Semiclassical Approximations in Phase Space with Coherent States

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Nineteen Dubious Ways to Compute the exponential of a matrix, twenty-five years later

1. Introduction. Mathematical models of many physical, biological, and economic processes involve systems of linear, constant coeﬃcient ordinary diﬀerential equations x ˙ (t) = Ax(t). Here A is a given, ﬁxed, real or complex n-by-n matrix. A solution vector x(t) is sought which satisﬁes an initial condition x(0) = x0 . In control theory, A is known as the state companion matrix and x(t) is the system response. In principle, the solution is given by x(t) = etA x0 , where etA can be formally deﬁned by the convergent power series etA = I + tA + t 2 AE MOLER AND CHARLES VAN LOAN
We will primarily be concerned with matrices whose order n is less than a few hundred, so that all the elements can be stored in the main memory of a contemporary computer. Our discussion will be less germane to t

太原理工大学理工科研究生专业英语词汇篇

Unit 1outcome结果statistics统计statistical 统计的toss投die骰子dice 掷骰子intuitive直觉的analogously类似的conceptually概念的simultaneous同时的collection集合identical同一的，同样的individual个别的ensemble集，总和ensemble average集平均variable变量random variable随机变量stationary平稳的ergodic个态历经的ergodic process各态历经过程deterministic确定性的normalize使标准化的normalized规格化的，归一化的expectation期望值product乘积truncate截断，截短periodic周期的covariance协方差uncorrelated不相关的uniform均匀的overlap部分重叠separation间隔，距离spacing间隔，间距random process随机过程on the average 平均，一般来说make measurement 量度sample function 样本函数statistical average 统计平均值probability density function概率密度函数statistical characteristic统计特征autocorrelation function 自相关函数in connection with 与，，有关physical significance物理意义Fourier transform傅里叶变换power spectral density 功率谱密度be at liberty to被允许gaussian process高斯过程physical fact外界存在的事实upper-frequency频率上限fall off下降in the limit在极限情况下delta function狄拉克函数random pulse随机脉冲time invariant时不变的parseval's theorem巴塞瓦尔定理statistically independent统计独立successive pulse连续脉冲be representative of表示，代表词组sample function样本函数ensemble average集合平均physical significance物理意义a fourier transform pair傅里叶变换对deterministic waveform 确定波形in the limit 在极限情况下time invariant时不变的an upper-frequency limit频率上限parseval，s theorem帕斯瓦尔定理random pulses随机脉冲随机过程random process统计平均statistical average随机变量random variable自相关函数autocorrelation function傅里叶变换fourier transform功率谱密度power spectral density概率密度函数probability density function 高斯过程gaussian process平稳过程stationary process统计独立statistically时间平均值time average统计特征statistical character各态历经过程ergodic processUnit 2abstraction抽象encompass包括devise设计whereby凭那个，由此arbitarily任意地theorem定理infrequent 罕见的override压倒thereby由此gaussian高斯的unity单一complementary补充的derivation推导formidable困难的undertake进行，从事presumed假定的presumably推测起来likelihood可能likely可能interval间隔rounding舍入，四舍五入abrupt，急剧的，不连续reliably可靠的，安全的heuristic启发式的intuitive直觉的intuitively直觉的contemplate预期的vice versa反之亦然quantize量化reception接受extreme极端的，偏激的attenuate衰减distort失真distortion失真equalizer均衡器recoverable可恢复的enter into参加，涉及due to 由于be something of 有一点in principle 原则上provided that假如on principle按照原则probability of error 误码率be close to接近bandlimited gaussian channel限带高斯信道physical system物理系统turn out结果lower bound下界upper limit 上限root-mean-square均方根root-mean-square error 均方误差amount of information 信息量ideal lowpass filter理想低通滤波器rise time上升时间as a matter of convenience为方便起见white gaussian noise 高斯白噪声on the other hand 另一面trade off交替换位tradeoff折中signal-to-noise ratio信号噪声比be free to 随意make up for 补偿词组physical system物理系统rise time 上升时间amount of information信息量in principle原则上gaussian channel高斯信道probability density概率密度root-mean-square均方根trade off折中lower bound下界equalizer均衡器vice versa反之亦然upper limit上限通信理论communication theory香农定理Shannon，s theorem信道带宽channel bandwidth信号波形signal waveform理想低通滤波器ideal lowpass filter自相关函数autocorrelation function无噪音高斯信道noiseless gaussian channel 通信信道communication channel信息速率information rate信噪比signal-to-noise ratio信道容量channel capacity双边功率谱密度two-side power误码率probability of error那奎斯特采样速率nyquist sampling rate限带高斯信道bandlimited gaussian channel 高斯白噪声White gaussian noiseUnit 3analog模拟quantization量化volt伏特discrete离散的quantum量unipolar单级的polar极性的on-off开关的encoding编码encoder编码器decoder解码器likewise同样的amplitude振幅margin余量trinary三倍的immune免疫的hereafter今后unavailable不能避免的reconstructed重建的evaluation计算span横跨interval间隔compandor压扩器companding压扩uniformly均匀的compress压缩compressor压缩器compressor ratio压缩比quantizer均匀量化器uniform quantizer均匀量化器equivalent相等的serially连续的reshape改造distortion失真interconnect使相互连接filter滤波transition转换，转变PCM: pulse code modulation脉冲编吗调制PAM: pulse amplitude modulation脉冲振幅调制PPM: pulse phase modulation 脉宽调制pulse train脉冲序列amount to等于round off舍入step size步长quantum step size 量化步长positive pulse 正脉冲refer to as称为quantization error 量化步长peak magnitude峰值overall performance总性能crest factor振幅因数root-mean-square均方根Differential Pulse Code Modulation 差分脉冲编码调制adaptive DPCM 自适应DPCMpredictive coding预测编码correspond to相应，符合in practice 实际中in an attempt to 力图词组overall performance总性能crest factor振幅因数nonlinear operation 非线性炒作inverse operation 逆运算RMS均方根PAM脉福调制maximum magnitude最大幅值error intervals误差间隔entropy墒round off四舍五入quantum level量化水平DPCM差分脉码调制正脉冲positive pulse脉冲编码调制PCM量化步长quantum step size 峰值peak magnitude线性函数linear function脉冲序列pulse train均匀量化器uniform quantizer 预测编码predictive coding 压扩器compandor压缩比compression ratio Unit 4selectively仔细挑选formula公式formulas公式化sacrifice牺牲redundant冗余的demonstrate证明orthogonal正交的arbitrarily任意地raw未加工的raw data 原始数据denote 指示Retransmission转发sophisticated复杂的impractical不切实际的parity奇偶校验encoder编码器codeword码字modulo以。

Negative concentrations(comsol中负浓度的处理方法)

Results With Small Unphysical ValuesWHERE AND WHY DO UNPHYSICAL VALUES APPEAR?In some models small unphysical values can occur due to numerical artifacts. Examples include: •Negativeconcentrations in mass transfer . •A temperature that is slightly higher than the initial condition in time dependent heat transfer studies. •Small reaction forces that appear in unloaded directions in structural mechanics models. •Small negative gaps in a contact analysis. •Small negative effective plastic strain values. •Stresses above the yield limit for an ideally plastic material in solid mechanics. Some reasons for why these unphysical values occur:•Numerical noise is a common cause. When the values of the dependent variables approach zero, the numerical noise can become relatively significant and cause some of the results to be slightly negative even if that is not physically possible. •Interpolation and extrapolation of values can cause some values to become unphysical. For example, results for an elastoplastic material are correct (within some tolerance) at the integration points (Gauss points) inside the finite elements, but values might become unphysical when extrapolating the data to the element boundaries. •Discontinuities in the model is another source of, for example, small negativeconcentrations due to a discontinuous initial value. With an initial value that is zero along a boundary for a convective transport models, for example, the physical interpretation is an initially sharp, gradually diffusing front moving away from the boundary. However , for the default shape function (second-order Lagrange elements), only continuous functions are admissible as solutions for the finite element model. The program then modifies the discontinuous initial value before the time stepping can begin. This often results in a small dip in the solution at the start time. In the example model that the following figure shows, the concentration is locally slightly negative at t = 0: • Lack of mesh resolution is another cause of unphysical values such as negativeconcentrations. The resulting convergence problems are often the underlying issue when negativeconcentrations are observed in high convection regimes (high Peclet number) and in those with large reaction terms or fast kinetics (high Damkohler number). •Incorrect physics in the model can also cause these types of problems. For mass transfer , for example, the use of a constant sink in a reaction term is an approximation that only works for large concentrations. When the concentration reaches zero, the reaction term continues to consume thespecies, finally resulting in a negative concentration. AVOIDING UNPHYSICAL VALUESThis section contains some ways to avoid computing or displaying unphysical values:•In some cases it is possible to add a baseline to the dependent variable so that the numerical noise does not affect the solution in the same way as when the values of the dependent variable approach zero. This scaling is not possible with, for example, a reaction term that depends on the concentration because then the scale and origin do matter . •Avoid discontinuities in the model by using, for example, the available smoothed step functions instead. •Formulate logarithmic variables as a way of eliminating mesh resolution problems and negative dips by using the logarithm of the original dependent variable (the concentration, for example) as the dependent variable. The reason for this is that a linearly varying mesh can sometimes not capture the exponential behavior of the changes in the dependent variable. In addition, modeling the logarithm of the dependent variable ensures that the real concentration, for example, cannot become negative during the solution process. •Avoid displaying small unphysical values due to numerical noise by clipping the values for the plot. You can do this by plotting, for example, c*(c>0) instead of c , which evaluates to 0 everywhere where c is smaller than 0. You can also adjust the range of the plot data and colors to only show values that are nonzero. In the case that the data range changes, parts of the plots where the values are outside the range become empty. •It can also be useful to check how the mesh affects the solution by refining the mesh and check if the problem with unphysical values gets better or worse. If it gets better , then continue to refine the mesh. If it gets worse, you probably need to check the physics of the model.。

物理学外文翻译

Effect of Quantum Confinement on Electrons and Phonons in Semiconductors We have studied the Gunn effect as an example of negative differential resistance(NDR).This effect is observed in semiconductors,such as GaAs,whose conduction band structure satisfies a special condition,namely,the existence of higher conduction minima separated from the band edge by about 0.2-0.4eV..As a way of achieving this condition in any semiconductor,Esaki and Tsu proposed in 1970 [9.1]the fabrication of an artificial periodic structure consisting of alternate layers of two dissimilar semiconductors with layer superlattice.They suggested that the artificial periodicity would fold the Brillouin zone into smaller Brillouin zones or “mini-zones”and therefore create higher conduction band minima with the requisite energies for Gunn oscillations.iWith the development of sophisticated growth techniques such as molecular beam epitaxy（MBE）and metal-organic chemical vapor deposition(MOCVD)discussed in Sect.1.2,it is now possible to fabricate the superlattices(to be abbreviated as SLs)envisioned by Esaki and Tsu[9.1].In fact,many other kinds of nanometer scale semiconductor structures(often abbreviated as nanostructures)have since been grown besides the SLs.A SL is only one example of a planar or two-dimensional nanostructure .Another example is the quantum well (often shortened to QW).These terms were introduced ｉｎＳｅｃｔｓ．１．２ａｎｄ７．１５ｂｕｔｈａｖｅｎｏｔｙｅｔｂｅｅｎｄｉｓｃｕｓｓｅｄｉｎｄｅｔｉａｌ．Ｔｈｅｐｒｏｐｏｓｅof this chapter is to study the electronic and vibrational properties of these two-dimensional nanostructures.Structures with even lower dimension than two have also been fabricated successfully and studied. For example,one-dimensional nanostructures are referred to as quantum wires.In the same spirit,nanometer-size crystallites are known as quantum dots.There are so many different kinds of nanostructures and ways to fabricate them that it is impossible to review them all in this introductory book. In some nanostructures strain may be introduced as a result of lattice mismatch between a substrate and its overlayer,giving rise to a so-called strained-layer superlattice.In this chapter we shall consider only the best-study nanostructures.Our purpose is to introduce readers to this fast growing field.One reason why nanostructures are of great interest is that their electronic and vibrational properties are modified as a result of their lower dimensions and symmetries.Thus nanostructures provide an excellent opportunity for applying the knowledge gained in the previous chapters to understand these new developments in the field of semiconductors physics.Due to limitations of space we shall consider in this chapter only the effects of spatial confinement on the electronic and vibrational properties of nanostructures and some related changers in their optical and transport properties.Our main emphasis will be on QWs,since at present they can be fabricated with much higher degrees of precision and perfection than all other structures.We shall start by defining the concept of quantum confinement and discuss its effect on the electrons and phonons in a crystal.This will be followed by a discussion of the interaction between confined electrons and phonons.Finally we shall conclude with a study of a device(known as a resonant tunneling device)based on confined electrons and the quantum Hall effect(QHE)in a two-dimensional electron gas.The latter phenomenon was discoveredby Klaus von Klitzing and coworkers in 1980 and its significance marked by the award of the 1985 Nobel Prize in physics to ｖｏｎ Klitzing for this discovery.Together with the fractional quantum Hall effect it is probably the most important development in semiconductor physics within the last two decades.Quantum Confinement and Density of StatesIn this book we have so far studied the properties of electrons ,phonons and excitons in either an infinite crystal or one with a periodic boundary condition(the cases of surface and interface states )In the absence of defects, these particles or excitations are described in terms of Bloch waves,which can propagate freely throughout the crystal.Suppose the crystal is finite and there are now two infinite barriers,separated by a distance L,which can reflect the Bloch waves along the z direction.These waves are then said to be spatially confined.A classical example of waves confined in one dimension by two impenetrable barriers is a vibrating string held fixed at two ends.It is well-known that the normal vibration modes of this string are standing waves whose wavelength λ takes on the discrete values given by Another classical example is a Fabry-Perot interferometer (which has been mentioned already in Set.7.2.6 in connection with Brillouin scattering).As a result of multiple reflections at the two end mirrors forming the cavity ，electromagnetic waves show maxima and minima in transmission through the interferometer at discrete wavelengths.If the space inside the cavity is filled with air,the condition for constructive interference is given by (9.1).At a transmission minimum the wave can be considered as “confined ”inside the interferometer.n λ=2L/n, n=1,2,3… .(9.1)For a free particle with effective mass *m confined in a crystal by impenetrablebarriers(i.e.,infinite potential energy)in the z direction,the allowed wavevectors z k of the Bloch waves are given byzn κ=2∏/n λ=n ∏/L, n=1,2,3… (9.2)And its ground state energy is increased by the amount E relative to the unconfined case:))(2(2222212Lm m k E z ∏==∆** (9.3)This increase in energy is referred to as the confinement energy of the particle.It is a consequence of the uncertainty principle in quantum mechanics. When the particle is confined within a distance L in space(along the z direction in this case)the uncertainty in the z component of its momentum increases by an amount of the order of /L.The corresponding increase in the particle ’s kinetic energy is then givenby(9.3).Hence this effect is known also as quantum confinement.In addition to increasing the minimum energy of the particle,confinement also causes its excited state energies to become quantized.We shall show later that for an infinite one-dimensional”square well”potential the excited state energies are given by n E∆2,where n=1,2,3…as in (9.2).It is important to make a distinction between confinement by barriers and localization via scattering with imperfections。

