Since the role that 𝓟𝓣-symmetric Hamiltonians could play in quantum mechanics theory was suggested in1998, much work has been conducted on the mathematical analysis of 𝓟𝓣-symmetry. Carl Benderchallenged the perceived wisdom of Quantum Mechanics that the Hamiltonian operator describing anyquantum mechanical system has to be Hermitian. He showed that Hamiltonians that are invariant undercombined parity-time 𝓟𝓣-symmetry transformations likewise can exhibit real eigenvalue spectra.However, 𝓟𝓣-symmetric systems began as an analytic extension of classical and quantum mechanical tothe complex domain. These findings had a particularly profound impact in the field of photonics, where PTsymmetricpotential landscapes can be implemented by introducing dissipative terms appropriatelydistributing gain and loss in open or coupled systems. The refractive index profile plays the role of thecomplex potential. Similar research activity was transferred to other physical settings such as electronicscircuits, microwaves and cavities resonators, acoustic, superconducting wires, and atomic systems.Unconventional characteristics compared with Hermitian ones were found and allow to engineer newdevices.Likewise, topological phases of matter are another new field of physics which have had a rapid growth inthe last years, based on the concept of Geometric or Berry phase. Topological Insulators andSuperconductors, Topological photonics and nanophotonics, Topological Computers are some of theapplications widely described in scientific journals.In Nature all the processes have to be described by a property in a discrete or iterative way instead of acontinuous way which usually is time. Continuous processes, being a mathematical idealization, has beenthe best tool to model observable properties in physics and is the conventional tool used in Classical andQuantum Mechanics. The formal way to model discrete systems is through difference equations which aresimilar to differential equations but not equal.Identification of dynamical discrete system with recurrence equations, give us the possibility to consider ananalytical continuation of its evolution variable, from discrete values to fractional one´s, in the solution ofthe system. Parametric solutions obtained by this procedure takes automatically complex values, avoidingto make any supposition about the possibility of their analytical extension to complex domain. Thisprocedure leads, in a very simple natural way, a rich variety of dynamical behaviors in the complex domainunknown in the real domain. This new mathematical tool is developed to enhance and provide a moresubtle description of the behavior of dynamical systems and provides a more general class of Non-Hermitian Hamiltonians than 𝓟𝓣-symmetric systems with real spectra, keeping also topological propertiesdue to a kind of geometric phase.
Since the role that 𝓟𝓣-symmetric Hamiltonians could play in quantum mechanics theory was suggested in1998, much work has been conducted on the mathematical analysis of 𝓟𝓣-symmetry. Carl Benderchallenged the perceived wisdom of Quantum Mechanics that the Hamiltonian operator describing anyquantum mechanical system has to be Hermitian. He showed that Hamiltonians that are invariant undercombined parity-time 𝓟𝓣-symmetry transformations likewise can exhibit real eigenvalue spectra.However, 𝓟𝓣-symmetric systems began as an analytic extension of classical and quantum mechanical tothe complex domain. These findings had a particularly profound impact in the field of photonics, where PTsymmetricpotential landscapes can be implemented by introducing dissipative terms appropriatelydistributing gain and loss in open or coupled systems. The refractive index profile plays the role of thecomplex potential. Similar research activity was transferred to other physical settings such as electronicscircuits, microwaves and cavities resonators, acoustic, superconducting wires, and atomic systems.Unconventional characteristics compared with Hermitian ones were found and allow to engineer newdevices.Likewise, topological phases of matter are another new field of physics which have had a rapid growth inthe last years, based on the concept of Geometric or Berry phase. Topological Insulators andSuperconductors, Topological photonics and nanophotonics, Topological Computers are some of theapplications widely described in scientific journals.In Nature all the processes have to be described by a property in a discrete or iterative way instead of acontinuous way which usually is time. Continuous processes, being a mathematical idealization, has beenthe best tool to model observable properties in physics and is the conventional tool used in Classical andQuantum Mechanics. The formal way to model discrete systems is through difference equations which aresimilar to differential equations but not equal.Identification of dynamical discrete system with recurrence equations, give us the possibility to consider ananalytical continuation of its evolution variable, from discrete values to fractional one´s, in the solution ofthe system. Parametric solutions obtained by this procedure takes automatically complex values, avoidingto make any supposition about the possibility of their analytical extension to complex domain. Thisprocedure leads, in a very simple natural way, a rich variety of dynamical behaviors in the complex domainunknown in the real domain. This new mathematical tool is developed to enhance and provide a moresubtle description of the behavior of dynamical systems and provides a more general class of Non-Hermitian Hamiltonians than 𝓟𝓣-symmetric systems with real spectra, keeping also topological propertiesdue to a kind of geometric phase.
