  Sensitive signal detection and processing in classical world, especially in
quantum regime, require nonreciprocal manipulation. In this paper we show how
to achieve nonreciprocal quantum synchronization for two magnon modes in a
two-sublattice antiferromagnet with strong isolation.The antiferromagnet is
trapped in a cavity with two posts so that the two magnon modes not only couple
to each other through a parametric-type interaction, but also interact with a
same cavity respectively in a beam splitter-type and parametric-type ways.
Under the condition of system's stability, we show that nonreciprocal quantum
synchronization between two magnon modes is mainly dependent on resonance
frequency of the cavity caused by direction of input currents. In
addition,quantum synchronization is enhanced by the increase of interaction
strength between two Bogoliubov modes and cavity mode. Moreover, numerical
simulation results with parameters commonly used in current experiments show
that the present scheme may be feasible.
