The classical probability theory initiated by Kolmogorov and its quantum counterpart, pioneered by von Neumann, were created at about the same time in the 1930s, but development of the quantum theory has trailed far behind. Although highly appealing, the quantum theory has a steep learning curve, requiring tools from both probability and analysis and a facility for combining the two viewpoints. This book is a systematic, self-contained account of the core of quantum probability and quantum stochastic processes for graduate students and researchers. The only assumed background is knowledge of the basic theory of Hilbert spaces, bounded linear operators, and classical Markov processes. From there, the book introduces additional tools from analysis, and then builds the quantum probability framework needed to support applications to quantum control and quantum information and communication. These include quantum noise, quantum stochastic calculus, stochastic quantum differential equations, quantum Markov semigroups and processes, and large-time asymptotic behavior of quantum Markov semigroups.

This book offers a didactic introduction to light-matter interactions at both the classical and semi-classical levels. Pursuing an approach that describes the essential physics behind the functionality of any optical element, it acquaints students with the broad areas of optics and photonics. Its rigorous, bottom-up approach to the subject, using model systems ranging from individual atoms and simple molecules to crystalline and amorphous solids, gradually builds up the reader's familiarity and confidence with the subject matter. Throughout the book, the detailed mathematical treatment and examples of practical applications are accompanied by problems with worked-out solutions. In short, the book provides the most essential information for any graduate or advanced undergraduate student wishing to begin their course of study in the field of photonics, or to brush up on important concepts prior to an examination.

Systems of trapped ions and systems of ultracold Rydberg atoms are used at the forefront of quantum physics research and they make strong contenders as platforms for quantum technologies. Trapped Rydberg ions are a new hybrid technology envisaged to have both the exquisite control of trapped ion systems and the strong interactions of Rydberg atoms. In this work a single trapped Rydberg ion is experimentally investigated. A trapped strontium ion is excited to Rydberg states using two ultraviolet lasers. Effects of the strong trapping electric fields on the highly-sensitive Rydberg ion are studied. After mitigating unwanted trap effects, the ion is coherently excited to Rydberg states and a quantum gate is demonstrated. This thesis lays much of the experimental groundwork for research using this novel system.

This volume contains two major articles, one providing a historical retrosp- tive of one of the great triumphs of nuclear physics in the twentieth century and the other providing a didactic introduction to one of the quantitative tools for understanding strong interactions in the twenty-first century. The article by Igal Talmi on "Fifty Years of the Shell Model - the Quest for the Effective Interaction," pertains to a model that has dominated nuclear physics since its infancy and that developed with astonishing results over the next five decades. Talmi is uniquely positioned to trace the history of the Shell Model. He was active in developing the ideas at the shell model's inception, he has been central in most of the subsequent initiatives which expanded, cl- ified and applied the shell model and he has remained active in the field to the present time. Wisely, he has chosen to restrict his review to the domin- ing issue: the choice of the effective interactions among valence nucleons that determine the properties of low lying nuclear energy levels. The treatment of the subject is both bold and novel for our series. The ideas pertaining to the effective interaction for the shell model are elucidated in a historical sequence.

This thesis lays the groundwork for producing a new class of ultracold molecule by associating an alkali-metal atom and a closed-shell alkaline-earth-like atom, specifically Cs and Yb. Such molecules exhibit both a magnetic dipole moment and an electric dipole moment in their ground state. This extra degree of freedom opens up new avenues of research including the study of exotic states of matter, the shielding of molecular collisions and the simulation of lattice spin models. In detail, the thesis reports the first and only ultracold mixture of Cs and Yb in the world, giving details of the methods used to cool such contrasting atomic species together. Using sensitive two-colour photoassociation measurements to measure the binding energies of the near-threshold CsYb molecular levels in the electronic ground state has allowed the previously unknown scattering lengths to be accurately determined for all the Cs-Yb isotopic combinations. As part of this work, the one-photon photoassociation of ultracold Cs*Yb is also studied, yielding useful information on the excited-state potential. Knowledge of the scattering lengths enables a strategy to be devised to cool both species to quantum degeneracy and, crucially, determines the positions of interspecies Feshbach resonances required for efficient association of ground-state CsYb molecules. With these results, the prospect of bringing a new molecule into the ultracold regime has become considerably closer.

