  It is often conjectured that quantum synchronisation and entanglement are two
independent properties which two coupled quantum systems may not exhibit at the
same time. However, as both these properties can be understood in terms of the
second order moments of a set of conjugate quadratures, there may exist
specific conditions for simultaneous existence of entanglement and quantum
synchronization. Here we present a theoretical scheme to achieve the same
between two mechanical oscillators, which are indirectly coupled with each
other via a coupling between two cavities. We show that in the presence of the
cavity-oscillator coupling, quadratically varying with their displacements,
these oscillators can be synchronized in the quantum sense and entangled as
well, at times much longer than the decay time-scale of the cavity modes.
Precisely speaking, we show that in the presence of quadratic coupling,
entanglement criterion and quantum synchronization measure are simultaneously
satisfied in steady state. This behaviour can be observed for a range of
quadratic coupling, temperature, and frequency difference of the two
oscillators.
