  Topological phases exhibit properties that are conserved for continuous
deformations, as demonstrated in topological protections in condensed-matter
physics and electromagnetic waves. Despite its ubiquitous nature and recent
extensions to synthetic dimensions, non-Hermitian Hamiltonians, and nonlinear
dynamics, topological protection has generally been described in spatial
lattices with the Chern number in the Brillouin zone, focusing on the
realization of backscattering-free wave transport. Here, we investigate a
different class of topological protection in parity-time-symmetric nonlinear
optical dynamics, exploiting the topological invariance of optical state
trajectories. For coupled nonlinear photonic systems composed of gain and loss
atoms, we classify the topology of equilibria separately for unbroken and
broken parity-time symmetry. Utilizing the immunity of topological phases
against temporal perturbations, we develop noise-immune laser modulation and
rectification with a parasitic nonlinear resonator based on oscillation
quenching mechanisms that are protected by parity-time symmetry. The connection
between topological photonics and parity-time symmetry through nonlinear
dynamics provides a powerful platform for noise-immune signal processing.
