<p>In this thesis we solve the Feshbach-Villars equations for spin-zero particles through use of matrix continued fractions. The Feshbach-Villars equations are derived from the Klein-Gordon equation and admit, for the Coulomb potential on an appropriate basis, a Hamiltonian form that has infinite symmetric band-matrix structure. The corresponding representation of the Green's operator of such a matrix can be given as a matrix continued fraction. Furthermore, we propose a finite dimensional representation for the potential operator such that it retains some information about the whole Hilbert space. Combining these two techniques, we are able to solve relativistic quantum mechanical problems of a spin-zero particle in a Coulomb-like potential with a high level of accuracy.

<p>The study of the quantum dynamics of many-particle systems has recently become the subject of intensive research, stimulated in part by enormous progress in experimental techniques, particularly the manipulation of ultracold atomic gases, which allow high tunability of artificial systems with decoherence and dissipation strongly suppressed. One of the simplest protocols for out of equilibrium dynamics is a quantum quench where the time-scale associated with an external variation is much smaller than the typical relaxation time of the system. Here we first study in detail the dynamics after a quantum quench in the one-dimensional sine-Gordon model in the phase where the boson spectrum remains gapless. We construct a Dyson equation to leading order in the cosine potential and show that the resulting quantum kinetic equation is atypical in that it involves multi-particle scattering processes. We also show that using an effective action, which generates the Dyson equation by a variational principle, the conserved stress-momentum tensor can be constructed. We solve the dynamics numerically by making a quasi-classical approximation that makes the quantum kinetic equation local in time while retaining the multi-particle nature of the scattering processes. At long times the system is found to thermalize, with a thermalization time that depends in a non-monotonic way on the amount of energy injected into the system by the quench. This non-monotonic behavior arises due to the competing effect of an increase of phase space for scattering on the one hand, and an enhancement of the orthogonality catastrophe on the other hand as the quench amplitude is increased. The approach to equilibrium is found to be purely exponential for large quench amplitudes but more complex for smaller ones. In the following chapter, the dynamics of interacting bosons in one dimension after a sudden switching on of a weak disordered potential is investigated. We find that on time scales before quasiparticles scatter, which correspond to the prethermalization regime, the dephasing from random elastic forward scattering causes the correlations to decay exponentially fast, while the system remains far from thermal equilibrium. For longer times however, the combined effect of disorder and interactions gives rise to inelastic scattering which eventually leads to thermalization. A novel quantum kinetic equation taking into account both disorder and interactions is employed to study the dynamics. It is found that thermalization becomes most effective close to the superfluid-Bose glass critical point where nonlinearities become increasingly important. The thermalization times obtained numerically are found to agree well with analytic estimates. In the last chapter we investigate the dynamics of a scalar field theory in spatial dimension d=4 after a quench close to a critical point, using renormalization-group methods. We show that after the system is quenched, but before eventually thermalizing due to dissipative effects, it approaches a different, thermal-like regime associated with a fixed-point describing a dynamical scaling behaviour. Within this regime the time dependence of the dynamical correlations is characterized by a novel short-time universal exponent. </p>

<p> The quantum non-demolition (QND) measurement process begins by entangling the system to be measured, a qubit for example, with an ancillary degree of freedom, usually a system with an infinite-dimensional Hilbert space. The ancilla is amplified to convert the quantum signal into a measurable classical signal. The continuous classical signal is recorded by a measurement apparatus; a discrete measurement outcome is recovered by thresholding the integrated signal record. Measurements play a central role in technologies based on quantum theory, like quantum computation and communication. They form the basis for a wide range of operations, ranging from state initialization to quantum error correction. Quantum measurements used for quantum computation must satisfy three essential requirements of being high fidelity, quantum non-demolition and efficient. Satisfying these criteria necessitates control over all the parts of the quantum measurement process, especially generating the ancilla, entangling it with the qubit and amplifying it to complete the measurement. </p><p> For superconducting quantum circuits, a promising platform for realizing quantum computation, a natural choice for the ancillae are modes of microwave-frequency electromagnetic radiation. In the paradigm of circuit quantum electrodynamics (cQED) with three-dimensional circuits, the most commonly used ancillae are coherent states, since they are easy to generate, process and amplify. Using these flying coherent states, we present results for achieving QND measurements of transmon qubits with fidelities of <i>F</i>> 0.99 and efficiencies of &eta; = 0.56 &plusmn; 0.01. By also treating the measurement as a more general quantum operation, we use the ancillae as carriers of quantum information to generate remote entanglement between two transmon qubits in separate cavities. By using microwave single photons as the flying qubits, it is possible to generate remote entanglement that is robust to loss since the generation of entanglement is uniquely linked to a particular measurement outcome. We demonstrate, in a single experiment, the ability to efficiently generate and detect single microwave photons and use them to generate robust remote entanglement between two transmon qubits. This operation forms a crucial primitive in modular architectures for quantum computation. The results of this thesis extend the experimental toolbox at the disposal to superconducting circuits. Building on these results, we outline proposals for remote entanglement distillation as well as strategies to further improve the performance of the various tools.</p><p>

