In recent and upcoming years new lasers are being constructed withever higher intensity. These lasers open up the possibility of probingthe high intensity regime of particle physics, which will lead to etherconrming our current models in this regime or the discovery of beyondstandard model physics. However most previous theoretical results in thisarea are based old assumptions about the intensity and shape of the laserpulse that are no longer valid. In this thesis we calculate the tree-levelprobabilities for multi photon emission from an electron propagating inan arbitrary plane wave electromagnetic eld. We show that the classicallimit of our result agrees with the purely classical description of the sameevent. We calculate the soft emission correction to non-linear Comptonscattering. We conclude that our results are infrared divergent and arguethat this will be solved by including loop contributions to the process. Ourresults provide an important component for the theoretical predictions forthe outcome of scattering experiments in high intensity background eld.This thesis will add to the understanding of high intensity QED.

In this thesis we investigate the role played by fundamental properties of QED in determining the non-perturbative fermion-boson vertex. These key features are gauge invariance and multiplicative renormalisability. We use the Schwinger-Dyson equations as the non- perturbative tool to study the general structure of the fermion-boson vertex in QED. These equations, being an infinite set, have to be truncated if they are to be solved. Such a truncation is made possible by choosing a suitable non-perturbative ansatz for the fermion-boson vertex. This choice must satisfy these key properties of gauge invariance and multiplicative renormalisability. In this thesis we develop the constraints, in the case of massless unquenched QED, that have to be fulfilled to ensure that both the fermion and photon propagators are multiplicatively renormalisable-at least as far as leading and subleading logarithms are concerned. To this end, the Schwinger-Dyson equations are solved perturbatively for the fermion and photon wave-function renormalisations. We then deduce the conditions imposed by multiplicative renormalisability for these renormalisation functions. As a last step we compare the two results coming from the solution of the Schwinger-Dyson equations and multiplicative renormalisability in order to derive the necessary constraints on the vertex function. These constitute the main results of this part of the thesis. In the weak coupling limit the solution of the Schwinger-Dyson equations must agree with perturbation theory. Consequently, we can find additional constraints on the 3- point vertex by perturbative calculation. Hence, the one loop vertex in QED is then calculated in arbitrary covariant gauges as an analytic function of its momenta. The vertex is decomposed into a longitudinal part, that is fully responsible for ensuring the Ward and Ward-Takahashi identities are satisfied, and a transverse part. The transverse part is decomposed into 8 independent components each being separately free of kinematic singularities in any covariant gauge in a basis that modifies that proposed by Ball and Chiu. Analytic expressions for all 11 components of the O(a) vertex are given explicitly in terms of elementary functions and one Spence function. These results greatly simplify in particular kinematic regimes. These are the new results of the second part of this thesis.

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L'électrodynamique quantique en guide d'onde étudie le comportement de circuits électriques supraconducteurs composés entre autres de jonctions Josephson et de lignes à transmission. Ces circuits présentent peu de pertes puisqu'ils sont supraconducteurs. De plus, grâce à la non-linéarité des jonctions Josephson, ils peuvent présenter des comportements typiquement quantiques. Dans cette thèse, nous élaborons un cadre théorique qui permet de traiter la connexion entre les lignes à transmission et les éléments de circuits localisés (lumped element). Nous présentons ensuite la théorie d'entrée-sortie dans le contexte de ce cadre théorique. Comme son nom l'indique, celle-ci lie les observables à la sortie du circuit à celles à son entrée et elle permet de faire des prédictions expérimentales. Nous obtenons aussi une équation maîtresse qui décrit le circuit lorsque l'information contenue dans les lignes à transmission est perdue ou ignorée. Nous utilisons le cadre théorique développé pour étudier la situation où deux circuits qui se comportent chacun comme un atome sont connectés à une ligne à transmission. Nous montrons que la physique dans ce type de système dépend de la distance entre les deux atomes artificiels. Lorsque la distance est telle que la phase [phi] acquise par le champ électromagnétique entre les deux atomes artificiels est un multiple entier de [pi], on observe qu'une superposition d'états particulière des atomes est couplée à la ligne à transmission. On dit que cet état est brillant tandis que l'autre état est dit sombre. Lorsque la phase [phi] acquise par le champ électromagnétique est un multiple impair de [pi]/2, on observe plutôt une interaction cohérente entre les deux atomes artificiels. Nous suggérerons des protocoles pour observer des signatures expérimentales de cette physique. Nous présentons des résultats expérimentaux obtenus suite à ces prédictions par nos collègues du groupe d'Andreas Wallraff à Zurich. Ces résultats confirment la théorie. Parmi ces données, on retrouve la première mesure d'une signature claire de l'interaction cohérente entre deux atomes. Nous utilisons aussi le cadre théorique développé pour étudier des circuits dans lesquels les inductances dépendent du temps. Nous nous intéressons à ces circuits puisqu'ils sont généralement non réciproque, ce qui en fait des candidats idéaux pour implémenter des circulateurs. Ces dispositifs qui permettent d'obtenir un couplage unidirectionnel entre deux circuits sont généralement réalisés à l'aide d'aimants. Ainsi, un défi important du domaine est de concevoir un circulateur qui peut s'intégrer à un circuit supraconducteur. On utilise notre cadre théorique pour décrire les circuits avec des inductances variables à l'aide d'un opérateur de transfert qui relient les entrées du circuit à ses sorties. Cet objet permet d'extraire les conditions sous lesquelles ce type de circuit se comporte comme un circulateur. On utilise aussi l'opérateur de transfert pour étudier un modèle de circuit qui sera testé sous peu par nos collaborateurs de JILA dans le but d'implémenter un des premiers circulateurs sans conversion de fréquence nette, sans pertes et sans ferrite. On montre que ce modèle de circuit se comporte bien comme un circulateur, avec une largeur de bande de l'ordre de 200 MHz et un niveau d'imperfections de -20 dB.

