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专利名称：用于调节半导体器件的载流子迁移率的结构和方法专利类型：发明专利
发明人：布鲁斯·B·多斯,杨海宁,朱慧珑
申请号：CN200410087007.8
申请日：20041022
公开号：CN1612327A
公开日：
20050504
专利内容由知识产权出版社提供
摘要：在制造互补的金属氧化物半导体(CMOS)场效应晶体管(包括nMOS和pMOS晶体管)的过程中，根据层和隔离应力层相互的以及在选定位置具有附加层的其它结构的性能，通过将各种不同的应力膜涂覆到nMOS或pMOS晶体管(或其两者)上来提高或调节载流子迁移率。
这样，可增大单一芯片或基片上的两种晶体管的载流子迁移率，从而改善CMOS器件和集成电路的性能。
申请人：国际商业机器公司
地址：美国纽约
国籍：US
代理机构：中国国际贸易促进委员会专利商标事务所
代理人：付建军
更多信息请下载全文后查看。
传送两原子直积态的腔qed方案(英文)Quantum electrodynamics (QED) is a theory of how electromagnetic interactions occur between particles at the microscopic level. It is one of the most successful theories in modern physics and plays an important role in understanding the behavior of atomic and molecular systems. In particular, QED has been used to explain many phenomena, such as the energy levels of atoms, the Zeeman effect and the Stark effect. Recently, it has been proposed that two-atom QED could be used to generate entangled states, which could have a wide range of applications in quantum information processing.Two-atom QED is a cavity version of the traditional QED system, in which two atoms interact with each other in a common electromagnetic field. In this system, the two atoms are positioned close to each other, and the electromagnetic field is confined to the space within the cavity. This allows the two atoms to interact more strongly than in the open space, leading to new and interesting effects. One of the most intriguing effects of two-atom QED is that it can create entangled states, which can then be used for a variety of applications.For instance, entangled states can be used for quantum computing, where information is encoded in the states of two or more qubits. Entangled states are also useful for quantum cryptography, where the state of two qubits is used to securely transmit sensitive information. Finally, entangled states can be used to measure quantum effects such as spincorrelations between two particles, which could be useful for understanding the behavior of complex quantum systems.In addition to these applications, two-atom QED may also be useful for sensitizing scanning tunneling microscopes or improving atom-based magnetometers. In general, two-atom QED is a promising opportunity for exploring the fascinating world of entanglement and quantum information processing. With further development, two-atom QED could potentially revolutionize our understanding of the fundamentals of quantum physics.。
Home Search Collections Journals About Contact us My IOPscienceCavity quantum electrodynamicsThis content has been downloaded from IOPscience. Please scroll down to see the full text.2006 Rep. Prog. Phys. 69 1325(/0034-4885/69/5/R02)View the table of contents for this issue, or go to the journal homepage for moreDownload details:IP Address: 202.127.153.42This content was downloaded on 23/09/2013 at 04:20Please note that terms and conditions apply.I NSTITUTE OF P HYSICS P UBLISHING R EPORTS ON P ROGRESS IN P HYSICS Rep.Prog.Phys.69(2006)1325–1382doi:10.1088/0034-4885/69/5/R02Cavity quantum electrodynamicsHerbert Walther1,Benjamin T H Varcoe2,Berthold-Georg Englert3and Thomas Becker1 1Max-Planck-Institut f¨u r Quantenoptik,Hans-Kopfermann-Str.1and Department of Physics,University of Munich,85748Garching,Germany2Department of Physics and Astronomy,University of Sussex,Brighton,East Sussex BN19QH,UK3Department of Physics,National University of Singapore,Singapore117542,SingaporeE-mail:herbert.walther@mpq.mpg.deReceived10February2006Published3April2006Online at /RoPP/69/1325AbstractThis paper reviews the work on cavity quantum electrodynamics of free atoms.In recent years, cavity experiments have also been conducted on a variety of solid-state systems resulting in many interesting applications,of which microlasers,photon bandgap structures and quantum dot structures in cavities are outstanding examples.Although these phenomena and systems are very interesting,discussion is limited here to free atoms and mostly single atoms because these systems exhibit clean quantum phenomena and are not disturbed by a variety of other effects.At the centre of our review is the work on the one-atom maser,but we also give a survey of the entirefield,using free atoms in order to show the large variety of problems dealt with.The cavity interaction can be separated into two main regimes:the weak coupling in cavity or cavity-like structures with low quality factors Q and the strong coupling when high-Q cavities are involved.The weak coupling leads to modification of spontaneous transitions and level shifts,whereas the strong coupling enables one to observe a periodic exchange of photons between atoms and the radiationfield.In this case,atoms and photons are entangled,this being the basis for a variety of phenomena observed,some of them leading to interesting applications in quantum information processing.The cavity experiments with free atoms reached a new domain with the advent of experiments in the visible spectral region.A review on recent achievements in this area is also given.(Somefigures in this article are in colour only in the electronic version)0034-4885/06/051325+58$90.00©2006IOP Publishing Ltd Printed in the UK13251326H Walther et alContentsPage1.Introduction13272.Modification of the spontaneous emission rate(weak coupling—real transitions)13283.Modification of atomic energies in confined space(weak coupling–virtual transitions)13294.Oscillatory exchange of photons between an atom and a cavityfield(strong coupling)—the one-atom maser or micromaser13344.1.Experimental set-up of the one-atom maser13374.2.One-atom maser as a source of non-classical light13384.3.Review of experiments on basic properties of the one-atom maser13404.4.Statistics of detector clicks13434.5.Trapping states13444.6.Trapping state stabilization13474.7.Fock states on demand13504.8.Dynamical preparation of n-photon states in a cavity13524.9.The one-atom maser spectrum13555.Other microwave cavity experiments13575.1.Collapse and revival of the Rabi oscillations in an injected coherentfield13585.2.Atom–photon and atom–atom entanglement13585.3.Atom–photon phase gate13595.4.Quantum nondestructive-measurement of a photon13595.5.Wigner function of a one-photon state13595.6.Multiparticle entanglement13595.7.Schr¨o dinger cats and decoherence13606.Slow atoms13606.1.Jaynes–Cummings dynamics for slow atoms13606.2.The mazer13657.Cavity QED experiments in the visible spectral region13677.1.The one-atom laser13687.2.Atoms pushed by a few photons13697.3.Single-photon sources13717.4.Single-atom laser using an ion trap13728.Conclusions and outlook1375Acknowledgments1377 References1377Cavity quantum electrodynamics1327 1.IntroductionThe quantization of the electromagneticfield in quantum electrodynamics is in general achieved by starting with afinite volume defined by conducting walls where thefield appears in a discrete, though infinite,set of modes(see,for example,[1]).Introducing canonical variables casts the Hamiltonian into a form similar to that for the harmonic oscillator which can easily be quantized by imposing canonical commutation relations.Finally,the customary continuous spectrum results when the volume is increased to infinity.For a long time this continuous and unlimited spectrum of the electromagneticfield has been exclusively considered.To obtain solutions for actual physical problems,the formalism of quantumfield theory together with