Scaling up the lattice dynamics of amorphous materials by orders of magnitude

We generalise the non-affine theory of viscoelasticity for use with large, well-sampled systems of arbitrary chemical complexity. Having in mind predictions of mechanical and vibrational properties of amorphous systems with atomistic resolution, we propose an extension of the Kernel Polynomial Method (KPM) for the computation of the vibrational density of states (VDOS) and the eigenmodes, including the $\Gamma$-correlator of the affine force-field, which is a key ingredient of lattice-dynamic calculations of viscoelasticity. We show that the results converge well to the solution obtained by direct diagonalization (DD) of the Hessian (dynamical) matrix. As is well known, the DD approach has prohibitively high computational requirements for systems with $N=10^4$ atoms or larger. Instead, the KPM approach developed here allows one to scale up lattice dynamic calculations of real materials up to $10^6$ atoms, with a hugely more favorable (linear) scaling of computation time and memory consumption with $N$

Advanced control systems for fast orbit feedback of synchrotron electron beams

Diamond Light Source is the UK’s national synchrotron facility that produces synchrotron radiation for research. At source points of synchrotron radiation, the electron beam stability relative to the beam size is critical for the optimal performance of synchrotrons. The current requirement at Diamond is that variations in the beam position should not exceed 10% of the beam size for frequencies up to 140Hz. This is guaranteed by the fast orbit feedback that actuates hundreds of corrector magnets at a sampling rate of 10kHz to reduce beam vibrations down to sub-micron levels. For the next-generation upgrade, Diamond-II, the beam stability requirements will be raised to 3% up to 1kHz. Consequently, the sampling rate will be increased to 100kHz and an additional array of fast correctors will be introduced, which precludes the use of the existing controller. This thesis develops two different control approaches to accommodate the additional array of fast correctors at Diamond-II: internal model control based on the generalised singular value decomposition (GSVD) and model predictive control (MPC). In contrast to existing controllers, the proposed approaches treat the control problem as a whole and consider both arrays simultaneously. To achieve the sampling rate of 100kHz, this thesis proposes to reduce the computational complexity of the controllers in several ways, such as by exploiting symmetries of the magnetic lattice. To validate the controllers for Diamond-II, a real-time control system is implemented on high-performance hardware and integrated in the existing synchrotron. As a first-of-its-kind application to electron beam stabilisation in synchrotrons, this thesis presents real-world results from both MPC and GSVD-based controllers, demonstrating that the proposed approaches meet theoretical expectations with respect to performance and robustness in practice. The results from this thesis, and in particular the novel GSVD-based method, were successfully adopted for the Diamond-II upgrade. This may enable the use of more advanced control systems in similar large-scale and high-speed applications in the future

Theory on the local electromagnetic field in crystalline dielectrics

Starting from the microscopic Maxwell equations, a self-contained theory of the local electromagnetic field in crystalline dielectrics and its relation to macroscopic electrodynamics is established. Applying the Helmholtz decomposition theorem to the microscopic Maxwell equations, two independent sets of equations determining the solenoidal (transverse) and irrotational (longitudinal) contributions of the local electromagnetic field are initially identified. This enables to restate the microscopic Maxwell equations in terms of equivalent inhomogeneous integral equations, where the entire matter and its response to the local electromagnetic field is fully taken into account by the current density. Implementing a phenomenological material model into this current density, where the local electric field is assumed to polarize individual atoms/ions or subunits of the material in reaction to an externally applied electric field, the inhomogeneous integral equation determining the local electric field is specified first with respect to crystalline dielectrics, solely with the crystal structure and the individual polarizabilities as an input into the theory. Afterwards, it is solved exactly by making use of a newly discovered orthonormal and complete system of non-standard Bloch eigenfunctions of the crystalline translation operator. The propagable modes within the dielectric crystal as well as their dispersion relations \omega_{n}\left(\mathbf{q}\right) , also called photonic band structure, are then obtained from the corresponding solvability condition of the associated homogeneous integral equation. Additionally, the impact of radiation damping and of a static external magnetic induction field on \omega_{n}\left(\mathbf{q}\right) is elucidated, followed by a discussion on the characteristics of the local electromagnetic field in crystalline dielectrics. Subsequently, the macroscopic electromagnetic field is consistently derived from the local electromagnetic field by applying a spatial low-pass filter to the latter one to eliminate all spatial variations that occur on length scales comparable to the lattice constant. The same filtering procedure is then deployed to the microscopic polarization, so that the dielectric tensor \varepsilon_{\Lambda}\left(\mathbf{q},\omega\right) can be introduced in a tried and tested way as that quantity that relates the low-pass filtered microscopic polarization with the macroscopic electric field. Next, the differential equations satisfied by the longitudinal and transverse parts of the macroscopic electric field describing electric field screening or wave propagation are deduced in terms of the longitudinal and transverse dielectric tensor \varepsilon^{\left(\text{L}\right)}\left(\mathbf{q},\omega\right) and \varepsilon^{\left(\text{T}\right)}\left(\mathbf{q},\omega\right) respectively, followed by a comparison of the local and macroscopic radiation field in dielectric crystals. Finally, the dielectric tensor and its transverse counterpart are discussed. It is shown, that the derived expression for the dielectric tensor \varepsilon_{\Lambda}\left(\mathbf{q},\omega\right) conforms in the static limit with general principles such as causality and thermodynamic stability and eventually reduces to the well-known Clausius-Mossotti equation when monatomic simple cubic crystals are considered. Furthermore, its frequency dependence is proven to comply well with the Lyddane-Sachs-Teller relation. In the end, the Taylor expansion of the transverse dielectric tensor \varepsilon^{\left(\text{T}\right)}\left(\mathbf{q},\omega\right) up to second order around \mathbf{q}=\mathbf{0} gives insight into various optical effects or quantities, that are related to non-locality (spatial dispersion) and retardation (chromatic dispersion), in full agreement with the phenomenological theory of Agranovich and Ginzburg. This includes the index of refraction, natural optical activity and spatial dispersion induced birefringence as well as their respective frequency dependencies. The utility of the presented theory is then proven by demonstrating, that the calculations regarding the previously mentioned optical effects or quantities coincide well with experimental data for a variety of highly diverse and complex crystal structures. In particular, the disruptive influence of spatial dispersion induced birefringence in cubic crystals is highlighted in view of the design of optical imaging systems for lithographic applications in the deep ultraviolet spectral region