Subthreshold Antiproton Spectra in Relativistic Heavy Ion Collisions

a rXiv:h ep-ph/959328v119Se p1995TPR-95-19Subthreshold Antiproton Spectra in Relativistic Heavy Ion Collisions Richard Wittmann and Ulrich Heinz Institut f¨u r Theoretische Physik,Universit¨a t Regensburg,D-93040Regensburg,Germany (February 1,2008)Abstract We study the structure of antiproton spectra at extreme subthreshold bom-barding energies using a thermodynamic picture.Antiproton production pro-cesses and ﬁnal state interactions are discussed in detail in order to ﬁnd out what can be learned about these processes from the observed spectra.Typeset using REVT E XI.INTRODUCTIONThere exist numerous examples for the production of particles in heavy ion collisions at bombarding energies well below the single nucleon-nucleon threshold[1].This phenomenon indicates collective interactions among the many participating nucleons and thus is expected to give information about the hot and dense matter formed in these collisions.At beam energies around1GeV per nucleon the most extreme of these subthreshold particles is the antiproton.Therefore,it is believed to be a very sensitive probe to collective behaviour in nucleus-nucleus collisions.However,presently neither the production mechanism nor theﬁnal state interactions of antiprotons in dense nucleonic matter are well understood.The antiproton yields measured at GSI and BEVALAC[2,3]seem to be described equally well by various microscopic models using diﬀerent assumptions about the production mechanism and particle properties in dense nuclear matter[4–7].This ambiguity raises the question which kind of information can be really deduced from subthreshold¯p spectra.In this paper we use a simple thermodynamic framework as a background on wich we can systematically study the inﬂuence of diﬀerent assumptions on theﬁnal¯p spectrum.In the next section we will focus on the production process.Following a discussion of theﬁnal state interactions of the antiproton in dense hadronic matter in Section III,we use in Section IV a one-dimensional hydrodynamic model for the explodingﬁreball to clarify which features of the production and reabsorption mechanisms should survive in theﬁnal spectra in a dynamical environment.We summarize our results in Section V.II.PRODUCTION OF ANTIPROTONS IN HEA VY ION COLLISIONSA.The Antiproton Production RateUnfortunately very little is known about the production mechanism for antiprotons in dense nuclear matter.Therefore,we are forced to use intuitive arguments to obtain aplausible expression for the production rate.As commonly done in microscopic models[8] we consider only two body collisions and take the experimentally measured cross sections for ¯p production in free NN collisions as input.The problem can then be split into two parts: the distribution of the two colliding nucleons in momentum space and the elementary cross sections for antiproton production.The procedure is later generalized to collisions among other types of particles(Section II C)using phase space arguments.Formally,the antiproton production rate P,i.e.the number of antiprotons produced in the space-time cell d4x and momentum space element d3p,is given by[9]P= i,j ds2w(s)d3σij→¯pδ(p2i−m2i)Θ(p0i).(2)(2π)3Finally,w(s,m i,m j)=√ofﬁnding two nucleons at a center-of-mass energys−4m)αIn order to obtain from the total cross section(3)a formula for the diﬀerential cross section,we assume like others that the momentum distribution of the produced particles is mainly governed by phase space.This leads to the simple relationshipd3σij→NNN¯pσij→NNN¯p(s).(4)R4(P)Here R n is the volume the n-particle phase space,which can be given analytically in the non-relativistic limit[12].P is the total4-momentum of the4-particleﬁnal state,which√reduces to(s min(p)/T.Hereρi,j are the densities of the incoming particles,that the cross section σij →NNN ¯p (s )is independent of the internal state of excitation of the colliding baryons in our thermal picture.The consequences of this assumption are quite obvious.While the distance to the ¯p -threshold is reduced by the larger rest mass of the resonances,the mean velocity of a heavy resonance state in a thermal system is smaller than that of a nucleon.Both factors counteract each other,and indeed we found that the total rate P is not strongly changed by the inclusion of resonances.The role of pionic intermediate states for ¯p -production in pp -collisions was pointed out by Feldman [15].As mesonsare created numerously in the course of a heavy ion collision,mesonic states gain even more importance in this case.In fact,Ko and Ge [13]claimed that ρρ→p ¯p should be the dominant production channel.Relating the ρρ-production channel to the p ¯p annihilation channel [13]byσρρ→p ¯p (s )= 2s −4m ρm σp ¯p →ρρ(s )(5)where S =1is the spin factor for the ρ,the production rate can be calculated straightfor-wardly from Eq.(1):P =g 2ρ(2π)516T .(6)The spin-isospin factor of the ρis g ρ=9,and E is the energy of the produced antiproton.The modiﬁed Bessel function K 1results from the assumption of local thermal equilibration for the ρ-distribution.Expanding the Bessel function for large values of 2E/T we see that the ”temperature”T ¯p of the ¯p -spectrum is only half the medium temperature:T ¯p =1w 2(s,m i ,m j )(8)Comparing this form with measured data onπ−p→np¯p collisions[16],a value ofσ0ij= 0.35mb is obtained.Due to the threshold behaviour of Eq.(8)and the rather large value ofσ0ij,it turns out[17]that this process is by far the most important one in a chemically equilibrated system.However,this chemical equilibration–if achieved at all–is reached only in theﬁnal stages of the heavy ion collision when cooling has already started.So it is by no means clear whether the dominance of the meson-baryon channel remains valid in a realistic collision scenario.This point will be further discussed in Section IV.III.FINAL STATE INTERACTION OF THE ANTIPROTONOnce an antiproton is created in the hot and dense hadronic medium,its state will be modiﬁed by interactions with the surrounding particles.Two fundamentally diﬀerent cases have to be distinguished:elastic scattering,which leads to a reconﬁguration in phase space, driving the momentum distribution towards a thermal one with the temperature of the surrounding medium,and annihilation.Each process will be considered in turn.A.Elastic ScatteringThe time evolution of the distribution function f(p,t)is generally described by the equation[18]f(p,t2)= w(p,p′;t2,t1)f(p′,t1)dp′(9) where w(p,p′;t2,t1)is the transition probability from momentum state p′at time t1to state p at t2.Because the number density of antiprotons is negligible compared to the total particle density in the system,the evolution of f(p,t)can be viewed as a Markov process.Assuming furthermore that the duration of a single scattering processτand the mean free pathλare small compared to the typical time scaleδt and length scaleδr which measure the variation of the thermodynamic properties of the system,τ≪δt,λ≪δr,Eq.(9)can be transformed into a master equation.Considering the structure of the diﬀer-ential p¯p cross section one notices that in the interesting energy range it is strongly peaked in the forward direction[19–21].Therefore,the master equation can be approximated by a Fokker-Planck equation[22]:∂f(p,t)∂p A(p)f(p,t)+1∂2pD(p)f(p,t).(10)For the evaluation of the friction coeﬃcient A and the diﬀusion coeﬃcient D we follow the treatment described by Svetitsky[23].For the diﬀerential cross section we took a form suggested in[19]:dσ|t|2D/A(e2At−1)+2mT0p2≡exp −p2/2mT eﬀ(t) .(14)This shows that the exponential shape of the distribution function is maintained throughout the time evolution,but that the slope T eﬀ(t)gradually evolves from T0to the value D/mAwhich,according to the Einstein relation(12),is the medium temperature T.Looking at Fig.3it is clear that after about10fm/c the spectrum is practically thermalized.Therefore initial structures of the production spectrum(like the ones seen in Fig.2)are washed out quite rapidely,and their experimental observation will be very diﬃcult.B.AnnihilationThe annihilation of antiprotons with baryons is dominated by multi-mesonﬁnal states X.For the parametrisation of the annihilation cross sectionσann(s)we take the form given in[14]for the process¯p+B−→X,B=N,∆, (15)Using the same philosophy as for the calculation of the production rate,a simple dif-ferential equation for the decrease of the antiproton density in phase space can be written down:dd3xd3p =−d6N(2π)3f i( x, p i,t)v i¯pσanni¯p≡−d6Nbecomes questionable.More reliable results should be based on a quantumﬁeld theoretic calculation which is beyond the scope of this paper.IV.ANTIPROTON SPECTRA FROM AN EXPLODING FIREBALLA.A Model for the Heavy Ion CollisionIn order to compare the results of the two previous sections with experimental data we connect them through a dynamical model for the heavy ion reaction.In the spirit of our thermodynamic approach the so-called hadrochemical model of Montvay and Zimanyi [24]is applied for the simulation of the heavy ion collision.In this picture the reaction is split into two phases,an ignition and an explosion phase,and particles which have at least scattered once are assumed to follow a local Maxwell-Boltzmann distribution.In addition, a spherically symmetric geometry is assumed for the explosion phase.The included particle species are nucleons,∆-resonances,pions andρ-mesons.As initial condition a Fermi-type density distribution is taken for the nucleons of the incoming nuclei,ρ0ρ(r)=collision with momentum p z=±700MeV in the c.m.system.One clearly sees that at the moment of full overlap of the two nuclei a dense zone with hot nucleons,resonances and mesons(not shown)has been formed.Only in the peripheral regions”cold”target and projectile nucleons can still be found.On the other hand,chemical equilibrium of the hot collision zone is not reached in the short available time before the explosion phase sets in; pions and in particularρmesons remain far below their equilibrium abundances[17].It is important to note that due to the arguments given in Section II¯p production is strongly suppressed in this initial stage of the reaction.In our simple model we have in fact neglected this early¯p production completely.The ignition phase is only needed to obtain the chemical composition of the hotﬁreball which is expected to be the dominant source for the creation of antiprotons.For the subsequent expansion of the spherically symmetricﬁreball into the surrounding vacuum analytical solutions can be given if the equation of state of an ideal gas is taken as input[25].If excited states are included in the model,an exact analytical solution is no longer possible.Because a small admixture of resonaces is not expected to fundamentally change the dynamics of the system,we can account for their eﬀect inﬁrst order by adjusting only the thermodynamic parameters of the explodingﬁreball,but not the expansion velocity proﬁles.There is one free parameter in the model[24],α,which controls the density and the temperature proﬁles,respectively.Small valuesα→0representδ-function like density proﬁles whereasα→∞corresponds to a homogeneous density distribution throughout the ﬁreball(square well proﬁle).The time-dependent temperature proﬁles for two representative values ofαare shown in Fig.5for diﬀerent times t starting at the time t m of full overlap of the nuclei.Clearly,a small value ofαleads to an unreasonably high temperature(T∼200MeV) in the core of theﬁreball at the beginning of the explosion phase,and should thus be considered unphysical.B.Antiproton Spectra from an Exploding FireballBased on the time-dependent chemical composition of this hadrochemical model we can calculate the spectrum of the antiprotons created in a heavy ion collision.Let usﬁrst con-centrate on the inﬂuence of the density distribution in theﬁreball characterized by the shape parameterα.Due to diﬀerent temperature proﬁles connected with diﬀerentαvalues(see Fig.5)the absolute normalization varies substantially when the density proﬁle is changed. For moreδ-like shapes an extremely hotﬁreball core is generated,whereas for increasingαboth density and temperature are more and more diﬀuse and spread uniformly over a wider area.Because of the exponential dependence upon temperature a small,but hot core raises the production rate drastically.This fact is illustrated in Fig.6for three values ofα.Not only the total normalization,but also the asymptotic slope of the spectrum is modiﬁed due to the variation of the core temperature withαas indicated in the Figure.Comparing the dotted lines,which give the pure production spectrum,to the solid lines representing the asymptotic spectrum at decoupling,the tremendous eﬀect of antiproton absorption in heavy ion collisions is obvious.As one intuitively expects absorption is more pronounced for low-energetic antiprotons than for the high-energetic ones which have the opportunity to escape the high density zone earlier.Therefore,theﬁnally observed spectrum isﬂatter than the original production spectrum.Interestingly,while the baryon-baryon and theρρchannels are comparable in their con-tribution to¯p production,the pion-baryon channel turned out to be much more eﬀective for all reasonable sets of parameters.This fact is indeed remarkable,because here,contrary to the discussion in Section II,the pions are not in chemical equilibrium;in our hadrochemical model the total time of the ignition phase is too short to saturate the pion channel.The meson-baryon channel is thus crucial for understanding¯p spectra.Only by including all channels reliable predictions about the antiprotons can be drawn.We did not mention so far that in our calculations we followed common practice and assumed aﬁnite¯p formation time ofτ=1fm/c;this means that during this time intervalafter a¯p-producing collision the antiproton is assumed to be not yet fully developed and thus cannot annihilate.However,there are some(although controversial)experimental indications of an extremely long mean free path for antiprotons beforeﬁnal state interactions set in[26].We have tested the inﬂuence of diﬀerent values for the formation timeτon the¯p spectrum.Fig.7shows that this highly phenomenological and poorly established parameter has a very strong inﬂuence in particular on the absolute normalization of the spectra,i.e. the total production yield.In the light of this uncertainty it appears diﬃcult to argue for or against the necessity for medium eﬀects on the antiproton production threshold based on a comparison between theoretical and experimental total yields only.parison with Experimental DataIn all the calculations shown above a bombarding energy of1GeV/A has been assumed. Experimental data are,however,only available at around2GeV/A.At these higher energies thermalization becomes more questionable[27],and our simple model may be stretching its limits.Especially,the temperature in theﬁreball core becomes extremely high.In order to avoid such an unrealistic situation and in recognition of results from kinetic simulations [4,6,7]we thus assume that only part of the incoming energy is thermalized–in the following a fraction of70%was taken.Fig.8shows calculations for the antiproton spectrum from Na-Na and Ni-Ni collisions at a kinetic beam energy of2GeV/nucleon.The calculation assumes a¯p formation time of τ=1fm/c,and takes for the density and temperature proﬁles the parameter valueα=1 which corresponds to an upside-down parabola for the density proﬁparing with the GSI data[2]we see our model features too weak a dependence on the size of the collision system;the absolute order of magnitude of the antiproton spectrum is,however,correctly reproduced by our simple hadrochemical model,without adjusting any other parameters. No exotic processes for¯p production are assumed.As mentioned in the previous subsection, the pion-baryon channel is responsible for getting enough antiprotons in our model,withoutany need for a reduced eﬀective¯p mass in the hot and dense medium[4,5].The existing data do not yet allow for a deﬁnite conclusion about the shape of the spectrum,and we hope that future experiments[28]will provide additional contraints for the model.V.CONCLUSIONSHeavy ion collisions at typical BEVALAC and SIS energies are far below the p¯p-production threshold.As a consequence,pre-equilibrium antiproton production in such collisions is strongly suppressed relative to production from the thermalized medium pro-duced in the later stages of the collision.Therefore,¯p production becomes important only when the heavy ion reaction is suﬃciently far progressed,in accordance with microscopic simulations[4].By assuming a local Maxwell-Boltzmann distribution for the scattered and produced particles forming the medium in the collision zone one maximizes the¯p production rate(see Fig.1).If,contrary to the assumptions made in this work,the extreme states in phase space described by the tails of the thermal Boltzmann distribution are not populated, the antiproton yield could be reduced substantially.We also found that the threshold behaviour of the¯p production cross section is not only crucial for the total¯p yield,but also introduces structures into the initial¯p spectrum.This might give rise to the hope that by measuring the¯p momentum spectrum one may obtain further insight into the¯p production mechanism.On the other hand we saw here,using a Fokker-Plank description for the later evolution of the distribution function f¯p(p,t)in a hot environment,that these structures are largely washed out by subsequent elastic scattering of the¯p with the hadrons in the medium.In addition,the large annihilation rate reduces the number of observable antiprotons by roughly two orders of magnitude relative to the initial production spectrum;the exact magnitude of the absorption eﬀect was found to depend sensitively on the choice of the¯p formation timeτ.We have also shown that meson(in particular pion)induced production channels con-tribute signiﬁcantly to theﬁnal¯p yield and should thus not be neglected.We were thus ableto reproduce the total yield of the measured antiprotons in a simple model for the reaction dynamics without including,for example,medium eﬀects on the hadron masses and cross sections[4,5].However,we must stress the strong sensitivity of the¯p yield on various unknown param-eters(e.g.the¯p formation time)and on poorly controlled approximations(e.g.the degree of population of extreme corners in phase space by the particles in the collision region),and emphasize the rapidly thermalizing eﬀects of elasticﬁnal state interactions on the¯p momen-tum spectrum.We conclude that turning subthreshold antiproton production in heavy ion collisions into a quantitative probe for medium properties and collective dynamics in hot and dense nuclear matter remains a serious challenge.ACKNOWLEDGMENTSThis work was supported by the Gesellschaft f¨u r Schwerionenforschung(GSI)and by the Bundesministerium f¨u r Bildung und Forschung(BMBF).REFERENCES[1]U.Mosel,Annu.Rev.Nucl.Part.Sci.1991,Vol.41,29[2]A.Schr¨o ter et al.,Z.Phys.A350(1994)101[3]A.Shor et al.,Phys.Rev.Lett.63(1989)2192[4]S.Teis,W.Cassing,T.Maruyama,and U.Mosel,Phys.Rev.C50(1994)388[5]G.Q.Li and C.M.Ko,Phys.Rev.C50(1994)1725[6]G.Batko et al.,J.Phys.G20(1994)461[7]C.Spieles et al.,Mod.Phys.Lett.A27(1993)2547[8]G.F.Bertsch and S.Das Gupta,Phys.Rep.160(1988)189[9]P.Koch,B.M¨u ller,and J.Rafelski,Phys.Rep.42(1986)167[10]B.Sch¨u rmann,K.Hartmann,and H.Pirner,Nucl.Phys.A360(1981)435[11]G.Batko et al.,Phys.Lett.B256(1991)331[12]burn,Rev.Mod.Phys.27(1955)1,and references therein[13]C.M.Ko and X.Ge,Phys.Lett.B205(1988)195[14]P.Koch and C.Dover,Phys.Rev.C40(1989)145[15]G.Feldman,Phys.Rev.95(1954)1697[16]Landolt-B¨o rnstein,Numerical Data and Functional Relationships in Science and Tech-nology,Vol.12a und Vol.12b,Springer-Verlag Berlin,1988[17]R.Wittmann,Ph.D.thesis,Univ.Regensburg,Feb.1995[18]N.G.van Kampen,Stochastic Processes in Physics and Chemistry,North-Holland Pub-lishing Company,Amsterdam1983[19]B.Conforto et al.,Nouvo Cim.54A(1968)441[20]W.Br¨u ckner et al.,Phys.Lett.B166(1986)113[21]E.Eisenhandler et al.,Nucl.Phys.B113(1976)1[22]S.Chandrasekhar,Rev.Mod.Phys.15(1943)1[23]B.Svetitsky,Phys.Rev.D37(1988)2484[24]I.Montvay and J.Zimanyi,Nucl.Phys.A316(1979)490[25]J.P.Bondorf,S.I.A.Garpman,and J.Zim´a nyi,Nucl.Phys.A296(1978)320[26]A.O.Vaisenberg et al.,JETP Lett.29(1979)661[27]ng et al.,Phys.Lett.B245(1990)147[28]A.Gillitzer et al.,talk presented at the XXXIII International Winter Meeting on NuclearPhysics,23.-28.January1995,Bormio(Italy)FIGURES468101214-810-710-610-510-410-310-210-110010s in GeVλt=0.2 fm/c t=2 fm/c t=5 fm/c ©©©2FIG.1.λ(s,t )at diﬀerent times t calculated from the model of Ref.[10].The starting point is a δ-function at s =5.5GeV 2.The dashed line is the asymptotic thermal distribution at t =∞),corresponding to a temperature T =133MeV.00.10.20.30.40.50.6-16-171010-1510-1410-1310-1210-1110-1010 E in GeVα = 1/2α = 3/2α = 5/2α = 7/2ÄÄÄÄPFIG.2.Antiproton production spectrum for diﬀerent threshold behaviour of the elementaryproduction process (x =12,52from top to bottom).203040506070 8090t in fm/cT e f f T = 10 MeV 0T = 50 MeV 0T = 70 MeV 00 246810i n M e V 10FIG.3.Eﬀective temperature T eﬀfor three Maxwell distributions with initial temperatures T 0=10MeV,50MeV and 70MeV,respectively.0z in fm12ρ / ρ0FIG.4.Density distributions ρ(0,0,z )along the beam axis of target and projectile nucleons for a 40Ca-40Cacollision,normalized to ρ0=0.15fm −3.The solid lines labelled by “incoming nuclei”show the two nuclei centered at ±5fm at time t 0=0.The two other solid lines denote the cold nuclear remnants at full overlap time t m ,centered at about ±3fm.Also shown are ﬁreball nucleons (long-dashed)and ∆-resonances (short-dashed)at time t m .α=0.20T i n G e V r in fm α=5r in fm024680.050.10.150.20.25T i n G e V t = 9 fm/c630FIG.5.Temperature proﬁles for α=0.2and α=5at four diﬀerent times t =0(=beginning of the explosion phase)and t =3,6,and 9fm/c (from top to bottom).E in GeV33d N / d p i n G e V -3α=0.2α=1α=500.10.20.30.40.50.6-11-10-9-8-7-6-5-4-3101010101010101010FIG.6.¯p -spectra for diﬀerent proﬁle parameters α.The dotted lines mark the initial pro-duction spectra.The asymptotic temperatures at an assumed freeze-out density ρf =ρ0/2corre-sponding to the solid lines are,from top to bottom,105MeV,87MeV and 64MeV,respectively.E in GeVd N / d p i n Ge V 33-10101010101010FIG.7.¯p -spectrum for diﬀerent formation times,for a proﬁle parameter α=1.The dashed line indicates the original production spectrum.E in GeVkin d p 3d σ3i n b /Ge V /10101010FIG.8.Diﬀerential ¯p spectrum for Na-Na and Ni-Ni collisions,for a shape parameter α=1.The data are from GSI experiments [2].。

