Quantum states on Harmonic lattices

一个含时谐振子的精确波函数和相干态(英文)A coherent state is a quantum state in which the oscillation of a harmonic oscillator is amazingly close tothe expected value of a classical harmonic oscillator. Coherent states are important in quantum mechanics, including a wide range of applications, from quantum information theory to quantum foundations and from quantum cryptography to quantum optics.A coherent state can be described as a wavefunction that obeys the Schrodinger equation for a harmonic oscillator. The wavefunction is a superposition of all the possible energy eigenstates and is highly localized in both space and time.It is often referred to as a 'clock state' because it accurately reflects the behavior of a classical harmonic oscillator.In quantum mechanics, coherent states can be used to describe the oscillations of a system with a specific frequency and amplitude. For example, the coherent states ofa harmonic oscillator are used to describe the motion of particles in a crystal lattice and the motion of particles in a cavity. In electro-optics, coherent states are used to describe the interference of light waves, where the lightfield is superimposed on the electric potential of the medium.In recent years, coherent states have also found application in quantum information theory. They allow for the manipulation of quantum states with great precision and accuracy, and can be used to generate and detect entangled states. Furthermore, coherent states are useful for therealization of quantum gates, which are gates of individual qubits, as well as for efficient error correction protocols.In summary, a coherent state is a quantum state in which the oscillations of a harmonic oscillator are amazingly close to the expected value of a classical harmonic oscillator. It is an important concept in quantum mechanics and finds application in a variety of areas, from quantum information theory to quantum optics. It is the basis of many important advances in quantum technologies.。

a r X i v :q u a n t -p h /0606222v 1 27 J u n 2006QUANTUM DECOHERENCE OF THE DAMPED HARMONIC OSCILLATORA.IsarInstitute of Physics and Nuclear Engineering,Bucharest-Magurele,Romaniae-mail:isar@theory.nipne.roAbstract In the framework of the Lindblad theory for open quantum systems,we de-termine the degree of quantum decoherence of a harmonic oscillator interacting with a thermal bath.It is found that the system manifests a quantum de-coherence which is more and more significant in time.We also calculate the decoherence time and show that it has the same scale as the time after which thermal fluctuations become comparable with quantum fluctuations.PACS numbers:03.65.Yz,05.30.-d 1Introduction The quantum to classical transition and classicality of quantum systems continue to be among the most interesting problems in many fields of physics,for both conceptual and experimental reasons [1,2].Two conditions are essential for the classicality of a quantum system [3]:a)quantum decoherence (QD),that means the irreversible,uncontrollable and persistent formation of a quantum correlation (entanglement)of the system with its environment [4],expressed by the damping of the coherences present in the quantum state of the system,when the off-diagonal elements of the density matrix decay below a certain level,so that this density matrix becomes approximatelydiagonal and b)classical correlations,expressed by the fact that the Wigner function of the quantum system has a peak which follows the classical equations of motion in phase space with a good degree of approximation,that is the quantum state becomes peaked along a classical trajectory.Classicality is an emergent property of open quantum systems,since both main features of this process –QD and classical correlations –strongly depend on the interaction between the system and its external environment[1,2].The role of QD became relevant in many interesting physical problems.In manycases one is interested in understanding QD because one wants to prevent decoherence from damaging quantum states and to protect the information stored in quantum states from the degrading effect of the interaction with the environment.Decoherence is also responsible for washing out the quantum interference effects which are desirableto be seen as signals in experiments.QD has a negative influence on many areas relying upon quantum coherence effects,in particular it is a major problem in quantum optics and physics of quantum information and computation[5].In this work we study QD of a harmonic oscillator interacting with an environ-ment,in particular with a thermal bath,in the framework of the Lindblad theory for open quantum systems.We determine the degree of QD and then we apply the criterion of QD.We consider different regimes of the temperature of environment and it is found that the system manifests a QD which in general increases with time and temperature.The organizing of the paper is as follows.In Sec.2we review the Lindblad master equation for the damped harmonic oscillator and solve the master equation in coordinate representation.Then in Sec.3we investigate QD and in Sec.4we calculate the decoherence time of the system.We show that this time has the same scale as the time after which thermalfluctuations become comparable with quantum fluctuations.A summary and concluding remarks are given in Sec.5.2Master equation and density matrixIn the Lindblad axiomatic formalism based on quantum dynamical semigroups,the irreversible time evolution of an open system is described by the following general quantum Markovian master equation for the density operatorρ(t)[6]:dρ(t)¯h [H,ρ(t)]+12(qp+pq),H0=12q2(2)and the operators V j,V†j,which model the environment,are taken as linear polynomials in coordinate q and momentum p.Then the master equation(1)takes the following form[7]:dρ¯h [H0,ρ]−i2¯h(λ−µ)[p,ρq+qρ]−D pp¯h2[p,[p,ρ]]+D pqcase when the asymptotic state is a Gibbs stateρG(∞)=e−H0kT,these coeffi-cients becomeD pp=λ+µ2kT,D qq=λ−µmωcoth¯hω2kT≥λ2(5)and the asymptotic valuesσqq(∞),σpp(∞),σpq(∞)of the dispersion(variance),re-spectively correlation(covariance),of the coordinate and momentum,reduce to[7]σqq(∞)=¯h2kT,σpp(∞)=¯h mω2kT,σpq(∞)=0.(6)We consider a harmonic oscillator with an initial Gaussian wave function(σq(0) andσp(0)are the initial averaged position and momentum of the wave packet)Ψ(q)=(14exp[−1¯hσpq(0))(q−σq(0))2+i 2mω,σpp(0)=¯h mω2√∂t=i¯h∂q2−∂22¯h(q2−q′2)ρ−1∂q−∂2(λ−µ)[(q+q′)(∂∂q′)+2]ρ−D pp∂q+∂∂q+∂a damping effect(exchange of energy with environment).The last three are noise (diffusive)terms and producefluctuation effects in the evolution of the system.D pp promotes diffusion in momentum and generates decoherence in coordinate q–it re-duces the off-diagonal terms,responsible for correlations between spatially separated pieces of the wave packet.Similarly D qq promotes diffusion in coordinate and gener-ates decoherence in momentum p.The D pq term is the so-called”anomalous diffusion”term and it does not generate decoherence.The density matrix solution of Eq.