In this paper we show how to give an intrinsic definition of a constructible circle on the sphere. The classical definition of constructible circle in the plane, using straight edge and compass is there by translated in ters of so called Lenart tools. The process by which we achieve our goal involves concepts from the algebra of Hermitian matrices, complex variables, and Sterographic projection. However, the discussion is entirely elementary throughout and hopefully can serve as a guide for teachers in advanced geometry.

The main purpose of this thesis is to explore the conditions of the existence and properties of the Euler line of a triangle in the hyperbolic plane. Poincaré's conformal disk model and Hermitian matrices were used in the analysis.ʹ

The scope of this thesis is analyzing and characterizing certain gapless states in tight-binding non-Hermitian systems. We start by providing a pedagogical introduction to tight-binding theory, topological phases of matter, Wannier functions as real-space duals of Bloch functions and their properties, non-Hermitian systems and associated differences from standard Hermitian systems. Subsequently we show the possibility of extending pre-existing concepts of Hermitian quantum mechanics to non-Hermitian settings without losing predicting power over some peculiar observables. We conclude by providing numerical evidence for existence of certain topological states in finite one-dimensional and two-dimensional systems, also testing their robustness against symmetry-breaking and disorder.

The ALPHA experiment at CERN is an ongoing project which tests fundamental symmetries between matter and antimatter by producing and trapping antihydrogen atoms in order to perform precision spectroscopic measurements. A logical next step is to form the antihydrogen molecular ion (consisting of one positron and two antiprotons). This system possesses net charge, and can therefore be trapped electrostatically, making repeated measurements possible. Moreover it has been suggested that the molecule has the potential to allow for higher-precision comparisons with ordinary matter than have been attained with the atom. Since both momentum and energy have to be conserved in a collision, a simple collision process between an antihydrogen atom (“Hbar”) and an antiproton (“pbar”) does not suffice in order to form the molecular ion. However it is possible, upon mixing of the two species, for a pbar colliding with an Hbar in the ground electronic state to form a metastable molecular state (i.e., a resonance) which is weakly coupled to a stable molecular state (i.e., a bound state) via spontaneous quadrupole transition. During the time a metastable ion exists, a second pbar can happen to undergo a Coulomb collision with the metastable molecular ion. The quadrupole electrostatic interaction with this secondary antiproton acts as a time-dependent perturbation on the molecular system which can strengthen the coupling between resonance and bound state. Hence a collision with a secondary pbar can induce a transition to a bound state whereby the excess energy is carried off by the secondary pbar. This work aims to determine the efficiency of the process just described. On the theoretical side, the following is done: a study is conducted on the topic of resonance scattering as it relates to the problem in consideration; building on this study a generalized time-dependent perturbation theory is constructed which is valid for transitions to and from resonant states as well as bound states. On the numerical side: the effective potential for pbar-Hbar scattering in the ground electronic state is obtained numerically within the adiabatic approximation; the energies and lifetimes of the resonant states of the molecular ion are estimated; a temperature-dependent rate coefficient is obtained for the process described which, in order to obtain a proper rate, needs to be multiplied by the square of the density of the antiproton plasma and by the number of antihydrogen atoms. It is concluded that at current capacity for trapping and storage of pbar and Hbar the process examined is not competitive with respect to other formation routes which have been proposed for the molecular ion.

<p>This dissertation focuses on emergent geometry from SU(1,1) dynamical symmetry and non-Hermitian physics. While the geometrical approach unifies distinct phenomena in Hermitian and non-Hermitian systems, it also provides distinct means of coherent control of quantum dynamics and simulating exotic spacetimes.</p>