Instabilities of the Small Black Hole a view from N=4 SYM

a r X i v :h e p -t h /0607111v 2 11 O c t 2006Preprint typeset in JHEP style -HYPER VERSIONTimothy Hollowood,S.Prem Kumar and Asad Naqvi Department of Physics,University of Wales Swansea,Swansea,SA28PP,UK.E-mail:t.hollowood,s.p.kumar,a.naqvi@ Abstract:We compute a one-loop eﬀective action for the constant modes of the scalars and the Polyakov loop matrix of N =4SYM on S 3at ﬁnite temperature and weak ’t Hooft coupling.Above a critical temperature,the eﬀective potential develops new unstable directions accompanied by new saddle points which only preserve an SO (5)subgroup of the SO (6)global R-symmetry.We identify this phenomenon as the weak coupling version of the well known Gregory-Laﬂamme localization instabil-ity in the gravity dual of the strongly coupled ﬁeld theory:The small AdS 5black hole when viewed as a ten dimensional,asymptotically AdS 5×S 5solution smeared on the S 5is unstable to localization on S 5.Our eﬀective potential,in a speciﬁc Lorentzian continuation,can provide a qualitative holographic description of the decay of the “topological black hole”into the AdS bubble of nothing.1.IntroductionTheﬁnite temperature behaviour of SU(N)gauge theories at large N provides a re-markable window into the physics of black holes and stringy gravity via the AdS/CFTcorrespondence[1].Features of the phase structure of large N,N=4supersymmet-ric Yang-Mills(SYM)theory on S3atﬁnite temperature are now known to mirroraspects of semiclassical gravity on asymptotically AdS5spacetimes[2,3,4,5,6].Importantly,the work of[6]has demonstrated that studying theﬁeld theory in thetractable regime of weak’t Hooft coupling may allow us to deduce qualitative physicsof the string theory dual emerging at intermediate and strong couplings.The weaklycoupledﬁeld theory exists in one of two thermodynamically stable phases separatedby aﬁrst order deconﬁnement transition;in addition there is the possibility of athermodynamically unstable saddle point[6,7,8].Indeed,the strong coupling grav-ity dual also exhibits aﬁrst order Hawking-Page transition[9]between two stablegeometries:thermal AdS space and the big AdS-Schwarzschild black hole,mediatedby the thermodynamically unstable Euclidean small AdS black hole bounce.1 The aim of this paper is to take the qualitative matching of thermodynamic phasestructure one step further and to explore certain questions which have dynamicalconsequences.Speciﬁcally,the small AdS5Schwarzschild black hole,when viewedas a ten dimensional asymptotically AdS5×S5solution smeared uniformly on the S5,has a classical dynamical instability(in addition to having a thermodynamicinstability).This is a Gregory-Laﬂamme instability to localization[12,13,14,15,16],wherein the small black hole originally smeared on the S5develops an instability asits horizon size in AdS5decreases below a critical radius(or equivalently,above acritical temperature).The unstable mode leads to the small black hole becomingnon-uniform and eventually point-like on the S5,breaking the associated SO(6)isometry down to SO(5).The question we aim to answer in this paper is how,if atall,this dynamical instability to localization may be seen within the framework of theholographically dual thermalﬁeld theory on a three-sphere.Remarkably,within theregime of validity of perturbation theory weﬁnd that above a critical temperatureat which the radius of the three-sphere becomes comparable to the Debye screeninglength,the weakly coupledﬁeld theory exhibits a clear signal of such an instability.2 The N=4theory on S3atﬁnite temperature has two scales,namely the radius R of the three-sphere and the temperature T.Consequently the theory possesses twotunable dimensionless parameters,the combination(T R)and the’t Hooft coupling λ=g2Y M N.On S3×S1,at weak’t Hooft couplingλ≪1,we calculate a one loop quantum eﬀective action as a function of homogeneous background expectation values for the Polyakov loop3and the six scalarﬁelds transforming in the adjoint representation of SU(N).More precisely,we compute aﬁnite temperature eﬀective action of the Coleman-Weinberg type on a slice of the full conﬁguration space pa-rameterized by the N eigenvalues of each of theseﬁelds.The one loop computation is valid for a wide range of temperatures at weak’t Hooft coupling10≤T R≪λwhich is well within the above range.At these temperatures the√radius of S3becomes comparable to the Debye screening length(λT)−1,ensures the validity of perturbation√theory at these length scales,albeit in the parameter3))−1(or slightly below it forﬁnite coupling)aﬁrst order deconﬁnement transition occurs beyond which a new global minimum appears with a gapped distribution for the Polyakov loop eigenvalues and vanishing scalarﬁelds.This is the“big AdS black hole”minimum.At temperatures above T H,the thermal AdS saddle point with vanishing scalar ﬁelds persists as a thermodynamically unstable saddle point,while the globally stable big black hole dominates the canonical ensemble.For all temperatures T H<T< R−1/√λhowever,something interesting begins to happen. The one loop contribution becomes comparable to the tree level mass while,crucially, the higher loop corrections remain perturbatively small,suppressed by powers of√λ,a new unstable mode,along the R-charged scalar directions,emerges at the thermodynamically unstable“thermal AdS”saddle point.In the large N limit this critical temperature is found to beT c=π2λR−1.(1.2)Beyond this temperature,new unstable saddle points appear with lower action than the“thermal AdS”conﬁguration which becomes unstable to rolling down to these new extrema.Importantly,at these extrema the scalarﬁelds of the N=4theory have non-zero values.In the large N theory these expectation values are left invariant only by an SO(5)subgroup of the full SO(6)R-symmetry of the theory.However, the“big black hole”conﬁguration continues to be the global minimum of the action with vanishing VEVs for all scalars which is consistent with expectations from the bulk gravitationalphysics. T InstabilityLocalizationHagedornSmall Black hole existenceFigure1:Qualitative plot of small black hole(SBH)existence lines,and extrapolation of weak coupling critical temperature for the Gregory-Laﬂamme localization instability as a function of ’t Hooft couplingλ.SBH comes into existence above the dashed line and below the Hagedorn temperature line.Localization occurs above the solid line.We identify this high temperature phenomenon as a continuation to weak’t Hooft coupling of the Gregory-Laﬂamme instability encountered in the gravity dual of the strongly coupled gauge theory.There too,a thermodynamically unstable saddle point,namely the small AdS black hole develops a new dynamical instability to localization on the S5breaking the SO(6)isometry to SO(5).The main diﬀerence√is that at weak coupling,at temperatures of O(R−1/√4For a discussion of these issues particularly from the viewpoint of the small black hole gravity solution,see[19].The outline of our paper is as follows.In Section2we review the general story of black hole instabilities in AdS space.Section3is devoted to a detailed calculation of the one loop eﬀective potential inﬁeld theory and determining its saddle points as a function of temperature.We present our conclusions,interpretations and questions for future study in Section4.The analysis of unstable directions of the eﬀective potential is presented in an Appendix.2.Instabilities and AdS Schwarzschild Black HolesWe begin by reviewing the physics of Schwarzschild black holes in asymptotically AdS spaces[9,3,2].Of particular interest is the theﬁve dimensional AdS-Schwarzschild black hole which is an asymptotically AdS5solution to the vacuum Einstein’s equa-tion with a negative cosmological constant:ds2=−V(r)dt2+dr2R2−r 2hR2)and R is the radius of AdS5.The black hole horizonis at r=r h where V(r)vanishes.This black hole emits Hawking radiation.Toﬁnd the associated Hawking temperature at which the black hole is in equilibrium with a thermal bath at that temperature,we employ the usual trick of going to the Euclidean section and requiring that there are no conical singularities.Taking t→−iχ,the resulting Euclidean metric isds2=V(r)dχ2+dr22r2h +R2.Asymptotically,as r→∞,the space isEuclidean AdS5with the Euclidean time direction being a circle of circumferenceβ. This implies an equilibrium Hawking temperatureT=12πr h R2.(2.3)Notice that for a certain range of T,there are two solutions for r h given by:r h=πT R21−2 21 224r h R2 Π1R TFigure 2:Temperature as a function of the horizon radius.there are two geometries corresponding to the two values of r h ,the small and the large AdS black holes.The speciﬁc heat C v ∼dT 2πR 2r 2h is negative for the small black hole and positive for the large black hole.This implies that the small black hole is in an unstable equilibrium with the radiation bath.At equilibrium,the Hawking radiation emitted and the radiation absorbed from the heat bath are equal.If the black hole emits a little more than it absorbs,it decreases in size,thereby increasing its temperature (because of the negative speciﬁc heat)which means it emits even more,implying the existence of a thermodynamic instability to decay to thermal AdS.If on the other hand,the small black hole emits inﬁnitesimally less than it ab-sorbs,it will grow in size until it becomes the large black hole.The large black hole is in stable thermal equilibrium by virtue of its positive speciﬁc heat.In the Euclidean setup,the thermodynamic instability of the small black hole manifests itself in the existence of a non-conformal negative mode in the small ﬂuctuation analysis.Although the small AdS 5Schwarzschild black hole is thermodynamically unsta-ble,it suﬀers from no classical dynamical instability,in the sense that there are no tachyonic small ﬂuctuation modes.This changes when we add extra dimensions along which the black hole is uniformly smeared.For example,in the context of type IIB string theory on asymptotically AdS 5×S 5background,the ﬁve dimensional small AdS black hole solution is smeared uniformly on the S 5.In what follows,we will review how this 10dimensional solution suﬀers from a classical dynamical instability to localization on the S 5[16].2.1Gregory-Laﬂamme instabilities and the Gubser-Mitra conjectureWhen black holes are smeared along some extra non compact “internal”direction,often the onset of a thermodynamic instability is accompanied by a classical dynam-ical instability signaled by the existence of tachyonic modes in a Lorentzian small ﬂuctuation analysis.In fact,this dynamical instability is of a Gregory-Laﬂamme type[12]:The instability is to localization in the extra directions.We will now review this in more detail.Such a link between thermodynamical and dynamical instabilitieswas conjectured by Gubser and Mitra[13,14].Further evidence for this connection was provided by Reall[15]and we brieﬂy review his argument by considering a black string in aﬁve dimensional asymptoticallyﬂat space[16].The black string inﬁve dimensions is translationally invariant in one spatial direction.It can be viewed as a four dimensional Schwarzschild black hole solution smeared uniformly along one extra non-compact direction.If we consider just the four dimensional black hole in the Euclidean section,the Euclidean Lichnerowicz operator∆(4)Ehas a negative eigen-mode∆(4)Ehµν=−η2hµν,(2.5) whereηis a real.The association of this Euclidean negative mode with the ther-modynamic instability signaled by a negative speciﬁc heat was made in[20].If we consider staticﬂuctuations in Lorentzian signature,the equations obtained are thesame as the Euclidean ones:∆(4)L =∆(4)Ewhen acting on staticﬂuctuations.However,(2.5)is not the relevant equation in the Lorentzian space.The lin-earized equation governing the smallﬂuctuations about a Lorentzian4D black hole solution is∆(4)Lhµν=0.(2.6) If,on the other hand,we are considering aﬁve-dimensional black string andﬂuc-tuations which carry momentum along the translationally invariantﬁfth direction, h(5)µν(x,z)=h(4)µν(x)e ikz,the relevantﬂuctuation equation involves aﬁve-dimensional Lichnerowicz operator:∆(5)L h(5)µν=(∆(4)Lh(4)µν+k2h(4)µν)e ikz=0.