(9)has the general Gaussian form<q|ρ(t)|q′>=(12exp[−12−σq(t))2−σ(t)¯hσqq (t)(q+q′¯hσp(t)(q−q′)],(10)whereσ(t)≡σqq(t)σpp(t)−σ2pq(t)is the Schr¨o dinger generalized uncertainty function. In the case of a thermal bath we obtain the following steady state solution for t→∞(ǫ≡¯hω/2kT):<q|ρ(∞)|q′>=(mω2exp{−mωcothǫ+(q−q′)2cothǫ]}.(11)3Quantum decoherenceAn isolated system has an unitary evolution and the coherence of the state is not lost–pure states evolve in time only to pure states.The QD phenomenon,that is the loss of coherence or the destruction of off-diagonal elements representing coherences between quantum states in the density matrix,can be achieved by introducing an interaction between the system and environment:an initial pure state with a density matrix which contains nonzero off-diagonal terms can non-unitarily evolve into afinal mixed state with a diagonal density matrix.Using new variablesΣ=(q+q′)/2and∆=q−q′,the density matrix(10) becomesρ(Σ,∆,t)= πexp[−αΣ2−γ∆2+iβΣ∆+2ασq(t)Σ+i(σp(t)2σqq(t),γ=σ(t)¯hσqq(t).(13)The representation-independent measure of the degree of QD[3]is given by the ratio of the dispersion1/√2/αof the diagonal elementρ(Σ,0,t):δQD(t)=1α24{e−4λt[1−(δ+1δ(1−r2)−2cothǫ)ω2−µ2cos(2Ωt)δ(1−r2))µsin(2Ωt)Ω2√2kT,(16)which for high T becomesδQD(∞)=¯hω/2kT.We see thatδQD decreases,and therefore QD increases,with time and temperature,i.e.the density matrix becomes more and more diagonal at higher T and the contributions of the off-diagonal elements get smaller and smaller.At the same time the degree of purity decreases and the degree of mixedness increases with T.For T=0the asymptotic(final)state is pure andδQD reaches its initial maximum value1.δQD=0when the quantum coherence is completely lost,and ifδQD=1there is no QD.Only ifδQD<1we can say that the considered system interacting with the thermal bath manifests QD,when the magnitude of the elements of the density matrix in the position basis are peaked preferentially along the diagonal q=q′.Dissipation promotes quantum coherences, whereasfluctuation(diffusion)reduces coherences and promotes QD.The balance of dissipation andfluctuation determines thefinal equilibrium value ofδQD.The initial pure state evolves approximately following the classical trajectory in phase space and becomes a quantum mixed state during the irreversible process of QD.4Decoherence timeIn order to obtain the expression of the decoherence time,we consider the coefficientγ(13),which measures the contribution of non-diagonal elements in the density matrix(12).For short times(λt≪1,Ωt≪1),we have:γ(t)=−mωδ(1−r2))cothǫ+µ(δ−r2δ√2[λ(δ+r2δ(1−r2))cothǫ−λ−µ−ωr1−r2].(18)The decoherence time depends on the temperature T and the couplingλ(dissipation coefficient)between the system and environment,the squeezing parameterδand the initial correlation coefficient r.We notice that the decoherence time is decreasing with increasing dissipation,temperature and squeezing.For r=0we obtain:t deco=12λ(δ−1).(20) We see that when the initial state is the usual coherent state(δ=1),then the deco-herence time tends to infinity.This corresponds to the fact that for T=0andδ=1 the coefficientγis constant in time,so that the decoherence process does not occur in this case.At high temperature,expression(18)becomes(τ≡2kT/¯hω)t deco=1δ(1−r2))+µ(δ−r24(λ+µ)δkT.(22) The generalized uncertainty functionσ(t)(15)has the following behaviour for short times:σ(t)=¯h2δ(1−r2))cothǫ+µ(δ−1This expression shows explicitly the contribution for small time of uncertainty that is intrinsic to quantum mechanics,expressed through the Heisenberg uncertainty prin-ciple and uncertainty due to the coupling to the thermal environment.From Eq.(23)we can determine the time t d when thermalfluctuations become comparable with quantumfluctuations.At high temperature we obtain1t d=)+µ(δ−1δ(1−r2)squeezing and correlation,it depends on temperature only.QD is expressed by theloss of quantum coherences in the case of a thermal bath atfinite temperature.(2)We determined the general expression of the decoherence time,which showsthat it is decreasing with increasing dissipation,temperature and squeezing.We havealso shown that the decoherence time has the same scale as the time after which thermalfluctuations become comparable with quantumfluctuations and the valuesof these scales become closer with increasing temperature and squeezing.After the decoherence time,the decohered system is not necessarily in a classical regime.There exists a quantum statistical regime in between.Only at a sufficiently high temperaturethe system can be considered in a classical regime.The Lindblad theory provides a self-consistent treatment of damping as a general extension of quantum mechanics to open systems and gives the possibility to extendthe model of quantum Brownian motion.The results obtained in the framework ofthis theory are a useful basis for the description of the connection between uncertainty, decoherence and correlations(entanglement)of open quantum systems with their en-vironment,in particular in the study of Gaussian states of continuous variable systemsused in quantum information processing to quantify the similarity or distinguishabilityof quantum states using distance measures,like trace distance and quantumfidelity. AcknowledgmentsThe author acknowledges thefinancial support received within the Project PN06350101/2006. References[1]E.Joos,H.D.Zeh,C.Kiefer,D.Giulini,J.Kupsch,I.O.Stamatescu,Decoherenceand the Appearance of a Classical World in Quantum Theory(Springer,Berlin,2003).[2]W.H.Zurek,Rev.Mod.Phys.75,715,2003.[3]M.Morikawa,Phys.Rev.D42,2929(1990).[4]R.Alicki,Open Sys.and Information Dyn.11,53(2004).[5]M.A.Nielsen and I.L.Chuang,Quantum Computation and Quantum Information(Cambridge Univ.Press,Cambridge,2000).[6]G.Lindblad,Commun.Math.Phys.48,119(1976).[7]A.Isar,A.Sandulescu,H.Scutaru,E.Stefanescu,W.Scheid,Int.J.Mod.Phys.E3,635(1994).[8]A.Isar,W.Scheid,Phys.Rev.A66,042117(2002).[9]B.L.Hu,Y.Zhang,Int.J.Mod.Phys.A10,4537(1995).[10]A.Isar,W.Scheid,Physica A,in press(2006).。

