Tunable quantum logic gate on photonic qubits with a ladder emitter

We present a scheme to implement a passive and deterministic controlled-variable phase gate on photonic qubits encoded in the frequency basis. Our gate employs a cascade system with the ground to first excited state interacting with the control photon of a given polarization, and the first to second excited state transition interacting with the target photon of the orthogonal polarization. By controlling the relative detuning between the target photon and the frequency of the transition between the first and second excited states of the cascade emitter, we enable any controlled-phase operation from 0 to $\pi$. This gate does not utilize any active control and needs only a single cascade emitter, enabling low-footprint and more efficient decomposition of quantum circuits, especially those rooted in the quantum Fourier transform.