Multi-Photon Multi-Channel Interferometry for Quantum Information Processing

TL;DR

This thesis advances the design, characterization, and simulation of multi-photon multi-channel interferometers, introducing algorithms for arbitrary unitary transformations, precise characterization methods, and group-theoretic analysis tools for linear optical systems.

Contribution

It introduces novel algorithms for designing arbitrary unitary transformations and accurate characterization procedures for multi-port interferometers, along with group-theoretic methods for analysis and simulation.

Findings

Developed an algorithm for arbitrary unitary transformations on combined spatial and internal modes.

Created a precise characterization method using one- and two-photon interference with error correction.

Connected immanants of submatrices to SU(n) D-functions, enabling group-theoretic analysis of linear optics.

Abstract

This thesis reports advances in the theory of design, characterization and simulation of multi-photon multi-channel interferometers. I advance the design of interferometers through an algorithm to realize an arbitrary discrete unitary transformation on the combined spatial and internal degrees of freedom of light. This procedure effects an arbitrary $n_{s}n_{p}\times n_{s}n_{p}$ unitary matrix on the state of light in $n_{s}$ spatial and $n_{p}$ internal modes. I devise an accurate and precise procedure for characterizing any multi-port linear optical interferometer using one- and two-photon interference. Accuracy is achieved by estimating and correcting systematic errors that arise due to spatiotemporal and polarization mode mismatch. Enhanced accuracy and precision are attained by fitting experimental coincidence data to a curve simulated using measured source spectra. The efficacy of our characterization procedure is verified by numerical simulations. I develop group-theoretic methods for the analysis and simulation of linear interferometers. I devise a graph-theoretic algorithm to construct the boson realizations of the canonical SU$(n)$ basis states, which reduce the canonical subgroup chain, for arbitrary $n$. The boson realizations are employed to construct $\mathcal{D}$-functions, which are the matrix elements of arbitrary irreducible representations, of SU$(n)$ in the canonical basis. I show that immanants of principal submatrices of a unitary matrix $T$ are a sum of the diagonal $\mathcal{D}(\Omega)$-functions of group element $\Omega$ over $t$ determined by the choice of submatrix and over the irrep $(\lambda)$ determined by the immanant under consideration. The algorithm for $\mathrm{SU}(n)$ $\mathcal{D}$-function computation and the results connecting these functions with immanants open the possibility of group-theoretic analysis and simulation of linear optics.