(2.7)This is the same as(2.5)with k2=η2.For this value of k,there is a“modulus”corre-sponding to this mode,which signals the onset of instability in Lorentzian signature. This was called the threshold unstable mode in[15].The above analysis may seem to imply that every small AdS black hole,when viewed as solution in AdS5×S5which is smeared on the S5should exhibit a Gregory-Laﬂamme like instability.However,the situation is more subtle if the internal smeared directions are compact,as was discussed in[16].This is simply because the momentum in the compact internal dimensions is quantized and this leads to a constraint on the negative Euclidean eigenvalue for the existence of a threshold un-stable mode.For the smeared AdS5×S5Schwarzschild case the negative eigenvalue has to be−l(l+4)unstable [16].This is smaller than r h =R2which is the largest possible horizonradius for the small black hole.In other words,although the small black hole exists for T ≥T 0=√πR =0.45R.2.2Hawking-Page transition and the big AdS Schwarzschild black hole The AdS/CFT correspondence relates the bulk string theory partition function on asymptotically AdS 5×S 5geometries to the partition function of the boundary SYM theory.In the semiclassical supergravity limit,the bulk partition function gets con-tributions from saddle points which are classical solutions to the equations of motion.Depending on the boundary geometry,there can be more than one bulk saddle points and in such a situation a careful sum [2,3]over all the saddle points is required.5For example,the appropriate boundary geometry to calculate the canonical en-semble partition function is S 3×S 1.For temperatures T >T 0=√πR ,there are three bulk saddle points with this boundary behavior.These are the two AdS Schwarzschild black holes (the small and the large)and the thermal AdS geome-try,which is Euclidean AdS with a periodic time direction.For temperatures lower than the critical temperature T 0,thermal AdS is the only saddle point.The diﬀerent saddle points correspond to diﬀerent phases,and the locally stable saddle point with the least action dominates the canonical ensemble.The actions of the three saddle points are formally inﬁnite because of the inﬁ-nite volume of thermal AdS and AdS Schwarzschild black holes.However,it turns 2 Π3 2ΠR T 0S G N Vol S 3Figure 3:The actions for the big and the small black holes as a function of temperatures.out that the diﬀerences between the actions of the diﬀerent solutions are ﬁnite and subtracting the thermal AdS action from the action of the AdS Schwarzschild black holes [9]yields ﬁnite results.Taking the thermal AdS action to be zero,the actionfor the AdS Schwarzschild black holes is given byS BH=Vol(S3)r3h(R2+2r2h).(2.8)It is easy to see that the action of the small black hole is always greater than both the action of the large black hole and that of thermal AdS(Figure3),so the small black hole never dominates the canonical ensemble.The action of the large black hole becomes less than that of thermal AdS at the Hawking-Page transition temper-ature T HP=3In the high temperature phase,although the uniform distribution remains asaddle point,it is not a minimum.The temperature at which the uniform distributionsaddle point ceases to be a minimum is the Hagedorn temperature,which correspondsto the temperature of the conﬁnement/deconﬁnement phase transition in the freeﬁeldlimit.Above the Hagedorn temperature,the minimum action eigenvalue distributionbecomes nonuniform and in particular,is gapped,i.e.the spectral densityρ(α)necessarily vanishes on a subset of the circle.This implies aﬁnite free energy forcolored external sources and is naturally associated to a deconﬁned phase.This isthe weak coupling version of the big AdS black hole.The small AdS black hole unstable saddle point cannot,however,be seen in thefreeﬁeld limit.It is expected to exist at weak,non-zero’t Hooft coupling[6,7,8].3.1Gregory-Laﬂamme inﬁeld theoryOur goal in this section is to see the analog of the Gregory-Laﬂamme instability inthe weakly coupled N=4SYM atﬁnite temperature on an S3.As we discussedin the previous section,this instability is associated with the localization of thesmeared small black hole on the S5,breaking the associated SO(6)isometry.In thedualﬁeld theory we would expect this phenomenon to manifest itself as an unstablesaddle point which breaks the SO(6)R-symmetry.Such symmetry breaking has tobe a subtle dynamical phenomenon for two reasons:i)The tree-level scalar potentialon S3×S1precludes a non-zero classical VEV for the scalarﬁelds in the N=4 multiplet;ii)ﬁeld theories on compact spaces do not usually exhibit spontaneoussymmetry breaking.To see the above phenomenon in weakly coupled N=4SYM we need to computea quantum eﬀective potential for the spatial zero modes of the scalarﬁelds whichare charged under the SO(6)R-symmetry.Thus in addition to the Polyakov loop,we will turn on the zero momentum modes of the six scalarﬁelds and obtain a jointeﬀective action for these degrees of freedom in perturbation theory.In addition,wewill also see that spontaneous breaking of the global R-symmetry can occur on acompact space,due to the large N limit.Finally,it is important to note that we will compute the quantum eﬀectivepotential along a special subspace in the space ofﬁeld conﬁgurations,namely onewhere all the scalarﬁelds and the Polyakov loop are simultaneously diagonalizable.This will be suﬃcient to explore the onset of the localization phenomenon which weare interested in.3.1.1The quantum eﬀective potentialIn the canonical ensemble,the thermal partition function of a quantumﬁeld theory is equal to the Euclidean path integral of the theory with a periodic time direction of periodβ=1R S1=RVol(S3) S3×S1A0,(3.2) andϕa=Tϕa are not zero modes of the classical theory since conformally coupled scalars have a mass of the order of the inverse radius of S 3,due to the curvature of the three-sphere.Tree level potential:g 2Y M d 4x √4F µνF µν+12R 2φa φa −1g 2Y M Tr −[α,ϕa ]2−1Computing the quantum corrections to the classical potential for generic homoge-neous background ﬁelds is technically diﬃcult.Instead,we will compute the quantum corrected potential for mutually commuting background ﬁeld conﬁgurations,namely those satisfying[α,ϕa ]=[ϕa ,ϕb ]=0.(3.7)This will allow us to explore only a slice of the full conﬁguration space.However it will be suﬃcient to address the issue of the presence of instabilities of the kind we are interested in.The classical potential on the space of these conﬁgurations isS (0)=βRπ21Hence,as indicated above,for the eﬀective potential calculation we shift φa →ϕa +φa where we assume that {ϕa }and the Polyakov loop αare simultaneously diagonal.The ﬂuctuations {φa }which will be integrated out are ﬂuctuations about the background,comprising of all the non-constant modes on S 3×S 1and oﬀ-diagonal components of the homogeneous modes.We denote indices along S 3as i =1,2,3(not to be confused with the gauge index)and that along S 1as 0.Unlike [6]who consider the case with vanishing background ﬁelds,we ﬁnd it easier to work in a covariant gauge.In particular,we choose a conventional “R ξgauge”of a spontaneously broken gauge theory.To this end,we add to the action the gauge ﬁxing term S gf =12ξd 4x √g 2Y M d 4x √2A 0(−˜D 20−∆(s )+ϕ2a )A 0+12φa (−˜D 20−∆(s )+R −2+ϕ2b )φa +¯c (−˜D 20−∆(s )+ϕ2a )c .(3.10)Here,∆(s )and ∆(v )are the Laplacians on S 3for scalar and gauge ﬁelds,respectively.The scalar Laplacian is simply ∇i ∇i acting on scalar functions whilst the vector Laplacian is deﬁned via the quadratic terms in the action: d 4x √4F µνF µν=(3.11) d 4x √2A µ(˜D 20+∆(v ))A µ−1where R ij is the Ricci tensor of S3.For the temporal component of the gaugeﬁeld, the Ricci tensor does not contribute and the vector Laplacian is equivalent to the scalar Laplacian:∆(v)A0=∆(s)A0.(3.13) The eigenvectors of the scalar Laplacian are spherical harmonics Yℓlabeled by angular momentumℓ∈Z≥0with∆(s)Yℓ=−ℓ(ℓ+2)R−2Yℓ,(3.14) and degeneracy(ℓ+1)2.The eigenvectors of the vector Laplacian can be split into two sets.Firstly,those in the image of∇i,i.e.∇i Yℓwithℓ>0,which satisfy∆(v)∇i Yℓ=−ℓ(ℓ+2)R−2∇i Yℓ,(3.15) and secondly those in kernel,∇i V iℓ=0,also labeled by the angular momentum ℓ>0,with∆(v)V iℓ=−(ℓ+1)2R−2V iℓ,(3.16) and degeneracy2ℓ(ℓ+2).We are now in position to integrate out all the bosonicﬂuctuations.For the vector modes,we mustﬁrst split the A i into the image and the kernel of∇i.To this end we write A i=B i+C i with∇i B i=0and C i=∇i f.Integrating these out along with A0,the ghosts and the six scalarﬂuctuations yields the following contribution to the one loop eﬀective potential from bosonic radiative corrections:S(1) b =12log detℓ>0(−˜D20−∆(s)+ϕ2)++(12log detℓ≥0(−˜D20−∆(s)+R−2+ϕ2a).(3.17) We now describe the various contributions in some detail.Theﬁrst two terms arise from integrating out the vectorﬂuctuations B i and C i respectively where the sub-scripts remind us to exclude theℓ=0mode of∆(s)as discussed above equations (3.15)and(3.16).The third term in the eﬀective potential results from the integrals over the ghost and A0ﬁelds,each of which contribute the same factor with weight −1and1S(1) b =12log detℓ=0(−˜D20+ϕ2a)+6×1ne−βεcos(nβα) .(3.19)Hereεis the energy of theﬂuctuation in question with x0interpreted as imaginary time and we have used the fact that the operator˜D0has eigenvalues(2πn/β+α)i; n∈Z.The remaining trace on the right hand side is to be taken over the modes on S3and the gauge group.The gauge trace can be made explicit by using the fact thatαandϕwere chosen to be simultaneously diagonalα=β−1diag(θ1,θ2,...θN),ϕ=diag(ϕ1,ϕ2,...ϕN),(3.20) which in turn yields in generalTr f(ϕ,α)=Nij=1f(ϕij,αij);ϕij=ϕi−ϕj,αij=β−1(θi−θj).(3.21)For SU(N)gauge group theθi andϕi must satisfyNi=1ϕi=0,N i=1θi=0mod2π.(3.22)These conditions are a consequence of the Hermiticity of the scalar and gaugeﬁelds. Theθi are each thermal Wilson lines for the Cartan components of the gaugeﬁeld. Hence they can be shifted by an integer multiple of2πby performing a topologically non-trivial gauge transformation which is single-valued in SU(N)/Z N.This must however be an invariance of the theory since there are noﬁelds charged under the center Z N of SU(N).This results in theθi being deﬁned only up to an integer multiple of2π.Including the explicit sums over the angular momenta on S3with the appropriate degeneracies,we can now write down the complete one-loop eﬀective action resulting from integrating out all the bosonicﬂuctuations,S(1) b =Nij=1 −1ϕ2aij+∞ n=11ϕ2aij cos(nθij)+ ℓ=12ℓ(ℓ+2) β(ℓ+1)2R−2+ϕ2aij−∞ n=11(ℓ+1)2R−2+ϕ2aij cos(nθij) +6 ℓ=0(ℓ+1)2 β(ℓ+1)2R−2+ϕ2aij−∞ n=11(ℓ+1)2R−2+ϕ2aij cos(nθij) .(3.23)Finally,to complete the calculation of the eﬀective action we have to include integrals over the four Weyl fermions.The eﬀect of the backgroundﬁelds is very simple on these modes,it simply induces a mass squaredϕ2a for the fermions via their Yukawa couplings.The fermions are eigenfunctions of the spinor Laplacian on S3which are also labeled by the angular momentumℓ>0with eigenvalue−(ℓ+12 2)2R−2+ϕ2aij−∞ n=1(−1)n+1(ℓ+1g2 Y MNi=1ϕ2ai,(3.26)ﬁeld angular mom.energy degeneracy weight (ℓ+1)2R −2+ϕ22ℓ(ℓ+2)1ℓ(ℓ+2)R −2+ϕ2(ℓ+1)21ℓ(ℓ+2)R −2+ϕ2(ℓ+1)2−1A 0ℓ≥0 2φaℓ≥0 2ψA αℓ>0 2)2R −2+ϕ22ℓ(ℓ+1)−1We now make the following observations about our result:•Firstly,when the scalar ﬁelds ϕa vanish we reproduce the results of[6].To see how this works,we note that the ﬁrst term in the one-loop eﬀective potential in the absence of the background scalars reduces tolog i<jsin2 θij 2 .(3.28)Here we have obtained this result following a diﬀerent route from that adopted by [6].So when the scalars vanish it is easy to see that we reproduce the eﬀective action written down in [6].•Our eﬀective potential was obtained by integrating out not only the non-zero momentum modes on S3×S1,but also the oﬀ-diagonalﬂuctuations ofαand of the zero modes(n=ℓ=0)ofφa.(In principle one also integrates out ﬂuctuations of the diagonal pieces but these give no nontrivial contributions, due to the adjoint action of the backgroundﬁelds on them).Note that the oﬀ-diagonal zero modes have massesϕ2aij actually cancels against an identical piece emerging from the inﬁnite sums overℓ.•The bosonic and fermionic determinants come with opposite signs as they must. It is,however,important to realize that they do not cancel against each other due both toﬁnite temperature eﬀects and the fact that the theory is formu-lated on a spatial three-sphere.A complete cancellation between bosonic and fermionicﬂuctuations only occurs inﬂat space and at zero temperature.We will come back to this point in the following subsection.•The eﬀective potential that we have obtained can be interpreted as a1PI-eﬀective action for the eigenvalues ofαand theϕa and yields the one loop partition function about a given conﬁguration of eigenvalues.At local minima of the action,we may interpret the partition function as a thermodynamic partition function whose logarithm is the thermodynamic free energy.For generic points in the conﬁguration space ofﬁeld eigenvalues,the system will not be in a stable or static conﬁguration.Nevertheless,we may formally deﬁnea free energy asF(ϕ,θ)=S eﬀ/β.(3.30)We are interested both in extrema of this eﬀective potential and the existence of and emergence of any new unstable directions as a function of temperature.•Finally,saddle points computed using the eﬀective potential will be true saddle points in the full conﬁgurations space,even though we are setting to zero the background values of the oﬀ-diagonal constant modes.The reason for this is that,in our chosen background of diagonalﬁelds,the oﬀ-diagonalﬂuctuations only appear quadratically and there are no terms linear in theseﬂuctuations.This means that there are extrema of the full eﬀective action where these can be consistently set to zero and these are the saddle points that we willﬁnd.However,there may well be other extrema where theﬁeldsϕandαare not simultaneously diagonal.3.2The zero temperature limitIn this section we discuss the detailed form of the one loop eﬀective potential at zero temperature.This is not relevant for the high temperature analysis which is our main focus.However,the limit of zero temperature will provide checks of our calculation and also important intuition about the behaviour of the one-loop corrections.The terms that survive in the zero temperature limit,β=∞,are basically Casimir contributions in the presence of background expectation values,and are identiﬁed as the terms without any Boltzmann suppression factors in the eﬀective potential.The total Casimir free energy is thereforeF0≡Nij=1C(ϕ2aij),(3.31)。