all its potentials to overcome divergences had to be developed.For some time the modification of atomic properties in the presence of conducting walls has been gaining considerable interest.Introducing conductors into a physical system imposes boundary conditions on the electromagneticfield and leads back to a discrete spectrum in the case of afinite volume enclosed in a cavity.In principle,there are two distinct cases to be discussed.Thefirst is the situation of an atom in close proximity to a conducting plate[2–6]. An oscillating atomic dipole induces image charges on the surface leading to a van der Waals type force adding up to the inner atomic forces,thus leading to position-dependent level shifts. Second,there are,in addition to these direct effects,the retarded phenomena from a discrete mode structure of the electromagneticfield inside a cavity due to its geometry.Of course,it is not possible to consider one of these phenomena without the other,but in most cases only one of the two produces the major influence.In what follows,we restrict our attention to the effects of the discrete mode structure of a cavity;this means we limit our discussion to the retarded case.The related phenomena are usually summarized under thefield of cavity quantum electrodynamics.The effects observed are(a)modification of the spontaneous emission rate of a single atom in a resonant cavity,(b)modification of the energy levels of atoms(modification of the Lamb shift)and(c)oscillatory energy exchange between a single atom and the cavity mode,showing purequantum behaviour such as,for example,the disappearance and revival of Rabi nutation induced in a single atom by a resonantfield.The phenomena under(a)and(b)can be observed in weak coupling between radiation and atom,whereas case(c)requires strong coupling in the ideal situation.The strong coupling is characterized by the fact that a real and periodic photon exchange occurs between the cavity field and the atom during the interaction time.In case(a)a real photon is emitted and absorbed by the surrounding cavity before it can interact with the atom again,whereas in case(b)a virtual interaction affects the atomic system.The situation for the experimental observation of those phenomena has drastically changed since frequency-tuneable lasers have become available,making it possible to create large populations of highly excited atomic states characterized by a high principal quantum number n of the valence electron.These states are generally called Rydberg states since their energy levels can be described by the simple Rydberg-type formula.Such excited atoms are very well suited to observing quantum effects in radiation–atom coupling for three reasons.Firstly, the states are very strongly coupled to the radiationfield(the induced transition rates between neighbouring levels scale as n4);secondly,transitions are in the millimetre wave region so that low-order mode cavities can be made large enough to allow rather long interaction times;finally,Rydberg states have relatively long lifetimes with respect to spontaneous decay[7,8]. In recent years,it has also become possible to use optical cavities with extremely small volume1328H Walther et al for these experiments so that resonance transitions of atoms also can now be investigated.Suchexperiments will also be discussed in this review.The strong coupling of Rydberg states to the radiation resonant with transitions toneighbouring levels can be understood in terms of the correspondence principle:withincreasing n the classical evolution frequency of the highly excited electron becomes identicalwith the transition frequency to the neighbouring level.The atom can therefore be seen as alarge dipole oscillating with resonance frequency;the dipole moment is very large since theatomic radius scales as n2.Historically,cavity QED started when Purcell published an abstract of a paper presentedat the1946Spring Meeting of the American Physical Society[9].It dealt with transitionsin nuclear magnetic resonance where the spins were coupled to a resonance circuit,and heshowed that the spontaneous emission probability is increased by a factor3Qλ3/4π2V,whereQ is a quality factor and V the volume of the resonator.For V≈λ3the increase in the emissionprobability corresponds roughly to the quality factor of the resonator.The influence of thevacuumfield on atoms is well summarized and described in a book on quantum electrodynamicsby Power[10].Thefirst experimental work on inhibited spontaneous emission was done byDrexhage,Kuhn and Sch¨a fer(the work is reviewed in[4]).In this work,thefluorescence ofa thin dyefilm near a mirror was investigated.A reduction in thefluorescence decay by up to25%results from the standing-wave pattern near the mirror.Much later,similar experimentswere conducted by de Martini et al[11].Early theoretical work on this has already beenmentioned above.This review is mainly focused on cavity quantum electrodynamics of free atoms becausethese systems provide the basis for the study of quantum phenomena in a very neat and largelyunperturbed way.Cavity quantum electrodynamics in other systems such as,for example,semiconductor devices is mentioned but not treated in detail.In the following,we would like to start with a brief review of the experiments on themodification of the spontaneous transition rate in confined space.2.Modification of the spontaneous emission rate(weak coupling—real transitions)In order to understand the modification of the spontaneous emission rate in an external cavity,we have to remember that in quantum electrodynamics this rate is determined by the density ofmodes of the electromagneticfield at the atomic transition frequencyω0,which depends on thesquare of the frequency.If the atom is not in free space but in a resonant cavity,the continuumof modes is changed into a spectrum of discrete modes,as discussed above,whereby one ofthem may be tuned into resonance with the atom.The spontaneous decay rate of the atom incavity will then be enhanced in relation to that in free space by a factor given by the ratio ofthe corresponding mode densities[9](see also the discussion above in connection with theproposal of Purcell):γc/γf=ρc(ω0)/ρf(ω0)=2πQ/V cω30=Qλ30/4π2V c,(2.1) where V c is the volume of the cavity and Q is the quality factor of the cavity,which expresses thesharpness of the mode.For low-order cavities in the microwave region one has V c≈λ30,which means that the spontaneous emission rate is increased by roughly a factor of Q.However,ifthe cavity is detuned,the decay rate will decrease.In this case,the atom cannot emit a photon,since the cavity is not able to accept it,and therefore the energy has to stay with the atom.In order to demonstrate experimentally the modification of the spontaneous decay rate,it is not necessary to consider single-atom densities in both cases.The experiments wherespontaneous emission is inhibited can also be performed with large atom numbers.However,Cavity quantum electrodynamics1329 in the opposite case,when the increase in the spontaneous rate is observed,a large number of excited atoms may disturb the experiment by induced transitions.Thefirst experimental work on inhibited spontaneous emission by Drexhage,Kuhn and Sch¨a fer(for a review see Drexhage[4])has already been mentioned above.Inhibited spontaneous emission was also observed by Gabrielse and Dehmelt[12].In these neat experiments,with a single