Reflection by two level system: Phase singularities on the Poincaré hypersphere

We consider the reflection of a photon by a two-level system in a quasi-one-dimensional waveguide. The waveguide polarisation at the location of the two-level system and the transition dipole are key determinants of the physics, controlling of the phase and amplitude of the scattered light in both directions. In most cases full control is possible by tuning only one of these two degrees of freedom. In reverse, this enables unique characterisation of the dipole from measurements of the scattered light. Phase singularities occur where the reflection coefficient is zero, with the (hyper-)spherical parameter space determining the dynamics of these singularities.