AB and Berry phases for a quantum cloud of charge
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量子光电器件及应用英文Quantum photonic devices and applications.Quantum photonic devices refer to devices that utilize the principles of quantum mechanics to manipulate and control light at the quantum level. These devices often involve the generation, manipulation, and detection of single photons, as well as the entanglement of photons for applications in quantum computing, quantum communication, and quantum cryptography.One important example of a quantum photonic device is the single-photon source, which is crucial for many quantum technologies. These sources are used in quantum key distribution systems, quantum metrology, and quantum information processing. They can be based on various physical platforms such as semiconductor quantum dots, trapped ions, or nonlinear optical processes.Another key area of research and development in quantumphotonic devices is quantum photodetectors, which are capable of detecting individual photons with highefficiency and low noise. These detectors are essential for applications such as quantum communication and quantum imaging.In addition to these foundational devices, there is ongoing research into more advanced quantum photonic devices, including quantum gates, quantum memories, and quantum repeaters. These devices are essential for the realization of large-scale quantum networks and quantum information processing systems.The applications of quantum photonic devices are wide-ranging. In quantum computing, for example, quantum photonic devices are used for the manipulation and storage of quantum information in the form of photons. In quantum communication, quantum photonic devices enable secure transmission of information through the quantum key distribution and quantum teleportation. Quantum photonic devices also have potential applications in high-precision sensing and metrology, as well as in the development ofquantum-enhanced imaging techniques.Overall, quantum photonic devices and their applications represent a rapidly growing and highly interdisciplinary field, with implications for both fundamental science and advanced technologies. As research in this area continues to advance, we can expect to see even more innovative quantum photonic devices and novel applications in the near future.。
Berry phase in electronic structure theoryIn quantum mechanics,Berry phase is a very important concept that describes the geometric properties of a system in parameter space.In electronic structure theory,Berry phase also plays an important role.It is not only of great significance for understanding the wave function and energy level of electrons,but also plays a crucial role in many physical phenomena.In the theory of electronic structure,Berry phase usually refers to the phase that the electron wave function evolves in the parameter space in a periodic lattice.This phase is dependent on system parameters and can affect the energy of electrons and the shape of wave functions.By calculating the Berry phase,one can gain a deeper understanding of the quantum behavior of electrons and the geometric properties of the system.In many physical phenomena,Berry phase plays an important role.For example,in spintronics, Berry phase can affect the spin state and magnetization direction of electrons.In topological insulators,Berry phase and topological properties are closely related and can affect the band structure and surface state of electrons.In addition,Berry phase can also affect optical and magnetic properties,making it widely applicable in materials science and physics.In recent years,with the continuous development of computer technology,calculating Berry phase has become a hot research field.Many numerical methods and computational software have been developed for calculating Berry phases and related physical quantities.These methods and software can not only be used for theoretical research,but also for the analysis and simulation of experimental data.In summary,Berry phase is a very important concept in electronic structure theory.It is not only of great significance for understanding the wave function and energy level of electrons,but also plays a crucial role in many physical phenomena.With the continuous development of computer technology,calculating Berry phase has become a hot research field,providing important tools and means for theoretical and experimental research.。
全文分为作者个人简介和正文两个部分:作者个人简介:Hello everyone, I am an author dedicated to creating and sharing high-quality document templates. In this era of information overload, accurate and efficient communication has become especially important. I firmly believe that good communication can build bridges between people, playing an indispensable role in academia, career, and daily life. Therefore, I decided to invest my knowledge and skills into creating valuable documents to help people find inspiration and direction when needed.