This book discusses the elementary ideas and tools needed for open quantum systems in a comprehensive manner. The emphasis is given to both the traditional master equation as well as the functional (path) integral approaches. It discusses the basic paradigm of open systems, the harmonic oscillator and the two-level system in detail. The traditional topics of dissipation and tunneling, as well as the modern field of quantum information, find a prominent place in the book. Assuming a basic background of quantum and statistical mechanics, this book will help readers familiarize with the basic tools of open quantum systems. Open quantum systems is the study of quantum dynamics of the system of interest, taking into account the effects of the ambient environment. It is ubiquitous in the sense that any system could be envisaged to be surrounded by its environment which could naturally exert its influence on it. Open quantum systems allows for a systematic understanding of irreversible processes such as decoherence and dissipation, of the essence in order to have a correct understanding of realistic quantum dynamics and also for possible implementations. This would be essential for a possible development of quantum technologies.

The De Gruyter Studies in Mathematical Physics are devoted to the publication of monographs and high-level texts in mathematical physics. They cover topics and methods in fields of current interest, with an emphasis on didactical presentation. The series will enable readers to understand, apply and develop further, with sufficient rigor, mathematical methods to given problems in physics. For this reason, works with a few authors are preferred over edited volumes. The works in this series are aimed at advanced students and researchers in mathematical and theoretical physics. They can also serve as secondary reading for lectures and seminars at advanced levels.

The nature of dark matter remains one of the preeminent mysteries in physics and cosmology. It appears to require the existence of new particles whose interactions with ordinary matter are extraordinarily feeble. One well-motivated candidate is the axion, an extraordinarily light neutral particle that may possibly be detected by looking for their conversion to detectable microwaves in the presence of a strong magnetic field. This has led to a number of experimental searches that are beginning to probe plausible axion model space and may reveal the axion in the near future. These proceedings discuss the challenges of designing and operating tunable resonant cavities and detectors at ultralow temperatures. The topics discussed here have potential application far beyond the field of dark matter detection and may be applied to resonant cavities for accelerators as well as designing superconducting detectors for quantum information and computing applications. This work is intended for graduate students and researchers interested in learning the unique requirements for designing and operating microwave cavities and detectors for direct axion searches and to introduce several proposed experimental concepts that are still in the prototype stage.

This book comprises selected peer-reviewed papers presented at the 7th Topical Conference of the Indian Society of Atomic and Molecular Physics, jointly held at IISER Tirupati and IIT Tirupati, India. The contributions address current topics of interest in atomic and molecular physics, both from the theoretical and experimental perspective. The major focus areas include quantum collisions, spectroscopy of atomic and molecular clusters, photoionization, Wigner time delay in collisions, laser cooling, Bose-Einstein condensates, atomic clocks, quantum computing, and trapping and manipulation of quantum systems. The book also discusses emerging topics such as ultrafast quantum processes including those at the attosecond time-scale. This book will prove to be a valuable reference for students and researchers working in the field of atomic and molecular physics.

This thesis presents and discusses recent optical low-temperature experiments on disordered NbN, granular Al thin-films, and the heavy-fermion compound CeCoIn5, offering a unified picture of quantum-critical superconductivity. It provides a concise introduction to the respective theoretical models employed to interpret the experimental results, and guides readers through in-depth calculations supplemented with supportive figures in order to both retrace the interpretations and span the bridge between experiment and state-of-the art theory.

