The primary aim of this thesis is to develop a minimal model Hamiltonian to describe the electronic properties of ceria (CeO$_{2}$) and its reduced phases. In order to do this, several energy scales of the problem were explored to determine their relative significance to the problem. These included the crystal electric field ($\Delta_{CEF}$), the spin-orbit coupling ($\lambda_{so}$), electron hopping ($t$), on-site Coulomb repulsion for the Ce $4f$ states ($U$), the reorganization energy ($\lambda_{0}$), the direct exchange ($J$) and the energy gap between the oxygen valence band level and the cerium $4f$ states ($\Gamma\epsilon$). Once the relative magnitudes of the various energy scales were determined, it was then possible to define a minimal set of degrees of freedom required to obtain a meaningful description of the system in the minimal model. The first task was to obtain baseline data for both CeO$_{2}$ and Ce$_{2}$O$_{3}$ as well as the metallic phases of Ce. Density Functional Theory (DFT) calculations were performed on these materials to obtain band structures as well as structural properties. The DFT results indicated that both the LDA and GGA functionals perform poorly for $\gamma$-Ce and Ce$_{2}$O$_{3}$. In the case of Ce$_{2}$O$_{3}$, both LDA and GGA give a metallic ground state contrary to the insulator observed in experiment. The first set of energy scales that were investigated included the crystal electric field and the spin-orbit coupling. Since these calculations, except for the presence of a crystal field, involve an isolated Ce ion, the energy scales associated with spin and charge fluctuations at a Ce site are ignored. In order to perform these calculations, it was necessary to determine the $f^{n}$ configuration at each Ce site in the various phases of the oxides. This was achieved by a simple empirical model called the bond valence model. The bond valence model provides a method for calculating site valences in crystals from bond length data alone. Apart from getting information about the $f^{n}$ configurations at the various Ce sites, the results of the bond valence analysis revealed that there was significant mixed valence in the different phases of the cerium oxies. In addition, it was possible to characterize the charge distribution in the local environment of the oxygen vacancies from the bond valence results. This analysis led to the important result that the charge prefers Ce sites farthest away from the oxygen vacancy. This is contrary to the widely accepted view that the extra charge resulting from oxygen vacancy formation localizes on Ce sites closest to the O vacancy. The LDA calculations which support the widely accepted view do not properly treat electron correlations. Thus this is an important result which emphasizes the role of strong electron correlations in the processes of oxidation and reduction in these materials. The results of the crystal field calculations were not quantitative because we could not find data on crystal field parameters which is required to evaluate the matrix elements. However, it is expected that the splitting of the $f$ manifold by the crystal fields will be smaller than that of the spin-orbit coupling energy which is of the order of $0.07\unit{eV}$. Results for the Hubbard $U$ parameter in Ce oxides were obtained from the literature but these vary widely and fall in the range $1.0 - 10.5\unit{eV}$. The reorganization energy reported from spectroscopic results of the intervalence transition band for reduced ceria is $1.4\unit{eV}$. Tight-binding calculations were performed for CeO$_{2}$ to obtain the energy band structure for this material. It was then possible to extract from these results the matrix elements for electron hopping. The sizes of the matrix elements obtained suggested that the direct $f$-$f$ hopping between Ce sites ($t_{ff}$) is of order $0.02\unit{eV}$ which is negligible compared to the indirect hopping via an O atom (i.e., $t_{eff}$) which is of order $0.2\unit{eV}$. The hopping matrix element between a Ce site and a neighbouring O atom ($t_{fp}$) is even higher $\sim 0.5\unit{eV}$. The role of multiple orbitals on a Ce site was explored to determine the minimal set of orbitals to include in the effective model Hamiltonian. The results indicated that the essential physics could be described well by replacing the $4f$ manifold with two orbitals on a Ce site. The magnitudes of the energy scales of the problem decrease in the order $U > \lambda_{0} > t_{fp}>t_{eff} >\lambda_{so} \gg t_{ff} \approx \Delta_{CEF} \approx J$ with a high uncertainty on the last two relationships. The results of the above analysis led us to the conclusion that the only relevant energy scales for the effective model Hamiltonian are the On-site Coulomb repulsion for the Ce $4f$ states ($U$) and electron hopping between a Ce site and a neighbouring O site ($t_{fp}$). A minimal model Hamiltonian was then constructed to take into account the spin and charge fluctuations on a Ce site due to electron hopping between the Ce site and neighbouring O atoms. In this description, the strongly correlated ground states of cerium oxides is then described by a variational wavefunction whose main feature is that double occupancy of a Ce $f$ orbital is prohibited.

