In the following work we investigate the dynamics of Bose-Einstein condensates(BECs) under the influence of time-dependent potentials. The response of such a sensitive system to external perturbations is a matter of increasing interest. This is because of the enormous growth in understanding the physics and emerging applications of BECs in many areas of physics such as sensing, microscopy and quantum information.

This thesis is concerned with aspects of quantum theory of fields in flat and curved spacetimes of arbitrary dimensions along with detecting bosons and fermions on these spacetimes. The thesis is divided into two main parts. In the first part, we analyse an Unruh-DeWitt particle detector that is coupled linearly to the scalar density of a massless Dirac field (neutrino field) in Minkowski spacetimes of dimension d ≥ 2 and on the two-dimensional static Minkowski cylinder, allowing the detector’s motion to remain arbitrary and working to leading order in perturbation theory. In d-dimensional Minkowski spacetime, with the field in the usual Fock vacuum, we show that the detector’s response is identical to that of a detector coupled linearly to a massless scalar field in 2d-dimensional Minkowski. In the special case of uniform linear acceleration, the detector’s response hence exhibits the Unruh effect with a Planckian factor in both even and odd dimensions, in contrast to the Rindler power spectrum of the Dirac field, which has a Planckian factor for odd d but a Fermi-Dirac factor for even d. On the two-dimensional cylinder, we set the oscillator modes in the usual Fock vacuum but allow an arbitrary state for the zero mode of the periodic spinor. We show that the detector’s response distinguishes the periodic and antiperiodic spin structures, and the zero mode of the periodic spinor contributes to the response by a state-dependent but well defined amount. Explicit analytic and numerical results on the cylinder are obtained for inertial and uniformly accelerated trajectories, recovering the d = 2 Minkowski results in the limit of large circumference. The detector’s response has no infrared ambiguity for d = 2, neither in Minkowski nor on the cylinder. In the second part, firstly, we give a thorough discussion for the Bogolubov transformation for Dirac field, and discuss pair creation in a non-stationary spacetime. Secondly, we derive the in and out vacua Wightman two-point functions for the Dirac field and the Klein-Gordon field for certain class of spatially flat Friedmann-Robertson-Walker (FRW) cosmological spacetimes wherein the two-point functions have the Hadamard form. We then establish the equivalence between the adiabatic vacuum of infinite order and the conformal vacuum in the massless limit. With the field in the conformal Fock vacuum, we then show that the detector’s response to an UDW particle detector coupled linearly to the scalar density of a massless Dirac field in the spatially flat FRW spacetimes in d-dimensions is identical to the response of a detector coupled to the massless scalar field in the spatially flat FRW spacetimes in 2d-dimensions. Lastly, we discuss a massive scalar field in the spatially compactified (1 + 1)-dimensional FRW spacetime. There, the issue of the conformal zero momentum mode arises. To resolve this issue, we develop a new scheme for quantizing the conformal zero-mode. This new quantization scheme introduces a family of two real parameters for every zero-momentum mode with an associated two-real-parameter set of in/out vacua. We then show that the zero momentum initial state’s wave functional corresponds to a two-real parameter set of Gaussian wave packets. For applications, we examine the finite-time detector’s response to a massive scalar field in the (1 + 1)-dimensional, spatially compactified Milne spacetime. Explicit analytic results are obtained for the comoving and inertially non-comoving trajectories. Numerical results are provided for the comoving trajectory. The numerical results suggest that when the in-vacuum is chosen to be very far from the conventional Minkowski vacuum state, then it contains particles. As result, spontaneous excitation of the comoving detector occurs.