electron stored in a Penning trap,they observed that cyclotron orbits show lifetimes which are up to10 times as large as that calculated for free space.The electrodes of the trap form a cavity which decouples the cyclotron motion from the vacuum radiationfield leading to a longer lifetime. Experiments with Rydberg atoms on inhibition of spontaneous emission have been conducted by Hulet et al[13]and Jhe et al[14].In the latter experiments a3.4µm transition was suppressed.The experiment by Hulet et al follows a much earlier proposal by Kleppner[15].Thefirst observation of enhanced atomic spontaneous emission in a resonant cavity was published by Goy et al[16].This experiment was conducted with Rydberg atoms of sodium excited in the23s state in a niobium superconducting cavity resonant at340GHz.Cavity tuning-dependent shortening of the lifetime was observed.The cooling of the cavity had the advantage of totally suppressing the black-bodyfield.The latter effect is completely absent if optical transitions are observed.Thefirst experiments on optical transitions were conducted by Feld and collaborators[17,18].They succeeded in demonstrating the enhancement of spontaneous transitions even in higher-order optical cavities.In modern semiconductor devices both electronic and optical properties can be tailored with a high degree of precision.Electron-hole systems producing recombination radiation analogous to radiating atoms can therefore be localized in cavity-like structures,e.g.in quantum wells.Optical microcavities of half or full wavelength size are thus obtained.Both suppression and enhancement of spontaneous emission in semiconductor microcavities were demonstrated in experiments by Yamamoto and collaborators[19].Similar structures were used by Yablonovitch et al[20–22],exhibiting a photonic band gap with spontaneous emission forbidden in certain frequency regions.3.Modification of atomic energies in confined space(weak coupling–virtual transitions) In the previous section,we focused on changes in radiation rates of atoms near conducting walls or in cavity-like structures.We now discuss the more subtle phenomenon of energy shifts.While radiation rates are modified by the component of thefield in phase quadrature with the atomic dipole,energy shifts are caused by the dispersive part of the interaction or,in other words,by thefield part in phase.Resonant and nonresonant phenomena have to be distinguished.The resonant self-energy shift of a decaying atomic dipole in the vicinity of a conducting wall can be determined from the average polarization energy produced by the image dipolefield.For distances z comparable to the wavelength the near-field condition is satisfied,resulting in the z−3dependence of the static dipole–dipole interaction characteristic for the van der Waals energy;under farfield conditions the distance dependence is given by z−4.The polarization of the atom by the nonresonant parts of the broadband electromagneticfield causes energy shifts,the Lamb shift being the most prominent one.In the spirit of the nonrelativistic treatment by Bethe[23]we can describe the major contribution of that shift as being a result of the emission and reabsorption of virtual photons.It is plausible that just as the real emission of a photon is modified in confined space, so also is the virtual process.The latter‘real’radiation energy shift is thus a consequence of vacuumfluctuations only.It is identical with the energy shift predicted by Casimir and Polder[24]and is analogous to the better-known result of Casimir[25]on the force between two plane neutral conducting plates.1330H Walther et al The question of modification of atomic energies in confined space has recently found considerable interest and many calculations of the phenomenon have been made(for reviews see[26,27]).Direct application to the energy shift of Rydberg atoms,which are of special interest for experimental studies,such as the one discussed below,was performed by Barton [28].In that paper,the direct electrostatic interaction with a conducting wall and the radiation-induced(retarded)effects was estimated.The result is that in the case of two parallel plates the electrostatic effect is dominant when the distance L between the conducting plates is small, L<n3a0/α(n is the principal quantum number,a0the Bohr radius andαthefine-structure constant),and the radiative effect plays a major role when large distances are used,L>n3a0/α.Experiments to measure the transition from the van der Waals type energy shift to the Casimir–Polder shift have been performed by the Hinds group[29,33].The results demonstrate reasonably well the change from a1/z3to a1/z4dependence of the energy shift.Since the pioneering work of Casimir and Polder[24,25]many authors have tried to improve the understanding of vacuum effects;particularly,shifts in atomic energy levels in a parallel-plate geometry have been calculated[28,30–32],some of which have already been mentioned above.The amount of theoretical work available is in contrast to the total absence of efforts on the experimental side,even though the planar cavity is considered to be the prototype geometry.Its special feature is the extreme selectivity of the wave vectors k and the creation of afield cutoff frequencyνcutoff for a particular polarization,with the result that only a discrete spectrum of the radiationfield with frequencies greater thanνcutoff is present inside the cavity.The modification of thefield between two parallel plates allows one to isolate the contribution to the atomic frequency shift resulting from a single k.This modification is resonantly enhanced when the plate distance reaches either d=λ0/2(cutoff)or small odd multiples ofλ0/2,whereλ0is the transition wavelength to a neighbouring atomic level.The restriction of the measurement to a particular contribution presents a special challenge owing to the tiny level shift expected.There have been experimental attempts to determine radiative shifts for other cavity geometries[18,33,34].In[18],a low-Q concentric resonator was used to observe a resonant frequency shift of1MHz in thefluorescence spectra of an optical transition of Ba atoms inside the resonator.The experiment of Hinds et al[33],where ground-state Na atoms were transmitted through a wedge,has already been mentioned above.From the atomic opacity of the wedge the authors derived information on the potential and used that to determine the energy shift.In[34]the authors used an open resonator with curved mirrors.The shift obtained from the dispersive interaction between superpositions of circular states in Rb was measured for different cavity detunings.Although the experiment was carried out at low temperature, the outcome included a non-negligible thermal influence which had to be corrected.In the following,we describe an experiment performed in our laboratory[35]using two plane copper plates to modify the vacuumfield.Rydberg atoms are suitable probes since the transitions to neighbouring levels are in the microwave region,resulting in a resonant plate distance in the mm range.With the analysis limited to atomic states between20S and30S, these atomic level changes are expected to be a few hundred hertz when thermal radiation at room temperature is additionally present within the cavity and is resonantly coupled to the relevant Rydberg transition.In a cavity without thermal radiation the effect is only of the order