The primary aim of this thesis is to develop a minimal model Hamiltonian to describe the electronic properties of ceria (CeO$_{2}$) and its reduced phases. In order to do this, several energy scales of the problem were explored to determine their relative significance to the problem. These included the crystal electric field ($\Delta_{CEF}$), the spin-orbit coupling ($\lambda_{so}$), electron hopping ($t$), on-site Coulomb repulsion for the Ce $4f$ states ($U$), the reorganization energy ($\lambda_{0}$), the direct exchange ($J$) and the energy gap between the oxygen valence band level and the cerium $4f$ states ($\Gamma\epsilon$). Once the relative magnitudes of the various energy scales were determined, it was then possible to define a minimal set of degrees of freedom required to obtain a meaningful description of the system in the minimal model. The first task was to obtain baseline data for both CeO$_{2}$ and Ce$_{2}$O$_{3}$ as well as the metallic phases of Ce. Density Functional Theory (DFT) calculations were performed on these materials to obtain band structures as well as structural properties. The DFT results indicated that both the LDA and GGA functionals perform poorly for $\gamma$-Ce and Ce$_{2}$O$_{3}$. In the case of Ce$_{2}$O$_{3}$, both LDA and GGA give a metallic ground state contrary to the insulator observed in experiment. The first set of energy scales that were investigated included the crystal electric field and the spin-orbit coupling. Since these calculations, except for the presence of a crystal field, involve an isolated Ce ion, the energy scales associated with spin and charge fluctuations at a Ce site are ignored. In order to perform these calculations, it was necessary to determine the $f^{n}$ configuration at each Ce site in the various phases of the oxides. This was achieved by a simple empirical model called the bond valence model. The bond valence model provides a method for calculating site valences in crystals from bond length data alone. Apart from getting information about the $f^{n}$ configurations at the various Ce sites, the results of the bond valence analysis revealed that there was significant mixed valence in the different phases of the cerium oxies. In addition, it was possible to characterize the charge distribution in the local environment of the oxygen vacancies from the bond valence results. This analysis led to the important result that the charge prefers Ce sites farthest away from the oxygen vacancy. This is contrary to the widely accepted view that the extra charge resulting from oxygen vacancy formation localizes on Ce sites closest to the O vacancy. The LDA calculations which support the widely accepted view do not properly treat electron correlations. Thus this is an important result which emphasizes the role of strong electron correlations in the processes of oxidation and reduction in these materials. The results of the crystal field calculations were not quantitative because we could not find data on crystal field parameters which is required to evaluate the matrix elements. However, it is expected that the splitting of the $f$ manifold by the crystal fields will be smaller than that of the spin-orbit coupling energy which is of the order of $0.07\unit{eV}$. Results for the Hubbard $U$ parameter in Ce oxides were obtained from the literature but these vary widely and fall in the range $1.0 - 10.5\unit{eV}$. The reorganization energy reported from spectroscopic results of the intervalence transition band for reduced ceria is $1.4\unit{eV}$. Tight-binding calculations were performed for CeO$_{2}$ to obtain the energy band structure for this material. It was then possible to extract from these results the matrix elements for electron hopping. The sizes of the matrix elements obtained suggested that the direct $f$-$f$ hopping between Ce sites ($t_{ff}$) is of order $0.02\unit{eV}$ which is negligible compared to the indirect hopping via an O atom (i.e., $t_{eff}$) which is of order $0.2\unit{eV}$. The hopping matrix element between a Ce site and a neighbouring O atom ($t_{fp}$) is even higher $\sim 0.5\unit{eV}$. The role of multiple orbitals on a Ce site was explored to determine the minimal set of orbitals to include in the effective model Hamiltonian. The results indicated that the essential physics could be described well by replacing the $4f$ manifold with two orbitals on a Ce site. The magnitudes of the energy scales of the problem decrease in the order $U > \lambda_{0} > t_{fp}>t_{eff} >\lambda_{so} \gg t_{ff} \approx \Delta_{CEF} \approx J$ with a high uncertainty on the last two relationships. The results of the above analysis led us to the conclusion that the only relevant energy scales for the effective model Hamiltonian are the On-site Coulomb repulsion for the Ce $4f$ states ($U$) and electron hopping between a Ce site and a neighbouring O site ($t_{fp}$). A minimal model Hamiltonian was then constructed to take into account the spin and charge fluctuations on a Ce site due to electron hopping between the Ce site and neighbouring O atoms. In this description, the strongly correlated ground states of cerium oxides is then described by a variational wavefunction whose main feature is that double occupancy of a Ce $f$ orbital is prohibited.