量子计算的基本原理和可能的应用领域1. Introduction1.1 Overview:Quantum computing is a rapidly advancing field that utilizes the fundamental principles of quantum mechanics to process and manipulate information in ways that are fundamentally different from classical computing systems. Unlike classical bits, which can exist in only two states (0 or 1), quantum bits or qubits can exist in multiple states simultaneously due to a property known as superposition. This ability to simultaneously represent and process multiple states forms the foundation of quantum computing.1.2 Importance of Quantum Computing:The potential impact of quantum computing extends beyond traditional computation, offering remarkable advantages in solving complex problems that are infeasible for classical computers. Quantum computers have the potential to significantly enhance computational speed, improve encryption methods, revolutionize data processing and simulation, and transform fields such as drug discovery and materialscience.1.3 Research Background:Over the past few decades, extensive research has been conducted worldwide to explore the principles and applications of quantum computing. Several key breakthroughs have been made, including advancements in qubit technologies, development of error correction techniques, and exploration of various computational algorithms designed specifically for quantum systems. These advancements have brought us closer to realizing the full potential of quantum computing and uncovering its vast possibilities.In this article, we will delve into the fundamental principles underlying quantum computing and explore its potential applications in various domains. We will also discuss the advantages offered by quantum computing over classical computation systems and address some of the challenges that need to be overcome for its widespread adoption. Finally, we will conclude by summarizing the current state of quantum computing research and providing insights into future developments and prospects within this exciting field.(Note: The content provided is written using plain text format.)2. 量子计算基础原理2.1 量子比特量子计算中的最基本单位是量子比特，也称为qubit。

芯片相干布居囚禁原子钟陈杰华1，2（1.中国科学院精密测量科学与技术创新研究院，湖北武汉430071；2.武汉量子技术研究院，湖北武汉 430206）摘要：对基于相干布居囚禁（CPT）态原子的芯片级原子钟技术进行了综述，包括CPT原子钟的基本原理、Ramsey⁃CPT原子钟技术、CPT原子钟的微小型化方案、CPT芯片原子钟的发展等，对其中的关键技术：激光频率调制、Ramsey技术、左右旋圆偏振光泵浦（push⁃pull）、激光和微波稳频以及微光机电加工系统（MOEMS）等技术进行分析讨论，最终总结出CPT原子钟向着低功耗、芯片化和时钟精度高的方向发展。

关键词：微波原子钟；相干布居囚禁；电磁感应透明；微光机电加工中图分类号：TB939 文献标志码：A 文章编号：1674-5795（2023）03-0053-07Chip⁃scale coherent population trapping atomic clockCHEN Jiehua1,2(1.Innovation Academy for Precision Measurement Science and Technology, Chinese Academy of Sciences,Wuhan 430071, China； 2.Wuhan Institute of Quantum Technology, Wuhan 430206, China) Abstract: The chip⁃scale atomic clock technology based on coherent population trapping (CPT) atoms is re⁃viewed, including the basic principle of CPT atomic clock, Ramsey⁃CPT atomic clock technology, schemes of CPT atomic clock suitable for miniaturization, and the development and current status of the chip⁃scale CPT atomic clock. The key technologies， such as laser frequency modulation, Ramsey technology, left and right circularly polar⁃ized light pumping (push⁃pull), laser and microwave frequency stabilization, and micro optical⁃electro⁃mechanical system (MOEMS), are analyzed and discussed. Finally, it is concluded that the CPT atomic clock is developing in the direction of low power consumption, chip⁃based and high clock precision.Key words: microwave atomic clock; coherent population trapping; electromagnetically induced transparency; micro optical⁃electro⁃mechanical system0　引言芯片级原子钟在小体积、低功耗的条件下可获得较高的时频稳定度，在很多对时频精度有较高要求的小型终端上有重要应用。

介绍量子计算与生活的关系英语作文全文共3篇示例，供读者参考篇1Quantum computing, as a revolutionary technology, has attracted increasing attention in recent years. It has been praised for its potential to solve complex problems at a speed unimaginable by traditional computers. But how does quantum computing relate to our daily lives? Let's explore the connection between quantum computing and everyday activities.First and foremost, quantum computing can significantly impact the field of medicine. With its ability to process vast amounts of data simultaneously, quantum computers can analyze genetic information, identify patterns, and develop new drugs more efficiently. This could lead to breakthroughs in personalized medicine and the treatment of diseases like cancer, Alzheimer's, and diabetes. In the future, quantum computing may enable doctors to provide more accurate diagnoses and tailored treatment plans based on individual genetic characteristics.Furthermore, quantum computing has the potential to revolutionize the financial sector. Traditional computers struggle to manage the immense amount of data in financial markets, leading to inefficiencies and delays in decision-making. Quantum computers, on the other hand, can process this data in real-time, enabling rapid analysis and prediction of market trends. This could enhance risk management strategies, optimize investment portfolios, and improve overall financial performance. In the future, quantum computing may redefine the way we conduct financial transactions and manage wealth.In addition, quantum computing could have a significant impact on cybersecurity. As our dependence on digital technology grows, the need for secure communication and data protection becomes more critical. Quantum computers have the potential to break traditional encryption methods used to safeguard sensitive information. On the other hand, quantum cryptography offers a new approach to secure communication that is virtually unbreakable. By harnessing the power of quantum mechanics, quantum computing can revolutionize the way we protect our digital assets and ensure data privacy.Moreover, quantum computing can accelerate scientific research and innovation across various fields. From climatemodeling and materials science to artificial intelligence and machine learning, quantum computers can tackle complex problems that are beyond the capabilities of classical computers. This could lead to new discoveries, advancements in technology, and improvements in our understanding of the universe. In the future, quantum computing may enable us to address some of the most pressing challenges facing humanity, such as climate change, energy sustainability, and healthcare.Overall, quantum computing has the potential to transform our lives in profound ways. By revolutionizing medicine, finance, cybersecurity, and scientific research, quantum computers can drive innovation, enhance efficiency, and lead to breakthroughs that were once thought impossible. As we continue to advance in the field of quantum computing, the possibilities are endless, and the impact on society is bound to be transformative. It is essential for us to embrace this technology and explore its applications to unlock its full potential for the benefit of humanity.