<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> I discuss the design, fabrication and measurement at millikelvin-temperatures of Al/AlO<i>_x</i>/Al Josephson junction-based transmon qubits coupled to superconducting thin-film lumped element microwave resonators made of aluminum on sapphire. The resonators had a center frequency of around 6GHz, and a total quality factor ranging from 15,000 to 70,000 for the various devices. The area of the transmon junctions was about 150 nm &times; 150 nm and with Josephson energy <i>E_J</i> such that 10GHz &le; <i>E_J</i> &le; 30 GHz. The charging energy of the transmons arising mostly from the large interdigital shunt capacitance, was <i>E_c/h</i> &ap; 300MHz. </p><p> I present microwave spectroscopy of the devices in the strongly dispersive regime of circuit quantum electrodynamics. In this limit the ac Stark shift due to a single photon in the resonator is greater than the linewidth of the qubit transition. When the resonator is driven coherently using a coupler tone, the transmon spectrum reveals individual "photon number'' peaks, each corresponding to a single additional photon in the resonator. Using a weighted average of the peak heights in the qubit spectrum, I calculated the average number of photons <i>n&macr;</i> in the resonator. I also observed a nonlinear variation of <i>n&macr;</i> with the applied power of the coupler tone <i>P_rf</i>. I studied this nonlinearity using numerical simulations and found good qualitative agreement with data. </p><p> In the absence of a coherent drive on the resonator, a thermal population of 5.474 GHz photons in the resonator, at an effective temperature of 120 mK resulted in a weak <i>n</i> = 1 thermal photon peak in the qubit spectrum. In the presence of independent coupler and probe tones, the <i> n</i> = 1 thermal photon peak revealed an Autler-Townes splitting. The observed effect was explained accurately using the four lowest levels of the dispersively dressed Jaynes-Cummings transmon-resonator system, and numerical simulations of the steady-state master equation for the coupled system. </p><p> I also present time-domain measurements on transmons coupled to lumped-element resonators. From <i>T</i>₁ and Rabi oscillation measurements, I found that my early transmon devices (called design LEv5) had lifetimes (<i>T</i>₁ &sim; 1 &mu;s) limited by strong coupling to the 50 &Omega; transmission line. This coupling was characterized by the the rate of change of the Rabi oscillation frequency with the change in the drive voltage (d<i>f_Rabi</i> /<i>dV</i>) &ndash; also termed the Rabi coupling to the drive. I studied the design of the transmon-resonator system using circuit analysis and microwave simulations with the aim being to reduce the Rabi coupling to the drive. By increasing the resonance frequency of the resonator &omega;<i>_r</i>/2&pi; from 5.4 GHz to 7.2 GHz, lowering the coupling of the resonator to the transmission line and thereby increasing the external quality factor <i>Q_e</i> from 20,000 to 70,000, and reducing the transmon-resonator coupling <i> g</i>/2&pi; from 70 MHz to 40 MHz, I reduced the Rabi coupling to the drive by an order of magnitude (&sim; factor of 20). The <i>T</i>₁ &sim; 4 &mu;s of devices in the new design (LEv6) was longer than that of the early devices, but still much shorter than the lifetimes predicted from Rabi coupling, suggesting the presence of alternative sources of noise causing qubit relaxation. Microwave simulations and circuit analysis in the presence of a dielectric loss tangent tan &delta; &sime; 5 &times; 10^-6 agree reasonably well with the measured <i>T</i>₁ values, suggesting that surface dielectric loss may be causing relaxation of transmons in the new designs.</p>

