Topological phases of matter have long been a central theme in condensed matter physics. When subject to certain symmetries, these phases are classified by topological invariants that often take fractional, quantized values. Despite significant theoretical advancements, major open questions remain regarding the computation of these invariants in systems with crystalline symmetry, nonzero Chern number, or fractional quantum Hall states with fractional Hall conductivity.

The first part of this dissertation establishes the necessary theoretical background, defining symmetry operators, geometrical measures, and topological actions used to determine crystalline topological invariants.

The second part focuses on extracting these invariants numerically for invertible fermionic states in multiple ways. We will discuss the properties of these invariants such as how they determines the universal charge response of crystalline defects and boundaries and how they depends on an origin o in real space. Concrete numerical methods are demonstrated using the Hofstadter model, which exhibit non-zero Chern number and magnetic field. These invariants, together with several established invariants in the literature, gives a complete classification of ground states and reveal novel colorings of the Hofstadter butterfly.

The third part of this dissertation extends the calculation of crystalline topological invariants to fractional quantum Hall states, such as the 1/2-Laughlin state constructed by projecting parton states onto the same position. We show that the numerically extracted values of these invariants align with theoretical predictions from conformal field theory and G-crossed braided tensor categories

The center of our Galaxy is an intriguing region in astrophysics, providing a unique

opportunity to study various astrophysical processes. However, our line of sight is ob-

scured by dense layers of dust and gas, making optical emission observation impossible.

Fortunately, we can observe other portions of the electromagnetic spectrum, such as radio,

X-rays, and gamma rays. The latter serve as probes of cosmic-ray acceleration to extremely

high energies and, in theory, for indirect Dark Matter (DM) detection. This dissertation an-

alyzes gamma-ray data from the Galactic Center (GC), obtained with the High-Altitude

Water Cherenkov (HAWC) Gamma-Ray Observatory.

The HAWC Observatory, situated on the Sierra Negra volcano in Mexico at an altitude

of 4,100 m, detects gamma rays with energies from 0.1 to greater than 100 TeV. It has a

wide field of view of about 2 sr, and with an operational duty cycle of over 95%, it observes

two-thirds of the sky daily. The observatory uses the water-Cherenkov detection technique

to detect Cherenkov emission from secondary air-shower particles. This study presents

improvements to the reconstruction algorithms that retrieve the properties of the primary

gamma rays. In particular, the angular resolution and background rejection have improved

by a factor of four for gamma-ray events coming from high-zenith angles and with energies

above 40 TeV. Thanks to these improvements, HAWC can detect the GC.

Although multiple sources of cosmic-ray acceleration to PeV energies, known as Pe-

Vatrons, have been proposed within the Galaxy, they remain insufficient to fully account

for the observed flux. In this context, the GC was proposed as one of the main Galactic

PeVatrons with a power-law spectrum that extends up to 50 TeV without a cutoff. Here,

we present, for the first time, observations of gamma rays with energies above 100 TeV,

which we suggest originated from PeV protons—accelerated in the GC—that interact with

the dense ambient gas. While the angular resolution of HAWC at this high zenith angle is

not enough to distinguish the PeVatron, we provide the first confirmation of its existence.

The GC is also one of the most promising candidates for indirectly detecting DM via

gamma rays. While the nature of DM remains a fundamental question in modern physics,

Weakly Interacting Massive Particles (WIMPs) are considered potential candidates for DM

and naturally emerge in several extensions of the Standard Model (SM) of particle physics.

The relic density of thermally produced WIMPs in the early Universe can account for all

the DM observed in the Universe, as measured from cosmological observations. In theory,

WIMPs self-annihilate in dense astrophysical environments, like the GC, producing gamma

rays in the final state from processes such as hadronization, radiation, and decay of SM

particles. We conduct a follow-up study of the above analysis and compare the spatial

and spectral morphology of the residual gamma-ray emission to the one theorized for DM

annihilation. Finding no significant emission, we place for the first time to date upper

limits at the 95% confidence level on the velocity-weighted cross section, for

DM particle with masses well above 70 TeV using GC gamma-ray data.

