  We show that Floquet chiral topological superconductivity arises naturally in
Josephson junctions made of magnetic topological insulator-superconductor
sandwich structures. The Josephson phase modulation associated with an applied
bias voltage across the junction drives the system into the anomalous Floquet
chiral topological superconductor hosting chiral Majorana edge modes in the
quasienergy spectrum, with the bulk Floquet bands carrying zero Chern numbers.
The bias voltage acts as a tuning parameter enabling novel dynamical
topological quantum phase transitions driving the system into a myriad of
exotic Majorana-carrying Floquet topological superconducting phases. Our theory
establishes a new paradigm for realizing Floquet chiral topological
superconductivity in solid-state systems, which should be experimentally
directly accessible.
