Tunable quantum logic gate on photonic qubits with a ladder emitter

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