半导体双极器件名词解释

1、Abrupt junction approximation （突变结近似）The assumption that there is an abrupt discontinuity in space charge density between the space charge region and neutral semiconductor region.认为从中性半导体区到空间电荷区的空间电荷密度有一个突然的不连续。

2、Depletion layer approximation （耗尽层近似）The number of carriers is almost zero due to the strong built-in electric field in the space charge region, that the charge in the space charge region is almost completely provided ionized impurity, this space charge region is called depletion layer.由于空间电荷区较强的内建电场，载流子的数量几乎为零，因此可以认为空间电荷区中的电荷几乎完全是由电离杂质所提供的，这种空间电荷区就称为耗尽层。

3、Built-in electric field （内建电场）An electric field due to the separation of positive and negative space charge densities in the depletion region.由于耗尽区正负空间电荷相互分离而形成的电场。

4、Built-in potential harrier （内建电势差）The electrostatic potential difference between the p and n regions of a pn junction in thermal equilibrium.热平衡状态下pn结内p区与n区的静电电势差。

振动方面的专业英语及词汇

振动方面的专业英语及词汇1 振动信号的时域、频域描述振动过程 (Vibration Process)简谐振动 (Harmonic Vibration)周期振动 (Periodic Vibration)准周期振动 (Ouasi-periodic Vibration)瞬态过程 (Transient Process)随机振动过程 (Random Vibration Process)各态历经过程 (Ergodic Process)确定性过程 (Deterministic Process)振幅 (Amplitude)相位 (Phase)初相位 (Initial Phase)频率 (Frequency)角频率 (Angular Frequency)周期 (Period)复数振动 (Complex Vibration)复数振幅 (Complex Amplitude)峰值 (Peak-value)平均绝对值 (Average Absolute Value)有效值 (Effective Value，RMS Value)均值 (Mean Value，Average Value)傅里叶级数 (FS，Fourier Series)傅里叶变换 (FT，Fourier Transform)傅里叶逆变换 (IFT，Inverse Fourier Transform)离散谱 (Discrete Spectrum)连续谱 (Continuous Spectrum)傅里叶谱 (Fourier Spectrum)线性谱 (Linear Spectrum)幅值谱 (Amplitude Spectrum)相位谱 (Phase Spectrum)均方值 (Mean Square Value)方差 (Variance)协方差 (Covariance)自协方差函数 (Auto-covariance Function)互协方差函数 (Cross-covariance Function)自相关函数 (Auto-correlation Function)互相关函数 (Cross-correlation Function)标准偏差 (Standard Deviation)相对标准偏差 (Relative Standard Deviation)概率 (Probability)概率分布 (Probability Distribution)高斯概率分布 (Gaussian Probability Distribution) 概率密度 (Probability Density)集合平均 (Ensemble Average)时间平均 (Time Average)功率谱密度 (PSD，Power Spectrum Density)自功率谱密度 (Auto-spectral Density)互功率谱密度 (Cross-spectral Density)均方根谱密度 (RMS Spectral Density)能量谱密度 (ESD，Energy Spectrum Density)相干函数 (Coherence Function)帕斯瓦尔定理 (Parseval''''s Theorem)维纳，辛钦公式 (Wiener-Khinchin Formula2 振动系统的固有特性、激励与响应振动系统 (Vibration System)激励 (Excitation)响应 (Response)单自由度系统 (Single Degree-Of-Freedom System) 多自由度系统 (Multi-Degree-Of- Freedom System) 离散化系统 (Discrete System)连续体系统 (Continuous System)刚度系数 (Stiffness Coefficient)自由振动 (Free Vibration)自由响应 (Free Response)强迫振动 (Forced Vibration)强迫响应 (Forced Response)初始条件 (Initial Condition)固有频率 (Natural Frequency)阻尼比 (Damping Ratio)衰减指数 (Damping Exponent)阻尼固有频率 (Damped Natural Frequency)对数减幅系数 (Logarithmic Decrement)主频率 (Principal Frequency)无阻尼模态频率 (Undamped Modal Frequency)模态 (Mode)主振动 (Principal Vibration)振型 (Mode Shape)振型矢量 (Vector Of Mode Shape)模态矢量 (Modal Vector)正交性 (Orthogonality)展开定理 (Expansion Theorem)主质量 (Principal Mass)模态质量 (Modal Mass)主刚度 (Principal Stiffness)模态刚度 (Modal Stiffness)正则化 (Normalization)振型矩阵 (Matrix Of Modal Shape)模态矩阵 (Modal Matrix)主坐标 (Principal Coordinates)模态坐标 (Modal Coordinates)模态分析 (Modal Analysis)模态阻尼比 (Modal Damping Ratio)频响函数 (Frequency Response Function)幅频特性 (Amplitude-frequency Characteristics)相频特性 (Phase frequency Characteristics)共振 (Resonance)半功率点 (Half power Points)波德图(Bodé Plot)动力放大系数 (Dynamical Magnification Factor)单位脉冲 (Unit Impulse)冲激响应函数 (Impulse Response Function)杜哈美积分(Duhamel’s Integral)卷积积分 (Convolution Integral)卷积定理 (Convolution Theorem)特征矩阵 (Characteristic Matrix)阻抗矩阵 (Impedance Matrix)频响函数矩阵 (Matrix Of Frequency Response Function) 导纳矩阵 (Mobility Matrix)冲击响应谱 (Shock Response Spectrum)冲击激励 (Shock Excitation)冲击响应 (Shock Response)冲击初始响应谱 (Initial Shock Response Spectrum)冲击剩余响应谱 (Residual Shock Response Spectrum)冲击最大响应谱 (Maximum Shock Response Spectrum)冲击响应谱分析 (Shock Response Spectrum Analysis3 模态试验分析模态试验 (Modal Testing)机械阻抗 (Mechanical Impedance)位移阻抗 (Displacement Impedance)速度阻抗 (Velocity Impedance)加速度阻抗 (Acceleration Impedance)机械导纳 (Mechanical Mobility)位移导纳 (Displacement Mobility)速度导纳 (Velocity Mobility)加速度导纳 (Acceleration Mobility)驱动点导纳 (Driving Point Mobility)跨点导纳 (Cross Mobility)传递函数 (Transfer Function)拉普拉斯变换 (Laplace Transform)传递函数矩阵 (Matrix Of Transfer Function)频响函数 (FRF，Frequency Response Function)频响函数矩阵 (Matrix Of FRF)实模态 (Normal Mode)复模态 (Complex Mode)模态参数 (Modal Parameter)模态频率 (Modal Frequency)模态阻尼比 (Modal Damping Ratio)模态振型 (Modal Shape)模态质量 (Modal Mass)模态刚度 (Modal Stiffness)模态阻力系数 (Modal Damping Coefficient)模态阻抗 (Modal Impedance)模态导纳 (Modal Mobility)模态损耗因子 (Modal Loss Factor)比例粘性阻尼 (Proportional Viscous Damping)非比例粘性阻尼 (Non-proportional Viscous Damping)结构阻尼 (Structural Damping，Hysteretic Damping)复频率 (Complex Frequency)复振型 (Complex Modal Shape)留数 (Residue)极点 (Pole)零点 (Zero)复留数 (Complex Residue)随机激励 (Random Excitation)伪随机激励 (Pseudo Random Excitation)猝发随机激励 (Burst Random Excitation)稳态正弦激励 (Steady State Sine Excitation)正弦扫描激励 (Sweeping Sine Excitation)锤击激励 (Impact Excitation)频响函数的H1 估计 (FRF Estimate by H1)频响函数的H2 估计 (FRF Estimate by H2)频响函数的H3 估计 (FRF Estimate by H3)单模态曲线拟合法 (Single-mode Curve Fitting Method)多模态曲线拟合法 (Multi-mode Curve Fitting Method)模态圆 (Mode Circle)剩余模态 (Residual Mode)幅频峰值法 (Peak Value Method)实频-虚频峰值法 (Peak Real／Imaginary Method)圆拟合法 (Circle Fitting Method)加权最小二乘拟合法 (Weighting Least Squares Fitting method) 复指数拟合法 (Complex Exponential Fitting method)1.2 振动测试的名词术语1 传感器测量系统传感器测量系统 (Transducer Measuring System)传感器 (Transducer)振动传感器 (Vibration Transducer)机械接收 (Mechanical Reception)机电变换 (Electro-mechanical Conversion)测量电路 (Measuring Circuit)惯性式传感器 (Inertial Transducer，Seismic Transducer)相对式传感器 (Relative Transducer)电感式传感器 (Inductive Transducer)应变式传感器 (Strain Gauge Transducer)电动力传感器 (Electro-dynamic Transducer)压电式传感器 (Piezoelectric Transducer)压阻式传感器 (Piezoresistive Transducer)电涡流式传感器 (Eddy Current Transducer)伺服式传感器 (Servo Transducer)灵敏度 (Sensitivity)复数灵敏度 (Complex Sensitivity)分辨率 (Resolution)频率范围 (Frequency Range)线性范围 (Linear Range)频率上限 (Upper Limit Frequency)频率下限 (Lower Limit Frequency)静态响应 (Static Response)零频率响应 (Zero Frequency Response)动态范围 (Dynamic Range)幅值上限 Upper Limit Amplitude)幅值下限 (Lower Limit Amplitude)最大可测振级 (Max．Detectable Vibration Level)最小可测振级 (Min．Detectable Vibration Level)信噪比 (S/N Ratio)振动诺模图 (Vibration Nomogram)相移 (Phase Shift)波形畸变 (Wave-shape Distortion)比例相移 (Proportional Phase Shift)惯性传感器的稳态响应 (Steady Response Of Inertial Transducer)惯性传感器的稳击响应 (Shock Response Of Inertial Transducer)位移计型的频响特性 (Frequency Response Characteristics Vibrometer)加速度计型的频响特性 (Frequency Response Characteristics Accelerometer) 幅频特性曲线 (Amplitude-frequency Curve)相频特性曲线 (Phase-frequency Curve)固定安装共振频率 (Mounted Resonance Frequency)安装刚度 (Mounted Stiffness)有限高频效应 (Effect Of Limited High Frequency)有限低频效应 (Effect Of Limited Low Frequency)电动式变换 (Electro-dynamic Conversion)磁感应强度 (Magnetic Induction， Magnetic Flux Density)磁通 (Magnetic Flux)磁隙 (Magnetic Gap)电磁力 (Electro-magnetic Force)相对式速度传 (Relative Velocity Transducer)惯性式速度传感器 (Inertial Velocity Transducer)速度灵敏度 (Velocity Sensitivity)电涡流阻尼 (Eddy-current Damping)无源微(积)分电路 (Passive Differential (Integrate) Circuit) 有源微(积)分电路 (Active Differential (Integrate) Circuit) 运算放大器 (Operational Amplifier)时间常数 (Time Constant)比例运算 (Scaling)积分运算 (Integration)微分运算 (Differentiation)高通滤波电路 (High-pass Filter Circuit)低通滤波电路 (Low-pass Filter Circuit)截止频率 (Cut-off Frequency)压电效应 (Piezoelectric Effect)压电陶瓷 (Piezoelectric Ceramic)压电常数 (Piezoelectric Constant)极化 (Polarization)压电式加速度传感器 (Piezoelectric Acceleration Transducer) 中心压缩式 (Center Compression Accelerometer)三角剪切式 (Delta Shear