<p>Over the past two decades, open systems that are described by a non-Hermitian Hamiltonian have become a subject of intense research. These systems encompass classical wave systems with balanced gain and loss, semi-classical models with mode selective losses, and lossy quantum systems. The rapidly growing research on these systems has mainly focused on the wide range of novel functionalities they demonstrate. In this thesis, I intend to present some intriguing properties of a class of open systems which possess parity (P) and time-reversal (T) symmetry with a theoretical background, accompanied by the experimental platform these are realized on. These systems show distinct regions of broken and unbroken symmetries separated by a special phase boundary in the parameter space. This separating boundary is called the PT-breaking threshold or the PT transition threshold.</p> <p>We investigate non-Hermitian systems in two settings: tight binding lattice models, and electrical circuits, with the help of theoretical and numerical techniques. </p> <p><br></p> <p>With lattice models, we explore the PT-symmetry breaking threshold in discrete realizations of systems with balanced gain and loss which is determined by the effective coupling between the gain and loss sites. In one-dimensional chains, this threshold is maximum when the two sites are closest to each other or the farthest. We investigate the fate of this threshold in the presence of parallel, strongly coupled, Hermitian (neutral) chains, and find that it is increased by a factor proportional to the number of neutral chains. These results provide a surprising way to engineer the PT threshold in experimentally accessible samples.</p> <p>In another example, we investigate the PT-threshold for a one-dimensional, finite Kitaev chain—a prototype for a p-wave superconductor— in the presence of a single pair of gain and loss potentials as a function of the superconducting order parameter, onsite potential, and the distance between the gain and loss sites. In addition to a robust, non-local</p> <p>threshold, we find a rich phase diagram for the threshold that can be qualitatively understood in terms of the band-structure of the Hermitian Kitaev model.</p> <p>Finally, with electrical circuits, we propose a protocol to study the properties of a PT-symmetric system in a single LC oscillator circuit which is contrary to the notion that these systems require a pair of spatially separated balanced gain and loss elements. With a dynamically tunable LC oscillator with synthetically constructed circuit elements, we demonstrate static and</p> <p>Floquet PT breaking transitions by tracking the energy of the circuit. Distinct from traditional mechanisms to implement gain and loss, our protocol enables parity-time symmetry in a minimal classical system.</p>

This Thesis is devoted to the study of topological phases of matter in optical platforms, focusing on non-Hermitian systems with gain and losses involving nonreciprocal elements, and fractional quantum Hall liquids where strong interactions play a central role.In the first part we investigated nonlinear Taiji micro-ring resonators in passive and active silicon photonics setups. Such resonators establish a unidirectional coupling between the two whispering-gallery modes circulating in their perimeter. We started by demonstrating that a single nonlinear Taiji resonator coupled to a bus waveguide breaks Lorentz reciprocity. When a saturable gain is added to a single Taiji resonator, a sufficiently strong unidirectional coupling rules out the possibility of lasing in one of the whispering-gallery modes with independence of the type of optical nonlinearity and gain saturation displayed by the material. This can be regarded as a dynamical time-reversal symmetry breaking. This effect is further enhanced by an optical Kerr nonlinearity. We showed that both ring and Taiji resonators can work as optical isolators over a broad frequency band in realistic operating conditions. Our proposal relies on the presence of a strong pump in a single direction: as a consequence four-wave mixing can only couple the pump with small intensity signals propagating in the same direction. The resulting nonreciprocal devices circumvent the restrictions imposed by dynamic reciprocity. We then studied two-dimensional arrays of ring and Taiji resonators realizing quantum spin-Hall topological insulator lasers. The strong unidirectional coupling present in Taiji resonator lattices promotes lasing with a well-defined chirality while considerably improving the slope efficiency and reducing the lasing threshold. Finally, we demonstrated that lasing in a single helical mode can be obtained in quantum spin-Hall lasers of Taiji resonators by exploiting the optical nonlinearity of the material. In the second part of this Thesis we dived into more speculative waters and explored fractional quantum Hall liquids of cold atoms and photons. We proposed strategies to experimentally access the fractional charge and anyonic statistics of the quasihole excitations arising in the bulk of such systems. Heavy impurities introduced inside a fractional quantum Hall droplet will bind quasiholes, forming composite objects that we label as anyonic molecules. Restricting ourselves to molecules formed by one quasihole and a single impurity, we find that the bound quasihole gives a finite contribution to the impurity mass, that we are able to ascertain by considering the first-order correction to the Born-Oppenheimer approximation. The effective charge and statistical parameter of the molecule are given by the sum of those of the impurity and the quasihole, respectively. While the mass and charge of such objects can be directly assessed by imaging the cyclotron orbit described by a single molecule, the anyonic statistics manifest as a rigid shift of the interference fringes in the differential scattering cross section describing a collision between two molecules.

