  Gate-tunable Josephson junctions embedded in a microwave environment provide
a promising platform to in-situ engineer and optimize novel superconducting
quantum circuits. The key quantity for the circuit design is the
phase-dependent complex admittance of the junction, which can be probed by
sensing an rf SQUID with a tank circuit. Here, we investigate a graphene-based
Josephson junction as a prototype gate-tunable element enclosed in a SQUID loop
that is inductively coupled to a superconducting resonator operating at 3 GHz.
With a concise circuit model that describes the dispersive and dissipative
response of the coupled system, we extract the phase-dependent junction
admittance corrected for self-screening of the SQUID loop. We decompose the
admittance into the current-phase relation and the phase-dependent loss and as
these quantities are dictated by the spectrum and population dynamics of the
supercurrent-carrying Andreev bound states, we gain insight to the underlying
microscopic transport mechanisms in the junction. We theoretically reproduce
the experimental results by considering a short, diffusive junction model that
takes into account the interaction between the Andreev spectrum and the
electromagnetic environment, from which we deduce a lifetime of ~17 ps for
non-equilibrium populations.