Major superconducting properties including zero resistance, Meissner effect, sharp phase change, flux quantization, excitation energy gap, Josephson effects are covered and microscopically explained, using quantum statistical mechanical calculations. First treated are the 2D superconductivity and then the quantum Hall effects. Included are exercise-type problems for each section. Readers can grasp the concepts covered in the book by following the worked-through problems. Bibliographies are included in each chapter and a glossary and list of symbols are given in the beginning of the book.

The book is based on the materials taught by S. Fujita for several courses in Quantum Theory of Solids, Advanced Topics in Modern Physics, and Quantum Statistical Mechanics.

Dirac cones are ubiquitous to non-trivial quantum matter and are expected to boost and reshape the field of modern electronics. Particularly relevant examples where these cones arise are topological insulators and graphene. From a fundamental perspective, this thesis proposes schemes towards modifying basic properties of these cones in the aforementioned materials. The thesis begins with a brief historical introduction which is followed by an extensive chapter that endows the reader with the basic tools of symmetry and topology needed to understand the remaining text. The subsequent four chapters are devoted to the reshaping of Dirac cones by external fields and delta doping. At all times, the ideas discussed in the second chapter are always a guiding principle to understand the phenomena discussed in those four chapters. As a result, the thesis is cohesive and represents a major advance in our understanding of the physics of Dirac materials.

The importance and the beauty of modern quantum field theory resides in the power and variety of its methods and ideas, which find application in domains as different as particle physics, cosmology, condensed matter, statistical mechanics and critical phenomena. This book introduces the reader to the modern developments in a manner which assumes no previous knowledge of quantum field theory. Along with standard topics like Feynman diagrams, the book discusses effective lagrangians, renormalization group equations, the path integral formulation, spontaneous symmetry breaking and non-abelian gauge theories. The inclusion of more advanced topics will also make this a most useful book for graduate students and researchers.

Playing a prominent role in communications, quantum science and laser physics, quantum nonlinear optics is an increasingly important field. This book presents a self-contained treatment of field quantization and covers topics such as the canonical formalism for fields, phase-space representations and the encompassing problem of quantization of electrodynamics in linear and nonlinear media. Starting with a summary of classical nonlinear optics, it then explains in detail the calculation techniques for quantum nonlinear optical systems and their applications, quantum and classical noise sources in optical fibers and applications of nonlinear optics to quantum information science. Supplemented by end-of-chapter exercises and detailed examples of calculation techniques in different systems, this book is a valuable resource for graduate students and researchers in nonlinear optics, condensed matter physics, quantum information and atomic physics. A solid foundation in quantum mechanics and classical electrodynamics is assumed, but no prior knowledge of nonlinear optics is required.

Providing a pedagogical introduction to the essential principles of path integrals and Hamiltonians, this book describes cutting-edge quantum mathematical techniques applicable to a vast range of fields, from quantum mechanics, solid state physics, statistical mechanics, quantum field theory, and superstring theory to financial modeling, polymers, biology, chemistry, and quantum finance. Eschewing use of the Schroedinger equation, the powerful and flexible combination of Hamiltonian operators and path integrals is used to study a range of different quantum and classical random systems, succinctly demonstrating the interplay between a system's path integral, state space, and Hamiltonian. With a practical emphasis on the methodological and mathematical aspects of each derivation, this is a perfect introduction to these versatile mathematical methods, suitable for researchers and graduate students in physics and engineering.

Professor Sir Roger Penrose's work, spanning fifty years of science, with over five thousand pages and more than three hundred papers, has been collected together for the first time and arranged chronologically over six volumes, each with an introduction from the author. Where relevant, individual papers also come with specific introductions or notes. Many important realizations concerning twistor theory occurred during the short period of this third volume, providing a new perspective on the way that mathematical features of the complex geometry of twistor theory relate to actual physical fields. Following on from the nonlinear graviton construction, a twistor construction was found for (anti-)self-dual electromagnetism allowing the general (anti-)self-dual Yang-Mills field to be obtained. It became clear that some features of twistor contour integrals could be understood in terms of holomorphic sheaf cohomology. During this period, the Oxford research group founded the informal publication, Twistor Newsletter. This volume also contains the influential Weyl curvature hypothesis and new forms of Penrose tiles.