Vision and revision: wavefront sensing from the image domain

An ideal telescope with no optical aberrations can achieve a resolution and contrast limited by the wave nature of light, such that the finest detail that can be resolved is of the order of the angle subtended by one wavelength over the diameter of the telescope. For telescopes operating close to this ideal case, however, it is rare that the full performance of the diffraction limit is achieved, as small optical imperfections cause speckles to appear in the image. These are difficult to calibrate, as they are often caused by thermal and mechanical variations in the optical path which vary slowly with time. The quasi-static speckles that they impose can mimic the real signal of a faint star or planet orbiting the primary target, and these therefore impose the principal limitation on the angular resolution and contrast of instruments designed to detect exoplanets and faint companions. These aberrations can be corrected by active optics, where a wavefront sensor is used to used to reconstruct a map of the distortions which can then be compensated for by a deformable mirror, but there is a problem with this also: differrential aberrations between the wavefront sensor and science camera are not detected. In this thesis, I will discuss a successful laboratory implementation of a recently-proposed technique for reconstructing a wavefront map using only the image taken with the science camera, which can be used to calibrate this non-common path error. This approach, known as the asymmetric pupil Fourier wavefront sensor, requires that the pupil not be centrosymmetric, which is easily achieved with a mask, with segment tilting, or with judiciously placed spiders to support the secondary mirror, and represents a promising way forward for characterizing and correcting segment misalignments on future missions such as the James Webb Space Telescope

Boundary Integral Operators in Linear and Second-order Nonlinear Nano-optics

Recent advances in the fabrication of nanoscale structures have enabled the production of almost arbitrarily shaped nanoparticles and so-called optical metamaterials. Such materials can be designed to have optical properties not found in nature, such as negative index of refraction. Noble metal nanostructures can enhance the local electric field, which is beneficial for nonlinear optical effects. The study of nonlinear optical properties of nanostructures and metamaterials is becoming increasingly important due to their possible uses in nanoscale optical switches, frequency converters and many other devices.The responses of nanostructures depend heavily on their geometry, which calls for versatile modeling methods. In this work, we develop a boundary element method for the modeling of surface second-harmonic generation from isolated nanoparticles of very general shape. The method is also capable of modeling spatially periodic structures by the use of appropriate Green’s function. We further show how to utilize geometrical symmetries to lower the computational time and memory requirements in the boundary element method even in cases where the incident field is not symmetrical.We validate the boundary element approach by the calculation of second-harmonic scattering from gold spheres of different radii. Comparison to analytical solution reveals that under one percent relative error is easily achieved. The method is then applied to model second-harmonic microscopy of single gold nanodots and second-harmonic generation from arrays of L- and T-shaped gold particles. The agreement between the calculations and measurements is shown to be excellent.To provide a more intuitive understanding of the optical response of nanostructures, we develop a full-wave spectral approach, which is based on boundary integral operators. We present a theory which proves that the resonances of a smooth scatterer are isolated poles that occur at complex frequencies. Other types of singularities, such as branch-cuts, may occur only via the fundamental Green function or material dispersion. We propose a definition of an eigenvalue problem at fixed real frequencies which gives rise to modes defined over the surface of the scatterer. We illustrate that these modes accurately describe the optical responses that are usually seen for certain particle shapes when using plane-wave excitations. With the spectral approach, the resonance frequencies and the modal responses of a scatterer can be found as intrinsic properties independent of any incident field. We show that the spectral theory is compatible with the Mie theory for pherical particles and with a previously studied quasi-static theory in the limit of zero frequency

