Analysis of the multiport beam splitter transfer matrix in photonic integrated devices

The use of silicon photonics in the context of quantum technologies has been widely discussed in the fields of communication, engineering, computation, biology, chemistry, and metrology due to its scalability, stability, room-temperature control, and system miniaturization capabilities. In this paper, in the context of using photonic chips for application in quantum information protocols, we focus on multiple waveguide encoding of qudits, that is, a multi-level quantum computational unit, instead of the usual two-level qubit. Given the complexity of constructing control gates and designing systems using qubits, we show a method where we use coupled-mode theory in the small perturbation approximation, so that we can describe the coupling matrix of n-waveguides that can be used in the context of interferometers in photonic chips, to find optimal parameters for describing operations using qudits.