<p> Bose-Einstein condensates (BECs) are a recent addition to the portfolio of quantum materials some of which have profound commercial and military applications e.g., superconductors, superfluids and light emitting diodes. BECs exist in the lowest motional modes of a trap and have the lowest temperatures achieved by mankind. With full control over the shape of the trap the experimentalist may explore an extremely diverse set of Hamiltonians which may be altered mid-experiment. These properties are particularly suited for realizing novel quantum systems.</p><p> This thesis explores interaction-driven domain formation and the subsequent domain coarsening for two immiscible BEC components. Because quantum coherences associated with interactions in BECs can be derived from low energy scattering theory we compare our experimental results to both a careful simulation (performed by Brandon Anderson) and an analytical prediction. This result very carefully explores the question of how a metastable system relaxes at the extreme limit of low temperature.</p><p> We also explore spin-orbit coupling (SOC) of a BEC which links the linear and discrete momentum transferable by two counterpropagating ''Raman'' lasers that resonantly couple the ground electronic states of our BECs. SOC is used similarly in condensed matter systems to describe coupling between charge carrier spin and crystal momentum and is a necessary component of the quantum spin Hall effect and topological insulators.</p><p> SOC links the linear and discrete momentum transferable by two counterpropagating ''Raman'' lasers and a subset of the ground electronic states of our BEC. The phases of an effective 2-spin component spin-orbit coupling (SOC) in a spin-1 BEC are described in Lin et al. (2011). We measure the phase transition between two phases of a spin-1 BEC with SOC which cannot be mimicked by a spin-1/2 system. The order parameter that describes transitions between these two phases is insensitive to magnetic field fluctuations.</p><p> I also describe a realistic implementation of Rashba SOC. This type of SOC is expected to exhibit novel many-body phases [Stanescu et al. 2008, Sedrakyan et al. 2012, Hu et al. 2011].</p>

<p> Despite the tremendous progress of quantum cryptography, efficient quantum communication over long distances (&ge; 1000km) remains an outstanding challenge due to fiber attenuation and operation errors accumulated over the entire communication distance. Quantum repeaters (QRs), as a promising approach, can overcome both photon loss and operation errors, and hence significantly speedup the communication rate. Depending on the methods used to correct loss and operation errors, all the proposed QR schemes can be classified into three categories (generations). First generation QRs use heralded entanglement generation for the correction of erasure errors and entanglement purification for the correction of operation errors. Second generation QRs use heralded entanglement generation for the correction of erasure errors and quantum error correction for the correction of operation errors. third-generation QRs use quantum error correction for the correction of both erasure and operation errors respectively. It is important to develop robust protocols for quantum repeaters, and systematically compare the performance of various QRs.</p><p> We investigate the use of efficient error correcting codes for third-generation QRs that make use of small encoding blocks to fault-tolerantly correct both loss and operation errors. Our schemes use quantum parity codes. quantum polynomial codes and quantum Reed-Solomon codes for encoding quantum information and use teleportation-based error correction to systematically correct erasure and operation errors in a fault-tolerant manner. We describe a way to optimize the resource requirements and system parameters for these QRs with the aim of generating a secure key. We perform a systematic comparison among these codes and identify the parameter regimes of operation errors where each code performs the best.</p><p> We then perform a systematic comparison of the three generations of QRs by evaluating the cost of both temporal and physical resources, and identify the optimized QR architecture for a given set of experimental parameters. Our work provides a roadmap for the experimental realization of highly efficient quantum networks over transcontinental distances.</p><p>