of100Hz[28,31,32].Measurement of these cavity-induced resonances were the main aim of the work described in the following.The spectroscopic technique involves Doppler-free two-photon absorption in combination with the method of successive oscillatoryfields[36].This high-precision spectroscopy needed to measure the expected small shifts requires a highly frequency-stabilized laser system.It is based on the phase modulation method and has a residual frequency noise on the hertz level[37].Cavity quantum electrodynamics1331Figure1.Scheme of the experimental set-up.The plate structure is placed between the two laserbeams crossing the atomic beam in the vertical direction.The laser beams form a standing-wavefield.Thefield ionization region is located downstream from the second laser beam.The insetson the right side show the Doppler-free two-photon absorption line with the Ramsey fringe(top)and the displacement of the Ramsey fringe on changing the plate distance(bottom).Note that theRamsey fringe is not in the centre of the two-photon absorption line because of its different acStark shift(measured to be8Hz for a reference intensity of1W cm−2)compared with that of theRamsey peak(220mHz for the same reference intensity).This difference results from the fact thatin the Ramsey two-field method the atom passes through the laser-free region most of the time.Thisfigure is taken from[35].The experimental scheme is shown infigure1[35,38].A beam of rubidium atoms is produced via an atomic oven collimated via liquid-nitrogen-cooled circular apertures located approximately2.6cm from thefirst laser interaction region formed by a standing wave.85Rb atoms are excited into a superposition of the5S1/2ground state and the n S1/2excited state with n 22.Afterwards the beam passes through the centre of the region containing the plate structure(grey rectangles in the picture)and then enters the second interaction region.Both excitations are in the low-saturation regime.After the second standing-wave laserfield the atoms are detected byfield ionization,with the ejected electron being multiplied by means of a channeltron.A spectrum is recorded by counting the channeltron pulses via a computer which also controls the laser frequency.Thefield ionization region is within a Faraday cage.This ensures that no electricfields from there can leak into the region with laser interaction and plate structure.Additionally,the whole area is shielded from external electric and magnetic fields by a copper tube surrounded by two layers of mu metal.The copper tube is in contact with a cryostat that can be cooled to either77or4K.Inchworms(piezo-driven devices) can move the plates perpendicularly to the direction of the atomic beam in a precise and reproducible manner which allows exact symmetric positioning of the plates relative to the beam.The interplate distances,ranging from0.5to3mm,can be monitored by means of an interferometric measuring device with a maximum uncertainty of±1µm.1332H Walther et al Besides the careful shielding of externalfields,it is also necessary to avoid any potential differences between the plates.To eliminate any shift from contact potentials,the two plates and their connection via thin and deformable lamellas were machined from a single piece of copper.To avoid adsorbates,the plates could be baked during the preparation stage of the experiment.A folded optical resonator with active stabilization provided a stable phase between the two Ramsey zones.This was achieved by a piezo-mounted mirror and the phase modulation method(similarly to the method used to stabilize the dye laser).Additionally,the resonator enhanced the laser power in the two interaction zones.Scanning the laser frequency and detecting the excited-state atoms reveal a two-photon peak with Ramsey fringes,as displayed in the upper inset offigure1.The width of the central peak of the Ramsey structure is limited by the time offlight between the two zones.The mean time offlight was28µs,which is slightly greater than the lifetime of the excited Rydberg states.Only the zero-order fringe is seen because a thermal atomic beam source was used,and higher-order fringes were therefore washed out owing to the longitudinal velocity distribution. The position of this Ramsey fringe was monitored as a function of the relative distance between the plates.The experimental protocol was as follows.The plates were initially set at a position which served as a reference distance d ref and the laser frequency was scanned across the central Ramsey fringe.Then the plates were moved to a new distance d and the laser was scanned again. This process was reiterated for other distances d relative to the same reference distance d ref. The central fringe position was determined by a least-squaresfitting routine which detected the peak centre with an accuracy between approximately20and100Hz,depending on the Ramsey fringe contrast,which was limited by the background pressure at different temperatures.Careful control of the drift of the reference frequency(which is provided by the supercavity in the locking scheme of the dye laser)in the course of the measurement had to be exercised. Because the measurement of the level shift is given relative to a reference frequency,its drift has to be considered.Although there were two temperature isolation stages,the residual thermal length change of the supercavity resulted in a drift ranging between zero and100Hz s−1.A measure of this drift as a function of time could be derived from the data obtained from the Ramsey spectra at the plate distance d ref.Retardation effects in the interaction of the atomic dipole with the cavity walls are dominant only when the distance between the plates is larger than the wavelength of the relevant Rydberg transition,i.e.d>n3a0/α,where n is the principal quantum number,a0the Bohr radius and αthefine-structure constant.In this regime the round-trip time for a photon leaving the atom, reaching the cavity walls and returning to the atom is larger than the oscillation period of the relevant transition,and the shift is caused by the interaction of the atom with the cavityfield. In the opposite case,d<n3a0/α,the atom would interact with its mirror image in the cavity walls,giving rise to a van der Waals interaction.Therefore,the condition d>n3a0/αhas to be satisfied.It is important to investigate the temperature influence on the energy shift in order to discriminate the thermal and vacuum contributions to the shift itself.This aim was pursued by scanning the plates at three different temperatures T=292,77and4K over interplate distances close to d res=3λ0/2=1.173mm,whereλ0is the wavelength associated with the transition24S1/2→23P3/2.The average number of thermal photons,n th per mode,isslightly above15at room temperature and thus a large thermal content is expected,whereas at T=4K,n th is only0.01.Of special interest are the level shifts at low temperature.Two different experiments using the transitions22S1/2→21P3/2and26S1/2→25P3/2were prepared at4K.The frequencyCavity quantum electrodynamics1333Figure parison of the level shift with theory for two different transitions resonantly coupledinto the cavity at T =4K:(a )26S 1/2→25P 3/2and (b )22S 1/2→21P 3/2.shift as a function of the plate distance for both cases is shown in figure 2.The resonance corresponding to 3/2of the wavelength of the transition 26S 1/2→25P 3/2has a peak value of about 100Hz.In figure 2(b )the measured resonance for the level 22S 1/2shows a shift of 120Hz.The theoretical treatment of quantum electrodynamic level shifts between parallel conducting plates in resonance with atomic transitions is very involved.According to[28,31,32,39]the relevant dependences of the shift on the distance and principal quantum number n are given (in atomic units)by the approximate formula≈4α23d n <n E 2n n R 2n n ln 2 cos 2 E n n d 2¯h c+ 2 1/2 ,(3.1)where E n n is the energy difference between the states n and n ,and R n n is the radial matrix element between the same states. =sinh (β/2)expresses the damping due to the attenuation of the field upon one reflection so that β=−ln (ρ/2),with ρbeing the reflectivity of the mirrors.In equation (3.1)the main contribution to the shift comes from the transitions to thefirst neighbouring levels and,because E 2n n ≈1/n 6and R 2n n ≈n 4,the changes in with。
