Passive $\mathcal{P}\mathcal{T}$-symmetric couplers without complex optical potentials

Abstract

In addition to the implementation of parity-time ($\mathcal{P}\mathcal{T}$)-symmetric optical systems by carefully and actively controlling the gain and loss, we show that a $2\times 2$ $\mathcal{P}\mathcal{T}$-symmetric Hamiltonian has a unitarily equivalent representation without complex optical potentials in the resulting optical coupler. Through the Naimark dilation in operator algebra, passive $\mathcal{P}\mathcal{T}$-symmetric couplers can thus be implemented with a refractive index of real values and asymmetric coupling coefficients. This opens up the possibility to implement general $\mathcal{P}\mathcal{T}$-symmetric systems with state-of-the-art asymmetric slab waveguides, dissimilar optical fibers, or cavities with chiral mirrors.