1,814 research outputs found
Multi-Photon Multi-Channel Interferometry for Quantum Information Processing
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
unitary matrix on the state of light in spatial and 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 basis states, which reduce the canonical
subgroup chain, for arbitrary . The boson realizations are employed to
construct -functions, which are the matrix elements of arbitrary
irreducible representations, of SU in the canonical basis. I show that
immanants of principal submatrices of a unitary matrix are a sum of the
diagonal -functions of group element over
determined by the choice of submatrix and over the irrep determined
by the immanant under consideration. The algorithm for
-function computation and the results connecting these functions
with immanants open the possibility of group-theoretic analysis and simulation
of linear optics.Comment: 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 co-authors. 145
pages, 31 figures, 11 algorithms and 4 tables. Comments are welcom
Fully discrete finite element data assimilation method for the heat equation
We consider a finite element discretization for the reconstruction of the
final state of the heat equation, when the initial data is unknown, but
additional data is given in a sub domain in the space time. For the
discretization in space we consider standard continuous affine finite element
approximation, and the time derivative is discretized using a backward
differentiation. We regularize the discrete system by adding a penalty of the
-semi-norm of the initial data, scaled with the mesh-parameter. The
analysis of the method uses techniques developed in E. Burman and L. Oksanen,
Data assimilation for the heat equation using stabilized finite element
methods, arXiv, 2016, combining discrete stability of the numerical method with
sharp Carleman estimates for the physical problem, to derive optimal error
estimates for the approximate solution. For the natural space time energy norm,
away from , the convergence is the same as for the classical problem with
known initial data, but contrary to the classical case, we do not obtain faster
convergence for the -norm at the final time
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