Information Briefing ■业界动态展，开发出了增强版的贴片，可以测量 婴幼儿其他方面的信息，如心脏颤动、活动和身体朝向、哭声大小和哭泣频率，当然还有生命体征。
研究人员表示，能够监测额外的 这些信息有助于及早发现与产伤、脑损 伤、疼痛或压力相关的并发症。
在一项 包括50名重症婴幼儿的试验中，研宄 团队发现，这种生物传感器贴片与标准 的临床监测系统相比，表现出高级别的 可靠性和准确性。
荷兰科学家发现用可编程光子新材料2020年3月31日，荷兰埃因霍芬理工大学（Eindhoven University of Technology)的研究人员宣布发现了 •种可转换的光学材料一一氢化非晶硅，能够加快光子集成电路的研发和生产。
研究项目负责人Oded R az表示，这是 第一个可编程的光子电路，研发人员可 以对光子材料本身进行编程和重新设 置，并且它不需要任何电力还保持自# 的编程状态，这将在一定程度上帮助工 程师加速幵发光子器件。
光子集成电路（PIC)是一种基于 晶态半导体晶圆，集成有源和无源光子 电路，以及单个微芯片上的电子元件，具有低能耗和高运行速度等性能优势。
但不足的是，目前P IC的制造方法存 在大量的可变性，因此许多生产出来的 光子器件与实际所需的规格有着轻微偏 差，从而也限制了产量。
而解决该问题 的一个潜在方法是开发可重新配置或可 编程的P1C，以帮助补偿制造过程中产 生的任何细微变化。
埃因霍芬理工大学的研究人员表示，他们发现的可转换光学材料能够避 免制造过程中可变性，以及切换性能损耗等不足。
这种材料名叫氢化非晶硅，目前主要用于薄膜硅太阳能电池。
Oded Raz表示，如果未来的研究能够增强转换效应的强度，那么P1C的产量也将得到显著提高。
他认为，以前的基本光子元件良率可能在10%〜20%之间，而可编程光学材料可将良率提高到 50%〜80%。
与此同时，产量的提高可反过来减少光子器件的生产制作时间。
尼曼半导体物理与器件
尼曼半导体物理与器件是固态电子学领域中的重要分支之一，主要研究半导体材料的物理性质和器件的设计及工艺制备。
在现代电子技术的发展中，尼曼半导体物理与器件发挥着重要的作用。
尼曼半导体物理与器件的研究对象主要包括半导体材料的物理特性和器件的性能。
半导体材料具有半导体和导体的特性，且在一定范围内具有可控制的电子特性，其导电性能可以通过控制材料的禁带宽度和掺杂浓度等参数来实现。
而尼曼半导体器件则是指基于半导体材料的电子器件，包括晶体管、二极管、太阳能电池等。
尼曼半导体物理与器件研究的一大应用领域是半导体器件的制造。
通过对半导体材料的物理特性进行研究和开发，可以设计出性能更加优异的半导体器件。
例如，随着人们对太阳能利用技术的不断探索和开发，尼曼半导体物理与器件的研究和应用也得到了大力推广。
太阳能电池能够将太阳能转化为电能，而半导体材料的导电性能可以让太阳能电池更加高效地转化太阳能为电能。
总的来说，尼曼半导体物理与器件在现代电子技术的发展中发挥着越来越重要的作用，有助于推动现代电子技术的发展和改进。
随着人们对电子技术需求的不断提升，尼曼半导体物理与器件的研究和应用将会成为未来电子技术领域的重要研究方向。
《类EIT太赫兹超材料与可调控极化转换器的设计与研究》篇一范文字幕：类EIT太赫兹超材料与可调控极化转换器设计与研究一、引言近年来，随着科技的不断进步，太赫兹波（THz wave）的应用日益广泛，尤其是在材料科学和光子学领域。
太赫兹超材料因其独特的电磁性质，为新一代的电子器件提供了可能。
本文旨在研究一种基于EIT（电磁诱导透明）效应的太赫兹超材料，并设计一种可调控的极化转换器。
二、EIT太赫兹超材料理论基础EIT现象是物理学中的一个重要概念，指的是在特定条件下，介质对光波的吸收和色散性质发生改变的现象。
在太赫兹波段，EIT超材料可以有效地控制电磁波的传播，实现特定的电磁功能。
本文将利用EIT效应设计一种新型的太赫兹超材料。
三、类EIT太赫兹超材料的设计设计过程中，我们采用了一种独特的亚波长结构，通过精确调控其几何参数和材料属性，实现了对太赫兹波的EIT效应。
设计过程中主要考虑了以下几个因素：1. 亚波长结构的形状和尺寸：通过调整结构的形状和尺寸，可以改变太赫兹波的传播速度和相位。
2. 材料的选择：选择具有特定电磁性质的材料，如高介电常数和高磁导率的材料，以增强EIT效应。
3. 几何参数的优化：通过优化结构的几何参数，如间距、厚度等，以实现最佳的EIT效果。
四、可调控极化转换器的设计为了实现极化转换器的可调控性，我们采用了液晶材料和电场控制技术。
液晶材料具有良好的电光响应特性，可以实时改变其光学性质。
电场控制技术则可以控制液晶分子的取向，从而改变极化转换器的性能。
具体设计如下：1. 液晶材料的选用：选择具有高双折射特性的液晶材料，以实现极化转换功能。
2. 电场控制技术：通过施加外部电场，改变液晶分子的取向，从而实现对极化转换器性能的调控。
3. 结构优化：结合太赫兹超材料的特性，优化极化转换器的结构，以提高其性能和稳定性。
五、实验研究及结果分析我们通过实验验证了设计的有效性。
首先，我们制备了类EIT太赫兹超材料样品，并对其进行了性能测试。
a r X i v :h e p -p h /9910526v 1 27 O c t 1999OKHEP–99–02Dual Quantum Electrodynamics:Dyon-Dyon and Charge-Monopole Scattering in a High-Energy ApproximationLeonard Gamberg ∗and Kimball ton †Department of Physics and Astronomy,University of Oklahoma,Norman,OK 73019USA(February 7,2008)We develop the quantum field theory of electron-point magnetic monopole interactions and moregenerally,dyon-dyon interactions,based on the original string-dependent “nonlocal”action of Diracand Schwinger.We demonstrate that a viable nonperturbative quantum field theoretic formulationcan be constructed that results in a string independent cross section for monopole-electron anddyon-dyon scattering.Such calculations can be done only by using nonperturbative approximationssuch as the eikonal and not by some mutilation of lowest-order perturbation theory.PACS number(s):14.80.Hv,11.80.Fv,12.20.Ds,11.30.FsI.INTRODUCTION The subject of magnetic charge in quantum mechanics has been of great interest since the work of Dirac [1],who showed that electric and magnetic charge could co-exist provided the quantization condition (in rationalized units)eg 2,N ∈Z ,(1.1)holds,where e and g are the electric and magnetic charges,respectively,Z being the integers.In addition,the existence of topological or “extended”magnetic charge has been demonstrated in non-Abelian gauge theories [2],where they can have profound effects,most notably in the infrared regime of the QCD vacuum.For example,in the illustrative scenario of color confinement proposed by Mandelstam and ’t Hooft [3]it is conjectured that the QCD vacuum behaves as a dual type II superconductor.Due to the condensation of magnetic monopoles which emerge from the Abelian projection [4]of the SU(3)color gauge group,the chromo-electric field acting between q ¯q pairs is squeezed into dual Abrikosov flux tubes [5].Very recent studies suggest that these flux tubes may determine the infrared properties of the gluon propagator [6].Also it has been suggested [7]that color monopoles are sufficient to trigger the spontaneous breakdown of chiral symmetry.The monopoles most commonly considered today as being potentially observable are those associated with the grand-unification symmetry breaking scale,monopoles having masses of order 1016GeV;however,much lower symmetry-breaking scales could be relevant,giving rise to monopoles of mass of order 10TeV [8].[It is worthwhile mentioning that the previously-believed prohibition (cf.Ref.[9])against topological monopoles in electroweak theory is apparently too restrictive [10],thus raising the possibility of the existence of monopoles at the electroweak-symmetry breakingscale.]Further,there is no reason why “elementary”Dirac monopoles should not exist;only experiment can settle this question.Since the early 1970s there have been many experimental searches for magnetic monopoles,ranging from seeking cosmological bounds to studies of lunar samples [11].Although none of these searches ultimately has yielded a positive signal,the arguments in favor of the existence of magnetic charge,whether elementary or “extended,”remain as cogent as ever.However,the quantum field theory of elementary or pointlike magnetic charges remains poorly developed,particularly at the phenomenological level.In view of the necessity of establishing a reliable estimate for monopole production in accelerators,for example,in order to set bounds on monopole masses in terrestrial experiments [12],it is important to put the theory on a firmer foundation.(This is especially so for limits set through virtual-monopole processes [13].For a critique of such limits see Ref.