  High quality factors, strong nonlinearities, and extensive design flexibility
make superconducting circuits an ideal platform to investigate synchronization
phenomena deep in the quantum regime. Recently~\cite{Loerch-2017}, it was
predicted that energy quantization and conservation can block the
synchronization of two identical, weakly coupled nonlinear self-oscillators.
Here we propose a Josephson junction circuit realization of such a system along
with a simple homodyne measurement scheme to observe this effect. We also show
that at finite detuning, where phase synchronization takes place, the two
oscillators are entangled in the steady state as witnessed by the positivity of
the logarithmic negativity.
