  Over the years, supersymmetric quantum mechanics has evolved from a toy model
of high energy physics to a field of its own. Although various examples of
supersymmetric quantum mechanics have been found, systems that have a natural
realization are scarce. Here, we show that the extension of the conventional
Cooper-pair box by a 4pi-periodic Majorana-Josephson coupling realizes
supersymmetry for certain values of the ratio between the conventional
Josephson and the Majorana- Josephson coupling strength. The supersymmetry we
find is a "hidden" minimally bosonized supersymmetry that provides a
non-trivial generalization of the supersymmetry of the free particle and relies
crucially on the presence of an anomalous Josephson junction in the system. We
show that the resulting degeneracy of the energy levels can be probed directly
in a tunneling experiment and discuss the various transport signatures. An
observation of the predicted level degeneracy would provide clear evidence for
the presence of the anomalous Josephson coupling.