正文:插上科学的翅膀飞时光穿梭机英语作文全文共3篇示例,供读者参考篇1Time Travel with the Wings of ScienceEver since I was a young child, I've been fascinated by the concept of time travel. The idea of journeying through the fabric of the past and future has captivated my imagination for as longas I can remember. What wonders would we discover by unlocking the secrets of the space-time continuum? Whatlong-forgotten civilizations could we explore? What terrible future catastrophes might we prevent? The possibilities seem endless and exhilarating.Of course, time travel has long been confined to the realms of science fiction, from H.G. Wells' seminal novel The Time Machine to the beloved Back to the Future film trilogy. Authors, filmmakers, and dreamers have spun incredible tales transporting us across the centuries. However, could the wings of science one day make this fantasy a reality? Might our technological progress eventually allow us to slip the bonds of the present? I certainly hope so.The core scientific concepts underlying time travel arise from Einstein's theories of relativity. The great physicist fundamentally altered our understanding of space and time, demonstrating that they are inextricably interwoven into a single continuum known as space-time. In this four-dimensional reality, time is no longer constant or absolute, but can dilate based on factors like velocity and gravity.The effects of time dilation predicted by relativity may seem minor in our daily lives on Earth, but they become extreme undermore significant gravitational forces or as objects approach the speed of light. A thought experiment can help illustrate this. Imagine twin paradox scenario where one identical twin remains on Earth while the other embarks on an interstellar voyage moving at an appreciable fraction of light speed. From the perspective of the Earth-bound twin, their sibling will have aged much more slowly due to the effects of time dilation.This bizarre consequence of relativity implies that by moving through space at sufficiently high velocities or by harnessing immense gravitational forces, we could theoretically propel ourselves forward through the river of time relative to another observer. In essence, we would be time traveling into the future, though not in the controlled manner typically depicted in science fiction tales of leaping centuries with technology like a "time machine."Still, even this limited form of time travel into the future demonstrated by Einstein's theories is a profound revelation overturning our classical notions of time as a constant, universal flow marching lockstep across the cosmos. If we can leverage these relativistic effects through future technological marvels like hyper-fast spaceships or artificially generated black holes, could we then possibly learn to navigate the timestream at will?A more daring notion inspired by quantum physics is that backward time travel might also be achievable through exploiting exotic properties of the universe like wormholes –hypothetical tunnels through space-time. While hotly debated, some interpretations of quantum theory leave open the possibility that under the correct conditions it may be possible to create traversable wormholes capable of looping back on themselves in four-dimensional space-time.If feasible engineering solutions could be found to stabilize these wormholes against collapse and usher travelers through their quantum gateways, they could provide portals into the past or future. The energy requirements predicted by calculations are absolutely staggering, however, and may forever remain science fiction. Some theorists have proposed that future civilizations perhaps trillions of years from now could possibly harness energies on that cosmic scale by exploiting exotic physical phenomena. For now, such notions can only serve as mathematical daydreams.The most speculative concepts for achieving time travel arise from fringe theories exploring the fundamental building blocks of reality. Perhaps our current models represent just the first baby steps in a grander unified theory fully describingspace-time. If discovered, such a "Theory of Everything" could potentially reveal loopholes in our present comprehension, allowing scientists to manipulate the cosmic fabric in currently unimaginable ways.While purely hypothetical at this stage, fringe thinkers have proposed such radical possibilities as using cosmic strings or constructing Traversable Acausal Retrohandled Hyperfinite (TARH) pathways looping through space-time to bypass entropy restrictions and accomplish causality violations. Without empirical evidence, however, such fanciful ideas remain the stuff of science fiction writers rather than legitimate theory. They remind us how little we may actually understand about deep aspects of reality.Despite the uncertainties of cutting-edge theorizing, history shows that making leaps into the unknown can unleash tremendous progress. The foundations of modern physics itself were seeded by a handful of wild ideas that flew in the face of prevailing scientific dogmas. Perhaps by following the wings of our curiosity to map the unexplored territory of space-time, we might eventually gain mastery over it. If so, could a fantastic age of time tourism one day open篇2Soaring on the Wings of Science Through a Time MachineEver since I was a young child, I've been fascinated by the concept of time travel. The idea of journeying through the cosmic ocean of the fourth dimension, transcending the linear constraints of chronology, has sparked an insatiable sense of wonder and curiosity within me. Time machines have long been the stuff of science fiction – the iconic DeLorean from Back to the Future, the intricate machinery of H.G. Wells' Time Machine, or the sleek, metaphysical wormholes that theoretical physicists speculate could breach the fabric of space-time itself.However, as I've delved deeper into the realms of science, particularly physics, I've come to realize that the prospect of time travel may not be as far-fetched as it seems. In fact, it might well be an inevitable consequence of our universe's fundamental laws, waiting to be unlocked by the boundless potential of human ingenuity and the relentless march of scientific