$\mathcal{P}\mathcal{T}$ symmetric phase transition and single-photon transmission in an optical trimmer system

The parity-time ($\mathcal{P}\mathcal{T}$) symmetric structures have exhibited potential applications in developing various robust quantum devices. In an optical trimmer with balanced loss and gain, we analytically study the $\mathcal{P}\mathcal{T}$ symmetric phase transition by investigating the spontaneous symmetric breaking. We also illustrate the single-photon transmission behaviors in both of the $\mathcal{P}\mathcal{T}$ symmetric and $\mathcal{P}\mathcal{T}$ symmetry broken phases. We find (i) the non-periodical dynamics of single-photon transmission in the $\mathcal{P}\mathcal{T}$ symmetry broken phase instead of $\mathcal{P}\mathcal{T}$ symmetric phase can be regarded as a signature of phase transition; and (ii) it shows unidirectional single-photon transmission behavior in both of the phases but comes from different underlying physical mechanisms. The obtained results may be useful to implement the photonic devices based on coupled-cavity system.