  Josephson junctions based on quantum dots offer a convenient tunability by
means of local gates. Here we analyze a Josephson junction based on a serial
double quantum dot in which the two dots are individually gated by
phase-shifted microwave tones of equal frequency. We calculate the
time-averaged current across the junction and determine how the phase shift
between the drives modifies the current-phase relation of the junction.
Breaking particle-hole symmetry on the dots is found to give rise to a finite
average anomalous Josephson current with phase bias between the superconductors
fixed to zero. This microwave gated weak link thus realizes a tunable "Floquet
$\varphi_{0}$-junction" with maximum critical current achieved for driving
frequencies slightly off-resonance with the energy cost of exciting a sub-gap
state on each dot. We provide numerical results supported by an analytical
analysis for infinite superconducting gap and weak inter-dot coupling. We
identify an interaction driven $0-\pi$ transition of anomalous Josephson
current as a function of driving phase difference. Finally, we show that this
junction can be tuned so as to provide for complete rectification of the
time-averaged Josephson current phase relation.