This proceedings volume contains peer-reviewed, selected papers and surveys presented at the conference Spectral Theory and Mathematical Physics (STMP) 2018 which was held in Santiago, Chile, at the Pontifical Catholic University of Chile in December 2018. The original works gathered in this volume reveal the state of the art in the area and reflect the intense cooperation between young researchers in spectral theoryand mathematical physics and established specialists in this field. The list of topics covered includes: eigenvalues and resonances for quantum Hamiltonians; spectral shift function and quantum scattering; spectral properties of random operators; magnetic quantum Hamiltonians; microlocal analysis and its applications in mathematical physics. This volume can be of interest both to senior researchers and graduate students pursuing new research topics in Mathematical Physics.

This edited collection provides new perspectives on some metaphysical questions arising in quantum mechanics. These questions have been long-standing and are of continued interest to researchers and graduate students working in physics, philosophy of physics, and metaphysics. It features contributions from a diverse set of researchers, ranging from senior scholars to junior academics, working in varied fields, from physics to philosophy of physics and metaphysics. The contributors reflect on issues about fundamentality (is quantum theory fundamental? If so, what is its fundamental ontology?), ontological dependence (how do ordinary objects exist even if they are not fundamental?), realism (what kind of realism is compatible with quantum theory?), indeterminacy (can the world itself exhibit ontological indeterminacy?). The book contains contributions from both physicists (including Nobel Prize winner Gerard 't Hooft), science communicators and philosophers.

Ultra-cold atomic ensembles have emerged in recent years as a powerful tool in many-body physics research, quantum information science and metrology. This thesis presents an experimental and theoretical study of the coherent properties of trapped atomic ensembles at high densities, which are essential to many of the aforementioned applications. The study focuses on how inter-particle interactions modify the ensemble coherence dynamics, and whether it is possible to extend the coherence time by means of external control. The thesis presents a theoretical model which explains the effect of elastic collision of the coherence dynamics and then reports on experiments which test this model successfully in the lab. Furthermore, the work includes the first implementation of dynamical decoupling with ultra-cold atomic ensembles. It is demonstrated experimentally that by using dynamical decoupling the coherence time can be extended 20-fold. This has a great potential to increase the usefulness of these ensembles for quantum computation.

Quantum mechanics is the foundation of modern technology, due to its innumerable applications in physics, chemistry and even biology. This second volume studies Schroedinger s equation and its applications in the study of wells, steps and potential barriers. It examines the properties of orthonormal bases in the space of square-summable wave functions and Dirac notations in the space of states. This book has a special focus on the notions of the linear operators, the Hermitian operators, observables, Hermitian conjugation, commutators and the representation of kets, bras and operators in the space of states. The eigenvalue equation, the characteristic equation and the evolution equation of the mean value of an observable are introduced. The book goes on to investigate the study of conservative systems through the time evolution operator and Ehrenfest s theorem. Finally, this second volume is completed by the introduction of the notions of quantum wire, quantum wells of semiconductor materials and quantum dots in the appendices.