Moiré transition metal dichalcogenide (TMD) bilayers have emerged as a versatile platform for exploring a wide range of correlated electronic phenomena. In optical studies of these semiconducting systems, excitons—bound electron-hole pairs—play a crucial role by linking electronic correlations to optical responses. This thesis develops theoretical frameworks to understand how excitonic behavior interplays with various correlated states in moiré TMD bilayers and the resulting implications for optical measurements.

We begin by investigating the interaction between a single moiré exciton and Wigner crystal states formed by doped electrons. Focusing on systems with repulsive electron–exciton interactions, we discuss a scenario where excitons move within the complementary lattice of the charge-ordered electrons. As different Wigner crystal configurations emerge at specific electron filling fractions, the effective exciton lattice changes accordingly, giving rise to distinct spectral and topological features. These characteristics can serve as optical signatures of the underlying charge order, and we propose a momentum- and polarization-resolved reflection experiment to detect them.

Next, we turn to undoped bilayers hosting multiple moiré excitons. While prior work often models these systems using a Bose-Hubbard framework, we show that their commutation relations deviate from those of conventional bosons. Instead, the excitons obey algebra similar to that of angular momentum operators, resulting in a finite Hilbert space and limiting local exciton occupancy to two or three, depending on the bilayer's specific parameters. We demonstrate that this occupancy constraint could manifest in high-intensity optical pumping experiments.

Finally, we explore exciton dynamics in a doped heterobilayer where the topmost valence moiré band forms an antiferromagnetic Mott insulator. We present a theoretical model describing the coupling between the exciton and the spin background, revealing that spin fluctuations substantially narrow the exciton’s effective bandwidth—by orders of magnitude compared to its non-interacting counterpart. This suppression in mobility provides a measurable contrast, which we suggest can be probed through exciton diffusion experiments.

Superconducting circuits are at the forefront of quantum computing and quantum sensing technologies, where accurate modeling and simulation are crucial for understanding and optimizing their performance. In this dissertation, we study modeling techniques and novel device designs to advance these technologies, focusing on efficient simulations, direct velocity measurement, and nonreciprocal devices for quantum information processing.  First, we investigate the use of discrete variable representations (DVRs) to numerically represent superconducting circuits, exploring their use and effectiveness in several prototypical examples. We find that not only are these DVRs capable of achieving decoherence-accurate simulation, i.e., accuracy at the resolution of experiments subject to decay, decoherence, and dephasing, they also demonstrate improvements in efficiency with smaller basis sizes and better convergence over current standard approaches, showing that DVRs are an advantageous alternative for representing superconducting circuits.

We then consider a specific quantum sensing application, direct velocity measurement in superconducting circuits. We propose and characterize theoretical models for backaction evading, direct velocity measurement that utilize traditional electric and magnetic transducers. We consider the readout of this signal via electric or magnetic field sensing by creating generic models analogous to the standard optomechanical position-sensing problem, thereby facilitating the assessment of measurement-added noise. Using simple models that characterize a wide range of transducers, we find that the choice of readout scheme --- voltage or current --- for each mechanical detector configuration implies access to either the position or velocity of the mechanical sub-system.

Finally, we explore the application of superconducting circuits in nonreciprocal devices, such as circulators. Commercial circulators in the microwave domain typically use ferromagnetic materials and wave interference, requiring large devices and large magnetic fields. However, quantum information devices for sensing and computation require small sizes, lower fields, and better on-chip integration. Equivalences to ferromagnetic order --- such as the XY model --- can be realized at much lower magnetic fields by using arrays of superconducting islands connected by Josephson junctions. Here we show that the quantum-coherent motion of a single vortex in such an array suffices to induce nonreciprocal behavior, enabling a small-scale, moderate-bandwidth, and low insertion loss circulator at very low magnetic fields and at microwave frequencies relevant for experiments with qubits.

