Space Time Reflection Symmetry in the Jones Formalism in Optics

We provide a description of the parity space reflection-time reversal (PT) operator acting on the two-dimensional polarization space of light represented by the linear algebra of Jones vectors and matrices. We establish the form of a PT-symmetric Jones matrix. We present two examples of laser resonators whose polarization eigenstates are described by PT-symmetric Jones matrices: one is based on the Faraday effect and a dichroic attenuation, while the other is made of twisted anisotropic mirrors. Both possess a control parameter that experimentally covers the exact, the broken PT-symmetry regions and their boundary, called an exceptional point, where the eigenstates of the resonator coalesce into a single state. The exact PT-symmetric region produces laser polarization modes emitting at the same frequency with different intracavity losses, while the broken PT-symmetric region features polarization modes emitting at distinct frequencies with the same intracavity losses. By applying unitary transformations, the concept of PT-symmetric Jones matrix is extended to matrices that commute with any antiunitary operator, thereby opening the prospect of a larger family of resonators geometries that also feature real or complex-conjugate spectra.