Accelerometer)压电方程 (Piezoelectric Equation)压电石英 (Piezoelectric Quartz)电荷等效电路 (Charge Equivalent Circuit)电压等效电路 (Voltage Equivalent Circuit)电荷灵敏度 (Charge Sensitivity)电压灵敏度 (Voltage Sensitivity)电荷放大器 (Charge Amplifier)适调放大环节 (Conditional Amplifier Section)归一化 (Uniformization)电荷放大器增益 (Gain Of Charge Amplifier)测量系统灵敏度 (Sensitivity Of Measuring System)底部应变灵敏度 (Base Strain Sensitivity)横向灵敏度 (Transverse Sensitivity)地回路 (Ground Loop)力传感器 (Force Transducer)力传感器灵敏度 (Sensitivity Of Force Transducer)电涡流 (Eddy Current)前置器 (Proximitor)间隙-电压曲线 (Voltage vs Gap Curve)间隙-电压灵敏度 (Voltage vs Gap Sensitivity)压阻效应 (Piezoresistive Effect)轴向压阻系数 (Axial Piezoresistive Coefficient)横向压阻系数 (Transverse Piezoresistive Coefficient)压阻常数 (Piezoresistive Constant)单晶硅 (Monocrystalline Silicon)应变灵敏度 (Strain Sensitivity)固态压阻式加速度传感器 (Solid State Piezoresistive Accelerometer) 体型压阻式加速度传感器 (Bulk Type Piezoresistive Accelerometer) 力平衡式传感器 (Force Balance Transducer)电动力常数 (Electro-dynamic Constant)机电耦合系统 (Electro-mechanical Coupling System)2 检测仪表、激励设备及校准装置时间基准信号 (Time Base Signal)李萨茹图 (Lissojous Curve)数字频率计 (Digital Frequency Meter)便携式测振表 (Portable Vibrometer)有效值电压表 (RMS Value Voltmeter)峰值电压表 (Peak-value Voltmeter)平均绝对值检波电路 (Average Absolute Value Detector)峰值检波电路 (Peak-value Detector)准有效值检波电路 (Quasi RMS Value Detector)真有效值检波电路 (True RMS Value Detector)直流数字电压表 (DVM，DC Digital Voltmeter)数字式测振表 (Digital Vibrometer)A/D 转换器 (A/D Converter)D/A 转换器 (D/A Converter)相位计 (Phase Meter)电子记录仪 (Lever Recorder)光线示波器 (Oscillograph)振子 (Galvonometer)磁带记录仪 (Magnetic Tape Recorder)DR 方式(直接记录式) (Direct Recorder)FM 方式(频率调制式) (Frequency Modulation)失真度 (Distortion)机械式激振器 (Mechanical Exciter)机械式振动台 (Mechanical Shaker)离心式激振器 (Centrifugal Exciter)电动力式振动台 (Electro-dynamic Shaker)电动力式激振器 (Electro-dynamic Exciter)液压式振动台 (Hydraulic Shaker)液压式激振器 (Hydraulic Exciter)电液放大器 (Electro-hydraulic Amplifier)磁吸式激振器 (Magnetic Pulling Exciter)涡流式激振器 (Eddy Current Exciter)压电激振片 (Piezoelectric Exciting Elements)冲击力锤 (Impact Hammer)冲击试验台 (Shock Testing Machine)激振控制技术 (Excitation Control Technique)波形再现 (Wave Reproduction)压缩技术 (Compression Technique)均衡技术 (Equalization Technique)交越频率 (Crossover Frequency)综合技术 (Synthesis Technique)校准 (Calibration)分部校准 (Calibration for Components in system)系统校准 (Calibration for Over-all System)模拟传感器 (Simulated Transducer)静态校准 (Static Calibration)简谐激励校准 (Harmonic Excitation Calibration)绝对校准 (Absolute Calibration)相对校准 (Relative Calibration)比较校准 (Comparison Calibration)标准振动台 (Standard Vibration Exciter)读数显微镜法 (Microscope-streak Method)光栅板法 (Ronchi Ruling Method)光学干涉条纹计数法 (Optical Interferometer Fringe Counting Method)光学干涉条纹消失法 (Optical Interferometer Fringe Disappearance Method) 背靠背安装 (Back-to-back Mounting)互易校准法 (Reciprocity Calibration)共振梁 (Resonant Bar)冲击校准 (Impact Exciting Calibration)摆锤冲击校准 (Ballistic Pendulum Calibration)落锤冲击校准 (Drop Test Calibration)振动和冲击标准 (Vibration and Shock Standard)迈克尔逊干涉仪 (Michelson Interferometer)摩尔干涉图象 (Moire Fringe)参考传感器 (Reference Transducer)3 频率分析及数字信号处理带通滤波器 (Band-pass Filter)半功率带宽 (Half-power Bandwidth)3 dB 带宽 (3 dB Bandwidth)等效噪声带宽 (Effective Noise Bandwidth)恒带宽 (Constant Bandwidth)恒百分比带宽 (Constant Percentage Bandwidth)1/N 倍频程滤波器 (1/N Octave Filter)形状因子 (Shape Factor)截止频率 (Cut-off Frequency)中心频率 (Centre Frequency)模拟滤波器 (Analog Filter)数字滤波器 (Digital Filter)跟踪滤波器 (Tracking Filter)外差式频率分析仪 (Heterodyne Frequency Analyzer) 逐级式频率分析仪 (Stepped Frequency Analyzer)扫描式频率分析仪 (Sweeping Filter Analyzer)混频器 (Mixer)RC 平均 (RC Averaging)平均时间 (Averaging Time)扫描速度 (Sweeping Speed)滤波器响应时间 (Filter Response Time)离散傅里叶变换 (DFT，Discrete Fourier Transform) 快速傅里叶变换 (FFT，Fast Fourier Transform)抽样频率 (Sampling Frequency)抽样间隔 (Sampling Interval)抽样定理 (Sampling Theorem)抗混滤波 (Anti-aliasing Filter)泄漏 (Leakage)加窗 (Windowing)窗函数 (Window Function)截断 (Truncation)频率混淆 (Frequency Aliasing)乃奎斯特频率 (Nyquist Frequency)矩形窗 (Rectangular Window)汉宁窗 (Hanning Window)凯塞-贝塞尔窗 (Kaiser-Bessel Window)平顶窗 (Flat-top Window)平均 (Averaging)线性平均 (Linear Averaging)指数平均 (Exponential Averaging)峰值保持平均 (Peak-hold Averaging)时域平均 (Time-domain Averaging)谱平均 (Spectrum Averaging)重叠平均 (Overlap Averaging)栅栏效应 (Picket Fence Effect)吉卜斯效应 (Gibbs Effect)基带频谱分析 (Base-band Spectral Analysis)选带频谱分析 (Band Selectable Sp4ctralAnalysis) 细化 (Zoom)数字移频 (Digital Frequency Shift)抽样率缩减 (Sampling Rate Reduction)功率谱估计 (Power Spectrum Estimate)相关函数估计 (Correlation Estimate)频响函数估计 (Frequency Response Function Estimate) 相干函数估计 (Coherence Function Estimate)冲激响应函数估计 (Impulse Response Function Estimate) 倒频谱 (Cepstrum)功率倒频谱 (Power Cepstrum)幅值倒频谱 (Amplitude Cepstrum)倒频率 (Quefrency)4 旋转机械的振动测试及状态监测状态监测 (Condition Monitoring)故障诊断 (Fault Diagnosis)转子 (Rotor)转手支承系统 (Rotor-Support System)振动故障 (Vibration Fault)轴振动 (Shaft Vibration)径向振动 (Radial Vibration)基频振动 (Fundamental Frequency Vibration)基频检测 (Fundamental Frequency Component Detecting) 键相信号 (Key-phase Signal)正峰相位 (+Peak Phase)高点 (High Spot)光电传感器 (Optical Transducer)同相分量 (In-phase Component)正交分量 (Quadrature Component)跟踪滤波 (Tracking Filter)波德图 (Bode Plot)极坐标图 (Polar Plot)临界转速 (Critical Speed)不平衡响应 (Unbalance Response)残余振幅 (Residual Amplitude)方位角 (Attitude Angle)轴心轨迹 (Shaft Centerline Orbit)正进动 (Forward Precession)同步正进动 (Synchronous Forward Precession)反进动 (Backward Precession)正向涡动 (Forward Whirl)反向涡动 (Backward Whirl)油膜涡动 (Oil Whirl)油膜振荡 (Oil Whip)轴心平均位置 (Average Shaft Centerline Position)复合探头 (Dual Probe)振摆信号 (Runout Signal)电学振摆 (Electrical Runout)机械振摆 (Mechanical Runout)慢滚动向量 (Slow Roll Vector)振摆补偿 (Runout Compensation)故障频率特征 (Frequency Characteristics Of Fault) 重力临界 (Gravity Critical)对中 (Alignment)双刚度转子 (Dual Stiffness Rotor)啮合频率 (Gear-mesh Frequency)间入简谐分量 (Interharmonic Component)边带振动 (Side-band Vibration)三维频谱图 (Three Dimensional Spectral Plot)瀑布图 (Waterfall Plot)级联图 (Cascade Plot)阶次跟踪 (Order Tracking)阶次跟踪倍乘器 (Order Tracking Multiplier)监测系统 (Monitoring System)适调放大器 (Conditional Amplifier)趋势分析 (Trend Analysis)倒频谱分析 (Cepstrum Analysis)直方图 (Histogram)确认矩阵 (Confirmation Matrix)通频幅值 (Over-all Amplitude)幅值谱 (Amplitude Spectrum)相位谱 (Phase Spectrum)报警限 (Alarm Level)。

PHASE COHERENCE PHENOMENA IN DISORDERED SUPERCONDUCTORS

ψ↑ (r) =
i
† ∗ γi↑ ui (r) − γi ↓ vi (r) ,
ψ↓ (r) =
i
† ∗ γi↓ ui (r) + γi ↑ vi (r)
the Hamiltonian can be brought to a diagonal form by choosing the spinor elements uα (r) and vα (r) to satisfy the coupled Bogoliubov-de Gennes (BdG)
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A. LAMACRAFT AND B. D. SIMONS
Before turning to the construction of the ﬁeld theoretic scheme, we will begin these notes with a qualitative discussion of phase coherence phenomena in the superconducting environment placing emphasis on the importance of fundamental symmetries. To close the introductory section, we will outline the quasi-classical theory which forms the basis of the ﬁeld theoretic scheme. In section 2 we will develop a quantum ﬁeld theory of the weakly disordered non-interacting superconducting system (i.e. in the mean-ﬁeld BCS approximation). To illustrate a simple application of this technique, we will explore the spectral properties of a normal quantum dot contacted to a superconducting terminal. Finally, in section 3, we will present a detailed study of the inﬂuence of magnetic impurities in the disordered superconducting system. This single application will emphasize a number of generic features of the phase coherent superconducting system including unusual spectral and localization properties and the importance of effects non-perturbative in the disorder. To orient our discussion, however, let us ﬁrst brieﬂy recapitulate the BCS mean-ﬁeld theory of superconductivity in order to establish some notations and deﬁnitions.