Weakly Interacting Massive Particles (WIMPs) remain one of the most-well motivated dark matter candidates. Liquid Xe (LXe) time projection chambers (TPC) are the leading technology for probing the WIMP mass range. The LUX-ZEPLIN (LZ) experiment is a 10 tonne LXe TPC searching for the elusive WIMP dark matter. As detector masses increase to maximize sensitivity, background sources in the Xe, such as 85Kr and 39Ar, must be reduced to meet background requirements. These species are more difficult to quantify and require more stringent monitoring compared to previous generations of experiments. In this work, the techniques used to monitor 85Kr and 39Ar during commissioning and through LZ's WIMP Search 2024 science run (WS2024) are discussed. The natural Kr concentration during WS2024 is measured to be 182 +/- 26 ppq (g/g) (Kr/Xe) using a mass spectrometry technique. A complementary search for an excited state decay of 85Kr measured a natural Kr equivalent concentration of 124 (+44) (-37) ppq (g/g) (Kr/Xe). The natural Ar concentration during WS2024 is measured to be 1.10 +/- 0.18 ppb (g/g) (Ar/Xe). LZ has set a world leading limit on the interaction cross section across the expected WIMP mass range with a minimum of 2.1 * 10^-48 cm^2 at 36 GeV/c^2.

Systems containing large numbers of electrons can exhibit surprisingly complex and rich dynamics. In this dissertation, we ask: What is the minimum necessary detail in measurement or data-driven modeling and simulation to capture complex dynamics manifesting from these systems? To answer this, we integrate experiment, simulation, and theory to understand their complex dynamics. In this dissertation, we examine two such systems: (i) a superparamagnetic tunnel junction (SMTJ) and (ii) charged-particle beam dynamics.

We first consider the deterministic-stochastic behavior of a constant-current driven SMTJ, where we create a measurement-driven overdamped Langevin model capturing statistical properties of the device. We show both how this model captures device statistics across time scales and how it can be refined to capture higher-order behavior.

We next examine the centroid motion of a charged-particle beam and propose a method for understanding and predicting it using an interpretable, data-driven approach whose output is directly identifiable to terms in underlying low-dimensional evolution equations. We derive the evolution equations solely on the basis of data—with no recourse to an underlying first principles model. We compare and contrast our methodology with both a machine learning technique and a first principles model, and we show that we can learn interpretable equations for nonlinear beam dynamics at lower computational cost while achieving comparable accuracy.

Lastly, we investigate the phase space evolution of a charged-particle beam. Accurate knowledge of the phase space at beam creation is crucial for understanding and predicting beam dynamics. We measure a velocity space modulation to initialize the phase space of first principle simulations and capture beam statistics and internal beam structure­­­­­­­­­­­­—resembling a cruciform—with high fidelity. This contrasts with both employed first principles models—which do not account for beam structure as they assume a uniform beam cross section—and simulations using ideal phase space distributions. Finally, we demonstrate sensitivity to beam and lattice parameters varied within experimental measurement error.

Superconductors host vortices when exposed to a magnetic field exceeding their first critical field Bc1. Understanding the dynamics of vortices is crucial for optimizing the performance of various applications of superconductors, including superconducting radio-frequency (SRF) cavities and superconducting digital and quantum circuits. In this thesis, a near-field magnetic microwave microscope is employed to locally stimulate superconductors with an intense rf magnetic field and measure the local nonlinear microwave response. Under the microscope probe, two distinct vortex-related phenomena are observed: the nucleation of rf vortices and the motion of pre-existing trapped vortices. To interpret the measured response, toy models of superconductors with local defects are introduced and analyzed using Time-Dependent Ginzburg-Landau (TDGL) simulations of probe/sample interactions.

This dissertation is divided into two parts. The first part investigates the nucleation of single/few rf vortices associated with surface defects by studying the third-harmonic response P3f produced by the superconductor under intense stimulus at frequency f. Seven Nb/Cu films, grown under different deposition conditions by collaborators at CERN, are measured. Their surface defect properties related to rf vortex nucleation are compared. The second part explores the dynamics of trapped vortices under oscillating magnetic fields by studying the second-harmonic response P2f. A superconducting Nb film with an antidot flux pinning array is measured. The results show that this measurement technique provides access to vortex dynamics at the micron scale, including depinning events of a small number of trapped vortices and spatially-resolved pinning properties. These findings contribute to a deeper understanding of microwave superconductivity and vortex-induced nonlinearities, shedding light on the fundamental interactions between rf fields, magnetic vortices, and defects in superconductors. Furthermore, they offer new insights into the design and optimization of superconducting devices for microwave applications.

