting coherence length ${\xi }_0$ this force is given by $F=-({\mathrm{\Delta }}_0\mathrm{/}\pi {\xi }_0)\ln ({\xi }_0\mathrm{/}L)\cos \phi$ per spin-degenerate transverse mode. Quantum fluctuations in states from the continuous spectrum outside the superconducting gap ${\mathrm{\Delta }}_0$ give the dominant contribution to this force, which may be understood as the superconducting counterpart of the electromagnetic Casimir force. We compare with earlier work that only included contributions from the discrete spectrum of Andreev levels.