篇2Quantum computing is an emerging technology that promises to revolutionize the way we solve complex problems and process information. Unlike classical computers, which usebits to represent information as either 0 or 1, quantum computers use quantum bits or qubits, which can exist in multiple states simultaneously through a property called superposition. This allows quantum computers to perform calculations at a much faster rate compared to classical computers.The potential applications of quantum computing are vast and varied, ranging from drug discovery and material science to cryptography and artificial intelligence. In our everyday lives, quantum computing has the potential to make a significant impact in several key areas:1. Drug discovery: Quantum computers have the ability to simulate complex molecular interactions with high accuracy and speed, which can greatly accelerate the process of drug discovery. This can lead to the development of new and more effective drugs to treat various diseases.2. Weather forecasting: Quantum computers can analyze large datasets and complex weather patterns much more efficiently than classical computers, leading to more accurate weather forecasts. This can help improve disaster preparedness and response efforts.3. Financial modeling: Quantum computers can process huge amounts of financial data and perform complex calculations in real-time, enabling more accurate risk assessments and investment decisions. This can help individuals and businesses make better financial choices.4. Cybersecurity: Quantum computers have the potential to break traditional encryption methods used to secure sensitive information, such as personal data and financial transactions. However, they can also be used to develop quantum-resistant encryption algorithms to enhance cybersecurity.5. Optimization problems: Quantum computers excel at solving optimization problems, such as finding the most efficient route for a delivery truck or minimizing energy consumption in a power grid. This can lead to more sustainable and cost-effective solutions in various industries.Overall, quantum computing has the potential to revolutionize our lives in ways we cannot yet fully comprehend. While the technology is still in its early stages of development, researchers and scientists are making significant progress in advancing quantum computing capabilities. As quantum computers become more powerful and accessible, we can expect to see even more innovative applications that will impact ourdaily lives in profound ways. It is an exciting time to be living in the age of quantum computing, and the possibilities are truly endless.篇3Quantum computing, a cutting-edge technology that harnesses the principles of quantum mechanics to perform complex calculations, has the potential to revolutionize many aspects of our lives. From boosting our cybersecurity to revolutionizing drug discovery, quantum computing holds great promise for the future.One of the key applications of quantum computing is in cryptography. With the exponential growth of data being transferred over the internet, traditional encryption methods are becoming increasingly vulnerable to attacks from quantum computers. Quantum cryptography offers a solution by using the principles of quantum mechanics to create secure communication channels that are virtually impossible to hack. This is particularly important for protecting sensitive data such as financial information, personal details, and government secrets.Another area where quantum computing has the potential to make a significant impact is in the field of drug discovery.Traditional drug discovery processes are time-consuming and expensive, often taking years to identify potential drug candidates. Quantum computing can greatly accelerate this process by simulating molecular interactions and predicting the effectiveness of potential drugs in a fraction of the time it would take using classical computers. This could lead to the development of new treatments for diseases that are currently considered incurable.Quantum computing also has the potential to revolutionize industries such as finance, logistics, and artificial intelligence. In the financial sector, quantum algorithms can optimize investment strategies and risk management, while in logistics, they can streamline supply chain operations and improve efficiency. In the field of artificial intelligence, quantum computing can enhance machine learning algorithms and enable the development of more sophisticated AI models that can solve complex problems.In our everyday lives, quantum computing may also have a direct impact in the near future. For example, quantum computers could enable the development of more accurate weather forecasting models, leading to better predictions of natural disasters such as hurricanes and earthquakes. Quantumcomputing could also revolutionize transportation systems by optimizing traffic flow and reducing congestion, leading to shorter commutes and lower carbon emissions.It is clear that quantum computing has the potential to revolutionize many aspects of our lives, from cybersecurity to drug discovery and beyond. However, there are still many challenges to overcome before quantum computing becomes widespread. These include developing more stable quantum systems, improving error correction techniques, and training a new generation of quantum engineers and scientists. Despite these challenges, the possibilities offered by quantum computing are truly exciting, and the impact on our daily lives could be profound. As we continue to explore the potential of this groundbreaking technology, we can look forward to a future where quantum computing is seamlessly integrated into our lives, transforming the way we work, communicate, and solve problems.。

More informationFundamentals of Photonic Crystal GuidingIf you’re looking to understand photonic crystals,this systematic,rigorous,and peda-gogical introduction is a must.Here you’llfind intuitive analytical and semi-analyticalmodels applied to complex and practically relevant photonic crystal structures.Y ou willalso be shown how to use various analytical methods borrowed from quantum mechanics,such as perturbation theory,asymptotic analysis,and group theory,to investigate manyof the limiting properties of photonic crystals,which are otherwise difficult to rationalizeusing only numerical simulations.An introductory review of nonlinear guiding in photonic lattices is also presented,as are the fabrication and application of photonic crystals.In addition,end-of-chapterexercise problems with detailed analytical and numerical solutions allow you to monitoryour understanding of the material presented.This accessible text is ideal for researchersand graduate students studying photonic crystals in departments of electrical engineering,physics,applied physics,and mathematics.Maksim Skorobogatiy is Professor and Canada Research Chair in Photonic Crystals atthe Department of Engineering Physics in´Ecole Polytechnique de Montr´e al,Canada.In2005he was awarded a fellowship from the Japanese Society for Promotion of Science,and he is a member of the Optical Society of America.Jianke Yang is Professor of Applied Mathematics at the University of Vermont,USA.Heis a member of the Optical Society of America and the Society of Industrial and AppliedMathematics.Fundamentals of Photonic Crystal GuidingMAKSIM SKOROBOGATIY 1JIANKE YANG 2´Ecole Polytechnique de Montr ´e al,Canada 1University of Vermont,USA2More informationMore informationcambridge university pressCambridge,New Y ork,Melbourne,Madrid,Cape Town,Singapore,S˜a o Paulo,DelhiCambridge University PressThe Edinburgh Building,Cambridge CB28RU,UKPublished in the United States of America by Cambridge University Press,New Y orkInformation on this title:/9780521513289C Cambridge University Press2009This publication is in copyright.Subject to statutory