[14].)It is the purpose of this paper to initiate a complete study of monopole scattering and production in relativistic quantum electrodynamics extended to include point magnetic charges,“dual QED.”Given ongoing experiments it is important to obtain reliable calculations of scattering processes.From the work of Dirac on the nonrelativistic and the relativistic quantum mechanics of magnetic monopoles[1,15], and the subsequent work of Schwinger[16–18]and later Zwanziger[19]on relativistic quantumfield theories of mag-netic monopoles,it is known that it is not possible to write down a theory of pointlike electric and magnetic currents interacting via the electromagneticfield described solely in terms of a local vector potential Aµ(x).Consistency of the Maxwell equations∂νFµν=jµand∂ν∗Fµν=∗jµ,(1.2) where∗Fµν=1= N4π1Upon separation of variables,the angular differential operator contains a contribution from the“intrinsic”field angular momentum carried by the interaction of the monopole with the photonfield,which is naturally quantized upon using the Dirac-Schwinger condition(1.4);that is,there is an additional effective magnetic quantum number m′=e a g b−e a g bHere the incoming momenta are p,k,and the outgoing momenta are p′,k′,respectively,while the initial andfinal electron spin projections are s and s′,and where the different symbols for the energies of the electron and monopole, E andω,respectively,refer to the different masses of the corresponding particles.Further,note that in this paper we use a metric with signature(−,+,+,+),so that q2>0represents a spacelike momentum transfer.The explicit dependence on the covariant string vector,nµ,does not disappear upon squaring the amplitude and multiplying by the appropriate phase space factors.To make matters worse,the value of the magnetic charge implied by(1.1),αg=g2/4π≈34N2,calls into question any approach based on a badly divergent perturbative expansion inαg. Although these earliest efforts using a Feynman-rule perturbation theory in one of a number of approaches to dual QED(see below)resulted in string-dependent cross–sections[25,26],subsequently ad hoc assumptions were invoked to render the resulting cross–sections string independent(e.g.Refs.[28]and[29]).Recently,in the context of calculating virtual monopole processes,De R´u jula[30]has advocated the notion that single photon-mediated monopole-antimonopole Drell-Yan production amplitudes are rendered string independent through the implausible argument of Deans[28],which amounts to dropping the pole terms in the photon propagator!2 Further,in a series of papers,Ignatiev and Joshi[31],while arguing that the prescription of Deans lacks believability, propose(as did Rabl[26])averaging the scattering amplitude over all possible directions of the string in order to eliminate the string dependence.In contrast to Deans,however,they arrive at a null result for the amplitude in lowest order.Consequently,they argue that general principles of quantumfield theory which do not rely upon the use of perturbation theory must be invoked;that is,they consider the constraints placed upon the scattering amplitude due to the requirements of discrete C and P symmetries(see also Tolkachev et al.[32]).In contrast,by studying the formal behavior of Green’s functions in the relativistic quantumfield theory of elec-trons and monopoles,both Schwinger[18,27]and Brandt,Neri and Zwanziger[33]demonstrated Lorentz and gauge invariance.In essence these demonstrations are nonperturbative and rely on the use of pointlike particle trajectories for the matterfields.In the Schwinger approach,however,classical particle currentsjµ(x)= e e dxµeδ(x−x e)∗jµ(x)= g g dxµgδ(x−x g)(1.9)are substituted for thefield theoretic sources,whereupon a change in the string trajectory gives rise to a change in the action which is a multiple of2πbecause of the quantization conditions(1.1)or(1.4).In the second approach,putting aside the issue of renormalization,Brandt et al.succeed in demonstrating the string independence of correlation functions of gauge-and Lorentz-invariant quantities using a functional path integral formalism,once again as a consequence of Eqs.(1.1)and(1.4),by converting the path integral overfields to one over closed particle trajectories (see Refs.[34–36]).However,while Lorentz invariance and string independence were formally demonstrated in dual QED(albeit with the above caveats),such demonstrations have been conspicuously absent in practical calculations.3This deficiency stems from the fact that in most phenomenological treatments of electron-monopole processes the“string indepen-dence”of the quantumfield theory and the strength of the coupling are treated as separate issues.This viewpoint (see e.g.Refs.[31,29])could not be more misleading.In fact,these two points are intimately related.4If any lesson is to be learned from the above-mentioned demonstrations of Lorentz and string invariance,it is this:The quanti-zation conditions(1.1)and(1.4),being intimately tied to the demonstration of Lorentz invariance,are intrinsically nonperturbative statements(and for that matter are obtained independently of any perturbative approximation).As such,attempts to demonstrate Lorentz invariance,whether formally or phenomenologically,which are based on a perturbative expansion in the coupling will most assuredly fail.While this is not a novel viewpoint(see Refs.[18,33]), it seems to have been overlooked in the more recent studies of point monopole processes(see also Ref.[29]).5In fact,there exists one instance in the literature of a successful calculation demonstrating string independence for a relativistic scattering cross section within dual QED.Utilizing Schwinger’s functional source theory[39,18]inthe context of a high energy,low momentum transfer(zeroth order eikonal)approximation(e.g.Refs.[40–44]and references therein),Urrutia[45]demonstrated string independence of the charge-monopole scattering cross section,although again in his treatment the currents were approximated by those of classical point particles as in Eq.(1.9). The outline of this paper is as follows.In Section II we establish the string-dependent dual-electrodynamic action.InSection III we quantize the dual potential action by solving the Schwinger-Dyson equations for the vacuum persistence amplitude for electron-monopole(and dyon-dyon)processes.Then,by taking the functional Fourier transform of thissolution,we are led unambiguously to the dual QED functional path integral that is compatible with the source theory formalism developed by Schwinger[18,27].In the process of this development,we enforce a gaugefixing condition thatsuggests that using the Dirac string formulation is not only entirely natural and consistent with the underlying gauge symmetry of the charge-monopole system,but in fact is preferable to the multi-valued potential of Ref.[20].Havingaccomplished this,in Section IV we calculate the dyon-dyon scattering cross section in the context of this relativistic string dependent version of dual QED and demonstrate that a nonperturbative approach in conjunction with(1.1)or (1.4)is in fact necessary to demonstrate Lorentz and gauge(string)invariance of physical observables.To accomplish this we use the nonperturbative functionalfield-theory formalism of Schwinger[35]and Symanzik[46].6Within thisformulation we are able to generalize the string-independent eikonal result of Urrutia[45]while treating the currents as constructed from quantumfields by invoking the quantization condition(1.4).Finally in Section V we draw someconclusions concerning the phenomenology of dual QED and outline our continuing efforts in this direction while in addition proposing future work.Appendices deal with some aspects of the path-integral formalism in dual QED.II.DUAL ELECTRODYNAMICSDecades after Dirac’s early quantum-mechanical treatment(both nonrelativistic[1]and relativistic[15])of the electron-monopole system,attempts to establish a consistent interacting quantumfield theory of(point)electronsand monopoles were carried out in two different approaches by Schwinger[16]and by Zwanziger[19]. Schwinger’s approach is based upon the construction of a nonlocal Hamiltonian which is a function of two transverse,string-dependent potentials,A T and B T,in addition to,say,spin-1/2electron and monopolefields.Second quan-tization is established by postulating the commutation rules between thefields and their conjugatefield momenta.In this formulation the two gaugefields have a nonvanishing commutator,and in fact are not linearly independent. Although this operator method is noncovariant,it has the merit of explicitly displaying the dual symmetry betweenthe Dirac equations for the spin-12 electron and monopolefields.The equivalence between these formalisms is shown in Refs.