This volume contains the proceedings of the First Ukrainian-French Romanian School "Algebraic and Geometric Methods in Mathematical Physics," held in Kaciveli, Crimea (Ukraine) from 1 September ti1114 September 1993. The School was organized by the generous support of the Ministry of Research and Space of France (MRE), the Academy of Sciences of Ukraine (ANU), the French National Center for Scientific Research (CNRS) and the State Committee for Science and Technologies of Ukraine (GKNT). Members of the International Scientific Committee were: J.-M. Bony (paris), A. Boutet de Monvel-Berthier (Paris, co-chairman), P. Cartier (paris), V. Drinfeld (Kharkov), V. Georgescu (Paris), J.L. Lebowitz (Rutgers), V. Marchenko (Kharkov, co-chairman), V.P. Maslov (Moscow), H. Mc-Kean (New-York), Yu. Mitropolsky (Kiev), G. Nenciu (Bucharest, co-chairman), S. Novikov (Moscow), G. Papanicolau (New-York), L. Pastur (Kharkov), J.-J. Sansuc (Paris). The School consisted of plenary lectures (morning sessions) and special sessions. The plenary lectures were intended to be accessible to all participants and plenary speakers were invited by the scientific organizing committee to give reviews of their own field of interest. The special sessions were devoted to a variety of more concrete and technical questions in the respective fields. According to the program the plenary lectures included in the volume are grouped in three chapters. The fourth chapter contains short communications."

Dynamics of billiard balls and their role in physics have received wide attention since the monumental lecture by Lord Kelvin at the turn of the 19th century. Billiards can nowadays be created as quantum dots in the microscopic world enabling one to envisage the so-called quantum chaos, i.e. quantum manifestation of chaos of billiard balls. In fact, owing to recent progress in advanced technology, nanoscale quantum dots, such as chaotic stadium and antidot lattices analogous to the Sinai Billiard, can be fabricated at the interface of semiconductor heterojunctions. This book begins its exploration of the effect of chaotic electron dynamics on ballistic quantum transport in quantum dots with a puzzling experiment on resistance fluctuations for stadium and circle dots. Throughout the text, major attention is paid to the semiclassical theory which makes it possible to interpret quantum phenomena in the language of the classical world. Chapters one to four are concerned with the elementary statistical methods (curvature, Lyapunov exponent, Kolmogorov-Sinai entropy and escape rate), which are needed for a semiclassical description of transport in quantum dots. Chapters five to ten discuss the topical subjects in the field, including the ballistic weak localization, Altshuler-Aronov-Spivak oscillation, partial time-reversal symmetry, persistent current, Arnold diffusion and Coulomb blockade.

This book presents concepts of theoretical physics with engineering applications. The topics are of an intense mathematical nature involving tools like probability and random processes, ordinary and partial differential equations, linear algebra and infinite-dimensional operator theory, perturbation theory, stochastic differential equations, and Riemannian geometry. These mathematical tools have been applied to study problems in mechanics, fluid dynamics, quantum mechanics and quantum field theory, nonlinear dynamical systems, general relativity, cosmology, and electrodynamics. A particularly interesting topic of research interest developed in this book is the design of quantum unitary gates of large size using the Feynman diagrammatic approach to quantum field theory. Through this book, the reader will be able to observe how basic physics can revolutionize technology and also how diverse branches of mathematical physics like large deviation theory, quantum field theory, general relativity, and electrodynamics have many common issues that provide the starting point for unifying the whole of physics, namely in the formulation of Grand Unified Theories (GUTS).

This book contains a mathematical exposition of analogies between classical (Hamiltonian) mechanics, geometrical optics, and hydrodynamics. This theory highlights several general mathematical ideas that appeared in Hamiltonian mechanics, optics and hydrodynamics under different names. In addition, some interesting applications of the general theory of vortices are discussed in the book such as applications in numerical methods, stability theory, and the theory of exact integration of equations of dynamics. The investigation of families of trajectories of Hamiltonian systems can be reduced to problems of multidimensional ideal fluid dynamics. For example, the well-known Hamilton-Jacobi method corresponds to the case of potential flows. The book will be of great interest to researchers and postgraduate students interested in mathematical physics, mechanics, and the theory of differential equations.

The book describes how the electrons in small "low-dimensional" structures interact with their surroundings. It contains a series of linked up to date review chapters as well as explanatory material and is written to be understandable to graduate students and newcomers to the field. All contributions come from leading scientists.