Rydberg-dressed ground state atoms are atoms with an electron off-resonantly excited to a very high energy state, i.e., a state of high principal quantum number n ≫ 1. This thesis investigates the quantum dynamics of interacting Rydberg-dressed ground state atoms trapped in several multi-well potential traps. Rydberg atoms are atoms with exaggerated properties. One of their most interesting properties is that they exhibit a strong and long-ranged interaction that can be tuned leading to a variety of different quantum behaviours. My work focuses on studying the effects of these interacting atoms when loaded in multi-well potential traps. Generally, multi-well systems are considered as the simplest example of a finite optical lattice structure. For this reason, this thesis covers three research topics that examine the effects of long-range interaction on Rydberg-dressed atoms trapped in several potential confinements. I begin, in the introduction, by discussing the theoretical background of relevance to this work. It starts with presenting the physics of Bose-Einstein condensate. Then, the fundamentals of the interaction between two-level atom and light are analytically studied. This study has the purpose of understanding both; the dressed interacting atoms and optical lattices. The definition, characteristics, and the nature of the interaction between Rydberg atoms are analysed afterwards. The second chapter examines the dynamics of an ensemble of interacting Rydberg- dressed atoms trapped in static, i.e., time-independent, multi-well potentials using a mean-field theoretical approach. I choose one-dimensional double- and triple-well in addition to a two-dimensional quadruple-well potentials. The time-dependent non-linear Gross-Pitaevskii equation is used to numerically explore the ensemble's quantum dynamics. Solving the dynamical differential equations along with tuning the strength of the applied long-range interaction shows that the behaviour of non-interacting Rydberg-dressed atoms does not differ conceptually according to the geometry of the trapping potential. However, this changes when the interactions are switched on where the shape of the confinement leads to interesting outcomes especially in the non-linear interacting limit, such as macroscopic quantum self-trapping. After investigating an ensemble of interacting Rydberg-dressed atoms in static multi-well potential traps, the second research topic examines the dynamical evolution of these atoms when loaded in a finite optical lattice using the extended Bose-Hubbard model. In this chapter, the atoms ensemble is assumed to be in a superfluid state where I investigate both, the order parameter when the Rydberg excitation laser is applied and the interference pattern of the condensates in different dimensions. The study shows the emerging long-range interactions lead to a rapid collapse of the superfluid order parameter and in general allow only for partial revivals. In addition, the interference experiments can directly reveal the interaction between Rydberg-dressed atoms. In the fourth chapter, the dynamics of Rydberg-dressed atoms trapped in a dynamical, i.e., time-dependent, potential confinement is presented. The dynamical trap is constructed such that it begins as a harmonic oscillator and ends as a double- well potential. The analysis investigates an ensemble of contact-interacting atoms via the time-dependent non-linear GP equation.

The research included in this thesis comes in two main bodies. In the first, the focus is on intensity interferometric schemes, and I attempt to identify the types of correlations dominant in their operation. This starts with the, now rather historical, Hanbury Brown and Twiss setup from the 1950s and progresses to more recent interests such as ghost imaging and a variant of `quantum illumination', which is a quantum-enhanced detection scheme. These schemes are considered in the continuous variable regime, with Gaussian states in particular. Intensity interferometry has been the cause of a number of disputes between quantum opticians over the past 60 years and I weigh in on the arguments using relatively recent techniques from quantum information theory. In the second half, the focus turns away from the optical imaging and detection schemes, and onto quantum estimation -- multiparameter quantum estimation to be precise. This is an intriguing area of study where one has to carefully juggle tradeoffs in choosing both the optimal measurement and optimal state for performing an estimation in two or more parameters. I lay out a framework for circumventing some of the difficulties involved in this and apply it to several physical examples, revealing some interesting and at times counterintuitive features of multiparameter estimation.

Thermal conductivity measurements have been used to study the low lying energy levels of Cr ions in GaAs. Strong resonant phonon scattering was observed in semi-insulating (SI) and p-type samples, which is attributed to Cr 2+ or Cr 3+ ions, while the scattering in the n-type samples additional to that in undoped material was very small. From the computer fits of the thermal conductivity, zero-field ground state splittings have been deduced. A splitting at ~ 23 cm -1 is attributed to Cr 3+ ions, others at ~ 0.7 cm and 4.9 cm are due to Cr 2+. The phonon scattering in the n-type samples did not show any magnetic field dependence while big effects were observed in SI and p-type ones. This seems consistent with the results of the zero-field work. The effect of uniaxial stress on the phonon scattering has been measured in the temperature range ~2-15 K. Again no effects were seen in n-type material. The results for SI and p-type material are interpreted in terms of a static Jahn-Tellereffect of Cr 2+ ions. A preliminary investigation was made of the effect on the phonon scattering of sub-band-gap illumination. In an-type sample, the decay in the increase in the thermal resistivity produced by photoexcitation showed two parts. The first part with a characteristic time of ~ 1 hour is attributed to electron-capture at Cr 2+ ions. The second decay was very slow (persistent) with a characteristic time T » 10 5 s. This effect has tentatively been attributed to the occurrence of large lattice relaxation. The phonon scattering by the Cr ions is found to be consistent with the 'double acceptor' model for Cr in GaAs. Another model where er can act as a hole trap is discussed. Finally the effect of high magnetic fields on the thermal conductivity of acceptors in Ge was measured. From this and previous results, the g-values describing the ra8 ground state were found to be much lower than the predicted ones. A Thermally Detected EPR apparatus was designed and constructed in an attempt to check on the ground state structure of p-Ge and also p-Si but no results were obtained. This is believed to be due to the very large line widths resulting from strain splitting of the ra8 ground state.