[48,49].However,the former covariant formalism of Schwinger is a natural generalization of Dirac’s(relativistic)quantum mechanics in the form of a covariant Lagrangianfield theoretic description of electron-monopole interactions.In this context it should come as no surprise that Schwinger’s source theory approach[27]laid the groundwork for thefirst consistent,gauge-invariant nonperturbative calculation of electron-monopole scattering[45].We also note that a“one potential”Hamiltonian formulation was developed later by Blagojevi´c[49]where a explicit set of Feynman rules was displayed.Finally we mention in passing the work of Deans[28],where a formalism based on Dirac’s monopole theory was quantized within the framework of Mandelstam’s gauge-invariant quantumfield theory[50].In Deans’paper,while a set of string-dependent Feynman rules was developed,they were used withoutcriticism of the badly divergent perturbation series;in addition,as we have seen,Lorentz-frame (string)dependence is argued away in a highly questionable fashion.A.The “Dual-Potential”String Dependent ActionWhile the completely general formalism for charge-monopole quantum field theory was developed by Schwinger in the context of source theory,for the sake of accessibility here we develop a string-dependent dual QED of monopole-electron interactionsina morefamiliar functionalformalism.7In order to facilitate the construction of the dual-QED formalism we recognize that the well-known continuous global U(1)dual symmetry [16,18,27]implied by Eqs.(1.2),(1.3),given byj ′∗j ′ = cos φsin φ−sin φcos φ j ∗j ,(2.1a) F ′∗F′ = cos φsin φ−sin φcos φ F ∗F ,(2.1b)suggests the introduction of an auxiliary vector potential B µ(x )dual to A µ(x ).In order to satisfy the Maxwell and charge conservation equations,Dirac modified the field strength tensor according toF µν=∂µA ν−∂νA µ+∗G µν,(2.2)where now Eqs.(1.2)and (1.3)give rise to the consistency condition on G µν(x )=−G νµ(x )∂ν∗F µν=−∂νG µν=∗j µ.(2.3)We then obtain the following inhomogeneous solution to the dual Maxwell’s equation (2.3)for the tensor G µν(x )in terms of the string function f µand the magnetic current,which for a spin-1/2monopole represented by a Dirac field χis ∗j µ(x )=g ¯χ(x )γµχ(x ):G µν(x )=(n ·∂)−1(n µ∗j ν(x )−n ν∗j µ(x ))= (dy )(f µ(x −y )∗j ν(y )−f ν(x −y )∗j µ(y )),(2.4)where use is made of Eqs.(1.3),(1.6),and (1.7).A minimal generalization of the QED Lagrangian including electron-monopole interactions readsL =−12 (dy )∗F µν(y )δG µν(y )7However,we emphasize that the source theory approach is in fact just that,the “source”of these ideas.8We regard G µν(x )as dependent on ¯χ,χbut not A µ.Thus,the dual Maxwell equation is given by the subsidiary condi-tion (2.3).It is straightforward to see from the Dirac equation for the monopole(2.7)and the construction(2.4)that introducing the auxiliary dualfield(which is a functional of Fµνand depends on the string function fµ)Bµ(x)=− (dy)fν(x−y)∗Fµν(y),(2.8) results in the following Dirac equation for the monopolefield(iγ∂+gγB(x)−m g)χ(x)=0.(2.9) Here we have chosen the string to satisfy the oddness condition(this is the“symmetric”solution)fµ(x)=−fµ(−x),(2.10) which is related to Schwinger’s integer quantization condition[22,51].Now(2.6)and(2.9)display the dual symmetry expressed in Maxwell’s equations(1.2)and(1.3).Noting that Bµsatisfies(dx′)fµ(x−x′)Bµ(x′)=0,(2.11)we see that Eq.(2.8)is a gauge-fixed vectorfield[52,53]defined in terms of thefield strength through an inversion formula(see Subsection III A).In terms of thesefields the“dual-potential”action can be re-expressed in terms of the vector potential Aµandfield strength tensor Fµν(where Bµis the functional(2.8)of Fµν)infirst-order formalism asW= (dx) −14Fµν(x)Fµν(x)+¯ψ(x)(iγ∂+eγA(x)−mψ)ψ(x)+¯χ(x)(iγ∂+gγB(x)−mχ)χ(x) ,(2.12a) or in terms of dual variables,W= (dx) −14∗Fµν(x)∗Fµν(x)+¯ψ(x)(iγ∂+eγA(x)−mψ)ψ(x)+¯χ(x)(iγ∂+gγB(x)−mχ)χ(x) .(2.12b)In Eq.(2.12a),Aµ(x)and Fµν(x)are the independentfield variables9and Bµ(x)is given by Eq.(2.8),while in Eq.(2.12b)the dualfields are the independent variables,in which case,Aµ(x)=− (dy)fν(x−y)Fµν(y)1=FµνFµν=−149Using Eq.(2.8),variations of the action,Eq.(2.12a),with respect to Aµ(x)and Fµν(x)yield Eqs.(1.2)and(2.2)where ∗G(x)is the dual of Eq.(2.4).µνIII.QUANTIZATION OF DUAL QED:SCHWINGER-DYSON EQUATIONSAlthough the various actions describing the interactions of point electric and magnetic poles can be described in terms of a set of Feynman rules which one conventionally uses in perturbative calculations,the large value ofαg or eg/4πrenders them useless for this purpose.In addition,as mentioned in Section I,calculations of physical processes using the perturbative approach from string-dependent actions such as Eqs.(2.12a)and(2.12b)have led only to string dependent results.In conjunction with a nonperturbative functional approach,however,the Feynman rules serve to elucidate the electron-monopole interactions.We express these interactions in terms of the“dual-potential”formalism as a quantum generalization of the relativistic classical theory of section II A.We use the Schwinger action principle[54]to quantize the electron-monopole system by solving the corresponding Schwinger-Dyson equations for the generating ing a functional Fourier transform of this generating functional in terms of a path integral for the electron-monopole system,we rearrange the generating functional into a form that is well-suited for the purpose of nonperturbative calculations.A.Gauge SymmetryIn order to construct the generating functional for Green’s functions in the electron-monopole system we must restrict the gauge freedom resulting from the local gauge invariance of the action(2.12a).The inversion formulae for Aµand Bµ,Eqs.(2.13)and(2.8)respectively,might suggest using the technique of gauge-fixedfields[52,50]as was adopted in[28].However,we use the technique of gaugefixing according to methods outlined by Zumino[55]and generalized by Zinn-Justin[56]in the language of stochastic quantization.The gaugefields are obtained in terms of the string and the gauge invariantfield strength,by contracting the field strength(2.2),(2.4)with the Dirac string,fµ(x),in conjunction with Eq.(1.6),yielding the following inversion formula for the equation of motion,Aµ(x)=− (dx′)fν(x−x′)Fµν(x′)+∂µ˜Λe(x),(3.1) where we use the suggestive notation,˜Λe(x)˜Λ(x)= (dx′)fν(x−x′)Aν(x′).(3.2)eIt is evident that Eq.(3.1)transforms consistently under gauge transformationAν(x)−→Aν(x)+∂νΛe(x),(3.3) while in addition we note that the Lagrangian(2.12a)is invariant under the gauge transformation,ψ→exp[ieΛe]ψ,Aµ→Aµ+∂µΛe,(3.4a) as is the dual action(2.12b)underχ→exp[igΛg]χ,Bµ→Bµ+∂µΛg.(3.4b) Assuming the freedom to choose˜Λe(x)=−Λe(x),we bring the vector potential into gauge-fixed form,10coinciding with Eq.