<p> The scaling of controlled quantum systems to large numbers of degrees of freedom is one of the long term goals of experimental quantum information science. Trapped-ion systems are one of the most promising platforms for building a quantum information processor with enough complexity to enable novel computational power, but face serious challenges in scaling up to the necessary numbers of qubits. In this thesis, I present both technical and operational advancements in the control of trapped-ion systems and their juxtaposition with photonic modes used for quantum networking. After reviewing the basic physics behind ion trapping, I then describe in detail a new method of implementing Raman transitions in atomic systems using optical frequency combs. Several dierent experimental setups along with simple theoretical models are reviewed and the system is shown to be capable of full control of the qubit-oscillator system. Two-ion entangling operations using optical frequency combs are demonstrated along with an extension of the operation designed to suppress certain experimental errors. I then give an overview of how spatially separated ions can be entangled using a photonic interconnect. Experimental results show that pulsed excitation of trapped ions provide an excellent single photon source that can be used as a heralded entangling gate between macroscopically separated systems. This heralded entangling gate is used to show a violation of a Bell inequality while keeping the detection loophole closed and can be used a source private random numbers. Finally, the coherent Coulomb force-based gates are combined with the probabilistic photon-based gates in a proof of concept experiment that shows the feasibility of a distributed ion-photon network.</p>

<p> Randomized benchmarking is an experimental procedure intended to demonstrate control of quantum systems. The procedure extracts the average error introduced by a set of control operations. When the target set of operations is intended to be the set of Clifford operators, the randomized benchmarking algorithm is particularly easy to perform and its results have an important interpretation with respect to quantum computation. The aim of the benchmark is to provide a simple, useful parameter describing the quality of quantum control with an experiment that can be performed in a standard way on any prospective quantum computer. This parameter can be used to fairly compare different experiments or to mark improvement in a single experiment. </p><p> In this thesis I discuss first the original randomized-benchmarking procedure and the importance of the Clifford operators for its implementation. I develop the statistical analysis of the results and the physical assumptions that are required for the simplest analysis to apply. The original procedure does not extend in an obvious way to benchmarking of more than one qubit, so I introduce a standardized procedure for randomized benchmarking that applies to any number of qubits. This new procedure also enables the benchmarking of an individual control operation. I describe two randomized-benchmarking experiments I helped to design: one involved a single qubit and utilized a variation of the original procedure and the second involved two qubits and demonstrated the new procedure. I conclude with several potential extensions to the original and new procedures that give them reduced experimental overhead, the ability to describe encoded operations, and fairer comparisons between experiments.</p>

<p> In quantum mechanics, the process of measurement is intrinsically probabilistic. As a result, continuously monitoring a quantum system will randomly perturb its natural unitary evolution. An accurate measurement record documents this stochastic evolution and can be used to reconstruct the quantum trajectory of the system state in a single experimental iteration. We use weak measurements to track the individual quantum trajectories of a superconducting qubit that evolves under the competing influences of continuous weak measurement and Rabi drive. We analyze large ensembles of such trajectories to examine their characteristics and determine their statistical properties. For example, by considering only the subset of trajectories that evolve between any chosen initial and final states, we can deduce the most probable path through quantum state space. Our investigation reveals the rich interplay between measurement dynamics, typically associated with wavefunction collapse, and unitary evolution. Our results provide insight into the dynamics of open quantum systems and may enable new methods of quantum state tomography, quantum state steering through measurement, and active quantum control.</p>

<p> We study quantum theory as an example of entropic inference. Our goal is to remove conceptual difficulties that arise in quantum mechanics. Since probability is a common feature of quantum theory and of any inference problem, we briefly introduce probability theory and the entropic methods to update probabilities when new information becomes available. Nelson's stochastic mechanics and Caticha's derivation of quantum theory are discussed in the subsequent chapters. </p><p> Our first goal is to understand momentum and angular momentum within an entropic dynamics framework and to derive the corresponding uncertainty relations. In this framework momentum is an epistemic concept &ndash; it is not an attribute of the particle but of the probability distributions. We also show that the Heisenberg's uncertainty relation is an osmotic effect. Next we explore the entropic analog of angular momentum. Just like linear momentum, angular momentum is also expressed in purely informational terms. </p><p> We then extend entropic dynamics to curved spaces. An important new feature is that the displacement of a particle does not transform like a vector. It involves second order terms that account for the effects of curvature . This leads to a modified Schr&ouml;dinger equation for curved spaces that also take into account the curvature effects. We also derive Schrodinger equation for a charged particle interacting with external electromagnetic field on general Riemannian manifolds. </p><p> Finally we develop the entropic dynamics of a particle of spin 1/2. The particle is modeled as a rigid point rotator interacting with an external EM field. The configuration space of such a rotator is <i>R</i>³ &times; <i>S</i>³ (<i>S</i>³ is the 3-sphere). The model describes the regular representation of <i>SU</i>(2) which includes all the irreducible representations (spin 0, 1/2, 1, 3/2,...) including spin 1/2.</p>