In this thesis we examine key structure-property relationships associated with eumelanin's photophysical properties. This has involved characterizing both the molecular and supramolecular structure of eumelanin, and examining how these relate to eumalanin's key optical properties that are relevant to their their role in the biosphere as photoprotectants. Using low-voltage high resolution transmission electron microscopy we definitively show that sheets of proto-molecules stack to form nanostructures. The inter-sheet spacings within these structures are between 3.7 and 4.0 Å consistent with non-covalent pie-pie stacking in heteroaromatic systems. Stacking interactions in similarly structured aromatic macromolecules play an important role in non-radiative energy dissipation and we propose that this may also be the case in the eumelanin system. We also examine the recently proposed hypothesis that excited state intramolecular proton transfer may play a role in the photophysics of 5,6-dihydoxyindole carboxylic acid, a key eumelanin monomer. The experimental results acquired in this study indicate that this hypothesis needs to be carefully re-examined and its justification would require more extensive experimental support. Key optical properties of 5,6-dihydoxyindole carboxylic acid are re-evaluated in an inert aprotic solvent and these have provided some insight into the electronic structure as well as the rates of radiative and non-radiative decay in this important eumelanin monomer. We go on to show how the structure of eumelanin can be manipulated to produce eumelanin thin films. These films display electrical conductivities comparable with amorphous silicon, as well as a host of other interesting and potentially useful optoelectronic properties. The results show great promise for eumelanin-based applications such as chemi-sensors (in a variety of architectures including organic field effect transistors with chemi-sensitive channels) and bolometric photon detectors.

In this thesis we examine key structure-property relationships associated with eumelanin's photophysical properties. This has involved characterizing both the molecular and supramolecular structure of eumelanin, and examining how these relate to eumalanin's key optical properties that are relevant to their their role in the biosphere as photoprotectants. Using low-voltage high resolution transmission electron microscopy we definitively show that sheets of proto-molecules stack to form nanostructures. The inter-sheet spacings within these structures are between 3.7 and 4.0 Å consistent with non-covalent pie-pie stacking in heteroaromatic systems. Stacking interactions in similarly structured aromatic macromolecules play an important role in non-radiative energy dissipation and we propose that this may also be the case in the eumelanin system. We also examine the recently proposed hypothesis that excited state intramolecular proton transfer may play a role in the photophysics of 5,6-dihydoxyindole carboxylic acid, a key eumelanin monomer. The experimental results acquired in this study indicate that this hypothesis needs to be carefully re-examined and its justification would require more extensive experimental support. Key optical properties of 5,6-dihydoxyindole carboxylic acid are re-evaluated in an inert aprotic solvent and these have provided some insight into the electronic structure as well as the rates of radiative and non-radiative decay in this important eumelanin monomer. We go on to show how the structure of eumelanin can be manipulated to produce eumelanin thin films. These films display electrical conductivities comparable with amorphous silicon, as well as a host of other interesting and potentially useful optoelectronic properties. The results show great promise for eumelanin-based applications such as chemi-sensors (in a variety of architectures including organic field effect transistors with chemi-sensitive channels) and bolometric photon detectors.

In this thesis we examine key structure-property relationships associated with eumelanin's photophysical properties. This has involved characterizing both the molecular and supramolecular structure of eumelanin, and examining how these relate to eumalanin's key optical properties that are relevant to their their role in the biosphere as photoprotectants. Using low-voltage high resolution transmission electron microscopy we definitively show that sheets of proto-molecules stack to form nanostructures. The inter-sheet spacings within these structures are between 3.7 and 4.0 Å consistent with non-covalent pie-pie stacking in heteroaromatic systems. Stacking interactions in similarly structured aromatic macromolecules play an important role in non-radiative energy dissipation and we propose that this may also be the case in the eumelanin system. We also examine the recently proposed hypothesis that excited state intramolecular proton transfer may play a role in the photophysics of 5,6-dihydoxyindole carboxylic acid, a key eumelanin monomer. The experimental results acquired in this study indicate that this hypothesis needs to be carefully re-examined and its justification would require more extensive experimental support. Key optical properties of 5,6-dihydoxyindole carboxylic acid are re-evaluated in an inert aprotic solvent and these have provided some insight into the electronic structure as well as the rates of radiative and non-radiative decay in this important eumelanin monomer. We go on to show how the structure of eumelanin can be manipulated to produce eumelanin thin films. These films display electrical conductivities comparable with amorphous silicon, as well as a host of other interesting and potentially useful optoelectronic properties. The results show great promise for eumelanin-based applications such as chemi-sensors (in a variety of architectures including organic field effect transistors with chemi-sensitive channels) and bolometric photon detectors.

