  In this work, we consider superconductor/flat band material/superconductor
(S/FB/S) Josephson junctions (JJs) where the flat band material possesses
isolated flat bands with exactly zero Fermi velocity. Contrary to conventional
S/N/S JJs where the critical Josephson current vanishes when the Fermi velocity
goes to zero, we show in this work that the critical current in the S/FB/S
junction is controlled by the quantum metric length $\xi_\mathrm{QM}$ of the
flat bands. Microscopically, when $\xi_\mathrm{QM}$ of the flat band is long
enough, the interface bound states originally localized at the two S/FB, FB/S
interfaces can penetrate deeply into the flat band material and hybridize to
form Andreev bound states (ABSs). These ABSs are able to carry long range and
sizable supercurrents. Importantly, $\xi_\mathrm{QM}$ also controls how far the
proximity effect can penetrate into the flat band material. This stands in
sharp contrast to the de Gennes' theory for S/N junctions which predicts that
the proximity effect is expected to be zero when the Fermi velocity of the
normal metal is zero. We further suggest that the S/FB/S junctions would give
rise to a new type of resonant Josephson transistors which can carry sizable
and highly gate-tunable supercurrent.
