The concept of a photonic statistical topological transition (PSTT) introduces a novel framework in which the statistical properties of photons, quantified by higher‑order correlation functions g^(n), undergo topological changes that give rise to new quantum photonic phases. This paper presents a comprehensive theoretical analysis of PSTT, establishes the mathematical conditions under which topological invariants associated with photon statistics can change, and discusses the physical implications of such transitions. We demonstrate that the second‑order correlation function g^(2) can evolve from values characteristic of classical light (g^(2) > 1) to those of non‑classical light (g^(2) < 1) via a topologically protected transition, and we generalize the argument to higher‑order correlations. The potential for discovering new quantum light states, which may have applications in quantum information processing, precision measurement, and fundamental tests of quantum mechanics, is highlighted.
The concept of a photonic statistical topological transition (PSTT) introduces a novel framework in which the statistical properties of photons, quantified by higher‑order correlation functions g^(n), undergo topological changes that give rise to new quantum photonic phases. This paper presents a comprehensive theoretical analysis of PSTT, establishes the mathematical conditions under which topological invariants associated with photon statistics can change, and discusses the physical implications of such transitions. We demonstrate that the second‑order correlation function g^(2) can evolve from values characteristic of classical light (g^(2) > 1) to those of non‑classical light (g^(2) < 1) via a topologically protected transition, and we generalize the argument to higher‑order correlations. The potential for discovering new quantum light states, which may have applications in quantum information processing, precision measurement, and fundamental tests of quantum mechanics, is highlighted.
The statistical properties of photons, quantified by the normalized correlation functions g^(n), have traditionally been treated as passive descriptors of quantum light. This paper introduces the concept of a Photonic Statistical Topological Transition (PSTT), whereby the photon correlation functions undergo discrete, topologically protected changes as system parameters are varied. We demonstrate that the second‑order correlation g^(2) can transition from a value of unity, characteristic of coherent light, to zero, indicative of perfect photon antibunching, in a manner that is accompanied by a non‑trivial winding of the phase of the two‑point correlation function across a parameter space. This behavior is shown to be governed by a topological invariant analogous to a winding number or Chern number in condensed‑matter systems. The theoretical framework is illustrated using a one‑dimensional array of coupled nonlinear cavities, modeled by the Bose‑Hubbard Hamiltonian with nearest‑neighbor hopping. The results suggest that PSTT constitutes a new class of quantum photonic phase transitions, with potential implications for the discovery of novel quantum light states and applications in quantum technologies.
Photon statistics is a branch of physics that involves observing the behavior of single or a small number of photons and then aggregating the results through statistical averaging. This process is essential to obtain meaningful and reliable data, especially in quantum optics research. Photon statistics provides valuable information about correlations in the electromagnetic field and characteristics of the type of light being measured. For example, second-order correlation function obtained via Hanbury Brown and Twiss experiment is the fundamental way to differentiate between coherent, chaotic and quantum light. Moreover, extensive investigations into the quantum properties of light resulted in significant progress that allows for precise control of quantum optical systems and sets the stage for genuine quantum engineering. This progress extends to applications in confirming the generation of entangled photons and exploring their properties for various purposes. The non-classical correlation between the entangled photons has resulted in various quantum optical techniques that enable the practical implementation of conceptual experiments that probe the core principles of quantum theory. This enhanced control over quantum phenomena offers exciting opportunities to explore novel information-processing approaches, with the potential to revolutionize technologies rooted in quantum information science. Driven by the significance and pivotal role of photon statistics, our research pursuits were directed toward two specific domains: the characterization of nanolasers and the characterization of time-bin entangled photons. In the first segment, we delve into the realm of metallic coaxial nanolasers, renowned for their intriguing threshold-less lasing behavior. Through comprehensive photon statistics analysis, we seek to unravel the coherence properties inherent in these nanolasers. Our focus also lies on understanding the elusive threshold-less operation, which has remained a subject of debate and intrigue. To evaluate this unique lasing characteristic, we conduct rigorous second-order coherence measurements alongside time-resolved second-order correlation analyses. The second segment of this study ventures into the generation and characterization of high-dimensional time-bin entangled states, a critical resource for quantum technological applications. We simulate a synthetic photonic lattice by employing a coupled fiber-loop system to perform various quantum walk schemes to generate and process high-dimensional time-bin entangled states. Our investigation focuses on the quantum interference phenomena through detailed photon statistics analysis, with an emphasis on their applications across diverse quantum technologies. Through these two interconnected research endeavors, this thesis demonstrates the profound significance of photon statistics as a versatile tool, bridging classical and quantum realms, and paving the way for advancements in diverse applications, ranging from nanolasers to quantum technologies.
