  Spectroscopy of a Josephson junction device with an embedded quantum dot
reveals the presence of a contribution to level splitting in external magnetic
field that is proportional to $\cos \phi$, where $\phi$ is the gauge-invariant
phase difference across the junction. To elucidate the origin of this
unanticipated effect, we systematically study the Zeeman splitting of spinful
subgap states in the superconducting Anderson impurity model. The magnitude of
the splitting is renormalized by the exchange interaction between the local
moment and the continuum of Bogoliubov quasiparticles in a variant of the
Knight shift phenomenon. The leading term in the shift is linear in the
hybridisation strength $\Gamma$ (quadratic in electron hopping), while the
subleading term is quadratic in $\Gamma$ (quartic in electron hopping) and
depends on $\phi$ due to spin-polarization-dependent corrections to the
Josephson energy of the device. The amplitude of the $\phi$-dependent part is
largest for experimentally relevant parameters beyond the perturbative regime
where it is investigated using numerical renormalization group calculations.
Such magnetic-field-tunable coupling between the quantum dot spin and the
Josephson current could find wide use in superconducting spintronics.