In this thesis, problems of structural optimisation are approached through analytic and computational techniques. A particular focus is the effect of hierarchical design. The first chapter forms an introduction for the reader. Chapter 2 investigates the optimisation of elastic support on a buckling rod. A cost function is associated with the strength of the total elastic support provided to a beam of uniform cross-section supporting a compressive load. Through a perturbative method, it is found that for a low cost of support, a single, centrally placed support is optimal; furthermore it is found, using simulational and analytic methods, that the optimal support placement undergoes a series of bifurcations as the cost increases. The nature of these bifurcations is non-trivial and, although the analogy is not complete, there exist similarities between the solution to this problem and Landau theory of second-order phase transitions. In Chapter 3, the theme of hierarchical design is introduced. By analysing all possible failure modes, it is shown that a hierarchical design is highly efficient for withstanding external pressure loading in the limit of low applied pressures. By changing the level of hierarchy, the scaling law for volume of material required for structural stability against the applied external pressure can be changed systematically. For a given applied pressure, a particular level of hierarchy is shown to be optimal. This optimal level of hierarchy increases without bound as the pressure decreases. The Hausdorff dimension of the optimal structure and its dependence on applied pressure is found. Two example structures are presented, although the design is applicable to any convex shape. The fourth chapter of this thesis investigates the use of hierarchical geometry for a highly efficient interface between two surfaces. It is proposed that for a given strength of surface interaction, alterations to the geometry of the interface play a strong role in determining the force that is required to separate the surfaces. In particular the case of two surfaces with one being very much more rigid than the second is investigated. Increasing the hierarchical order of the design is seen to change the scaling relationship between the interface interaction strength and failure load. In Chapter 5, a hierarchical design for high mechanical efficiency under compressive loading is fabricated and mechanically tested. The particular design has previously been shown to be highly efficient under compressive loading. The scaling of material required to build a stable structure against a specific loading has previously been shown to be dependent on the level of hierarchy. A second order design is fabricated using rapid prototyping techniques. Additionally, a similar design based on hollow tubes rather than solid beams is proposed and is shown to make further savings on volume when compared to the original design. The final investigation presented in this thesis focuses on the role of imperfections in determining the buckling load of a hierarchical design. A two-dimensional design is proposed before simple, single beam, imperfections are added to the structure. The dependence of the structure on the magnitude of the imperfections is calculated analytically for the generation-1 and 2 designs. In the generation-1 structure, the magnitude of the imperfection is related to the reduction in failure load by a one-half power-law. The behaviour of a generation-2 frame with a single beam perturbed in thickness is found to be dominated by the behaviour of the generation-1 subframe. The behaviour found analytically is confirmed with finite element simulations for the generation-1 structure.

Surface plasmons (SP) are guided electromagnetic wave propagating along the surface of metal. The properties of SP are affected by the material attached to the metallic surface so they can be used as a very sensitive sensor capable of detecting the deposition of subnanometric layers of dielectric. SP has been widely investigated for biosensor applications and the theory is well established. Although SP sensors have been well studied, integrating the SP to a microscope is a relatively young field. Since the SPs are surface waves; microscopy techniques to optimise the SP microscope performance will require totally different techniques to non-surface wave microscopy. This thesis develops a theoretical framework to understand different types of SP microscope setups through the rigorous diffraction theory. The framework analyses the diffraction process through rigorous wave coupled analysis (RCWA) and a software package processes the diffracted orders to recover the microscope response for a range of different systems. In this thesis I will investigate the non-interferometric SP microscope, interferometric SP microscope and confocal SP microscope. I will show that the non-interferometric system exhibits a trade-off between lateral resolution and sensitivity, where an image obtained with a good contrast will have low lateral resolution. In order to get around the trade-off, the interferometric system can be employed; however, the main challenge for the interferometric setup is its optical alignment. I will show that a confocal SP microscope, which has been developed as a part of this thesis, can simplify the complexity of the interferometric system and give similar measurement performance. For the interferometric and confocal systems, the SP measurements are normally carried out through the interference signal, which is interference between a reference beam and the SP. I will suggest a method to extract SP propagation parameters from the interference signal by employing a spatial light modulator and also show that the SP propagation parameters do not only give us some insight to the SP effect for the interferometric system, but also gives us a new imaging mode to improve the resolution.