<p> As a subset of quantum optics, single photons are one of the competing physical resources for quantum information processing. When used as carriers of quantum information, they have no equal. For the processing of quantum information, single photons have proven difficult to scale beyond the order of ~10 photons. The lack of single-photon-level interaction has led to creative approaches which rely on post-selection to filter the possible measured outcomes to those which appear as though interaction occurred. This approach of post-selection leans heavily on the ability to not only generate but also detect scores of single photons simultaneously and with near perfect efficiency. Our work relaxes the emphasis which has been placed on single photons for quantum information processing to that of states with, in principle, an arbitrary number of photons. Central moment expectations on two-mode squeezed states are shown to exhibit post-selection behavior which reflects the single-photon counterpart. These measures are proven to be robust to loss and return entangled state statistics on average. With naive estimation of the central moment, states with ~20 modes are within reach with current technology, closing the gap between quantum states which can and cannot be classically simulated.</p><p>

The present work is concerned with de Broglie's quantum theory of light. It is assumed that the photon is described by a Hermitian wave function with 16 components. Using this wave function it is shown that the 32 of de Broglie's equations are reduced to one set of 16 equations in the form: [equation] where the [alpha]'[rho] are Dirac matrices and [psi] is a matrix with 16 components. The electromagnetic quantities associated with the photon are described by means of the Dirac matrices operating on [psi] in a specified way. It is shown also that these electromagnetic quantities satisfy Maxwell's equations as a result of the equation the interaction between an electron and a photon is developed and the matrix elements for the radiation transitions are calculated. It is further shown that the above wave equation can be considered as the superposition of two similar wave equations, one for the positive energy photons and the other for the negative energy photons. To each of these states there corresponds electromagnetic quantities defined by the above method. It is the superposition of these fields which gives rise to the reality of the electromagnetic field, found in experience. The wave mechanics of the positive energy photon is discussed and the method of second quantization is applied to its wave function, from which we deduce the commutation relations for the complex fields.

Several topics of quantum field theory are discussed within the algebraic context. It is shown that for the charged Bose field there are two natural ways of defining the local field algebras; however, these are relatively antilocal in the sense of Segal and Goodman. We define the charge sectors and show that although they are unitarily inequivalent representations of the observable algebra, they are physically (and, in fact, strongly locally) equivalent. This is a partial justification of the use of abstract algebras. The converse problem, that of constructing charge carrying fields given the observable algebra in the charge zero sector, is then tackled for the case of a massless boson field in two dimensional space time. This is achieved by applying the techniques of Doplicher, Haag and Roberts, viz, the use of localised automorphisms. The specific localised automorphisms used are suggested by consideration of Skyrme's model for zero mass. Finally, we discuss the time evolution co-responding to a bounded interaction density in an arbitrary number of space dimensions. This extends a result of Guenin. A. condition on the interaction in order that the resulting time evolution be causal is given.

The BCS free energy for ³P₂ paired neutron matter is derived taking account of relativistic effects. It is found that the values taken by the Ginzburg-Landau parameters are always in the region of the phase diagram correponding to a unitary phase. Phase transitions in the early universe are also discussed with inclusion of the effects of Higgs scalar chemical potentials as well as fermionic chemical potentials. The conditions for equilibrium, and the critical density to prevent symmetry restoration at high temperatures are studied. It is observed that the decay of pre-existing Higgs scalar asymmetries could greatly reduce baryon number and lepton number to entropy ratios from their initial values. Phase transitions in supersymmetric theories and the phenomenom of symmetry anti-restoration in a supersymmetric model with a U(1) gauge symmetry are studied at finite density.