  We study the nonreciprocal transmission of a single-photon in a cavity
optomechanical system, in which the cavity supports a clockwise and a
counter-clockwise circulating optical modes, the mechanical resonator (MR) is
excited by a weak coherent driving, and the signal photon is made up of a
sequence of pulses with exactly one photon per pulse. We find that, if the
input state is a single-photon state, it is insufficient to study the
nonreciprocity only from the perspective of the transmission spectrums, since
the frequencies where the nonreciprocity happens are far away from the peak
frequency of the single-photon. So we show the nonreciprocal transmission
behavior by comparing the spectrums of the input and output fields. In our
system, we can achieve a transformation of the signal transmission from
unidirectional isolation to unidirectional amplification in the single-photon
level by changing the amplitude of the weak coherent driving. The effects of
the mechanical thermal noise on the single-photon nonreciprocal transmission
are also discussed.