PT-symmetric quantum mechanics (PTQM) has become a hot area of research and investigation. Since its beginnings in 1998, there have been over 1000 published papers and more than 15 international conferences entirely devoted to this research topic. Originally, PTQM was studied at a highly mathematical level and the techniques of complex variables, asymptotics, differential equations and perturbation theory were used to understand the subtleties associated with the analytic continuation of eigenvalue problems. However, as experiments on PT-symmetric physical systems have been performed, a simple and beautiful physical picture has emerged, and a PT-symmetric system can be understood as one that has a balanced loss and gain. Furthermore, the PT phase transition can now be understood intuitively without resorting to sophisticated mathematics. Research on PTQM is following two different paths: at a fundamental level, physicists are attempting to understand the underlying mathematical structure of these theories with the long-range objective of applying the techniques of PTQM to understanding some of the outstanding problems in physics today, such as the nature of the Higgs particle, the properties of dark matter, the matter-antimatter asymmetry in the universe, neutrino oscillations and the cosmological constant; at an applied level, new kinds of PT-synthetic materials are being developed, and the PT phase transition is being observed in many physical contexts, such as lasers, optical wave guides, microwave cavities, superconducting wires and electronic circuits. The purpose of this Theme Issue is to acquaint the reader with the latest developments in PTQM. The articles in this volume are written in the style of mini-reviews and address diverse areas of the emerging and exciting new area of PT-symmetric quantum mechanics.
More than a decade ago, it was shown that non-Hermitian Hamiltonians with combined parity (P) and time-reversal (T ) symmetry exhibit real eigenvalues over a range of parameters. Since then, the field of PT symmetry has seen rapid progress on both the theoretical and experimental fronts. These effective Hamiltonians are excellent candidates for describing open quantum systems with balanced gain and loss. Nature seems to be replete with examples of PT -symmetric systems; in fact, recent experimental investigations have observed the effects of PT symmetry breaking in systems as diverse as coupled mechanical pendula, coupled optical waveguides, and coupled electrical circuits. Recently, PT -symmetric Hamiltonians for tight-binding lattice models have been extensively investigated. Lattice models, in general, have been widely used in physics due to their analytical and numerical tractability. Perhaps one of the best systems for experimentally observing the effects of PT symmetry breaking in a one-dimensional lattice with tunable hopping is an array of evanescently-coupled optical waveguides. The tunneling between adjacent waveguides is tuned by adjusting the width of the barrier between them, and the imaginary part of the local refractive index provides the loss or gain in the respective waveguide. Calculating the time evolution of a wave packet on a lattice is relatively straightforward in the tight-binding model, allowing us to make predictions about the behavior of light propagating down an array of PT -symmetric waveguides. In this thesis, I investigate the the strength of the PT -symmetric phase (the region over which the eigenvalues are purely real) in lattices with a variety of PT - symmetric potentials. In Chapter 1, I begin with a brief review of the postulates of quantum mechanics, followed by an outline of the fundamental principles of PT - symmetric systems. Chapter 2 focuses on one-dimensional uniform lattices with a pair of PT -symmetric impurities in the case of open boundary conditions. I find that the PT phase is algebraically fragile except in the case of closest impurities, where the PT phase remains nonzero. In Chapter 3, I examine the case of periodic boundary conditions in uniform lattices, finding that the PT phase is not only nonzero, but also independent of the impurity spacing on the lattice. In addition, I explore the time evolution of a single-particle wave packet initially localized at a site. I find that in the case of periodic boundary conditions, the wave packet undergoes a preferential clockwise or counterclockwise motion around the ring. This behavior is quantified by a discrete momentum operator which assumes a maximum value at the PT -symmetry- breaking threshold. In Chapter 4, I investigate nonuniform lattices where the parity-symmetric hop- ping between neighboring sites can be tuned. I find that the PT phase remains strong in the case of closest impurities and fragile elsewhere. Chapter 5 explores the effects of the competition between localized and extended PT potentials on a lattice. I show that when the short-range impurities are maximally separated on the lattice, the PT phase is strengthened by adding short-range loss in the broad-loss region. Consequently, I predict that a broken PT symmetry can be restored by increasing the strength of the short-range impurities. Lastly, Chapter 6 summarizes my salient results and discusses areas which can be further developed in future research.
Modern physicists incorporate into their statistical analyses additional degrees of freedom, parameters that were unknown to the nineteenth-century scientists. The true black body intensity distribution is continuous, but the model of discrete radiators provides a good approximation for the observed wavelengths of light and stimulated a paradigm shift in the thinking of physicists about microphysics. Niels Bohr model was a source of consternation to the contemporary physics community. As with thermodynamics, quantum theory is extraordinarily useful and has been applied with tremendous ingenuity by solid-state physicists, engineers, and others who are able to harness its guidance for technological innovation. The disconcerting quantum jumps were explained away by re-envisioning the localized planetary electron as a distributed cloud surrounding the atomic nucleus. The technique of Fourier analysis was Werner Heisenberg’s path to the development of a quantum theory, which he ultimately expressed in the mathematical form of matrices.