exceptionand to the provisions of relevant collective licensing agreements,no reproduction of any part may take place withoutthe written permission of Cambridge University Press.First published2009Printed in the United Kingdom at the University Press,CambridgeA catalog record for this publication is available from the British LibraryLibrary of Congress Cataloging in Publication dataSkorobogatiy,Maksim,1974–Fundamentals of photonic crystal guiding/by Maksim Skorobogatiy and Jianke Y ang.p.cm.Includes index.ISBN978-0-521-51328-91.Photonic crystals.I.Y ang,Jianke.II.Title.QD924.S562008621.36–dc222008033576ISBN978-0-521-51328-9hardbackCambridge University Press has no responsibility for the persistence oraccuracy of URLs for external or third-party internet websites referred toin this publication,and does not guarantee that any content on suchwebsites is,or will remain,accurate or appropriate.More informationM.Skorobogatiy dedicates this book to his family.He thanks his parentsAlexander and Tetyana for never-ceasing support,encouragement,andparticipation in all his endeavors.He also thanks his wife Olga,his children,Alexander junior and Anastasia,andhis parents for their unconditional love.J.Yang dedicates this book to his family.More informationContentsPreface page xiAcknowledgements xii1Introduction11.1Fabrication of photonic crystals21.2Application of photonic crystals41.2.1Photonic crystals as low-loss mirrors:photonicbandgap effects41.2.2Photonic crystals for out-of-bandgap operation10References112Hamiltonian formulation of Maxwell’s equations(frequency consideration)142.1Plane-wave solution for uniform dielectrics162.2Methods of quantum mechanics in electromagnetism182.2.1Orthogonality of eigenstates192.2.2Variational principle202.2.3Equivalence between the eigenstates of twocommuting Hamiltonians222.2.4Eigenstates of the operators of continuous anddiscrete translations and rotations232.3Properties of the harmonic modes of Maxwell’s equations302.3.1Orthogonality of electromagnetic modes322.3.2Eigenvalues and the variational principle322.3.3Absence of the fundamental length scale in Maxwell’sequations342.4Symmetries of electromagnetic eigenmodes352.4.1Time-reversal symmetry352.4.2Definition of the operators of translation and rotation352.4.3Continuous translational and rotational symmetries382.4.4Band diagrams432.4.5Discrete translational and rotational symmetries44More informationviii Contents2.4.6Discrete translational symmetry and discreterotational symmetry522.4.7Inversion symmetry,mirror symmetry,and other symmetries532.5Problems553One-dimensional photonic crystals–multilayer stacks593.1Transfer matrix technique593.1.1Multilayer stack,TE polarization593.1.2Multilayer stack,TM polarization613.1.3Boundary conditions623.2Reflection from afinite multilayer(dielectric mirror)633.3Reflection from a semi-infinite multilayer(dielectricphotonic crystal mirror)643.3.1Omnidirectional reflectors I683.4Guiding in afinite multilayer(planar dielectric waveguide)693.5Guiding in the interior of an infinitely periodic multilayer703.5.1Omnidirectional reflectors II803.6Defect states in a perturbed periodic multilayer:planarphotonic crystal waveguides823.7Problems864Bandgap guidance in planar photonic crystal waveguides934.1Design considerations of waveguides with infinitelyperiodic reflectors934.2Fundamental TE mode of a waveguide with infinitelyperiodic reflector964.3Infinitely periodic reflectors,field distribution in TM modes984.3.1Case of the core dielectric constantεc<εhεl/(εh+εl)984.3.2Case of the core dielectric constantεl≥εc>εhεl/(εh+εl)1014.4Perturbation theory for Maxwell’s equations,frequencyformulation1034.4.1Accounting for the absorption losses of the waveguidematerials:calculation of the modal lifetime and decay length1044.5Perturbative calculation of the modal radiation loss in aphotonic bandgap waveguide with afinite reflector1064.5.1Physical approach1064.5.2Mathematical approach1085Hamiltonian formulation of Maxwell’s equations for waveguides(propagation-constant consideration)1105.1Eigenstates of a waveguide in Hamiltonian formulation1105.1.1Orthogonality relation between the modes of a waveguide madeof lossless dielectrics111More informationContents ix5.1.2Expressions for the modal phase velocity1145.1.3Expressions for the modal group velocity1145.1.4Orthogonality relation between the modes of a waveguide madeof lossy dielectrics1155.2Perturbation theory for uniform variations in a waveguide dielectric profile1165.2.1Perturbation theory for the nondegenerate modes:example ofmaterial absorption1185.2.2Perturbation theory for the degenerate modes coupled byperturbation:example of polarization-mode dispersion1205.2.3Perturbations that change the positions of dielectric interfaces1235.3Problems126References127 6Two-dimensional photonic crystals1296.1T wo-dimensional photonic crystals with diminishingly smallindex contrast1296.2Plane-wave expansion method1326.2.1Calculation of the modal group velocity1346.2.2Plane-wave method in2D1346.2.3Calculation of the group velocity in the case of2Dphotonic crystals1356.2.4Perturbative formulation for the photonic crystallattices with small refractive index contrast1386.2.5Photonic crystal lattices with high-refractive-index contrast1426.3Comparison between various projected band diagrams1426.4Dispersion relation at a band edge,density of states andVan Hove singularities1446.5Refraction from photonic crystals1476.6Defects in a2D photonic crystal lattice1486.6.1Line defects1486.6.2Point defects1586.7Problems167References171 7Quasi-2D photonic crystals1727.1Photonic crystalfibers1727.1.1Plane-wave expansion method1727.1.2Band diagram of modes of a photonic crystalfiber1767.2Optically induced photonic lattices1777.2.1Light propagation in low-index-contrast periodicphotonic lattices1787.2.2Defect modes in2D photonic lattices with localized defects1817.2.3Bandgap structure and diffraction relation for the modes of auniform lattice182More informationx Contents7.2.4Bifurcations of the defect modes from Bloch band edges forlocalized weak defects1857.2.5Dependence of the defect modes on the strength oflocalized defects1887.2.6Defect modes in2D photonic lattices with nonlocalized defects1927.3Photonic-crystal slabs1957.3.1Geometry of a photonic-crystal slab1957.3.2Eigenmodes of a photonic-crystal slab1977.3.3Analogy between the modes of a photonic-crystal slab and themodes of a corresponding2D photonic crystal2007.3.4Modes of a photonic-crystal slab waveguide2047.4Problems207References208 8Nonlinear effects and gap–soliton formation in periodic media2108.1Solitons bifurcated from Bloch bands in1D periodic media2118.1.1Bloch bands and bandgaps2118.1.2Envelope equations of Bloch modes2128.1.3Locations of envelope solitons2158.1.4Soliton families bifurcated from band edges2168.2Solitons bifurcated from Bloch bands in2D periodic media2188.2.1T wo-dimensional Bloch bands and bandgaps of linearperiodic systems2198.2.2Envelope equations of2D Bloch