  The density of states of proximitized normal nanowires interrupting
superconducting rings can be tuned by the magnetic flux piercing the loop.
Using these as the contacts of a single-electron transistor allows to control
the energetic mirror asymmetry of the conductor, this way introducing
rectification properties. In particular, we show that the system works as a
diode that rectifies both charge and heat currents and whose polarity can be
reversed by the magnetic field and a gate voltage. We emphasize the role of
dissipation at the island. The coupling to substrate phonons enhances the
effect and furthermore introduces a channel for phase tunable conversion of
heat exchanged with the environment into electrical current.

Mesoscale and Nanoscale Physics