Coherent frequency down-conversions and entanglement generation in a Sagnac interferometer

A waveguide loop coupled to two external line waveguides by a 50/50 beam splitter forms a Sagnac interferometer. We consider the situation where two Λ-type three-level emitters are symmetrically coupled to the loop of a Sagnac interferometer and a single photon is input through one end of the line waveguides. Since the incoming photon is always in a superposition of the clockwise and counterclockwise modes of the loop and the two emitters are positioned symmetrically with respect to the input port of photon, the processes of photon scattering at the two emitters are symmetric and coherent. When the separation of the emitters and the coupling strengths of the emitters with the waveguide loop take some special values, due to quantum interference, a frequency down-conversion can certainly happen at one of the two emitters during the photon scattering but one cannot know at which emitter the frequency down-conversion takes place. This indistinguishability of the coherent frequency down-conversion processes can result in the generation of the symmetric or antisymmetric two-qubit maximally entangled states of the emitters. In the present scheme, a single photon comes in and goes out of the waveguide loop, and no photon localization modes exists. The entangled states result from the coherent frequency down-conversion processes of the emitters. Thus, the resulting entangled states are stable if the two lower-lying states of the emitters have no decay. We also investigate the influence of the dissipation of the emitters and the finite bandwidth of an input photon wavepacket on the success probability of entanglement generation, and find that the present scheme is robust to these effects and feasible with current available technologies.