progress.The theoretical underpinnings of time travel find their roots in Albert Einstein's revolutionary theory of relativity. According to this paradigm-shifting framework, time is not an absolute, universal constant, but rather a malleable dimension inextricably intertwined with space, matter, and energy. The very fabric of space-time can be warped and distorted by the presence ofmassive gravitational fields, opening up tantalizing possibilities for traversing the temporal domain.One of the most intriguing concepts arising from Einstein's theories is that of the "closed timelike curve" – a hypothetical trajectory in space-time that loops back on itself, allowing an object or traveler to theoretically return to their own past. While the precise mechanics of such a phenomenon remain shrouded in mystery, it has captured the imaginations of physicists and science fiction enthusiasts alike.Another intriguing avenue for potential time travel lies in the realm of wormholes – hypothetical tunnels or shortcuts through the cosmic fabric that could, in theory, connect two distant regions of space-time. Traversing a wormhole could potentially enable a traveler to bypass the conventional flow of time, effectively traveling into the future or even the past, depending on the wormhole's properties.Of course, the realization of time travel is fraught with mind-bending paradoxes and logical conundrums that have perplexed philosophers and scientists for decades. The infamous "grandfather paradox," for instance, poses a seemingly insurmountable logical obstacle: if you were to travel back in time and inadvertently (or perhaps intentionally) prevent yourgrandparents from meeting, you would effectively erase your own existence from the timeline – a self-contradictory scenario that challenges our very notions of causality and free will.Despite these daunting challenges, the pursuit of time travel remains an irresistible lure for the human intellect, driving us to push the boundaries of our understanding and to unravel the deepest mysteries of the cosmos. After all, if we were to achieve even the slightest degree of temporal maneuverability, the implications would be nothing short of revolutionary.Imagine being able to witness pivotal moments in human history firsthand, to walk alongside luminaries like Socrates, Leonardo da Vinci, or Marie Curie, and to gain invaluable insights into the triumphs and tribulations that have shaped our collective journey. Or consider the tantalizing prospect of peering into the future, glimpsing the technological marvels and societal transformations that await us, and using that knowledge to steer humanity towards a brighter, more sustainable path.Of course, such power would also carry immense responsibility, as the potential for abuse or unintended consequences could be catastrophic. Any successful time travel endeavor would necessitate a profound ethical framework,rigorously developed and adhered to, to ensure that the delicate tapestry of causality is not irreparably disrupted.As a student of science, I find myself both awed and humbled by the audacious quest for time travel. It represents the pinnacle of human curiosity and intellectual daring, a bold venture into realms once deemed utterly fanciful and impossible. Yet, it is precisely this unquenchable thirst for knowledge, this relentless drive to push against the boundaries of the known, that has propelled humanity's greatest achievements throughout history.From the rudimentary tools of our prehistoric ancestors to the awe-inspiring marvels of modern technology, our species has consistently defied the limitations imposed by our finite comprehension, venturing forth into uncharted territories with a spirit of fearless exploration. The pursuit of time travel is simply the latest, and perhaps the most ambitious, chapter in this grand narrative of human discovery.As I stand on the precipice of adulthood, poised to embark on my own scientific journey, I cannot help but feel a profound sense of excitement and anticipation. The challenges that lie ahead are daunting, the obstacles seemingly insurmountable,but it is in the crucible of such adversity that true innovation is forged.Perhaps, one day, I will have the privilege of contributing, even in the smallest of ways, to the realization of this age-old dream – to soar on the wings of science, transcending the shackles of linear time, and unlocking the secrets of the cosmic tapestry that binds us all. For now, I can only marvel at the audacity of such an endeavor and embrace the endless possibilities that await us at the forefront of human knowledge.Time travel may yet remain a tantalizing fantasy, a thought experiment to be pondered and debated. But in theever-expanding realm of science, where the impossible is routinely transmuted into reality, one can never discount the power of human ingenuity and the boundless potential that lies waiting to be unveiled. As I gaze skyward, I see not merely the vast expanse of the cosmos, but a canvas upon which the most extraordinary dreams of humanity may one day be etched – a tapestry woven from the threads of curiosity, perseverance, and an unwavering commitment to pushing the frontiers of knowledge ever further.And who knows? Perhaps, in some distant future, or even some long-forgotten past, a traveler from another era willstumble upon these very words, a testament to the enduring spirit of human inquiry and our eternal quest to unravel the mysteries of time itself.篇3Soaring Through Time with the Wings of ScienceEver since I was a young child, my imagination has been captivated by the concept of time travel. The idea of journeying through the cosmic ocean of the past and future has kindled an insatiable curiosity within me. However, as I matured and delved deeper into the realms of science, I realized that this fantasy might not be as implausible as it seems. With the wings of scientific advancement, we may one day conquer the barriers of time itself.The notion of time travel has long been a subject of fascination for scientists, philosophers, and storytellers alike. From H.G. Wells' seminal novel "The Time Machine" to the mind-bending scientific theories of Albert Einstein, the concept has transcended mere fiction and entered the realm of theoretical possibility. Einstein's theory of relativity introduced the groundbreaking idea that time is not an absolute constant,but rather a malleable dimension intricately intertwined with space and matter.This revolutionary understanding paved the way for further exploration into the nature of time and its potential manipulability. Physicists have proposed various hypothetical mechanisms for time travel, including wormholes, cosmic strings, and even the exploitation of the quantum realm. While these concepts may seem outlandish, they are grounded in the fundamental principles of modern physics and have sparked intense scientific debate and investigation.One particularly intriguing avenue of research is the study of wormholes – hypothetical tunnels in the fabric of spacetime that could potentially connect distant regions of the universe or even different eras. Although the existence of traversable wormholes remains purely theoretical, some scientists have proposed methods to stabilize them using exotic matter or cosmic strings. The implications of such a discovery would be nothing short of revolutionary, allowing us to transcend the linear constraints of time and explore the vast tapestry of the cosmos.Another tantalizing possibility lies in the realm of quantum mechanics, where the strange and counterintuitive behavior of subatomic particles defies our classical understanding of reality.Some theories suggest that quantum entanglement, a phenomenon where particles become inextricably linked regardless of distance, could potentially facilitate a form of time travel through the manipulation of information. While the practical applications of such concepts are still the subject of intense speculation, they open up a fascinating realm of possibilities that challenge our fundamental assumptions about the nature of time.Beyond the realm of theoretical physics, technological advancements in fields such as nanotechnology, quantum computing, and advanced propulsion systems may also play a pivotal role in our quest to conquer time. As our understanding of the universe deepens and our capabilities expand, we inch closer to the possibility of engineering solutions that could one day make time travel a tangible reality.Of course, the implications of such a monumental achievement extend far beyond mere scientific curiosity. Time travel could revolutionize our understanding of history, allowing us to witness pivotal moments firsthand and unravel the mysteries of the past. It could also provide invaluable insights into the future, enabling us to anticipate and prepare for potential challenges or disasters before they occur. Furthermore,the ability to traverse time could have profound implications for fields such as medicine, archaeology, and even space exploration, opening up new avenues of discovery and understanding.Yet, as we contemplate the exhilarating prospects of time travel, we must also confront the ethical and philosophical quandaries that accompany such a transformative technology. The potential for abuse or unintended consequences is not to be taken lightly. Would altering the past irrevocably alter the present? Could knowledge of the futureundermine the very fabric of human agency and free will? These are but a few of the complex questions that must be grappled with as we inch closer to this incredible feat.Despite these challenges, the allure of time travel remains undeniable. It represents the pinnacle of human curiosity and ambition, a testament to our relentless pursuit of knowledge and understanding. As a student of science, I am both awed and humbled by the prospect of one day soaring through the vast expanse of time, carried aloft by the wings of our collective scientific endeavors.While the path ahead is shrouded in uncertainty, one thing remains clear: the quest to unlock the secrets of time travel is a testament to the boundless potential of the human mind and ourunwavering determination to push the boundaries of what is possible. With each new discovery, with each theoretical breakthrough, we inch closer to realizing this age-old dream, and I am honored to be a part of this incredible journey.As I stand on the precipice of a future where the constraints of time may be transcended, I am filled with a profound sense of awe and anticipation. The wings of science have carried us this far, and I have no doubt that they will continue to propel us towards even greater heights of understanding and exploration. Time travel may once have been the stuff of dreams and fanciful tales, but today, it stands as a tantalizing reality, beckoning us to take flight and soar through the vast expanse of the cosmic tapestry.。
量子点具有量子力学的英文回答:Quantum dots exhibit quantum mechanical effects due to their nanoscale dimensions. These effects include:Quantization of energy levels: The energy levels of electrons in quantum dots are discrete, meaning they can only occupy certain specific energies. This is in contrast to the continuous energy levels of electrons in bulk materials.Tunable bandgap: The bandgap of a quantum dot is the energy difference between the valence band and the conduction band. The bandgap of a quantum dot can be tuned by changing the size of the dot. This allows quantum dots to be used in a variety of optoelectronic applications.Enhanced optical properties: Quantum dots have enhanced optical properties, such as high photoluminescenceefficiency and narrow emission spectra. These properties make quantum dots ideal for use in applications such as light-emitting diodes (LEDs), lasers, and solar cells.中文回答:量子点由于其纳米尺度的尺寸而表现出量子力学效应。
量子力学照亮前程英文英文回答:Quantum mechanics, the study of matter and energy at atomic and subatomic levels, has profoundly illuminated our understanding of the world and continues to shape our technological advancements.Quantum mechanics has revolutionized our comprehension of the fundamental nature of reality. It has revealed that particles, such as electrons and photons, can exhibit wave-like properties and that particles can exist in multiple states simultaneously. These insights have led to the development of new theories in physics, such as quantum field theory, which describes the interactions of particles at the subatomic level.The principles of quantum mechanics have also been applied to develop many transformative technologies that have revolutionized various fields. For example, the laser,which is based on the amplification of stimulated emission of radiation, has had a profound impact on diverse areas such as medicine, manufacturing, and communication.Quantum computers, which harness the principles of quantum mechanics to perform complex computations, hold immense potential for solving problems that are intractable for classical computers. These computers couldrevolutionize fields such as materials science, drug discovery, and cryptography.Quantum mechanics has also played a pivotal role in the development of advanced imaging techniques, such as magnetic resonance imaging (MRI) and positron emission tomography (PET). These techniques have revolutionized the diagnosis and treatment of diseases by providing detailed images of the body's internal structures.In addition, quantum mechanics has inspired the development of novel materials, such as graphene and topological insulators, which exhibit extraordinary electronic properties. These materials hold promise forapplications in electronics, energy storage, and computing.中文回答:量子力学,对原子和亚原子级别物质和能量的研究,深刻地阐明了我们对世界的理解,并持续塑造着我们的技术进步。
T7噬菌体DNA的提取及其反向遗传拯救方法的建立徐海;王义伟;陈瑾;郑其升;侯继波【摘要】从T7噬菌体培养液中粗提噬菌体颗粒,经热裂解后用苯酚、氯仿抽提进而获得纯净的T7噬菌体DNA.用PCR、酶切法鉴定T7噬菌体DNA的完整性.通过对不同感受态细菌浓度、T7噬菌体DNA用量、电转化电压条件的优化,建立了T7噬菌体反向遗传拯救方法.结果显示,提取的DNA结构完整,能够被特异性酶切割,多克隆位点序列正确.T7噬菌体的反向遗传拯救方法最优化条件为200 ng T7噬菌体DNA、1 ml 5×109感受态细菌、1.5 kV电转化电压,在此条件下获得的拯救效率为3.5×105 PFU/ng (DNA).%The T7 phage DNA was purified with benzene polyphenol and chloroform from T7 phage particles after heat cracking. The integrity of T7 phage DNA was identified by PCR and enzemy digestion, and a reverse genetic rescue system for the purified T7 DNA was established through the optimization of the conditions, such as T7 phage DNA input dosage, density of complete cells, and electrotransformation voltage. The results showed that the purified T7 DNA could be digested with EcoR I or Hind Ⅲ, and the multiclon site was correct. The reverse genetics rescue system was successfully established. The highest rescue efficiency was obtained under the conditions of 200 ng T7 phage DNA,5×l09 complete cell per millili ter and 1.5 kV electrotransformation voltage, with a output of 3.5×l05 PFU/ng (DNA).【期刊名称】《江苏农业学报》【年(卷),期】2012(028)002【总页数】4页(P355-358)【关键词】T7噬菌体;DNA提取;电转化;反向遗传拯救【作者】徐海;王义伟;陈瑾;郑其升;侯继波【作者单位】江苏省农业科学院国家兽用生物制品工程技术研究中心,江苏南京210014;江苏省农业科学院国家兽用生物制品工程技术研究中心,江苏南京210014;江苏省农业科学院国家兽用生物制品工程技术研究中心,江苏南京210014;江苏省农业科学院国家兽用生物制品工程技术研究中心,江苏南京210014;江苏省农业科学院国家兽用生物制品工程技术研究中心,江苏南京210014【正文语种】中文【中图分类】S432.4+1T7噬菌体是感染大肠杆菌的烈性噬菌体,在大肠杆菌的胞浆内组装,成熟的T7噬菌体通过细胞裂解而释放,现已完成其全序列分析,遗传背景清楚,病毒颗粒结构复杂。
英文原版量子论科普If you're interested in reading about quantum theory in English, here are some recommendations:1. "Quantum Mechanics: The Theoretical Minimum" by Leonard Susskind and Art Friedman. This textbook provides a thorough and accessible introduction to quantum mechanics, starting from the basics and building up to more advanced concepts. It's written in a lively and engaging style, making it suitable for self-study or as a classroom textbook.2. "Quantum Physics: A First Encounter" by John Taylor. This book is aimed at undergraduates and covers the key ideas of quantum theory, including wavefunctions, operators, measurement, and entanglement. It provides plenty of examples and exercises to help readers understand and apply the theory.3. "Quantum Computation and Quantum Information" by Michael Nielsen and Isaac Chuang. This textbook provides a comprehensive introduction to the field of quantum information science, covering quantum computing, quantum algorithms, quantum error correction, and quantum cryptography. It's suitable for graduate students and researchers in the field.4. "The Quantum World" by Christopher French and Carlo Michelli. This book provides a broad overview of modern quantum theory and its applications, including quantum computing, quantum cryptography, and quantum metrology. It's written in a clear and accessible style, making it suitable for non-experts who want to understand the basics of quantum theory.Remember that reading about quantum theory can be challenging because it involves concepts that are counterintuitive and打破常识。
a rXiv:h ep-th/9312131v115D ec1993TAUP 2110-93February 1,2008A-B AND BERRY PHASES FOR A QUANTUM CLOUD OF CHARGE Yakir Aharonov,a b Sidney Coleman,c Alfred S.Goldhaber,d Shmuel Nussinov,a Sandu Popescu,e Benni Reznik,a Daniel Rohrlich,a and Lev Vaidman a ABSTRACT We investigate the phase accumulated by a charged particle in an extended quantum state as it encircles one or more magnetic fluxons,each carrying half a flux unit.A simple,essentially topological analysis reveals an interplay between the Aharonov-Bohm phase and Berry’s phase.PACS numbers:3.65.-w,3.65.Ge,74.20.KkThe Aharonov-Bohm(AB)phase[1]ΦAB=(q/¯h) C A·d r collected by a charge q,moving in a closed path C about a line of magneticfluxφ,is purely topological:ΦAB=2πn(q/e)(φ/φ0),with n the winding number of C around thefluxon,e the elementary charge,andφ0the correspondingflux unitφ0=2π¯h/e.The AB phase is independent of the shape of the path C and of the history of motion along it. If the charge is not pointlike,or thefluxon is not linelike,they may overlap;what then happens to the AB phase?As long as the charge andflux are distributed classically,the answer is straightforward:a system of charges moving in a closed path through a classical magneticfield collects an AB phaseΦAB=(1/¯h) q iφi, withφi theflux enclosed by the path of the i-th charge.Here,however,we consider charges distributed by quantum smearing.The phase of a quantum charge is no simple sum over the undeformed charge distribution.If we compute it via the Born-Oppenheimer approximation,wefind a remarkable interplay between the AB phase and Berry’s phase that determines the overall topological phase.Consider a single electron bound to a heavy“nucleus”(assumed neutral,for simplicity)in the presence of an infinitely longflux line.Both the nucleus and thefluxon may move.If thefluxon makes a closed path around the nucleus,it may encircle some parts of the quantum charge distribution(the electron“cloud”) and not others.Suppose that the time in which thefluxon crosses the“atom”, multiplied by typical electronic frequencies,is much smaller than1.In this limit, the initial electronic wave functionΨ0transforms intoΨ1=e iΦABΠCΦ0+(1−ΠC)Φ0,whereΠC projects onto the part ofΨ0that thefluxon encircles.The electron has no time to move between the two parts ofΨ0.But,except in this limit,we cannot assign parts of the wave function distinct phases.Consider now the opposite limit,of adiabatic motion.In this limit,another phase effect comes into play.Berry’s phase[2]arises when parameters for a quantum system vary adiabatically in a closed path.Applying the Born-Oppenheimer approximation to thefluxon-atom system,and for definitenessfixing the nucleus,we obtain both an AB phase and a Berry phase.An AB phase arises from motion of thefluxon with respect to the instantaneous charge distribution,while a Berry phase arises fromrearrangement of the electronic wave function.There is a subtle interplay of these two phases,which is purely topological for special values of thefluxφcarried by the fluxon.For example,whenφ=nφ0,the phases completely cancel[3]as expected since thefluxon is a pure gauge artifact.Here,we consider the more interesting case offluxons carrying half aflux unit(“half-fluxons”or“semifluxons”).Topological analysis,with no computations,reveals the interplay of the AB phase and Berry’s phase.Let us begin with the electron(mass m1)bound at the origin with a potential V( r1)and thefluxon(mass m2)free to move but constrained to remain parallel to the z-axis.The Hamiltonian isH=( p1+e A)22m2;(1)for a half-fluxon we take A=(¯h/2e) ∇2ϕ12withϕ12the angle of thefluxon in polar coordinates with the electron at the origin.[4]Consider a limited time reversal operation T sending p i→− p i but leaving A unchanged.T sends( p i±e A)2→( p i∓e A)2;since A=− A,T seems not to be a symmetry of H.However,for the special case of a half-fluxon,the difference between A and− A amounts to a pure gauge transformation: A=− A+∇2ΛwithΛ≡(¯h/e)ϕ12;so T is a symmetry of H.Thus there is a gauge in which we can choose the eigenstates of H real.Suppose m2>>m1.If we apply the Born-Oppenheimer approximation to Eq.(1),the effective Hamiltonian for thefluxon will contain an induced vector potential due to adiabatic transport of a real electron wave function,thus it will preserve the time reversal symmetry.Let an initial state for the half-fluxon evolve according to this effective Hamiltonian and move adiabatically around a loop C. The state accumulates a phase factor which may include a geometric as well as a dynamical phase.Time reversal symmetry implies that the state acquires the same phase factor if it moves around C in the opposite sense.The dynamical phase is the same in the two cases but the geometric phaseΦ(C)changes sign. Since the overall phase factor remains unchanged,we conclude that the geometric phase acquired by the electron-fluxon system can be only0orπ.What,then,becomes of the geometrical phaseΦ(C)as we deform the path C?Let us assume the electronic wave function to be restricted to afinite region S.Fig.1shows a closedfluxon path C1which lies completely outside the region S without encircling it.For this path,the AB phase is zero.Furthermore,Berry’s phase also vanishes.Now let us gradually distort the path C1until it becomes a large loop C2that encircles the region S without touching it.For this loop Berry’s phase vanishes,but the AB phase isΦAB(C2)=π,since all the charge has been encircled once.[5]We can distort C1into C2by many steps which enlarge the loop by an infinitesimal region.Naively,we would expect the phaseΦof the loop to vary smoothly from0toπbut,as noted,Φcan only be0orπ.Thus,we conclude that some infinitesimal region contains a“singular point”P so thatΦjumps when this infinitesimal region is annexed.The electronic wave function yields a vector potential that is always bounded,and so an infinitesimal region cannot lead to a jump in the AB phase.Therefore,the jump inΦis due to Berry’s phase.The significance of P is clear:P is a point such that if a half-fluxon is introduced there, the electron wave function becomes degenerate.(Only a degeneracy can cause such a jump in Berry’s phase.)The feature that we exhibit with this indirect argument, namely that such a point P exists(even if V( r1)=V(r1)),would be hard to see from a direct study of Schr¨o dinger’s equation.Conversely,suppose we suspect that two states become degenerate at a point P. Near P,we can truncate the Hilbert space for the system to the subspace spanned by the two states,and write the effective Hamiltonian as a sum of Pauli matrices (plus a constant)H0(x,y)+H1(x,y)σ1+H2(x,y)σ2+H3(x,y)σ3.For a generic fluxon,the degeneracy condition involves three equations with two parameters x,y (the coordinates of thefluxon),so that there are no solutions.However,for the special case of a half-fluxon,the eigenstates and thus the effective Hamiltonian can always be chosen real.Then H2(x,y)vanishes.The degeneracy point P=(x∗,y∗) isfixed by requiring H1(x∗,y∗)=H3(x∗,y∗)=0;these two equations naturally lead to isolated points of degeneracy.The actual location of P depends on the stateΨ0and relevant potential.Whenthe potential is spherically symmetric,V( r)=V(r),the point P corresponds to a fluxon through the z-axis.The Hamiltonian retains azimuthal symmetry.If states depend onϕas e imϕfor integer m,introducing thefluxon is equivalent to shifting the angular momentum L z by half a unit:−i(∂/∂ϕ)→−i(∂/∂ϕ)−1/2or L z→L z−¯h/2.Initially the energy is proportional to m2.All the energy levels are doubly degenerate except for the ground state.The shift m→m′=m−1/2rearranges all the levels into degenerate pairs.In particular,the ground state m=0becomes degenerate with m=1(since m=0→m′=−1/2and m=1→m′=1/2).This degeneracy occurs only for a half-fluxon.There could be any odd number of degeneracy points.Indeed,consider the m′=±3/2(degenerate)states of a rotationally symmetric potential with a half-fluxon at the center.By adding a perturbation V′=λcos3ϕwhich connects these two states,the degeneracy is lifted.To restore the degeneracy,we must move the half-fluxon away from the origin.The problem is now invariant under rotations of2π/3and so,by symmetry,there will be three degeneracy points.A similar argument with m′=±(2k+1)/2and V′=λcos(2k+1)ϕleads to2k+1 symmetrically situated points.[6]Let us now determine the phase collected by an atom which slowly moves in the presence of two semifluxons.When the undisturbed ground state is spherically symmetric and thefluxons arefixed,we can map this problem to an equivalent one, replacing the spherical charge distribution by a point charge located at its center, and thefluxons by“shadow”fluxons.The shadowfluxons are defined as points such that when the center of the atom coincides with one of them,a degeneracy results.The winding number of the path of the point charge around the shadow fluxons gives the phase accumulated by the atom.Consider two straight and parallel semifluxon lines situated a distance L apart.Two extreme cases are easily solved.When the distance between thefluxons is much larger than the size of the atom,we can move the atom in the vicinity of one of thefluxons without the electron cloud crossing the otherfluxon.In this case the atom collects a phase of πeach time its center encircles thefluxon,exactly as if the otherfluxon were notpresent.The“shadow”fluxons coincide therefore with the originalfluxons.On the other hand,for L=0the two semifluxons are at the same point,adding up to an integerfluxon with no effect on the energy levels of the electron,and therefore no“shadow”fluxons can exist.When thefluxons are slightly separated,they do affect the energy.However,by continuity,an infinitesimal separation of thefluxons cannot produce a degeneracy;rather,a minimal distance L∗>0is required.Thus we arrive at the conclusion that in an adiabatic quantum process(say,an atom in a specific state moving slowly)the geometric phase due to two half-fluxons will always be zero once their separation L is less than some L∗>0.We may now interpolate between L=L∗and L→∞.Instead of considering the atom as moving byfixed semifluxons,let usfix the atom and one semifluxon and allow the second semifluxon to move.Let the center of the atom be at O and a fluxon F1at P1,and let us determine the phase accumulated by a secondfluxon F2 as it slowly moves along various closed paths(Fig.2).Again,this phase can only be0orπ;thus there must be a point P2such that when F2encircles P2,the phase jumps byπ.Insertion of thefluxon F2at the point P2produces a degeneracy.The connection with the“shadows”is that here the point O corresponds to a shadow fluxon.Let us assume that the points P1and P2are related by a continuous function.By symmetry,P1,P2and O must form a straight line.We claim that P1and P2lie on opposite sides of O.Let us examine P1as a function of P2.If P2is located in the region where the wave function vanishes,P1must be situated at the atom’s center O.As P2enters the electron cloud and moves towards O, P1must move either towards P2or in the opposite direction.Thefirst possibility must be discarded:in this case either P1and P2will collide,or P1will reverse direction and eventually return to O to avoid collision with P2.Both alternatives are inconsistent.If the two half-fluxons collide,they form an integerfluxon with no degeneracy.If P1reverses direction,we obtain an“accidental”degeneracy with P1at O and P2inside the electron cloud,where a degeneracy cannot arise.