Guaranteed Minimum-Rank Solutions of Linear Matrix Equations via Nuclear Norm Minimization

The affine rank minimization problem consists of finding a matrix of minimum rank that satisfies a given system of linear equality constraints. Such problems have appeared in the literature of a diverse set of fields including system identification and control, Euclidean embedding, and collaborative filtering. Although specific instances can often be solved with specialized algorithms, the general affine rank minimization problem is NP-hard. In this paper, we show that if a certain restricted isometry property holds for the linear transformation defining the constraints, the minimum rank solution can be recovered by solving a convex optimization problem, namely the minimization of the nuclear norm over the given affine space. We present several random ensembles of equations where the restricted isometry property holds with overwhelming probability. The techniques used in our analysis have strong parallels in the compressed sensing framework. We discuss how affine rank minimization generalizes this pre-existing concept and outline a dictionary relating concepts from cardinality minimization to those of rank minimization

Efficient ab initio many-body calculations based on sparse modeling of Matsubara Green's function

This lecture note reviews recently proposed sparse-modeling approaches for efficient ab initio many-body calculations based on the data compression of Green's functions. The sparse-modeling techniques are based on a compact orthogonal basis representation, intermediate representation (IR) basis functions, for imaginary-time and Matsubara Green's functions. A sparse sampling method based on the IR basis enables solving diagrammatic equations efficiently. We describe the basic properties of the IR basis, the sparse sampling method and its applications to ab initio calculations based on the GW approximation and the Migdal-Eliashberg theory. We also describe a numerical library for the IR basis and the sparse sampling method, irbasis, and provide its sample codes. This lecture note follows the Japanese review article [H. Shinaoka et al., Solid State Physics 56(6), 301 (2021)].Comment: 26 pages, 10 figure

Molecular Structure and Modeling of Water-Air and Ice-Air Interfaces Monitored by Sum-Frequency Generation.

From a glass of water to glaciers in Antarctica, water-air and ice-air interfaces are abundant on Earth. Molecular-level structure and dynamics at these interfaces are key for understanding many chemical/physical/atmospheric processes including the slipperiness of ice surfaces, the surface tension of water, and evaporation/sublimation of water. Sum-frequency generation (SFG) spectroscopy is a powerful tool to probe the molecular-level structure of these interfaces because SFG can specifically probe the topmost interfacial water molecules separately from the bulk and is sensitive to molecular conformation. Nevertheless, experimental SFG has several limitations. For example, SFG cannot provide information on the depth of the interface and how the orientation of the molecules varies with distance from the surface. By combining the SFG spectroscopy with simulation techniques, one can directly compare the experimental data with the simulated SFG spectra, allowing us to unveil the molecular-level structure of water-air and ice-air interfaces. Here, we present an overview of the different simulation protocols available for SFG spectra calculations. We systematically compare the SFG spectra computed with different approaches, revealing the advantages and disadvantages of the different methods. Furthermore, we account for the findings through combined SFG experiments and simulations and provide future challenges for SFG experiments and simulations at different aqueous interfaces

Compressing the two-particle Green's function using wavelets: Theory and application to the Hubbard atom

Precise algorithms capable of providing controlled solutions in the presence of strong interactions are transforming the landscape of quantum many-body physics. Particularly exciting breakthroughs are enabling the computation of non-zero temperature correlation functions. However, computational challenges arise due to constraints in resources and memory limitations, especially in scenarios involving complex Green's functions and lattice effects. Leveraging the principles of signal processing and data compression, this paper explores the wavelet decomposition as a versatile and efficient method for obtaining compact and resource-efficient representations of the many-body theory of interacting systems. The effectiveness of the wavelet decomposition is illustrated through its application to the representation of generalized susceptibilities and self-energies in a prototypical interacting fermionic system, namely the Hubbard model at half-filling in its atomic limit. These results are the first proof-of-principle application of the wavelet compression within the realm of many-body physics and demonstrate the potential of this wavelet-based compression scheme for understanding the physics of correlated electron systems.Comment: 25 pages, 16 figures, 2 table