The Refractive Index of Curved Spacetime the Fate of Causality in QED

$The Refractive Index of Curved Spacetime the Fate of Causality in QED$

a r X i v :0707.2303v 2 [h e p -t h ] 29 O c t 2007Preprint typeset in JHEP style -HYPER VERSIONTimothy J.Hollowood and Graham M.Shore Department of Physics,University of Wales Swansea,Swansea,SA28PP,UK.E-mail:t.hollowood@,g.m.shore@ Abstract:It has been known for a long time that vacuum polarization in QED leads to a superluminal low-frequency phase velocity for light propagating in curved spacetime.Assuming the validity of the Kramers-Kronig dispersion relation,this would imply a superluminal wavefront velocity and the violation of causality.Here,we calculate for the ﬁrst time the full frequency dependence of the refractive index using world-line sigma model techniques together with the Penrose plane wave limit of spacetime in the neighbourhood of a null geodesic.We ﬁnd that the high-frequency limit of the phase velocity (i.e.the wavefront velocity)is always equal to c andcausality is assured.However,the Kramers-Kronig dispersion relation is violated due to a non-analyticity of the refractive index in the upper-half complex plane,whose origin may be traced to the generic focusing property of null geodesic congruences and the existence of conjugate points.This puts into question the issue of micro-causality,i.e.the vanishing of commutators of ﬁeld operators at spacelike separated points,in local quantum ﬁeld theory in curved spacetime.1.IntroductionQuantumﬁeld theory in curved spacetime is by now a well-understood subject.How-ever,there remain a number of intriguing puzzles which hint at deeper conceptual implications for quantum gravity itself.The best known is of course Hawking radia-tion and the issue of entropy and holography in quantum black hole physics.A less well-known eﬀect is the discovery by Drummond and Hathrell[1]that vacuum po-larization in QED can induce a superluminal phase velocity for photons propagating in a non-dynamical,curved spacetime.The essential idea is illustrated in Figure1. Due to vacuum polarization,the photon may be pictured as an electron-positron pair, characterized by a length scaleλc=m−1,the Compton wavelength of the electron. When the curvature scale becomes comparable toλc,the photon dispersion relation is modiﬁed.The remarkable feature,however,is that this modiﬁcation can induce a superluminal1low-frequency phase velocity,i.e.the photon momentum becomes spacelike.Figure1:Photons propagating in curved spacetime feel the curvature in the neighbourhood of their geodesic because they can become virtual e+e−pairs.Atﬁrst,it appears that this must be incompatible with causality.However, as discussed in refs.[2–4],the relation of causality with the“speed of light”is far more subtle.For our purposes,we may provisionally consider causality to be the requirement that no signal may travel faster than the fundamental constant c deﬁning local Lorentz invariance.More precisely,we require that the wavefront velocity v wf, deﬁned as the speed of propagation of a sharp-fronted wave pulse,should be less than,or equal to,c.Importantly,it may be shown[2,4,5]that v wf=v ph(∞),the high-frequency limit of the phase velocity.In other words,causality is safe even if the low-frequency2phase velocity v ph(0)is superluminal provided the high-frequency limit does not exceed c.This appears to remove the potential paradox associated with a superluminal v ph(0).However,a crucial constraint is imposed by the Kramers-Kronig dispersion relation3(see,e.g.ref.[6],chpt.10.8)for the refractive index,viz.Re n(∞)−Re n(0)=−2ωIm n(ω).(1.1)where Re n(ω)=1/v ph(ω).The positivity of Im n(ω),which is true for an absorptive medium and is more generally a consequence of unitarity in QFT,then implies that Re n(∞)<Re n(0),i.e.v ph(∞)>v ph(0).So,given the validity of the KK dispersion relation,a superluminal v ph(0)would imply a superluminal wavefront velocity v wf= v ph(∞)with the consequent violation of causality.We are therefore left with three main options[4],each of which would have dramatic consequences for our established ideas about quantumﬁeld theory: Option(1)The wavefront speed of light v wf>1and the physical lightcones lie outside the geometric null cones of the curved spacetime,inapparent violation of causality.It should be noted,however,that while this would certainly violate causality for theories in Minkowski spacetime,it could still be possible for causality to be preserved in curved spacetime if the eﬀective metric characterizing the physical light cones deﬁned by v wf nevertheless allow the existence of a global timelike Killing vectorﬁeld. This possible loophole exploits the general relativity notion of“stable causality”[8,9] and is discussed further in ref.[2].Option(2)Curved spacetime may behave as an optical medium ex-hibiting gain,i.e.Im n(ω)<0.This possibility was explored in the context ofΛ-systems in atomic physics in ref.[4], where laser-atom interactions can induce gain,giving rise to a negative Im n(ω)and superluminal low-frequency phase velocities while preserving v wf=1and the KKdispersion relation.However,the problem in extending this idea to QFT is that the optical theorem,itself a consequence of unitarity,identiﬁes the imaginary part of forward scattering amplitudes with the total cross section.Here,Im n(ω)should be proportional to the cross section for e+e−pair creation and therefore positive.A negative Im n(ω)would appear to violate unitarity.Option(3)The Kramers-Kronig dispersion relation(1.1)is itself vio-lated.Note,however,that this relation only relies on the analyticity ofn(ω)in the upper-half plane,which is usually considered to be a directconsequence of an apparently fundamental axiom of local quantumﬁeldtheory,viz.micro-causality.Micro-causality in QFT is the requirement that the expectation value of the com-mutator ofﬁeld operators 0|[A(x),A(y)]|0 vanishes when x and y are spacelike separated.While this appears to be a clear statement of what we would understand by causality at the quantum level,in fact its primary rˆo le in conventional QFT is as a necessary condition for Lorentz invariance of the S-matrix(see e.g.ref.[6], chpts.5.1,3.5).Since QFT in curved spacetime is only locally,and not globally, Lorentz invariant,it is just possible there is a loophole here allowing violation of micro-causality in curved spacetime QFT.Despite these various caveats,unitarity,micro-causality,the identiﬁcation of light cones with geometric null cones and causality itself are all such fundamental elements of local relativistic QFT that any one of these options would represent a major surprise and pose a severe challenge to established wisdom.Nonetheless,it appears that at least one has to be true.To understand how QED in curved spacetime is reconciled with causality,it is therefore necessary to perform an explicit calculation to determine the full frequency dependence of the refractive index n(ω)in curved spacetime.This is the technical problem which we solve in this paper.The remarkable result is that QED chooses option(3),viz.analyticity is violated in curved spacetime.Weﬁnd that in the high-frequency limit,the phase velocity always approaches c,so we determine v wf= 1.Moreover,we are able to conﬁrm that where the background gravitationalﬁeld induces pair creation,γ→e+e−,Im n(ω)is indeed positive as required by unitarity. However,the refractive index n(ω)is not analytic in the upper half-plane,and the KK dispersion relation is modiﬁed accordingly.One might think that this implies a violation of microcausality,however,there is a caveat in this line of argument which requires a more ambitious oﬀ-shell calculation to settle deﬁnitively[7].–3–In order to establish this result,we have had to apply radically new techniques to the analysis of the vacuum polarization for QED in curved spacetime.The original Drummond-Hathrell analysis was based on the low-energy,O(R/m2)eﬀective action for QED in a curved background,L=−1m2 aRFµνFµν+bRµνFµλFνλ+cRµνλρFµνFλρ +···.(1.2) derived using conventional heat-kernel or proper-time techniques(see,for example, [10–14].A geometric optics,or eikonal,analysis applied to this action determines the low-frequency limit of the phase velocity.Depending on the spacetime,the photon trajectory and its polarization,v ph(0)may be superluminal[1,15,16].In subsequent work,the expansion of the eﬀective action to all orders in derivatives,but still at O(R/m2),was evaluated and applied to the photon dispersion relation[11,12,17, 18].However,as emphasized already in refs.[2,3,18],the derivative expansion is inadequate toﬁnd the high-frequency behaviour of the phase velocity.The reason is that the frequencyωappears in the on-shell vacuum polarization tensor only in the dimensionless ratioω2R/m4.The high-frequency limit depends non-perturbatively on this parameter4and so is not accessible to an expansion truncated atﬁrst order in R/m2.In this paper,we instead use the world-line formalism which can be traced back to Feynman and Schwinger[19,20],and which has been extensively developed in recent years into a powerful tool for computing Green functions in QFT via path integrals for an appropriate1-dim world-line sigma model.(For a review,see e.g.ref.[21].) The power of this technique in the present context is that it enables us to calculate the QED vacuum polarization non-perturbatively in the frequency parameterω2R/m4 using saddle-point techniques.Moreover,the world-line sigma model provides an extremely geometric interpretation of the calculation of the quantum corrections to the vacuum polarization.In particular,we are able to give a very direct interpretation of the origin of the Kramers-Kronig violating poles in n(ω)in terms of the general relativistic theory of null congruences and the relation of geodesic focusing to the Weyl and Ricci curvatures via the Raychoudhuri equations.A further key insight is that to leading order in R/m2,but still exact inω2R/m4, the relevant tidal eﬀects of the curvature on photon propagation are encoded in thef(ωm2Penrose plane-wave limit[22,23]of the spacetime expanded about the original null geodesic traced by the photon.This is a huge simpliﬁcation,since it reduces the problem of studying photon propagation in an arbitrary background to the much more tractable case of a plane wave.In fact,the Penrose limit is ideally suited to this physical problem.As shown in ref.[24],where the relation with null Fermi normal coordinates is explained,it can be extended into a systematic expansion in a scaling parameter which for our problem is identiﬁed as R/m2.The Penrose expansion therefore provides us with a systematic way to go beyond leading order in curvature.The paper is organized as follows.In Section2,we introduce the world-line formalism and set up the geometric sigma model and eikonal approximation.The relation of the Penrose limit to the R/m2expansion is then explained in detail, complemented by a power-counting analysis in the appendix.The geometry of null congruences is introduced in Section3,together with the simpliﬁed symmetric plane wave background in which we perform our detailed calculation of the refractive index. This calculation,which is the heart of the paper,is presented in Section4.The interpretation of the result for the refractive index is given in Section5,where we plot the frequency dependence of n(ω)and prove that asymptotically v ph(ω)→1. We also explain exactly how the existence of conjugate points in a null congruence leads to zero modes in the sigma model partition function,which in turn produces the KK-violating poles in n(ω)in the upper half-plane.The implications for micro-causality are described in Section6.Finally,in Section7we make some further remarks on the generality of our results for arbitrary background spacetimes before summarizing our conclusions in Section8.2.The World-Line FormalismFigure2:The loop xµ(τ)with insertions of photon vertex operators atτ1andτ2.–5–In the world-line formalism for scalar QED5the1-loop vacuum polarization is given byΠ1-loop=αT3 T0dτ1dτ2Z V∗ω,ε1[x(τ1)]Vω,ε2[x(τ2)] .(2.1)The loop with the photon insertions is illustrated in Figure(2).The expectation value is calculated in the one-dimensional world-line sigma model involving periodic ﬁelds xµ(τ)=xµ(τ+T)with an actionS= T0dτ 15Since all the conceptual issues we address are the same for scalars and spinors,for simplicity we perform explicit calculations for scalar QED in this paper.The generalization of the world-line formalism to spinor QED is straightforward and involves the addition of a further,Grassmann,ﬁeld in the path integral.For ease of language,we still use the terms electron and positron to describe the scalar particles.6This will require some appropriate iǫprescription.In particular,the T integration contour should lie just below the real axis to ensure that the integral converges at inﬁnity.7In general,one has to introduce ghostﬁelds to take account of the non-trivial measure for the ﬁelds, [dxµ(τ)of geometric optics where Aµ(x)is approximated by a rapidly varying exponential times a much more slowly varying polarization.Systematically,we haveAµ(x)= εµ(x)+ω−1Bµ(x)+··· e iωΘ(x).(2.4) We will need the expressions for the leading order piecesΘandε.This will necessitate solving the on-shell conditions to theﬁrst two non-trivial orders in the expansion in R1/2/ω.To leading order,the wave-vector kµ=ωℓµ,whereℓµ=∂µΘis a null vector (or more properly a null1-form)satisfying the eikonal equation,ℓ·ℓ≡gµν∂µΘ∂νΘ=0.(2.5) A solution of the eikonal equation determines a family or congruence of null geodesics in the following way.9The contravariant vectorﬁeldℓµ(x)=∂µΘ(x),(2.6) is the tangent vector to the null geodesic in the congruence passing through the point xµ.In the particle interpretation,kµ=ωℓµis the momentum of a photon travelling along the geodesic through that particular point.It will turn out that the behaviour of the congruence will have a crucial rˆo le to play in the resulting behaviour of the refractive index.The general relativistic theory of null congruences is considered in detail in Section3.Now we turn to the polarization vector.To leading order in the WKB approxima-tion,this is simply orthogonal toℓ,i.e.