As artificial intelligence (AI) systems grow increasingly powerful and permeate every aspect of our lives, their impact on both individuals and society is an urgent concern. Questions of safety and robustness in AI stem largely from our limited understanding of deep learning. Research in this domain has traditionally followed two parallel paths: an empirical approach that prioritizes practical advancements and a theoretical approach that seeks a mathematical understanding from first principles. Despite notable progress, a significant gap remains between deep learning practice and its theoretical underpinnings. This dissertation advocates for a phenomenological approach to understanding AI systems -- one that integrates empirical observations with theoretical model-building. This methodology has been instrumental in the physical sciences, and it holds similar promise for advancing the science of deep learning. Over two broad parts, this work demonstrates the effectiveness of this approach in characterizing model architectures and their emergent capabilities.

In the first part, we explore how signal propagation analysis in large-N limits can inform the design and initialization of model architectures. We develop a diagnostic observable that distinguishes between ordered and chaotic behaviors in neural networks, guiding optimal parameter initialization for training. Our analysis establishes the theoretical soundness of this observable in simple networks and confirms its empirical utility in state-of-the-art architectures. The findings reveal an architecture design paradigm that eliminates the need for careful initialization, shedding light on widely used heuristic practices. Additionally, we introduce an algorithm that automates initialization across diverse model architectures, enhancing their trainability.

In the second part, we highlight the importance of the systems identification approach for characterizing AI systems. We explore several stylized setups where model capabilities emerge as a function of compute, data quantity, and data diversity. Using arithmetic and cryptographic tasks as examples, we demonstrate that emergent abilities such as grokking and in-context learning arise alongside the formation of interpretable structures within the model’s parameters, hidden representations, and outputs. Through targeted experiments, we identify these structures using (i) black-box probing, which examines model responses to characteristic inputs, and (ii) open-box analysis, which leverages curated task-specific observables and metrics to study internal model states.

This dissertation promotes a paradigm for understanding deep learning that complements both heuristic-driven and hypothesis-driven approaches. By integrating experimental methodologies and analytical tools from established scientific disciplines, this framework has the potential to steer the field toward safer, more robust, and more efficient AI systems.

In recent years, quantum experiments have become increasingly precise, fast, and capable of high resolution. Particular interest has been given to quantum control, which aims to prepare, manipulate, and steer quantum states toward desired outcomes. Common applications of quantum control include state preparation for quantum computing algorithms, protocols to implement nanoscale machines, and feedback to guide a system's evolution. Feedback involves collecting information from quantum measurements, then acting on the system based on measurement outcomes. The standard measurement model in quantum mechanics is the projective measurement, which destroys quantum coherence by causing the wave function to collapse to a subspace spanned by the eigenstates of the measured operator.

This thesis explores the theory of weak measurement processes, a class of measurement protocols that extract information from a quantum system while (partially) preserving coherence. The weak measurement protocol has a tunable parameter that controls the information obtained per measurement cycle and the disturbance (decoherence) introduced into the quantum system. Using this nondestructive form of measurement, one can extract information during the system's evolution and apply real-time feedback to drive the system's evolution to specific target states. A general master equation is derived to describe continuous feedback using weak measurements with general filtering processing. Particular cases of low-pass and band-pass filters are studied in detail and applied to a harmonic oscillator cooling protocol. Results show that ground-state cooling of the quantum harmonic oscillator can be achieved.

Finally, this dissertation discusses an experimental and computational project that uses machine learning to estimate the temperature and the number of atoms of a cold atomic cloud. The goal is to use non-destructive measurements to infer hidden properties of the atomic ensemble without disturbing the atomic trap. Results show that reasonable accuracy can be achieved using various neural network architectures, depending on the complexity of the input data. The accuracy and responsiveness of the trained models make them suitable for real-time estimators that can be used in closed-loop feedback.

