Models of beam splitters for frequency-encoded photonic qubits

The ubiquity of off-the-shelf optical elements has made the photonic platform a viable and flexible route towards large-scale quantum devices. By encoding the qubits in the frequency of the photons, the hardware footprint of this architecture can be greatly reduced, thus facilitating the on-chip realization of these devices. In this encoding, generating bit-flips and phase-shifts of the qubits, easily implementable for the spatial-mode encoding via linear optical elements, i.e., beam splitters and phase shifters, now requires external drives as one must change the photon's energy. We present simple designs for frequency-mode beam splitters of variable ratio based on driven arrays of ring resonators. Using the SLH formalism for quantum input-output networks we develop a methodology to construct effective transfer matrices for these beam splitters. We investigate in detail the properties of the 2- and 4-frequency mode beam splitters, and present general arguments on the possibility of natively generating N-mode beam splitters with these driven resonator arrays.