$\mathrm{PT}$-phase diagram with quantum jump in a non-Hermitian photonic structure

Quantum jumps induced by gain and loss have different quantum behaviors in non-Hermitian photonics. However, the jump effect on global quantum $\mathrm{PT}$ properties has not been comprehensively understood yet. Here, with considering quantum jump of loss and gain in a photonic dimer structure, we analytically obtained the $\mathrm{PT}$-phase diagram under the steady-state condition and defined a Hermitian exchange operator to characterize the $\mathrm{PT}$-symmetry or -broken phase. When we input Fock states into a $\mathrm{PT}$-broken bi-waveguide splitting system, most photons will concentrate in the dominant waveguide with some state distributions. Especially in Hong-Ou-Mandel interferences, if gain is added, whether loss exists or not, $g^{\left(2\right)}$ is always larger than zero because loss cannot compensate gain as their different quantum jump effects. The quantum $\mathrm{PT}$-phase diagram paves the way to the quantum state engineering, quantum interferences, and logic operations in non-Hermitian photonic systems.