  We explore the chiral dynamics in a parity-time-symmetric system consisting
of a giant Kerr cavity nonlocally coupled to a one-dimensional waveguide. By
tuning the phase difference between the two coupling points to match the
propagation phase at the driving frequency, chiral cavity-waveguide
interactions are achieved, enabling the deterministic generation of photons
with nontrivial statistics only for a single incident direction. This
nontrivial-statistical photons can be produced even in the strong dissipation
regime due to the interference between reflected and transmitted photons
propagating between the coupling points. Our investigation encompasses a broad
range of distances between the coupling points, incorporating non-Markovian
effects. Notably, at a phase difference of $\pi/2$, the system's dynamics
become exactly Markovian, while the output field retains non-Markovian
characteristics. Under these conditions, we analyze nonreciprocal dissipative
phase transitions driven by a strong external field and elucidate the influence
of the non-Markovian effect. Our results offer valuable insights for the
advancement of nonreciprocal photon devices and deterministic photon
generations, providing a deeper understanding of dissipative phase transitions.