(2.13),Aµ(x)=− (dy)fν(x−y)Fµν(y)(3.5)where the gauge choice is equivalent to a string-gauge condition11(dx′)fµ(x−x′)Aµ(x′)=0.(3.7) More generally,the fact that a gauge function exists,such thatΛe(x)=−˜Λe(x)[cf.Eq.(3.2)],implying that we have the freedom to consistentlyfix the gauge,is in fact not a trivial claim.If this were not true,it would certainly derail the consistency of incorporating monopoles into QED while utilizing the Dirac string formalism.On the contrary, the string gauge condition,Eq.(3.7),is in fact a class of possible consistent gauge conditions characterized by the symbolic operator function(1.7)depending on a unit vector nµ(which may be either spacelike or timelike).In a similar manner,given the dualfield strength(2.14)the dual vector potential takes the following form[cf.Eq.(2.8)]Bµ(x)=− (dx′)fν(x−x′)∗Fµν(x′)+∂µ˜Λg,(3.8) where˜Λ(x)= (dx′)fµ(x−x′)Bµ(x′).(3.9)gIn order to quantize this system we must divide out the equivalence class offield values defined by a gauge trajectory infield space;in this sense the gauge condition restricts the vector potential to a hypersurface offield space which is embodied in the generalization of Eq.(3.7)(dx′)fµ(x−x′)Aµ(x′)=Λe(x),(3.10)where hereΛe is any function defining a unique gaugefixing hypersurface infield space.12In a path integral formalism,we enforce the condition(3.10)by introducing aδfunction,symbolically written as δ(fµAµ−Λe)= [dλe]exp i (dx)λe(x) (dx′)fµ(x−x′)Aµ(x′)−Λe(x) ,(3.11) or by introducing a Gaussian functional integralΦ(fµAµ−Λe)= [dλe]exp −i11Taking the divergence of Eq.(3.5)and using Eq.(1.2),the gauge-fixed condition(3.5)can be written as∂µAµ= (dy)fµ(x−y)jµ(y),(3.6)which is nothing other than the gauge-fixed condition of Zwanziger in the two-potential formalism[19].12Choosing a different functionΛe(which the gauge freedom permits us to do)merely yields a different section offield space under the restriction that it cut each equivalence class offield values once.∂νF µν(x )−(dx ′)λe (x ′)f µ(x ′−x )=j µ(x ),(3.14a)∂ν∗F µν(x )− (dx ′)λg (x ′)f µ(x ′−x )=∗j µ(x ).(3.14b)where the second equation refers to a similargauge fixing in the dual sector.Taking the divergence of Eqs.(3.14a)implies λe =0from Eqs.(1.6)and (1.3),which consistently yields the gauge condition (3.10).Using our freedom to make a transformation to the gauge-fixed condition (3.5),Λe =0,the equation of motion (3.14a)for the potential becomes−g µν∂2+∂µ∂ν+κn µ(n ·∂)−2n ν A ν(x )=j µ(x )+ǫµνστn ν(n ·∂)+n 2 1−1n 2 ∂µ∂ν−∂2−iǫδ(x ).(3.18)This in turn enables us to rewrite Eq.(3.15)as an integral equation,expressing the vector potential in terms of the electron and monopole currents,A µ(x )= (dx ′)D µν(x −x ′)j ν(x ′)+ǫνλστ (dx ′)(dx ′′)D µν(x −x ′)f λ(x ′−x ′′)∂′′σ∗j τ(x ′′).(3.19)The steps for B µ(x )are analogous.B.Vacuum Persistence Amplitude and the Path IntegralGiven the gauge-fixed but string-dependent action we are prepared to quantize this theory of dual QED.Quanti-zation using a path integral formulation of such a string dependent action is by no means straightforward;therefore we will develop the generating functional making use of a functional ing the quantum action principle (cf.Ref.[54])we write the generating functional for Green functions (or the vacuum persistence amplitude)in the presence of external sources JZ (J )= 0+|0− J ,(3.20)for the electron-monopole system.Schwinger’s action principle states that under an arbitrary variationδ 0+|0− J =i 0+|δW (J )|0− J ,(3.21)where W (J )is the action given in Eq.(2.12a)externally driven by the sources,J ,which for the present case are given by the set {J,∗J,¯η,η,¯ξ,ξ}:W (J )=W +(dx ) J µA µ+∗J µB µ+¯ηψ+¯ψη+¯ξχ+¯χξ .(3.22)The one-point functions are then given byδ0+|0− J ,δ 0+|0− J,δ 0+|0− J ,δ 0+|0− J .(3.23)Using Eqs.(3.23)we canwrite down derivatives with respect to the charges 13in terms of functional derivatives [57–59]with respect to the external sources;∂δ˜A µ(x )δ∂g 0+|0− J =i 0+ (dx )∗j µ(x )B µ(x ) 0− J=−i (dx ) δδ∗J µ(x )0+ 0− J .(3.24)Here we have introduced an effective source to bring down the electron and monopole currents,δi δδ¯η,δi δδ¯ξ.(3.25)These first order differential equations can be integrated with the result0+|0− J =exp −ig (dx ) δδ∗J ν(x ) −ie (dx ) δδJ µ(x ) 0+|0− J 0,(3.26)where 0+|0− J 0is the vacuum amplitude in the absence of interactions.By construction,the vacuum amplitude and Green’s functions for the coupled problem are determined by functional derivatives with respect to the external sources J of the uncoupled vacuum amplitude,where 0+|0− J 0is the product of the separate amplitudes for the quantized electromagnetic and Dirac fields since they constitute completely independent systems in the absence of coupling,that is,0+|0− J 0= 0+|0− (¯η,η,¯ξ,ξ)00+|0− (J,∗J )0.(3.27)First we consider 0+|0− J 0as a function of J and ∗J δ13Here we redefine the electric and magnetic currents j →ej and ∗j →g ∗j .Note that the changes in the action due to induced changes in the fields vanish by virtue of the stationary principle.Using Eq.(3.15)we arrive at the equivalent gauge-fixed functional equation,−g µν∂2+∂µ∂ν+κn µ(n ·∂)−2n νδδJ ν0+|0− J 0=0,(3.31a)or(dx ′)f ν(x −x ′)δiδJ ν(x )0+|0− J=exp −ig (dy )δδ∗J α(y ) −ie (dy )δδJ α(y )× J µ(x )+ǫµνστ(dx ′)f ν(x −x ′)∂′σ∗J τ(x ′) 0+|0− J 0.(3.32)Commuting the external currents to the left of the exponential on the right side of Eq.(3.32)and using Eqs.(3.23),we are led to the Schwinger-Dyson equation for the vacuum amplitude,where we have restored the meaning of thefunctional derivatives with respect to ˜A,˜B given in Eq.(3.25), −g µν∂2+∂µ∂ν+κn µ(n ·∂)−2n νδiδη(x )γµδiδξ(x ′)γτδiδ∗J µ(x )0+|0− J 0= 0+|B µ(x )|0− J0,(3.34)we obtain the functional equation (which is consistent with duality)−g µν∂2+∂µ∂ν+κn µ(n ·∂)−2n νδiδξ(x )γµδiδη(x ′)γτδδ∗J µ(x ′)0+|0− J =0.(3.36)In a straightforward manner we obtain the functional Dirac equationsiγ∂+eγµδiδ¯η(x )0+|0− J =−η(x ) 0+|0− J ,(3.37a)iγ∂+gγµδiδ¯ξ(x )0+|0− J =−ξ(x ) 0+|0− J .(3.37b)In order to obtainageneratingfunctional for Green’s functions we must solve the set of equations (3.33),(3.35),(3.37a),(3.37b)subject to Eqs.(3.31b)and (3.36)for 0+|0− J .In the absence of interactions,we can immediately integrate the Schwinger-Dyson equations;in particular,(3.35)then integrates to0+|0− J 0=N (J )expi2(dx )(dx ′)J µ(x )D µν(x −x ′)J ν(x ′),(3.39)resulting in the generating functional for the photonic sector0+|0− (J,∗J )0=expi 2(dx )(dx ′)∗J µ(x )D µν(x −x ′)∗J ν(x ′)−i(dx )(dx ′)J µ(x )˜Dµν(x −x ′)∗J ν(x ′′) ,(3.40)where we use the shorthand notation for the “dual propagator”that couples magnetic to electric charge˜Dµν(x −x ′)=ǫµνστ(dx ′′)D +(x −x ′′)∂′′σf τ(x ′′−x ′).(3.41)The term coupling electric and magnetic sources has the same form as in Eq.(1.5);here,we have replaced D κµ→g κµD +,as we may because of the appearance of the Levi-Civita symbol in Eq.(3.41).In an even more straightforward manner Eqs.(3.37a),(3.37b)integrate to0+|0− (¯η,η,¯ξ,ξ)0=exp i(dx )(dx ′) ¯η(x )G ψ(x −x ′)η(x ′)+¯ξ(x )G χ(x −x ′)ξ(x ′) ,(3.42)where G ψand G χare the free propagators for the electrically and magnetically charged fermions,respectively,G ψ(x )=1−iγ∂+m χδ(x ).(3.43)In the presence of interactions the coupled equations (3.33),(3.35),(3.37a),(3.37b)are solved by substitutingEqs.(3.40)and (3.42)into Eq.(3.26).The resulting generating function isZ (J )=exp −ie(dx )δiδJ µ(x )δδξ(y )γµδδ¯ξ(y ) Z 0(J ).(3.44)。