Understanding the collective behavior of a quantum many-body system, a system composed of a large number of interacting microscopic degrees of freedom, is a key aspect in many areas of contemporary physics. However, as a direct consequence of the difficultly of the so-called many-body problem, many exotic quantum phenomena involving extended systems, such as high temperature superconductivity, remain not well understood on a theoretical level. Entanglement renormalization is a recently proposed numerical method for the simulation of many-body systems which draws together ideas from the renormalization group and from the field of quantum information. By taking due care of the quantum entanglement of a system, entanglement renormalization has the potential to go beyond the limitations of previous numerical methods and to provide new insight to quantum collective phenomena. This thesis comprises a significant portion of the research development of ER following its initial proposal. This includes exploratory studies with ER in simple systems of free particles, the development of the optimisation algorithms associated to ER, and the early applications of ER in the study of quantum critical phenomena and frustrated spin systems.

One of the current hot topics in physics is quantum information, which, broadly speaking, is concerned with exploring the information-processing and storing tasks that can be performed in quantum mechanical systems. Besides driving forward our experimental control and understanding of quantum systems, the field is also in the early stages of developing revolutionary new technology of far reaching implication. As part of these endeavors, this thesis presents some results in experimental quantum information. Specifically, we develop several new tools for performing quantum information processing in optical quantum systems, and use them to explore a number of applications and novel physical phenomena. A central theme, and one of the most sought after applications of quantum information, is the pursuit of a programmable quantum computer. This thesis is divided into 3 parts. In Part I we develop some new optical quantum logic gates, which are tools for manipulating quantum information and the fundamental building blocks of a quantum computer. We also develop a new technique for simplifying the construction of quantum logic circuits, by exploiting multi-level quantum systems, that has the potential for application in any physical encoding of quantum information. In Part II we use these tools to perform some of the first demonstrations of quantum algorithms. Each of these could, in principle, efficiently solve an important problem that is thought to be fundamentally intractable using conventional `classical' techniques. Firstly we implement a simplified version of the quantum algorithm for factoring numbers, and demonstrate the core processes, coherent control, and resultant entangled states required for a full-scale implementation. Secondly we implement an algorithm for calculating the energy of many-body quantum systems. Specifically, we calculate the energy spectrum of the Hydrogen molecule, in a minimal basis. Finally we demonstrate an algorithm for a novel model of quantum computing that uses mixed states. Here we perform the first characterisation of intrinsically non-classical correlations between fully separable quantum systems, captured by the 'discord'---a measure of quantum correlations in mixed states that goes beyond entanglement. Part III presents a technique that extends experimental control over biphotons---the novel quantum information carriers formed by the polarisation of two photons in the same spatial and temporal mode. We also generate and explore new forms of entanglement: producing the first instance of qubit-qutrit entanglement, by entangling the polarisation of a photon and a biphoton, and developing a technique that enables full control over the level of `W-class' of multi-partite entanglement between the polarisation of three photons.