[7]As claimed,then,the points P1and P2lie on opposite sides of O.Thus,the shadow fluxon associated with each semifluxon is shifted towards the other semifluxon.It is amusing to consider various patterns of half-fluxons and resulting shadows. Even in the case of a single half-fluxon,the shadow need not coincide with the original,if thefluxon line is not straight.For two half-fluxon lines intersecting at an acute angle,we expect tofind shadowfluxons in the plane of the half-fluxons,located near the latter but shifted towards a more acute angle.Then from continuity,we expect“hyperbolic”shadowfluxons as shown in Fig.3.For the case of n half-fluxon lines in a plane intersecting symmetrically at one point,the shadow fluxons will be identical with the half-fluxons and will induce simple degeneracies. The intersection could be a point of higher degeneracy.Finally,we discuss the case of N semifluxons and an electron cloud of arbi-trary shape.For simplicity we consider a two-dimensional problem.The set of points(P1,P2,...,P N)such that if in each of them a semifluxon is introduced, the initial wave function of the electron becomes degenerate,constitutes a(2N−2)-dimensional hypersurfaceΣ.Indeed,for any given points P1,P2,...,P N−1there always exists at least one corresponding point P N.As proof,we introduce a semi-fluxon in each of the N−1points P1,...,P N−1and consider the phase accumulated by the N-th semifluxon as it takes various paths.Similar arguments to those above for one and two semifluxons lead to the conclusion that at some point P N,the Berry phase jumps byπ.Introducing semifluxons at P1,...,P N−1,P N therefore results in a degeneracy.We may describe the locations of the Nfluxons by a point(x1,y1,...,x N,y N). To every configuration of N−1fluxons there corresponds a location x N,y N where the N-thfluxon induces a degeneracy:x N=f(x1,y1,...,x N−1,y N−1)y N=g(x1,y1,...,x N−1,y N−1).(2)Eq.(2)then defines the(2N−2)-dimensional hypersurfaceΣ.Suppose that N semifluxons move slowly and after a certain time all return to their initial positions. What is the topological phase in this case?Thefluxons describe a closed path C in the2N-dimensional space(x1,y1,...,x N,y N).The phase accumulated by thefluxons as they move through the charge distribution is simply nπ,where n is the winding number of the closed path C aroundΣ.ACKNOWLEDGEMENTSWe thank Aharon Casher for discussions.This research was supported in part by grant425-91-1of the Basic Research Foundation,administered by the Israel Academy of Sciences and Humanities,by the Program in Alternative Thinking at Tel-Aviv University,and by grants PHY8807812and PHY90-8936of the National Science Foundation.REFERENCES1.Y.Aharonov and D.Bohm,Phys.Rev.115,485(1959).2.M.V.Berry,Proc.Roy.Soc.London A392,45(1984).3.B.Reznik and Y.Aharonov,Phys.Lett.B315,386(1993).4.See Y.Aharonov and A.Casher,Phys.Rev.Lett.33,319(1984).TheAharonov-Casher(AC)effect is dual to the AB effect:a neutron interacting with a line of charge is equivalent to an electron interacting with afluxon.Eq.(1)is an effective two-dimensional Hamiltonian for both the AB and AC effects,and all our conclusions hold for both cases.5.LetΨ( r1, r2)represent a localizedfluxon wave packet encircling the electronwithout entering S.Since the wave packet is localized,the wave function Ψ′defined byΨ=e(i/2)ϕ12Ψ′is single-valued.NoteΨ′solves Eq.(1)with A=0.The remaining factor e(i/2)ϕ12inΨyields the phaseπfor a circuit about S.6.By contrast,V′=λcos(2kϕ)does not connect any degenerate pair and thisargument fails–as it must,since the number of degeneracy points cannot be even.7.P.A.M.Dirac,Proc.Roy.Soc.A133,60(1931)showed that the wavefunction for a charged particle,in the presence of a magnetic monopole, must have a line of zeros extending from the monopole to an antipole or to infinity.An analogous argument shows that,in the presence of a semifluxon,a nondegenerate stateΨ0must have a line of zeros extending from thesemifluxon to another semifluxon or to infinity.(A surface of zeros issues from a three-dimensionalfluxon,but we refer to a line for the effective two-dimensional problem.)In the ground state only one such null line issues from each semifluxon.We can choose the vector potential A to be singular on the null line and zero elsewhere.Approaching the“accidental”degeneracy configuration,the semifluxons carry null lines which,by symmetry,must liealong their common line.A null line can connect thefluxons,or two null lines can issue from them in opposite directions.The corresponding wave functions are not degenerate because the two null lines are more constraining than one.FIGURE CAPTIONSFig1.The shaded region S indicates where the wave function is non-zero, and C1and C2are limiting paths.Insertion of a half-fluxon at point P induces degeneracy.Fig.2.The center of the atom is O;at points P1,P2simultaneous insertion of half-fluxons induces degeneracy.C1and C2are limiting paths of the half-fluxon F2.Fig. 3.Two half-fluxons intersecting at an acute angle and the resulting shadowfluxons.11This figure "fig1-1.png" is available in "png" format from: /ps/hep-th/9312131v1This figure "fig1-2.png" is available in "png" format from: /ps/hep-th/9312131v1This figure "fig1-3.png" is available in "png" format from: /ps/hep-th/9312131v1。