MultiPhoton MultiChannel Interferometry for Quantum Information Processing
Abstract
This thesis reports advances in the theory of design, characterization and simulation of multiphoton multichannel 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 multiport linear optical interferometer using one and twophoton 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 grouptheoretic methods for the analysis and simulation of linear interferometers. I devise a graphtheoretic 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 grouptheoretic analysis and simulation of linear optics.
 Publication:

arXiv eprints
 Pub Date:
 March 2016
 arXiv:
 arXiv:1603.07476
 Bibcode:
 2016arXiv160307476D
 Keywords:

 Quantum Physics;
 Mathematical Physics;
 Physics  Optics
 EPrint:
 PhD thesis submitted and defended successfully at the University of Calgary. This thesis is based on articles arXiv:1403.3469, arXiv:1507.06274, arXiv:1508.00283, arXiv:1508.06259 and arXiv:1511.01851 with coauthors. 145 pages, 31 figures, 11 algorithms and 4 tables. Comments are welcome