In this thesis we study the physics of quantum many-body systems confined to one-dimensional geometries. The work was motivated by the recent success of experimentalists in developing atom traps, capable of restricting the motion of the individual atoms to a single spatial dimension. Specifically, we look at aspects of the one-dimensional Bose gas including; excitation spectrum, correlation functions, and dynamical behaviour. In Chapter \ref{ch:excitation1D} we consider the Lieb-Liniger model of interacting bosons in one-dimension. We numerically solve the equations arising from the Bethe ansatz solution for the exact many-body wave function in a finite-size system of up to twenty particles for attractive interactions. We discuss novel features of the solutions, including deviations from the well-known string solutions due to finite size effects. We present excited state string solutions in the limit of strong interactions and discuss their physical interpretation, as well as the characteristics of the quantum phase transition that occurs as a function of interaction strength in the mean-field limit. Our results are compared to those obtained via exact diagonalization of the Hamiltonian in a truncated basis. In Chapter \ref{ch:g2} we analytically calculate the spatial nonlocal pair correlation function for an interacting uniform one dimensional Bose gas at finite temperature and propose an experimental method to measure nonlocal correlations. Our results span six different physical realms, including the weakly and strongly interacting regimes. We show explicitly that the characteristic correlation lengths are given by one of four length scales: the thermal de Broglie wavelength, the mean interparticle separation, the healing length, or the phase coherence length. In all regimes, we identify the profound role of interactions and find that under certain conditions the pair correlation may develop a global maximum at a finite interparticle separation due to the competition between repulsive interactions and thermal effects. In Chapter \ref{ch:casimirdrag} we study the drag force below the critical velocity for obstacles moving in a superfluid. The absence of drag is well established in the context of the mean-field Gross-Pitaevskii theory. We calculate the next order correction due to quantum and thermal fluctuations and find a non-zero force acting on a delta-function impurity moving through a quasi-one-dimensional Bose-Einstein condensate at all subcritical velocities and at all temperatures. The force occurs due to an imbalance in the Doppler shifts of reflected quantum fluctuations from either side of the impurity. Our calculation is based on a consistent extension of Bogoliubov theory to second order in the interaction strength, and finds new analytic solutions to the Bogoliubov-de Gennes equations for a gray soliton. In Chapter \ref{ch:solitons} we study the effect of quantum noise on the stability of a soliton. We find the soliton solutions exactly define the reflectionless potentials of the Bogoliubov-de Gennes equations. This results in complete stability of the solitons in a purely one dimensional system. We look at the modifications to the density profile of a black soliton due to quantum fluctuations.

In this thesis we study the physics of quantum many-body systems confined to one-dimensional geometries. The work was motivated by the recent success of experimentalists in developing atom traps, capable of restricting the motion of the individual atoms to a single spatial dimension. Specifically, we look at aspects of the one-dimensional Bose gas including; excitation spectrum, correlation functions, and dynamical behaviour. In Chapter \ref{ch:excitation1D} we consider the Lieb-Liniger model of interacting bosons in one-dimension. We numerically solve the equations arising from the Bethe ansatz solution for the exact many-body wave function in a finite-size system of up to twenty particles for attractive interactions. We discuss novel features of the solutions, including deviations from the well-known string solutions due to finite size effects. We present excited state string solutions in the limit of strong interactions and discuss their physical interpretation, as well as the characteristics of the quantum phase transition that occurs as a function of interaction strength in the mean-field limit. Our results are compared to those obtained via exact diagonalization of the Hamiltonian in a truncated basis. In Chapter \ref{ch:g2} we analytically calculate the spatial nonlocal pair correlation function for an interacting uniform one dimensional Bose gas at finite temperature and propose an experimental method to measure nonlocal correlations. Our results span six different physical realms, including the weakly and strongly interacting regimes. We show explicitly that the characteristic correlation lengths are given by one of four length scales: the thermal de Broglie wavelength, the mean interparticle separation, the healing length, or the phase coherence length. In all regimes, we identify the profound role of interactions and find that under certain conditions the pair correlation may develop a global maximum at a finite interparticle separation due to the competition between repulsive interactions and thermal effects. In Chapter \ref{ch:casimirdrag} we study the drag force below the critical velocity for obstacles moving in a superfluid. The absence of drag is well established in the context of the mean-field Gross-Pitaevskii theory. We calculate the next order correction due to quantum and thermal fluctuations and find a non-zero force acting on a delta-function impurity moving through a quasi-one-dimensional Bose-Einstein condensate at all subcritical velocities and at all temperatures. The force occurs due to an imbalance in the Doppler shifts of reflected quantum fluctuations from either side of the impurity. Our calculation is based on a consistent extension of Bogoliubov theory to second order in the interaction strength, and finds new analytic solutions to the Bogoliubov-de Gennes equations for a gray soliton. In Chapter \ref{ch:solitons} we study the effect of quantum noise on the stability of a soliton. We find the soliton solutions exactly define the reflectionless potentials of the Bogoliubov-de Gennes equations. This results in complete stability of the solitons in a purely one dimensional system. We look at the modifications to the density profile of a black soliton due to quantum fluctuations.