modes2208.2.3Families of solitons bifurcated from2D band edges2238.3Soliton families not bifurcated from Bloch bands2268.4Problems227References228Problem solutions230Chapter2230Chapter3236Chapter5244Chapter6246Chapter7257Chapter8260 Index263More informationPrefaceThefield of photonic crystals(aka periodic photonic structures)is experiencing anunprecedented growth due to the dramatic ways in which such structures can control,modify,and harvest theflow of light.The idea of writing this book came to M.Skorobogatiy when he was developingan introductory course on photonic crystals at the Ecole Polytechnique de Montr´e al/University of Montr´e al.Thefield of photonic crystals,being heavily dependent onnumerical simulations,is somewhat challenging to introduce without sacrificing thequalitative understanding of the underlying physics.On the other hand,exactly solvablemodels,where the relation between physics and quantitative results is most transpar-ent,only exist for photonic crystals of trivial geometries.The challenge,therefore,wasto develop a presentational approach that would maximally use intuitive analytical andsemi-analytical models,while applying them to complex and practically relevant pho-tonic crystal structures.We would like to note that the main purpose of this book is not to present the latestadvancements in thefield of photonic crystals,but rather to give a systematic,logical,andpedagogical introduction to this vibrantfield.The text is largely aimed at students andresearchers who want to acquire a rigorous,while intuitive,mathematical introductioninto the subject of guided modes in photonic crystals and photonic crystal waveguides.The text,therefore,favors analysis of analytically or semi-analytically solvable problemsover pure numerical modeling.We believe that this is a more didactical approach whentrying to introduce a novice into a newfield.To further stimulate understanding of thebook content,we suggest many exercise problems of physical relevance that can besolved analytically.In the course of the book we extensively use the analogy between the Hamiltonian for-mulation of Maxwell’s equations and the Hamiltonian formulation of quantum mechan-ics.We present both frequency and propagation-constant based Hamiltonian formula-tions of Maxwell’s equations.The latter is particularly useful for analyzing photoniccrystal-based linear and nonlinear waveguides andfibers.This approach allows us touse a well-developed machinery of quantum mechanical semi-analytical methods,suchas perturbation theory,asymptotic analysis,and group theory,to investigate many ofthe limiting properties of photonic crystals,which are otherwise difficult to investigatebased only on numerical simulations.M.Skorobogatiy has contributed Chapters2,3,4,5,and6of this book,and J.Y anghas contributed Chapter8.Chapters1and7were co-authored by both authors.More informationAcknowledgementsM.Skorobogatiy would like to thank his graduate and postgraduate program mentors,Professor J.D.Joannopoulos and Professor Y.Fink from MIT,for introducing him intothefield of photonic crystals.He is grateful to Professor M.Koshiba and ProfessorK.Saitoh for hosting him at Hokkaido University in2005and for having many excitingdiscussions in the area of photonic crystalfibers.M.Skorobogatiy acknowledges theCanada Research Chair program for making this book possible by reducing his teachingload.J.Y ang thanks the funding support of the US Air Force Office of Scientific Research,which made many results of this book possible.He also thanks the Zhou Pei-Yuan Centerfor Applied Mathematics at Tsinghua University(China)for hospitality during his visit,where portions of this book were written.Both authors are grateful to their graduate andpostgraduate students for their comments and help,while this book was in preparation.Especially,J.Y ang likes to thank Dr.Jiandong Wang,whose help was essential for hisbook writing.。

Hamiltonian (quantum mechanics)From Wikipedia, the free encyclopediaIn quantum mechanics, the Hamiltonian is the operator corresponding to the total energy of the system. It is usually denoted by H, also Ȟ or Ĥ. Its spectrum is the set of possible outcomes when one measures the total energy of a system. Because of its close relation to the time-evolution of a system, it is of fundamental importance in most formulations of quantum theory.The Hamiltonian is named after Sir William Rowan Hamilton (1805 – 1865), an Irish physicist, astronomer, and mathematician, best known for his reformulation of Newtonian mechanics, now called Hamiltonian mechanics.Contents1 Introduction2 The Schrödinger Hamiltonian2.1 One particle2.2 Many particles3 Schrödinger equation4 Dirac formalism5 Expressions for the Hamiltonian5.1 General forms for one particle5.2 Free particle5.3 Constant-potential well5.4 Simple harmonic oscillator5.5 Rigid rotor5.6 Electrostatic or coulomb potential5.7 Electric dipole in an electric field5.8 Magnetic dipole in a magnetic field5.9 Charged particle in an electromagnetic field6 Energy eigenket degeneracy, symmetry, and conservation laws7 Hamilton's equations8 See also9 ReferencesIntroductionThe Hamiltonian is the sum of the kinetic energies of all the particles, plus the potential energy of the particles associated with the system. For different situationsor number of particles, the Hamiltonian is different since it includes the sum ofkinetic energies of the particles, and the potential energy function corresponding tothe situation.The Schrödinger HamiltonianOne particleBy analogy with classical mechanics, the Hamiltonian is commonly expressed as the sum of operators corresponding to the kinetic and potential energies of a system in the formwhereis the potential energy operator andis the kinetic energy operator in which m is the mass of the particle, the dot denotes the dot product of vectors, andis the momentum operator wherein ∇ is the gradient operator. The dot product of ∇ with itself is the Laplacian ∇2. In three dimensions using Cartesian coordinates the Laplace operator isAlthough this is not the technical definition of the Hamiltonian in classical mechanics, it is the form it most commonly takes. Combining these together yields the familiar form used in the Schrödinger equation:which allows one to apply the Hamiltonian to systems described by a wave function Ψ(r, t). This is the approach commonly taken in introductory treatments of quantum mechanics, using the formalism of Schrödinger's wave mechanics.Many particlesThe formalism can be extended to N particles:whereis the potential energy function, now a function of the spatial configuration of the system and time (a particular set of spatial positions at some instant of time defines a configuration) and;is the kinetic energy operator of particle n, and ∇n is the gradient for particle n,∇n2 is the Laplacian for particle using the coordinates:Combining these yields the Schrödinger Hamiltonian for the N-particle case:However, complications can arise in the many-body problem. Since