Since the 1973 announcement of the initial Vela detection of Gamma Ray Bursts (GRBs) in 1967, these powerful events have been of significant interest. The study of GRBs has advanced our knowledge of plasma physics, particle physics, cosmology, among many other fields. However, even with the advancement in understanding gained through various detections, both multi-wavelength and multi-messenger, there is still much to be learned by observing these transients in as many different ways as possible. This dissertation will explore the development of instrumentation (RIMAS and PRIME), operational software, data reduction software, and observations aimed at furthering the study of GRBs.

RIMAS is a near-infrared imager spectrometer operating in wavelengths from 0.97 to 2.37 µm, and will be installed on the 4.3 m Lowell Discovery Telescope (LDT) in Happy Jack, Arizona. Among other features, RIMAS has four broad band filters (Y, J, H, K), and three spectral modes (R30, R300 and R5000). Completing this instrument required the development, integration and testing of multiple detector readout systems, cryogenic systems, spectral and imaging modes, communication systems, and cooling systems. Furthermore, software needed to be developed to reduce RIMAS data so that it will be ready to contribute to scientific discoveries shortly after commissioning. Due to the development of this software, observers will quickly obtain high signal-to-noise exposures, astronomically and photometrically calibrated stacked images, and one-dimensional spectra that is flux and wavelength calibrated.

PRIME is a 1.8 m telescope with 1.56 square degree FOV (0.5"/pixel) operating in the NIR (0.82-1.78 µm) located in Sutherland, South Africa at the South African Astronomical Observatory (SAAO). The primary science objectives for PRIME are to perform a time domain survey of the Galactic Bulge for the purpose of observing microlensing events caused by transiting exoplanets. In doing so, it will support the Nancy Roman Space Telescope (RST) mission by taking baseline measurements of this field before RST launch, while testing the performance of RST package H4RG-10 detectors. After RST launch, PRIME will perform simultaneous observation with RST of the Galactic Bulge to perform parallax measurements. PRIME is also continuously available for Target of Opportunity (ToO) observations, making it a powerful tool for observing GRB and GW counterparts. Since commissioning in October 2022, PRIME has been observing a time domain survey of the Galactic Bulge searching for microlensing events, observing high energy transients (GRBs, X-ray bursts, gravitational waves, and supernovae), and performing an all-sky survey in J-band. During the course of this thesis work, the development of PRIME's camera greatly mirrors that of RIMAS, with lessons learned shared between them. Most notable in this development is the integration and testing of PRIME's detectors and detector readout system. This is the first instance of these detectors and electronics being used on-sky, and as such required the development of custom software and thorough testing.

Beyond these instrumentation efforts, photometric and spectroscopic observations were performed in the optical for various GRB afterglows, GRB host galaxies, and GW counterpart candidates using LDT's optical imager, LMI, and optical spectrograph, DeVeny. Multiple high energy transients (GRBs, X-ray Bursts, LVK probability fields) were also observed using the recently commissioned PRIME telescope. These efforts have resulted in many GCN circulars, and contributed to multiple papers.

The study of quantum computers and quantum algorithms has captured the attention of physics globally in recent decades. While some researchers are working towards constructing fault-tolerant large-scale quantum computers, others are theorizing about the capabilities of these new computers and developing quantum algorithms to leverage their increased abilities. This dissertation engages with the second area of research concerning quantum algorithms. The first part of this dissertation discusses quantum algorithms for state preparation, with a focus on thermal state preparation. A selection of other state preparation methods is briefly summarized before a novel method of thermal state preparation is laid out wherein a so-called heat pump is employed in a technique of active cooling. The second part of this dissertation discusses quantum algorithms that aim to address the problem of gauge violation during the simulation of lattice gauge theories. The problem of gauge violation in both quantum analog and digital simulations is presented before two related methods are delineated. These methods leverage the quantum Zeno effect with frequent measurements on a system's physical subspace to keep the system's state physical throughout the course of a simulation. Furthermore, the properties of gauge transformations are utilized to help curtail the growth of unwanted unphysical amplitudes.