  We propose how to realize nonreciprocity for a weak input optical field via
nonlinearity and synthetic magnetism. We show that the photons transmitting
from a linear cavity to a nonlinear cavity (i.e., an asymmetric nonlinear
optical molecule) exhibit nonreciprocal photon blockade but no clear
nonreciprocal transmission. Both nonreciprocal transmission and nonreciprocal
photon blockade can be observed, when one or two auxiliary modes are coupled to
the asymmetric nonlinear optical molecule to generate an artificial magnetic
field. Similar method can be used to create and manipulate nonreciprocal
transmission and nonreciprocal photon blockade for photons bi-directionally
transport in a symmetric nonlinear optical molecule. Additionally, a photon
circulator with nonreciprocal photon blockade is designed based on nonlinearity
and synthetic magnetism. The combination of nonlinearity and synthetic
magnetism provides us an effective way towards the realization of quantum
nonreciprocal devices, e.g., nonreciprocal single-photon sources and
single-photon diodes.