the potential energy depends on the spatial arrangement of the particles, the kinetic energy will also depend on the spatial configuration to conserve energy. The motion due to any one particle will vary due to the motion of all the other particles in the system. For this reason cross terms for kinetic energy may appear in the Hamiltonian; a mix of the gradients for two particles:where M denotes the mass of the collection of particles resulting in this extra kinetic energy. Terms of this form are known as mass polarization terms, and appear in the Hamiltonian of many electron atoms (see below).For N interacting particles, i.e. particles which interact mutually and constitute a many-body situation, the potential energy function V is not simply a sum of the separate potentials (and certainly not a product, as this is dimensionally incorrect). The potential energy function can only be written as above: a function of all the spatial positions of each particle.For non-interacting particles, i.e. particles which do not interact mutually and move independently, the potential of the system is the sum of the separate potential energy for each particle,[1] that isThe general form of the Hamiltonian in this case is:where the sum is taken over all particles and their corresponding potentials; the result is that the Hamiltonian of the system is the sum of the separate Hamiltonians for each particle. This is an idealized situation - in practice the particles are usually always influenced by some potential, and there are many-body interactions. One illustrative example of a two-body interaction where this form would not apply is for electrostatic potentials due to charged particles, because they certainly do interact with each other by the coulomb interaction (electrostatic force), shown below.Schrödinger equationThe Hamiltonian generates the time evolution of quantum states. If is the stateof the system at time t, thenThis equation is the Schrödinger equation. It takes the same form as the Hamilton–Jacobi equation, which is one of the reasons H is also called the Hamiltonian. Given the state at some initial time (t = 0), we can solve it to obtain the state at any subsequent time. In particular, if H is independent of time, thenThe exponential operator on the right hand side of the Schrödinger equation is usually defined by the corresponding power series in H. One might notice that taking polynomials or power series of unbounded operators that are not defined everywhere may not make mathematical sense. Rigorously, to take functions of unbounded operators, a functional calculus is required. In the case of the exponential function, the continuous, or just the holomorphic functional calculus suffices. We note again, however, that for common calculations the physicists' formulation is quite sufficient.By the *-homomorphism property of the functional calculus, the operatoris a unitary operator. It is the time evolution operator, or propagator, of a closed quantum system. If the Hamiltonian is time-independent, {U(t)} form a one parameter unitary group (more than a semigroup); this gives rise to the physical principle of detailed balance.Dirac formalismHowever, in the more general formalism of Dirac, the Hamiltonian is typically implemented as an operator on a Hilbert space in the following way:The eigenkets (eigenvectors) of H, denoted , provide an orthonormal basis for theHilbert space. The spectrum of allowed energy levels of the system is given by the set of eigenvalues, denoted {E a}, solving the equation:Since H is a Hermitian operator, the energy is always a real number.From a mathematically rigorous point of view, care must be taken with the above assumptions. Operators on infinite-dimensional Hilbert spaces need not have eigenvalues (the set of eigenvalues does not necessarily coincide with the spectrum of an operator). However, all routine quantum mechanical calculations can be done using the physical formulation.Expressions for the HamiltonianFollowing are expressions for the Hamiltonian in a number of situations.[2] Typical ways to classify the expressions are the number of particles, number of dimensions, and the nature of the potential energy function - importantly space and time dependence. Masses are denoted by m, and charges by q.General forms for one particleFree particleThe particle is not bound by any potential energy, so the potential is zero and this Hamiltonian is the simplest. For one dimension:and in three dimensions:Constant-potential wellFor a particle in a region of constant potential V = V0 (no dependence on space or time), in one dimension, the Hamiltonian is:in three dimensionsThis applies to the elementary "particle in a box" problem, and step potentials. Simple harmonic oscillatorFor a simple harmonic oscillator in one dimension, the potential varies with position (but not time), according to:where the angular frequency, effective spring constant k, and mass m of the oscillator satisfy:so the Hamiltonian is:For three dimensions, this becomeswhere the three-dimensional position vector r using cartesian coordinates is (x, y, z), its magnitude isWriting the Hamiltonian out in full shows it is simply the sum of the one-dimensional Hamiltonians in each direction:Rigid rotorFor a rigid rotor – i.e. system of particles which can rotate freely about any axes, not bound in any potential (such as free molecules with negligible vibrational degrees of freedom, say due to double or triple chemical bonds), Hamiltonian is:where I xx, I yy, and I zz are the moment of inertia components (technically the diagonalelements of the moment of inertia tensor), and , and are the total angular momentum operators (components), about the x, y, and z axes respectively.Electrostatic or coulomb potentialThe Coulomb potential energy for two point charges q1 and q2 (i.e. charged particles, since particles have no spatial extent), in three dimensions, is (in SI units - rather than Gaussian units which are frequently used in electromagnetism):However, this is only the potential for one point charge due to another. If there are many charged particles, each charge has a potential energy due to every other point charge (except itself). For N charges, the potential energy of charge q j due to all other charges is (see also Electrostatic potential energy stored in a configuration of discrete point charges):[3]where φ(r i) is the electrostatic potential of charge q j at r i. The total potential of the system is then the sum over j:so the Hamiltonian is:Electric dipole in an electric fieldFor an electric dipole moment d constituting charges of magnitude q, in a uniform, electrostatic field (time-independent) E, positioned in one place, the potential is:the dipole moment itself is the operatorSince the particle is stationary, there is no translational kinetic energy of the dipole, so the Hamiltonian of the dipole is just the potential energy:Magnetic dipole in a magnetic fieldFor a magnetic dipole moment μ in a uniform, magnetostatic field (time-independent) B, positioned in one place, the potential is:Since the particle is stationary, there is no translational kinetic energy of the dipole, so the Hamiltonian of the dipole is just the potential energy:For a Spin-½ particle, the corresponding spin magnetic moment is:[4]where g s is the spin gyromagnetic ratio (aka "spin g-factor"), e is the electron charge, S is the spin operator vector, whose components are the Pauli matrices, henceCharged particle in an electromagnetic fieldFor a charged particle q in an electromagnetic field, described by the scalar potential φ and vector potential A, there are two parts to the Hamiltonian to substitute for.