Spatially distributed PT symmetric refractive index using four-wave-mixing in a double-Λ setup

We propose a scheme for designing Parity-Time (PT) spatially symmetric structures of the refractive index for a generated probe field propagating in a double-Λ (DL) atomic medium. By creating position-dependent atomic coherences and manipulating them controllably—via applied control fields and one-photon detuning— we achieve PT symmetry of the refractive index. This causes the refractive index components to become symmetric/antisymmetric functions of the transverse coordinate. We demonstrate that, in order to achieve PT condition for DL, a four-wave mixing process can be employed, with one of the control fields and detuning being anti-symmetric functions of the transverse coordinates, while the other control field being either symmetric or constant. Furthermore, we illustrate that this PT symmetry holds for different distances along the propagation direction and analyze its dependence on the phase matching conditions.