Tunable quantum logic gate on photonic qubits with a ladder emitter

We describe how a ladder emitter can implement tunable quantum logic gate on photonic qubits encoded in the frequency basis. The ground-to-first excited state of interacts with control photon, and first-to-second transition target photon. By controlling relative detuning between photon emitter, we enable any controlled-phase operation from 0 to π. derive analytical formulas for performance through S-matrix formalism as well mechanism intuitively. This is deterministic, does not utilize active control, needs only single enabling low-footprint more efficient decomposition circuits, especially Fourier transform. suggest multiple potential systems physical realization our proposal, such lanthanide ions embedded Purcell-enhanced cavities. expect these results motivate further interest information processing designer emitters.