Abstract: This study proposes a unified framework that integrates topological quantum field theories with statistical mechanics to provide a comprehensive understanding of quantum states and dynamics. The problem of reconciling the quantum mechanical description of microscopic systems with macroscopic statistical behavior is addressed by exploring the relationship between topological invariants and thermodynamic properties in quantum systems. The methodology involves the development of a mathematical model that combines the abstract principles of topological quantum field theory with the probabilistic approaches of statistical mechanics, utilizing tensor calculus and advanced mathematical physics tools. Key results indicate that the proposed framework offers new insights into quantum phase transitions, critical phenomena, and the thermodynamic behavior of quantum systems, especially in systems exhibiting topological phases. The framework is further evaluated through a series of theoretical simulations, demonstrating its applicability to quantum information science, condensed matter physics, and quantum technology applications. The implications of this research are significant, providing a foundation for the development of new quantum technologies, such as quantum computing and quantum materials, and contributing to a deeper understanding of the connections between topological properties and statistical mechanics in quantum systems. Keywords: Topological Quantum Field Theory, Statistical Mechanics, Quantum States, Quantum Dynamics, Phase Transitions, Critical Phenomena, Quantum Computing, Quantum Materials, Thermodynamics, Topological Phases, Quantum Information Science.
Topological phases set themselves apart from other phases since they cannot be understood in terms of the usual Landau theory of phase transitions. This fact, which is a consequence of the property that topological phase transitions can occur without breaking symmetries, is reflected in the complicated form of topological order parameters. While the mathematical classification of phases through homotopy theory is known, an intuition for the relation between phase transitions and changes to the physical system is largely inhibited by the general complexity. In this thesis we aim to get back some of this intuition by studying the properties of the Chern number (a topological order parameter) in two scenarios. First, we investigate the effect of electronic correlations on topological phases in the Green's function formalism. By developing a statistical method that averages over all possible solutions of the manybody problem, we extract general statements about the shape of the phase diagram and investigate the stability of topological phases with respect to interactions. In addition, we find that in many topological models the local approximation, which is part of many standard methods for solving the manybody lattice model, is able to produce qualitatively correct phase transitions at low to intermediate correlations. We then extend the statistical method to study the effect of the lattice, where we evaluate possible applications of standard machine learning techniques against our information theoretical approach. We define a measure for the information about particular topological phases encoded in individual lattice parameters, which allows us to construct a qualitative phase diagram that gives a more intuitive understanding of the topological phase. Finally, we discuss possible applications of our method that could facilitate the discovery of new materials with topological properties.
This paper develops a rigorous C*-algebraic framework for understanding and classifying topological quantum phases through the lens of spectral entanglement. Topological quantum phases represent a frontier in condensed matter physics, characterized by robust properties that are insensitive to local perturbations and often manifest exotic excitations. Concurrently, entanglement, a quintessential quantum mechanical phenomenon, serves as a crucial resource for quantum information processing and a powerful diagnostic tool for quantum phases of matter. By integrating the formalisms of C*-algebras and spectral theory, we propose a robust mathematical language to describe the algebraic structure of observables in systems exhibiting topological order. Our approach leverages the spectral properties of density matrices and entanglement Hamiltonians within this algebraic setting to define and quantify spectral entanglement, revealing its intrinsic connection to topological invariants. We demonstrate how this framework provides a unified perspective on the interplay between local algebras of observables, their spectral characteristics, and the non-local correlations that define topological order. The paper outlines theoretical constructs for computing entanglement measures and topological indices directly from the C*-algebraic properties, thereby offering new avenues for classifying these phases and exploring their underlying quantum information content with potential implications for fault-tolerant quantum computing and novel electronic devices.
This paper introduces the concept of Quantum Causal Topological Phase (QCTP), a novel framework for understanding topological states in quantum materials. Traditional topological materials are primarily characterized by their band topology, leading to protected surface states. QCTP expands this concept by proposing that causal relationships within the material's quantum dynamics can also generate topological phases, distinct from and potentially synergistic with spatial topology. This framework posits that the flow of quantum information, governed by interactions and correlations, creates a "causal topology" that contributes to the overall topological state. The paper explores the theoretical foundations of QCTP, highlighting the potential for identifying new phases and materials with enhanced topological properties. The concept is initially presented through a theoretical lens, emphasizing the mathematical formalism and potential implications. Further exploration involves examining how engineered quantum systems, mimicking aspects of natural materials, could be utilized to realize and investigate QCTP. The potential impact of QCTP, including its potential for Nobel Prize recognition and the discovery of new quantum phases of matter, is discussed. ---