In this thesis, we present a theoretical study of the effects that a current-carrying Two-Dimensional Electron Gas (2DEG) produce on a neighbouring magnetically trapped Bose-Einstein Condensate of alkali atoms (BEC). We suggest that technology used for magnetic micro-controlling of cold gases could be improved by replacing or combining the metallic wires used in such structures with 2DEG-based conductors or quantum electronic devices. All calculations presented in this thesis consider parameters attainable with present technology, suggesting that experimental realization of the scenarios proposed here is already feasible. In Chapter one we present the general context in which this thesis is developed. It includes a definition of the Bose-Einstein condensate state, a description of the principles of magnetic trapping and a brief review of the developments in the area of micromanipulation of atomic BECs. Chapter two is devoted to describe the characteristics of the heterojunction considered in the thesis, and a simple model used to evaluate the electron flow in the 2DEG it contains. Chapter three shows in detail the properties of magnetic trapping configurations considered afterwards. We study two simple applications that can be developed by bringing a BEC near to a current carrying 2DEG. Firstly, in Chapter four, we demonstrate the feasibility of creating magnetic trapping potentials using a current-carrying 2DEG and an external magnetic field. We identify the advantages of such a 2DEG-based trap over traditional metal-based traps and the conditions needed for operability. Recently developed techniques of magnetic field microscopy with BECs motivate our second considered application, namely, using a Bose-Einstein Condensate to probe the electron transport in 2DEGs and structural characteristics of the heterojunction. In Chapter five, we demonstrate how the quantization of conductance occurring in Quantum Point Contacts (QPC) fabricated from the heterojunction in the 2DEG, can be detected through a localized depletion of the BEC caused by a small inhomogeneity of the magnetic field that originates from a current through the QPC. In addition, we show that the electron density fluctuations in the 2DEG can be measured by detecting the corresponding inhomogeneous magnetic field produced when current flows, via modulation of the BEC’s density. We establish the conditions under which a sensitive response of the BEC to the magnetic field is possible. We also derive a general relation between the modulation of the magnetic field affecting the BEC and the distribution of ionized donors in the heterojunction. Creating semiconductor-cold-atom hybrid systems, where electrons in the semiconductor and atoms in the BEC are coupled to each other, requires a full understanding of the properties of both systems. Since typical micro-traps have an elongated geometry, in Chapter six we present a study of phase correlations of BECs in such geometries. To do this, we use a recently proposed effective one-dimensional equation that takes into account the 3D character of the BEC. Finally, in Chapter seven, we conclude and identify directions for future work emerging from this thesis.

This thesis describes theoretical investigations into the dynamics of superfluid films and the effects of disorder on the roughening transition of crystal surfaces. The dynamic theory of superfluid helium films, due to Ambegaokar et al., is refined to improve the precision of the predictions made. A detailed comparison is made between the predictions of the modified theory and the results from experiments on helium films and on superconducting systems. It is found that, despite the modifications in the theory, agreement with experiments on helium films remains only qualitative. Consideration is then given to the effects on the roughening transition of disorder arising from screw dislocations. A crystal surface which is threaded by screw dislocation pairs may be in one of three different states depending on the temperature of the system and the way in which screw pairs are distributed. At high temperatures the interface is rough: it is not pinned to the lattice. At low temperatures the state of the interface depends on how the screw dislocations are distributed: when distributed as closely spaced pairs they lead to a faceted state with a single ground state energy; when distributed randomly they lead to a state of the interface which, though pinned to the underlying crystal lattice, has a degenerate ground state. It is then shown that the dynamic sine-Gordon formulation of the roughening transition can be used, via a Hubbard-Stratonovich transformation, to model the dynamic behaviour of superfluid systems. This method provides a re-normalization group framework within which the a.c. linear response can be studied. The ways in which the approach could be extended to study the effects of disorder and atomic layering are also discussed.

This thesis describes the behaviour of cold atoms in optical lattices. In particular, it explores how transport through the energy bands of the optical lattice can be used to study quantum chaos and Bose-Einstein condensation. Firstly, this study examines the dynamics of ultra-cold sodium atoms in a one-dimensional optical lattice and a three-dimensional harmonic trap, using both semi-classical and quantum-mechanical analyses. The atoms show mixed stable-chaotic classical dynamics, which originate from the intrinsically quantum-mechanical nature of the energy band. The quantised energy levels exhibit Gutzwiller fluctuations, and the wavefunctions are scarred by an unstable periodic orbit. Distinct types of wavefunction are identified and related directly to particular parts of the classical phase space via a Wigner function analysis. Secondly, this report studies the dynamics of a rubidium Bose-Einstein condensate in a one-dimensional optical lattice and three-dimensional harmonic trap. The condensates are set in motion by displacing the trap and initially follow simple semi-classical paths, shaped by the lowest energy band. Above a critical displacement, the condensate undergoes Bragg reflection, and performs Bloch oscillations. After multiple Bragg reflections, solitons and vortices form which damp the centre-of-mass motion. Finally, the dynamics of Bose-Einstein condensates in optical lattices are investigated for different parameter regimes, as realised in recent experiments. The results reveal how the experiments can be understood, and identify regimes in which vortices trigger explosive expansion of the condensate.