[1] The momentum operator must be replaced by the kinetic momentum operator, which includes a contribution from the A field:where is the canonical momentum operator given as the usual momentum operator:so the corresponding kinetic energy operator is:and the potential energy, which is due to the φ field:Casting all of these into the Hamiltonian gives:Energy eigenket degeneracy, symmetry, and conservation lawsIn many systems, two or more energy eigenstates have the same energy. A simple example of this is a free particle, whose energy eigenstates have wavefunctions that are propagating plane waves. The energy of each of these plane waves is inversely proportional to the square of its wavelength. A wave propagating in the x direction is a different state from one propagating in the y direction, but if they have the same wavelength, then their energies will be the same. When this happens, the states are said to be degenerate.It turns out that degeneracy occurs whenever a nontrivial unitary operator U commutes with the Hamiltonian. To see this, suppose that is an energy eigenket. Then is an energy eigenket with the same eigenvalue, sinceSince U is nontrivial, at least one pair of and must represent distinct states. Therefore, H has at least one pair of degenerate energy eigenkets. In the case of the free particle, the unitary operator which produces the symmetry is the rotation operator, which rotates the wavefunctions by some angle while otherwise preserving their shape.The existence of a symmetry operator implies the existence of a conserved observable. Let G be the Hermitian generator of U:It is straightforward to show that if U commutes with H, then so does G:Therefore,In obtaining this result, we have used the Schrödinger equation, as well as its dual,Thus, the expected value of the observable G is conserved for any state of the system. In the case of the free particle, the conserved quantity is the angular momentum. Hamilton's equationsHamilton's equations in classical Hamiltonian mechanics have a direct analogy in quantum mechanics. Suppose we have a set of basis states , which need not necessarily be eigenstates of the energy. For simplicity, we assume that they are discrete, and that they are orthonormal, i.e.,Note that these basis states are assumed to be independent of time. We will assume that the Hamiltonian is also independent of time.The instantaneous state of the system at time t, , can be expanded in terms of these basis states:whereThe coefficients a n(t) are complex variables. We can treat them as coordinates which specify the state of the system, like the position and momentum coordinates which specify a classical system. Like classical coordinates, they are generally not constant in time, and their time dependence gives rise to the time dependence of the system as a whole.The expectation value of the Hamiltonian of this state, which is also the mean energy,iswhere the last step was obtained by expanding in terms of the basis states.Each of the a n(t)'s actually corresponds to two independent degrees of freedom, since the variable has a real part and an imaginary part. We now perform the following trick: instead of using the real and imaginary parts as the independent variables, we use a n(t) and its complex conjugate a n*(t). With this choice of independent variables, we can calculate the partial derivativeBy applying Schrödinger's equation and using the orthonormality of the basis states,this further reduces toSimilarly, one can show thatIf we define "conjugate momentum" variables πn bythen the above equations becomewhich is precisely the form of Hamilton's equations, with the s as the generalizedcoordinates, the s as the conjugate momenta, and taking the place of theclassical Hamiltonian.See alsoHamiltonian mechanicsOperator (physics)Bra-ket notationQuantum stateLinear algebraConservation of energyPotential theoryMany-body problemElectrostaticsElectric fieldMagnetic fieldLieb–Thirring inequalityReferences1. ^ a b Quantum Physics of Atoms, Molecules, Solids, Nuclei and Particles (2nd Edition), R.Resnick, R. Eisberg, John Wiley & Sons, 1985, ISBN 978-0-471-87373-02. ^ Quanta: A handbook of concepts, P.W. Atkins, Oxford University Press, 1974, ISBN 0-19-855493-13. ^ Electromagnetism (2nd edition), I.S. Grant, W.R. Phillips, Manchester Physics Series, 2008ISBN 0-471-92712-04. ^ Physics of Atoms and Molecules, B.H. Bransden, C.J.Joachain, Longman, 1983, ISBN 0-582-44401-2Retrieved from "/w/index.php?title=Hamiltonian_(quantum_mechanics)&oldid=641030087"Categories: Hamiltonian mechanics Operator theory Quantum mechanics Quantum chemistry Theoretical chemistry Computational chemistryThis page was last modified on 5 January 2015, at 02:40.Text is available under the Creative Commons Attribution-ShareAlike License; additional terms may apply. By using this site, you agree to the Terms of Use and Privacy Policy. Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc., a non-profit organization.。

单量子态的探测及相互作用英文回答：Detection of Single Quantum States.The detection of single quantum states is a fundamental task in quantum information science. It enables the preparation, manipulation, and measurement of quantum bits (qubits), which are the building blocks of quantum computers. Single-quantum-state detection is also essential for quantum communication, quantum sensing, and quantum cryptography.There are various techniques for detecting single quantum states, depending on the type of qubit being used. For example, in the case of superconducting qubits, the state of the qubit can be detected by measuring its resonant frequency. In the case of trapped ions, the state of the qubit can be detected by measuring its fluorescence.Interaction of Single Quantum States.Single quantum states can interact with each other in a variety of ways. These interactions can be mediated by photons, phonons, or other quantum systems. The interaction between single quantum states can lead to entanglement, which is a fundamental resource for quantum computing and quantum communication.The interaction of single quantum states can also be used to create quantum gates, which are the basic building blocks of quantum circuits. Quantum gates can be used to perform operations on qubits, such as rotations and measurements.中文回答：单量子态的探测。