Detecting the relative phase between different frequency components of a photon using a three-level $\Lambda$ atom coupled to a waveguide

We study the scattering of a single photon propagating along a waveguide in an arbitrary superposition state two frequencies with a single three-level $\Lambda$ atom in a superposition of two non-degenerate ground states where the atom is coupled to a waveguide. We find that the scattering depends on both the relative phase between the photon frequencies and the relative phase between the atomic ground states. Our results show that a three-level $\Lambda$ atom coupled to a waveguide can be used as photon phase filter that could detect the relative phase between the two frequencies of the photon superposition state.