Unitary Decomposition and Optimal Design of Universal Multiport Interferometers : Overview, Theory and Application

The aim of the project was to understand and provide a simple template for building programmable, universal multiport interferometer (a linear optics setup) given an initial unitary matrix. Starting with an N × N unitary matrix Û of finite dimension, how do we construct a linear optics setup that performs the operation on the state of photons? An answer can be found with a decomposition algorithm. First, we define the mathematical description of the photons. Second, we build up our understanding of the decomposition. Third, we explore the mathematics of the decomposition and provide proofs for some critical properties. Fourth, we briefly explain the steps one might take in order to fabricate the setup. The decomposition consists of separating the unitary matrix into a product of simpler factors using Clements et al.’s algorithm. Each factor represents one component of the setup. The final multiport interferometer is a rectangle mesh consisting of variable phase shifters and 50/50 beam splitters, which can be interpreted as Mach-Zehnder interferometers in a zig-zag-like pattern. The next steps would be to fabricate and test the interferometer. We encode the information in photon path, which can be implemented on various platforms (for example, photonic integrated circuits). However, the mathematics is applicable to all systems with interference-like behaviour.