<p> Standard Quantum Mechanics, although successful in terms of calculating and predicting results, is inherently diffcult to understand and can suffer from misinterpretation. Entropic Dynamics is an epistemic approach to quantum mechanics based on logical inference. It incorporates the probabilities that naturally arise in situations in which there is missing information. It is the author's opinion that an advantage of this approach is that it provides a clearer mental image with which to picture quantum mechanics. This may provide an alternate means of presenting quantum mechanics to students. After a theory is presented to students, an instructor will then work through the paradigmatic examples that demonstrate the theory. In this thesis, we will be applying Entropic Dynamics to some of those paradigmatic examples. We begin by reviewing probability theory and Bayesian statistics as tools necessary for the development of Entropic Dynamics. We then review the topic of entropy, building from an early thermodynamic interpretation to the informational interpretation used here. The development of Entropic Dynamics involves describing a particle in terms of a probability density, and then following the time evolution of the probability density based on diffusion-like motion and the maximization of entropy. At this point, the review portion of the thesis is complete. </p><p> We then move on to applying Entropic Dynamics to several of the paradigmatic examples used to explain quantum mechanics. The rst of these is wave packet expansion. The second is interference, which is the basis behind many of the important phenomena in quantum mechanics. The third is the double slit experiment, which provides some interesting insight into the subject of interference. In particular, we look at the way in which minima can occur without a mechanism for destructive interference, since probabilities only add. The idea of probability flow is very apparent at this point in the discussion. The next example is that of the harmonic oscillator. This leads to an interesting insight concerning rotation and angular momentum as it corresponds to the flow of probability. The last example explored is that of entanglement. The discussion begins with a review of EPR, but then comes to the interesting conclusion that many of the problems inherent in the traditional approach to entanglement do not exist in Entropic Dynamics. The last topic covered in this thesis consists of some remarks concerning the state of education research as it pertains to quantum mechanics and the ways in which Entropic Dynamics might address them.</p><p>

Electronic band structure of various crystal orientations of relaxed and strained bulk, 1D and 2D confined semiconductors are investigated using nonlocal empirical pseudopotential method with spin-orbit interaction. For the bulk semiconductors, local and nonlocal pseudopotential parameters are obtained by fitting transport-relevant quantities, such as band gap, effective masses and deformation potentials, to available experimental data. A cubic-spline interpolation is used to extend local form factors to arbitrary q and the resulting transferable local pseudopotential V(q) with correct work function is used to investigate the 1D and 2D confined systems with supercell method. Quantum confinement, uniaxial and biaxial strain and crystal orientation effects of the band structure are investigated. Regarding the transport relavant quantities, we have found that the largest ballistic electron conductance occurs for compressively-strained large-diameter [001] wires while the smallest transport electron effective mass is found for larger-diameter [110] wires under tensile stress.

<p> Engineered quantum systems are poised to revolutionize information science in the near future. A persistent challenge in applied quantum technology is creating controllable, quantum interactions while preventing information loss to the environment, decoherence. In this thesis, we realize mesoscopic superconducting circuits whose macroscopic collective degrees of freedom, such as voltages and currents, behave quantum mechanically. We couple these mesoscopic devices to microwave cavities forming a cavity quantum electrodynamics (QED) architecture comprised entirely of circuit elements. This application of cavity QED is dubbed Circuit QED and is an interdisciplinary field seated at the intersection of electrical engineering, superconductivity, quantum optics, and quantum information science. Two popular methods for taming active quantum systems in the presence of decoherence are discrete feedback conditioned on an ancillary system or quantum reservoir engineering. Quantum reservoir engineering maintains a desired subset of a Hilbert space through a combination of drives and designed entropy evacuation. Circuit QED provides a favorable platform for investigating quantum reservoir engineering proposals. A major advancement of this thesis is the development of a quantum reservoir engineering protocol which maintains the quantum state of a microwave cavity in the presence of decoherence. This thesis synthesizes strongly coupled, coherent devices whose solutions to its driven, dissipative Hamiltonian are predicted a <i>priori</i>. This work lays the foundation for future advancements in cavity centered quantum reservoir engineering protocols realizing hardware efficient circuit QED designs. </p>

<p> After an introduction to the variational principle of three body systems via the Helium atom, we present general analytical formulas for the radial parts of integrals that occur when three body systems are described using wave functions that consist of powers and exponentials in all three interparticle Hylleraas coordinates [Hylleraas1929]. This work is an extension of integrals given by Harris, Frolov and Smith, Jr. [Harris2004]. Specifically included are radial integrals encountered in calculations involving the dipole moment matrix element in Hylleraas coordinates that contain a function <i>f(kr </i>₁) (such as a spherical Bessel function) in addition to a plane wave, a hydrogenic orbital and exponentials in all three interparticle coordinates.</p>

<p> As the LHC enters its second run at 13 TeV, new parameter space will become available that will allow for a more extensive search for supersymmetric partners. This thesis explores limits on a baryon number violating R-parity-violating (RPV) extension of the s-channel production of top squarks, examining the experimental signature for the R-parity conserving decay of the top squark into the lightest neutralino and a hadronically decaying top quark. Using Monte Carlo simulations I calculate upper bounds for the RPV coupling parameters &lambda;'' for a range of top squark and neutralino masses that would allow for its existence. </p><p>