We consider electron dynamics in strong electromagnetic fields, such as those expected from the next generation of high-intensity laser facilities. Beginning with a review of constant classical fields, we demonstrate that the electron motion (as given by the Lorentz force equation) can be divided into one of four Lorentz invariant cases. Parameterising the field tensor in terms of a null tetrad, we calculate the radiative energy spectrum for an electron in crossed fields. Progressing to an infinite plane wave, we demonstrate how the electron orbit in the average rest frame changes from figure-of-eight to circular as the polarisation changes from linear to circular. To move beyond a plane wave one must resort to numerics. We therefore present a novel numerical formulation for solving the Lorentz equation. Our scheme is manifestly covariant and valid for arbitrary electromagnetic field configurations. Finally, we reconsider the case of an infinite plane wave from a strong field QED perspective. At high intensities we predict a substantial redshift of the usual kinematic Compton edge of the photon emission spectrum, caused by the large, intensity dependent effective mass of the electrons inside the laser beam. In addition, we find that the notion of a centre-of-mass frame for a given harmonic becomes intensity dependent.

We investigate the coupling of a single trapped ion to a miniature optical cavity operating in the ultraviolet. Our cavity provides a source of single photons at a high rate into a single spatial mode. Using our apparatus, we have demonstrated the highest atom-cavity coupling rate achieved with a single ion by an order of magnitude. When the ion is continuously excited, we observe phase-sensitive correlations between emission into free-space and into the cavity mode, which can be explained by a cavity induced back-action effect on a driven dipole. We demonstrate coherent manipulation of a hyperfine qubit and ultra-short optical π rotations, which are essential tools for creation and detection of spin-photon entanglement. To this end, we have developed optical fibre-based Fabry-Perot cavities in the ultraviolet spectral range. These cavities operate near the primary dipole transition of Yb at 370 nm, and allow us to couple a pure atomic two-level system offered by a single trapped ion to the cavity mode. A new Paul trap apparatus in an ultra-high vacuum chamber has been built which allows for the integration of these cavities at very small ion-mirror separations. In order for independent operation of the trap, a compact system of diode lasers has been built which are stabilised to low-drift optical reference cavities. Coherent control of the hyperfine qubit in Yb 171 is achieved through application of microwave radiation, and ultra-short optical π rotations are performed with resonant light pulses derived from a frequency-doubled mode-locked titanium-sapphire laser. The experiment is controlled through a system of hardware and software which has been developed in a modular fashion and will allow for efficient control on the nanosecond time-scale when several such systems are interconnected. The success of our system opens the door to future experiments with trapped ions which will reach the strong coupling regime with a single ion. Furthermore, when operated in the fast-cavity regime, systems based on our approach will enable high-efficiency collection of photons from the ion into the single mode of an optical fibre. These systems will allow for the generation of distributed entanglement and will prove ideal as nodes in a larger quantum network of trapped ions.

To utilize a single atom as a quantum bit for a quantum computer requires exquisite control over the internal and external degrees of freedom. This thesis develops techniques for controlling the external degrees of freedom of individual atoms. In the first part of this thesis, individual atoms are trapped and detected non-destructively by the addition of cooling beams in an optical lattice. This non-destructive imaging technique led to atomic storage times of two minutes in an optical lattice. The second part of thesis incorporated the individual atoms into a high finesse cavity. Inside this optical cavity, atoms are cooled and non-destructively observed for up to 10 seconds.

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The thesis analyses the process of pair annihilation into one photon in a laser pulse. The theory of how to include pulse shapes in Strong Field QED and the resulting cross section is presented. The cross section is calculated and estimated for lasers of ELI and XFEL facilites. It is found that the effect may be experimentally verifiable at high frequency XFEL facilities for very finely tuned particle kinematics, but negligible at high intensity optical laser facilities such as ELI.