ε·ℓ=0.Notice that this does not determine the overall normalization ofε,the scalar amplitude,which will be a space-dependent function in general.It is useful to splitεµ=Aˆεµ,whereˆεµis unit normalized.At the next order,the WKB approximation requires thatˆεµis parallel transported along the geodesics:ℓ·Dˆεµ=0.(2.7) The remaining part,the scalar amplitude A,satisﬁes1ℓ·D log A=−εµD·ℓ.(2.9)2Since the polarization vector is deﬁned up to an additive amount of k,there are two linearly independent polarizationsεi(x),i=1,2.Since there are two polarization states,the one-loop vacuum polarization is ac-tually a2×2matrixΠ1-loop ij =αT3 T0dτ1dτ2Z× εi[x(τ1)]·˙x(τ1)e−iωΘ[x(τ1)]εj[x(τ2)]·˙x(τ2)e iωΘ[x(τ2)] .(2.10)In order for this to be properly deﬁned we must specify how to deal with the zero mode of xµ(τ)in the world-line sigma model.Two distinct–but ultimately equiv-alent–methods for dealing with the zero mode have been proposed in the litera-ture[25–29].In theﬁrst,the position of one particular point on the loop is deﬁned as the zero mode,while in the other,the“string inspired”deﬁnition,the zero mode is deﬁned as the average position of the loop:xµ0=1Now notice that the exponential pieces of the vertex operators in(2.1)act as source terms and so the complete action including these ism2S=−T+can always be brought into the formds2=2du dΘ−C(u,Θ,Y a)dΘ2−2C a(u,Θ,Y b)dY a dΘ−C ab(u,Θ,Y c)dY a dY b.(2.14) It is manifest that dΘis a null1-form.The null congruence has a simple description as the curves(u,Θ0,Y a0)forﬁxed values of the transverse coordinates(Θ0,Y a0).The geodesicγis the particular member(u,0,0,0).It should not be surprising that the Rosen coordinates are singular at the caustics of the congruence.These are points where members of the congruence intersect and will be described in detail in the next section.With the form(2.14)of the metric,oneﬁnds that the classical equations of motion of the sigma model action(2.13)have a solution with Y a=Θ=0whereu(τ)satisﬁes¨u=−2ωTm2δ(τ).(2.15)More general solutions with constant but non-vanishing(Θ,Y a)are ruled out by the constraint(2.12).The solution of(2.15)is˜u(τ)=−u0+ 2ωT(1−ξ)τ/m20≤τ≤ξ2ωTξ(1−τ)/m2ξ≤τ≤1.(2.16)where the constantu0=ωTξ(1−ξ)/m2(2.17) ensures that the constraint(2.12)is satisﬁed.The solution describes a loop which is squashed down onto the geodesicγas illustrated in Figure(3).The electron and positron have to move with diﬀerent world-line velocities in order to accommodate the fact that in generalξis not equal to1Now that we have deﬁned the Rosen coordinates and found the classical saddle-point solution,we are in a position to set up the perturbative expansion.The idea is to scale the transverse coordinatesΘand Y i in order to remove the factor of m2/T in front of the action.The aﬃne coordinate u,on the other hand,will be left alone since the classical solution˜u(τ)is by deﬁnition of zeroth order in perturbation theory. The appropriate scalings are precisely those needed to deﬁne the Penrose limit[22]–in particular we closely follow the discussion in[23].The Penrose limit involvesﬁrst a boost(u,Θ,Y a)−→(λ−1u,λΘ,Y a),(2.18) whereλ=T1/2/m,and then a uniform re-scaling of the coordinates(u,Θ,Y a)−→(λu,λΘ,λY a).(2.19) As argued above,it is important that the null coordinate along the geodesic u is not aﬀected by the combination of the boost and re-scaling;indeed,overall(u,Θ,Y a)−→(u,λ2Θ,λY a).(2.20) After these re-scalings,the sigma model action(2.13)becomesS=−T+1m2Θ(ξ)+ωT4 10dτ 2˙u˙Θ−C ab(u,0,0)˙Y a˙Y b −ωT m2Θ(0)+···.(2.22) The leading order piece is precisely the Penrose limit of the original metric in Rosen coordinates.Notice that we must keep the source terms because the combination ωT/m2,or more precisely the dimensionless ratioωR1/2/m2,can be large.However, there is a further simplifying feature:once we have shifted the“ﬁeld”about the clas-sical solution u(τ)→˜u(τ)+u(τ),it is clear that there are no Feynman graphs with-out externalΘlines that involve the vertices∂n u C ab(˜u,0,0)u n˙Y a˙Y b,n≥1;hence, we can simply replace C ab(˜u+u,0,0)consistently with the background expression C ab(˜u,0,0).This means that the resulting sigma model is Gaussian to leading order in R/m2:S(2)=14 10dτC ab(˜u,0,0)˙Y a˙Y b,(2.23)whereﬁnally we have dropped the˙u˙Θpiece since it is just the same as inﬂat space and the functional integral is normalized relative toﬂat space.This means that all the non-trivial curvature dependence lies in the Y a subspace transverse to the geodesic.10It turns out that the Rosen coordinates are actually not the most convenient co-ordinates with which to perform explicit calculations.For this,we prefer Brinkmann coordinates(u,v,y i).To deﬁne these,weﬁrst introduce a“zweibein”in the subspace of the Y a:C ab(u)=δij E i a(u)E j b(u),(2.24) with inverse E a i.This quantity is subject to the condition thatΩij≡dE iadE ia2E a j.(2.29)du2We have introduced these coordinates at the level of the Penrose limit.However, they have a more general deﬁnition for an arbitrary metric and geodesic.They are in fact Fermi normal coordinates.These are“normal”in the same sense as the more common Riemann normal coordinates,but in this case they are associated to the geodesic curveγrather than to a single point.This description of Brinkmann coordinates as Fermi normal coordinates and their relation to Rosen coordinates and the Penrose limit is described in detail in ref.[24].In particular,this reference givestheλexpansion of the metric in null Fermi normal coordinates to O(λ2).To O(λ) this isds2=2du dv−R iuju y i y j du2−dy i2+λ −2R uiuv y i v du2−43R uiuj;k y i y j y k du2 +O(λ2),(2.30)which is consistent with(2.28)since R iuju=−h ij for a plane wave.It is worth pointing out that Brinkmann coordinates,unlike Rosen coordinates,are not singular at the caustics of the null congruence.One can say that Fermi normal coordinates (Brinkmann coordinates)are naturally associated to a single geodesicγwhereas Rosen coordinates are naturally associated to a congruence containingγ.In Brinkmann coordinates,the Gaussian action(2.23)for the transverse coordi-nates becomesS(2)=−12m2Ωij y i y j τ=ξ−ωTexplicitly.In doing so,we discover many surprising features of the dispersion relation that will hold in general.The symmetric plane wave metric is given in Brinkmann coordinates by(2.28), with the restriction that h ij is independent of u.This metric is locally symmetric in the sense that the Riemann tensor is covariantly constant,DλRµνρσ=0,and can be realized as a homogeneous space G/H with isometry group G.12With no loss of generality,we can choose a basis for the transverse coordinates in which h ij is diagonal:h ij y i y j=σ21(y1)2+σ22(y2)2.(3.1) The sign of these coeﬃcients plays a crucial role,so we allow theσi themselves to be purely real or purely imaginary.For a general plane-wave metric,the only non-vanishing components of the Rie-mann tensor(up to symmetries)areR uiuj=−h ij(u).(3.2) So for the symmetric plane wave,we have simplyR uu=σ21+σ22,(3.3)R uiui=−σ2iand for the Weyl tensor,1C uiui=−σ2i+12Notice that,contrary to the implication in ref.[4,18],the condition that the Riemann tensor is covariantly constant only implies that the spacetime is locally symmetric,and not necessarily maximally symmetric[13,23].A maximally symmetric space has Rµνρσ=1plane wave background,then explain how the key features are described in the gen-eral theory of null congruences.The geodesic equations for the symmetric plane wave(2.28),(3.1)are:¨u=0,¨v+2˙u2i=1σ2i y i˙y i=0,¨y i+˙u2σ2i y i=0.(3.5)We can therefore take u itself to be the aﬃne parameter and,with the appropriate choice of boundary conditions,deﬁne the null congruence in the neighbourhood of, and including,γas:v=Θ−122i=1σi tan(σi u+a i)y i2.(3.9)The tangent vector to the congruence,deﬁned asℓµ=gµν∂νΘ,is therefore ℓ=∂u+1The polarization vectors are orthogonal to this tangent vector,ℓ·εi=0,and are further constrained by(2.9).Solving(2.7)for the normalized polarization(one-form) yields13ˆεi=dy i+σi tan(σi u+a i)y i du.(3.11) The scalar amplitude A is determined by the parallel transport equation(2.8),from which we readilyﬁnd(normalizing so that A(0)=1)A=2i=1 cos(σi u+a i)(3.12)The null congruence in the symmetric plane wave background displays a number of features which play a crucial role in the analysis of the refractive index.They are best exhibited by considering the Raychoudhuri equation,which expresses the behaviour of the congruence in terms of the optical scalars,viz.the expansionˆθ, shearˆσand twistˆω.These are deﬁned in terms of the covariant derivative of the tangent vector as[30]:ˆθ=1112Rµνℓµℓν,Ψ0=Cµρνσℓµℓνmρmσ.14As demonstrated in refs.[31],the eﬀectof vacuum polarization on low-frequency photon propagation is also governed by the two curvature scalarsΦ00andΨ0.Indeed,many interesting results such as the polarization sum rule and horizon theorem[31,32]are due directly to special properties ofΦ00andΨ0.As we now show,they also play a key rˆo le in the world-line formalism in determining the nature of the full dispersion relation.√2R uu=1 2(C u1u1−C u2u2)=1By its deﬁnition as a gradientﬁeld,it is clear that D[µℓν]=0so the null con-gruence is twist-freeˆω=0.The remaining Raychoudhuri equations can then be rewritten as∂u(ˆθ+ˆσ)=−(ˆθ+ˆσ)2−Φ00−|Ψ0|,∂u(ˆθ−ˆσ)=−(ˆθ−ˆσ)2−Φ00+|Ψ0|.(3.15) The eﬀect of expansion and shear is easily visualized by the eﬀect on a circular cross-section of the null congruence as the aﬃne parameter u is varied:the expansionˆθgives a uniform expansion whereas the shearˆσproduces a squashing with expansion along one transverse axis and compression along the other.The combinationsˆθ±ˆσtherefore describe the focusing or defocusing of the null rays in the two orthogonal transverse axes.We can therefore divide the symmetric plane wave spacetimes into two classes, depending on the signs ofΦ00±|Ψ0|.A Type I spacetime,whereΦ00±|Ψ0|are both positive,has focusing in both directions,whereas Type II,whereΦ00±Ψ0 have opposite signs,has one focusing and one defocusing direction.Note,however, that there is no“Type III”with both directions defocusing,since the null-energy condition requiresΦ00≥0.For the symmetric plane wave,the focusing or defocusing of the geodesics is controlled byeq.(3.6),y i=Y i cos(σi u+a i).Type I therefore corresponds toσ1and σ2both real,whereas in Type II,σ1is real andσ2is pure imaginary.The behaviour of the congruence in these two cases is illustrated in Figure(4).1y21y2Figure4:(a)Type I null congruence with the special choiceσ1=σ2and a1=a2so that the caustics in both directions coincide as focal points.(b)Type II null congruence showing one focusing and one defocusing direction.To see this explicitly in terms of the Raychoudhuri equations,noteﬁrst that the curvature scalarsΦ00−|Ψ0|=σ21,Φ00+|Ψ0|=σ22are simply the eigenvalues of h ij.The optical scalars areˆθ=−12 σ1tan(σ1u+a1)−σ2tan(σ2u+a2) (3.16)and we easily verify∂uˆθ=ˆθ2−ˆσ2−12(σ21−σ22).(3.17)It is clear that provided the geodesics are complete,those in a focusing direction will eventually cross.In the symmetric plane wave example,with y i=Y i cos(σi u+ a i),these“caustics”occur when the aﬃne parameterσi u=π(n+115This does not necessarily mean that the conjugate points are joined by more than one actual geodesic,only that an inﬁnitesimal deformation ofγter we shall see that the existence of conjugate points relies on the existence of zero modes of a linear problem.Conversely,the existence of a geodesic other thanγjoining p and q does not necessarily mean that p and q are conjugate[8,33].16Whether these deformed geodesics become actual geodesics is the question as to whether they lift from the Penrose limit to the full metric.4.World-line Calculation of the Refractive IndexIn this section,we calculate the vacuum polarization and refractive index explicitly for a symmetric plane wave.As we mentioned at the end of Section 2,the explicit calculations are best performed in Brinkmann coordinates.We will need the expres-sions for Θand εi for the symmetric plane wave background:these are in eqs.(3.9),(3.11)and (3.12).From these,we have the following explicit expression for the vertex operator 17V ω,εi [x µ(τ)]= ˙y i +σi tan(σi ˜u +a i )˙˜u y i 2 j =1 cos(σj ˜u +a j )×exp iω v +1410dτ ˙y i 2−˙˜u 2σ2i yi 2 −ωT σi −det g [x (τ)]which can be exponenti-ated by introducing appropriate ghosts [25–29].However,in Brinkmann coordinatesafter the re-scaling (2.27),det g =−1+O (λ)and so to leading order in R/m 2the determinant factor is simply 1and so plays no rˆo le.The same conclusion would not be true in Rosen coordinates.The y i ﬂuctuations satisfy the eigenvalue equation¨y i+˙˜u 2σ2i y i −2ωT σi 17Notice that at leading order in R/m 2we are at liberty to replace u (τ)by its classical value ˜u (τ).The argument is identical to the one given in Section 2.。

PhysRevLett.105.133201 Cold Heteronuclear Atom-Ion Collisions

Cold Heteronuclear Atom-Ion CollisionsChristoph Zipkes,Stefan Palzer,Lothar Ratschbacher,Carlo Sias,*and Michael Ko¨hl Cavendish Laboratory,University of Cambridge,JJ Thomson Avenue,Cambridge CB30HE,United Kingdom(Received20May2010;published23September2010)We study cold heteronuclear atom-ion collisions by immersing a trapped single ion into an ultracold atomic ing ultracold atoms as reaction targets,our measurement is sensitive to elastic collisions with extremely small energy transfer.The observed energy-dependent elastic atom-ion scattering rate deviates signiﬁcantly from the prediction of Langevin but is in full agreement with the quantum mechanical cross section.Additionally,we characterize inelastic collisions leading to chemical reactions at the single particle level and measure the energy-dependent reaction rate constants.The reaction products are identiