This thesis is a study of the phenomenon of superconductivity.

Non-contact atomic force microscopy allows us to directly probe the interactions between atoms and molecules. When operated in UHV and at low temperatures, a host of experiments, uniquely possible with the technique, can be carried out. The AFM allows us to characterise the forces present on a surface, resolve the atomic structure of molecules, measure the force required to move an atom, and even directly measure molecular pair potentials. Generally speaking, it is the interaction between the outermost tip and surface atoms that we measure. Therefore, in each of these experiments, understanding, or controlling, the tip termination is essential. As NC-AFM experiments become increasingly sophisticated, the combination of experiment and simulation has become critical to understand, and guide the processes at play. In this thesis, I focus on semiconductor surfaces and investigate the role of tip structure in a variety of situations with both DFT simulations and NC-AFM experiments. The clean Si(100) surface consists of rows of dimers, which can be manipulated between two different states using an NC-AFM. In order to understand the manipulation process, detailed DFT and NEB simulations were conducted to examine the energy balance of ideal and defective surfaces, with or without the presence of an AFM tip. We show that an explanation can only be reached when we consider both the AFM tip and variations in the PES caused by surface defects. NC-AFM experiments were also conducted on Si(100):H. We find that on this surface we regularly cultivate chemically passivated, hydrogen-terminated, tip apices which lead to distinct inverted image contrasts in our AFM images. Following a thorough characterisation of the tip apex, we conduct preliminary experiments designed to investigate surface defect structures, and to chemically modify the tip termination. Detailed DFT simulations show that this type of tip engineering, however, critically depends on the larger tip structure, significantly complicating the chances of success. Additionally, we investigate the structure and stability of silicon tip apices using DFT. Even with relatively simple tip structures, we observe complex behaviours, such as tip-dependent dissipation and structural development. These processes provide interesting information regarding tip stability, and commonly observed experimental behaviour. We also model an experiment in which we functionalise the tip apex with a C60 molecule, revealing for the first time that submolecular resolution is possible in the attractive regime.

This thesis is devoted to the study of modified gravity theories, especially, the scalar-tensor theories. A theorem due to Weinberg which states, that the equivalence principle is a necessary consequence of Lorentz invariance in a gravitational theory described by spin-2 massless particles is presented in Chapter 2. In view of this theorem modified gravity models either attempt to make \textit{graviton} massive or add other spin degrees of freedom. Scalar tensor theories are a simple and natural choice. An overview of some important scalar-tensor theories such as, Brans-Dicke model, DGP theory (although not a scalar-tensor theory it reduces to one in the so called \textit{decoupling} limit as we would see in chapter 2), Galileon model, Horndeski theory is also given in Chapter 2. The Hamiltonian analysis of the Galileon model is presented in Chapter 3. Chapter 4 presents the boundary terms and junction conditions of the Horndeski theory in the presence of codimension-1 branes. A generalised multiple-scalar-tensor theory analogous to Horndeski theory is developed in Chapter 5. We conclude with the proof of the most general multiple scalar field theory in arbitrary dimensions and flat-space time in Chapter 6. Chapters 3,4,5,6 are original work where the first 3 are based on the following journal articles.

Atom chips are a great tool for creation of low dimensional magnetic trapping geometries via micro-structures on the chip surface. Such structures allow the creation of time-dependent magnetic and electric potentials with highly accurate spatial and temporal dependency. As part of this thesis we have investigated the coherence dynamics in one-dimensional Bose-Einstein condensate while creating a sudden change in the atomic trapping potential. Such sudden changes create phase perturbations of the wave function, which leads to density perturbations. Analyzing these changes enables studies of the evolution of the coherence in a one-dimensional Bose gas with dynamically changing boundary conditions. Of particular interest is the study of prethermalization which can be understood in an integrable systems as so-called generalized Gibbs state. This state does not decay, but in case that there are perturbations that break integrability, this state relaxes further to a thermal state. To get a good understanding of such 1D systems we first investigated the transition from 3D to 1D Bose gas by observing both in situ and time of flight density profiles and analyzing the spatial variations in atom number as a function of temperature, geometry, and atomic density. High quality imaging is essential in these types of atomic physics experiments, and therefore a whole chapter is devoted to a new optimization method of absorption imaging. In this method we have taken into account the quantum nature of both the atomic medium and imaging light. Last, we have outlined an experiment that utilizes one-dimensional Bose-Einstein condensate as an analogue model of quantum field theory, in particular the dynamical Casimir effect and Hawking radiation. We do so by dynamically splitting a condensate along its long axis to a Y-like shape and measure the differential phase between the branches.

In this thesis two single molecule magnets based on the dodecamanganese (III, IV) cluster, with either benzoate or terphenyl-4-carboxylate ligands have been studied on the Au(111) and rutile TiO2(110) surfaces. We have used in situ electrospray deposition to produce a series of surface coverages from a fraction of a monolayer to multilayer films in both cases. X-ray absorption spectroscopy measured at the Mn L-edge (Mn 2p) has been used to study the effect of adsorption on the oxidation states of the manganese atoms in the core. In the case of the enzoate-functionalized complex, reduction of the manganese metal centres is observed due to the interaction of the manganese core with the underlying surface. In the case of terphenyl-4 carboxylate, the presence of this much larger ligand prevents the magnetic core from interacting with either the gold or the titanium dioxide surfaces and the characteristic Mn3+ and Mn4+ oxidation states necessary for magnetic behaviour are preserved. In contrast to the single molecule magnets where no charge transfer between the molecules and the substrates or within the molecules themselves were wanted, the molecules of bi isonicotinic acid and the giant zinc porphyrin nanorings have been studied on rutile TiO2(110) and Au(111) surfaces in the pursuit of charge transfer. In the case of the bi-isonicotinic acid it is studied on the rutile TiO2(110) where the technique of resonant inelastic X-ray scattering was been employed. Here we introduce the core-hole clock implementation to estimate the charge transfer from the molecule to the substrate. We verify previous results of ultrafast charge transfer in the sub-femtosecond regime (2.9 ± 0.3 femtoseconds) out of the LUMO+1 orbital. When the higher lying state of the LUMO+2 state is probed charge transfer out of this state and to the substrate is possibly there, but it is not possible to resolve it since it is masked by other effects originating from the inelastic scattering of the system. Furthermore, we see potential charge transfer within the molecule itself and new states observed in the inelastic scattering. Finally, zinc porphyrin nanorings were investigated on two surfaces of rutile TiO2(110) and Au(111). The techniques used here were X-ray photoemission spectroscopy and resonant photoemission spectroscopy. When the rutile TiO2(110) surface was employed hardly any participator decay was present suggesting charge transfer within the molecule itself or to the surface. This is further backed up by the fact that all of the core-excited unoccupied states are found to overlap energetically with the unoccupied states of the substrate, facilitating charge transfer out all the core-excited states. In the case of the Au(111) surface somewhat similar results are found, having all the core-excited states of the molecule located within the unoccupied states of the substrate, which again will facilitate charge transfer out all the core-excited states of the molecule. When the Au(111) substrate was employed the technique of near edge X-ray absorption fine structure was used to investigate the geometric orientation of the molecule on the surface. With the result of 86◦ ± 10◦ to the surface normal we verify previous scanning tunneling microscopy measurement that the zinc porphyrin nanorings will take a at lying orientation on the gold substrate.

This thesis investigates the dynamical properties of equilibrium and nonequilibrium systems, both quantum and classical, under the guise of a thermodynamic formalism. Large deviation functions associated with the generating functions of time-integrated observables play the role of dynamical free energies and thus determine the trajectory phase structure of a system. The 1d Glauber-Ising chain is studied using the time-integrated energy as the dynamical order parameter and a whole curve of second order trajectory transitions are uncovered in the complex counting field plane. The leading dynamical Lee-Yang zeros of the associated generating function are extracted directly from the time dependent high order cumulants. Resolving the cumulants into constituent contributions the motion of each contribution’s leading Lee-Yang zeros pair allows one to infer the positions of the trajectory transition points. Contrastingly if one uses the full cumulants only the positions of those closest to the origin, in the limit of low temperatures, can be inferred. Motivated by homodyne detection schemes this thermodynamic approach to trajectories is extended to the quadrature trajectories of light emitted from open quantum systems. Using this dynamical observable the trajectory phases of a simple “blinking” 3-level system, two weakly coupled 2-level systems and the micromaser are studied. The trajectory phases of this observable are found to either carry as much information as the photon emission trajectories or in some cases capture extra dynamically features of the system (the second example). Finally, the statistics of the time-integrated longitudinal and transverse magnetization in the 1d transverse field quantum Ising model are explored. In both cases no large deviation function exists but the generating functions are still calculable. From the singularities of these generating functions new transition lines emerge. These were shown to be linked to: (a) the survival probability of an associated open system, (b) PT-symmetry, (c) the temporal scaling of the cumulants and (d) the topology of an associated set of states.

In this work we mainly focus on two main aspects of interest within the field of Relativistic Quantum Information. We first expand on the current knowledge of the effects of relativity on entanglement between global field modes. Within this aspect, we focus on two topics: we address and revise the single mode approximation commonly used in the literature. We study the nonlocal correlations of charged bosonic field modes and the degradation of entanglement initially present in maximally entangled states as a function of acceleration, when one observer is accelerated. In the second part of this work we introduce, develop and exploit a method for confining quantum fields within one (or two) cavities and analyzing the effects of motion of one cavity on the entanglement initially present between cavity field modes. One cavity is always allowed to undergo arbitrary trajectories composed of segments of inertial motion and uniform acceleration. We investigate how entanglement is degraded, conserved and created as a function of the parameters describing the motion and we provide the analytical tools to understand how these effects occur. We conclude this work by analyzing the effects of the change of spatial topology on the nonlocal correlations present in the Hawking-Unruh radiation in the topological geon analogue of black hole spacetimes.

We describe what can be called the “universal” phase space of 2+1 AdS gravity, in which the moduli spaces of globally hyperbolic AdS spacetimes with compact Cauchy surface, as well as the moduli spaces of multi black hole spacetimes are realized as submanifolds. Importantly our phase space also includes all Brown-Henneaux excitations on the conformal boundary of asymptotically AdS spacetimes, with Diff+(S1)/SL(2,R)xDiff+(S1)/SL(2,R) contained as a submanifold. Our description of the universal phase space is obtained from results on the correspondence between maximal surfaces in AdS3 and quasi-symmetric homeomorphisms of the unit circle. We find that the phase space can be parametrized by two copies of the universal Teichmuller space T(D), or equivalently by the cotangent bundle over T(D). This yields a symplectic map from T*T(D) to T(D)xT(D) generalizing the well-known Mess map in the compact spatial surface setting. We also relate our parametrization to the Chern-Simons formulation of 2+1 gravity and, infinitesimally, to the holographic (Fefferman-Graham) description. In particular, we relate the charges arising in the holographic description (such as the mass and angular momentum of asymptotically AdS spacetimes) to the periods of holomorphic quadratic differentials arising via the Bers embedding of T(D)xT(D).

This thesis examines the electrical transport properties of a series of n-type GaAs/(AIGa)As double barrier resonant tunnelling devices with well widths between 50 angstrem and 2400 angstrem . The current-voltage characteristics show peak-to-valley ratios as high as 25:1 and as many as seventy resonances, with clear evidence of quantum interference effects at room temperature. The application of a high magnetic field parallel to the current flow produces magnetooscillations in the transport properties which allow the sheet charge density in the accumulation layer to be determined. The energy level in the well over a wide range of bias is obtained from analysis of thermal activation of resonant tunnelling. The contributions of elastic scattering and LO phonon emission to the valley current are investigated spectroscopically with a magnetic field and two phonon modes of the (AIGa)As barrier are observed. The buildup of space charge in the quantum well at resonance leads to intrinsic bistability in the current and differential capacitance of an asymmetric structure. Magnetoquantum oscillations due to a degenerate electron gas in the well are used to measure this charge buildup and demonstrate that the tunnelling process is truly sequential. The bistability is dramatically enhanced at high magnetic fields when the lowest energy Landau level of the well can accommodate a high electron density. In a strongly asymmetric sample, a new kind of bistability is observed where the off-resonant current exceeds the resonant current due to enhancement of charge buildup by intersubband scattering. The modulation of the scattering rate by a magnetic field produces periodic oscillations in the width of the bistability. In a magnetic field applied perpendicular to the current, the resonances are broadened as a consequence of the conservation of canonical momentum. The transition from electric to magnetic quantisation in wide wells is investigated and tunnelling into interfacial Landau levels is observed. The angular dependence of the resonances is used to probe conduction band anisotropy. In a tilted field, a completely new type of magneto-oscillations is observed.

In this thesis, we investigate analytically and numerically the existence and stability of discrete solitons governed by discrete nonlinear Schrödinger (DNLS) equations with two types of nonlinearity, i.e., cubic and saturable nonlinearities. In the cubic-type model we consider stationary discrete solitons under the effect of parametric driving and combined parametric driving and damping, while in the saturable-type model we examine travelling lattice solitons. First, we study fundamental bright and dark discrete solitons in the driven cubic DNLS equation. Analytical calculations of the solitons and their stability are carried out for small coupling constant through a perturbation expansion. We observe that the driving can not only destabilise onsite bright and dark solitons, but also stabilise intersite bright and dark solitons. In addition, we also discuss a particular application of our DNLS model in describing microdevices and nanodevices with integrated electrical and mechanical functionality. By following the idea of the work above, we then consider the cubic DNLS equation with the inclusion of parametric driving and damping. We show that this model admits a number of types of onsite and intersite bright discrete solitons of which some experience saddle-node and pitchfork bifurcations. Most interestingly, we also observe that some solutions undergo Hopf bifurcations from which periodic solitons (limit cycles) emerge. By using the numerical continuation software Matcont, we perform the continuation of the limit cycles and determine the stability of the periodic solitons. Finally, we investigate travelling discrete solitons in the saturable DNLS equation. A numerical scheme based on the discretization of the equation in the moving coordinate frame is derived and implemented using the Newton-Raphson method to find traveling solitons with non-oscillatory tails, i.e., embedded solitons. A variational approximation (VA) is also applied to examine analytically the travelling solitons and their stability, as well as to predict the location of the embedded solitons.