Quantum and classical frameworks form two perspectives for describing physical systems. Their formulation also presents interesting isomorphism: for example, the Schrodinger equation can find its classical correspondence in the paraxial Helmholtz equation, and coherent atomic population transfers is analogous to coupling dynamics in waveguides. In classical coupled mode systems, quantum notion can be manifested in the following ways: (1) adiabatic (i.e., sufficiently slow) evolution of the Hamiltonian enables robust mode conversion and light transfer, where the dynamics is carried out in predominantly one eigenmode; (2) non-Hermitian Hamiltonians give rise to peculiar singularities known as exceptional points (EPs), associated with not only degenerate eigenvalues but coalescent eigenvectors. In this dissertation, we explore the above principles in light manipulation, sensing, and photonic emulation. First, we numerically demonstrate two examples of photonic devices based on adiabatic evolution engineering. We present a coupled waveguide system analogous to the atomic physics process of stimulated Raman adiabatic passage, where the principle of adiabaticity not only allows high-extinction polarization mode splitting, but also counterintuitively mitigates the losses from the plasmonic structure involved. We show a modal hybridization effect in rib waveguide geometry that allows the mode to adiabatically evolve from one polarization to its orthogonal state upon electro-optic modulation in thin film lithium niobate, enabling an actively switchable polarization converter. We propose a generic EP emulator based on programmable photonics to tackle the challenging implementation of EP. Our approach combines on-chip operations of coupling, loss and detuning based on generic photonic modules (Mach-Zehnder interferometers), and a discrete scheme for mapping Hamiltonians to common mesh architecture. We demonstrate multiple exemplary EP functionalities, including loss-induced transparency, encircling second-order EPs in the PT and anti-PT symmetry picture, and a third-order EP. The proposed EP emulator marks a new paradigm for discrete, \textit{in situ} programming of EPs and multi-functional, repurposable EP devices. We also present our preliminary work on NV center-induced EPs. In contrast to conventional fluorescence-based schemes for addressing NV centers, we leverage NV centers' absorption to bring a coupled ring resonator system to an EP and numerically demonstrate the emerging dynamics. Our primary numerical results promise proof-of-concept magnetometry, combining NV centers' response to magnetic and microwave fields with the sensitivity enhancing nature of EP. This dissertation sheds light on unconventional photonics inspired by quantum-like principles. / 2025-08-29T00:00:00Z

Krylov subspace iterative methods provide an effective tool for reducing the solution of large linear systems to a size for which a direct solver may be applied. However, the problems of limited storage and speed are still a concern. Therefore, in this dissertation work, we present iterative Krylov subspace algorithms for non-Hermitian systems which do have fixed memory requirements and have favorable convergence characteristics. This dissertation describes three projects. The first project concerns short-term recurrence Krylov subspace methods for nearly-Hermitian linear systems. In 2008, Beckermann and Reichel introduced a short-term recurrence progressive GMRES algorithm for nearly-Hermitian linear systems. However, we have found this method to be unstable. We document the instabilities and introduce a different fixed-memory algorithm to treat nearly-Hermitian problems. We present numerical experiments demonstrating that the performance of this algorithm is competitive. The other two projects involve extending a strategy called Krylov subspace recycling, introduced by Parks and colleagues in 2005. This method requires more overhead than other subspace augmentation methods but offers the ability to recycle subspace information between cycles for a single linear system and recycle information between related linear systems. In the first project, we extend subspace recycling to the block Krylov subspace setting. A block Krylov subspace is a generalization of Krylov subspace where a single starting vector is replaced with a block of linearly independent starting vectors. We then apply our method to a sequence of matrices arising in a Newton iteration applied to fluid density functional theory and present some numerical experiments. In the second project, we extend the methods of subspace recycling to a family of linear systems differing only by multiples of the identity. These problems arise in the theory of quantum chromodynamics, a theory of the behavior of subatomic particles. We wish to build on the class of Krylov methods which allow the simultaneous solution of all shifted linear systems while generating only one subspace. However, the mechanics of subspace recycling complicates this situation and interferes with our ability to simultaneously solve all systems using these techniques. Therefore, we introduce an algorithm which avoids this complication and present some numerical experiments demonstrating its effectiveness. / Mathematics

Indiana University-Purdue University Indianapolis (IUPUI) / Open quantum systems have become a rapidly developing sector for research. Such systems present novel physical phenomena, such as topological chirality, enhanced sensitivity, and unidirectional invisibility resulting from both their non-equilibrium dynamics and the presence of exceptional points. We begin by introducing the core features of open systems governed by non-Hermitian Hamiltonians, providing the PT -dimer as an illustrative example. Proceeding, we introduce the Lindblad master equation which provides a working description of decoherence in quantum systems, and investigate its properties through the Decohering Dimer and periodic potentials. We then detail our preferred experimental apparatus governed by the Lindbladian. Finally, we introduce the Liouvillian, its relation to non-Hermitian Hamiltonians and Lindbladians, and through it investigate multiple properties of open quantum systems.