The turn of the XXth century witnessed a revolution in physics comparable to Isaac Newton’s discovery of the universal laws of mechanics and of gravitation three centuries earlier. The world required to be described in novel terms, as the immutable, deterministic view of our familiar universe had given way to a new world picture, one which featured chance, flux, and an incessant upsurge of waves of matter. Such a worldview was so radically new and counterintuitive that it gave rise to strong debates, to the effect that Albert Einstein himself tried to oppose it on the grounds that “God does not play dice”. In spite of the intense debates that accompanied its emergence, quantum mechanics quickly proved an incredibly efficacious new tool to understand and to predict a wide array of new phenomena. It was so successful that in no time it broke free from the environment of research labs to become part of daily life, making it possible, for example, to understand why some materials were conductors, while others were insulators. Along with it, came, too, the discovery of transistors, on which much of modern electronics rests. It also led to understand how novel materials known as superconductors allow the transport of electricity with no loss, thus paving the way for new developments in the fields of medical imagery or energy distribution. Other aspects of the quantum theory led to the development of atomic clocks of astounding accuracy, which would be wrong by no more than fifteen seconds, had they been set at the beginning of the universe. A hundred years later, at the turn of the XXIst century, Quantum mechanics has lost none of its astounding power. Contemporary research has undertaken the task of exploring its less immediately perceptible aspects. Groundbreaking developments have ensued, such as the teleportation of grains of light or the possibility, once predicted by the great physicist Richard Feynman, to build, one day, novel computers which, unlike the ones we are familiar with today, will be able to process innumerable numbers of operations in parallel. The Wright Colloquium will be for us the occasion to explore, in the company of five internationallyknown specialists in the field, some of the fascinating aspects of quantum mechanics. We will appraise how efficiently quantum physics can describe our world, and confront its limitation when it is faced with the infinitely small (in relation to recent experiments carried out at the CERN) as with the incommensurably large scale of sidereal spaces. We will appreciate the extent to which quantum physics already has impacted our everyday lives, and evoke the way in which novel fields such as quantum computers and quantum information will entail profound changes in the future. The quantum adventure has only just begun! Monday 15 November 2010 - 18h30 Jochen Mannhart Center for Electronic Correlations and Magnetism, University of Augsburg, Germany Quantum Physics on the Scale of Daily Life Tuesday 16 November 2010 - 18h30 Wolfgang Ketterle Nobel Laureate 2001 (Physics), Department of Physics, Massachusetts Institute of Technology, Cambridge, U.S.A. When freezing cold is not cold enough New forms of matter close to absolute zero temperature Wednesday 17 November 2010 - 18h30 David Gross Nobel Laureate 2004 (Physics), Kavli Institute for Theoretical Physics, University of California, Santa-Barbara, U.S.A. Quantum Mechanics of the Very, Very Small and the Very, Very Large Thursday 18 November 2010 - 18h30 Alain Aspect CNRS senior scientist and Professor Institut d'Optique and Ecole Polytechnique Palaiseau, France. From Einstein’s intuition to quantum bits: a new quantum age? Vendredi 19 novembre 2010 - 18h30 Rainer Blatt Institute of Quantum Optics and Quantum Information, Austrian Academy of Sciences and University of Innsbruck, Austria The Quantum Way of Doing Computations
Analysis of fundamentally open systems which exhibit both spatial reflection (parity) and time-reversal symmetry, called PT-symmetric [1] systems, has played an important role in understanding the dynamics of a broad class of non-Hermitian systems. Such systems support distinct regimes of stability and instability respectively, and the transition between these plays an important role in the system dynamics. We investigate a variety of non-Hermitian systems which exhibit a PT-symmetry breaking transition in both classical and quantum contexts using an array of theoretical and numerical approaches which are directly amenable to physical experiments. Organizationally, we divide these studies into those done on classical and quantum systems. Specifically for classical systems we analyze several interesting phenomena in the context of optical waveguide arrays. These provide an important testing ground for much of the discrete, PT-symmetric theory. In our case, we analyze PT-symmetric extensions of several interesting lattice models. Firstly, we explore single gain/loss pair, PT-symmetric extensions of the Su-Schreiffer-Heeger (SSH) model [2] and the Aubry-Andre-Harper (AAH) model [3,4]. In both cases, we find a veiled symmetry which is responsible for the PT-symmetry. Secondly we expand upon the Rudner-Levitov (RL) model [5,6], a non-Hermitian extension of the SSH model with multiple gain/loss pairs, and we explore the dependence of topological phenomena on changes in the initial state as well as their robustness in the presence of gain-induced nonlinearity. For quantum systems, we explore the rich PT-symmetric phase diagram resulting from time-periodic, Floquet PT-symmetry [7,8]. This is experimentally implementable in ultracold atomic systems [9], and features a reentrant PT-symmetric phase. Secondly, we examine quantum multi-photon statistics for a simple non-Hermitian model which is transformed and mapped to a system with unitary evolution via a unitary dilation, and is fully realizable on a small-scale quantum simulator [10].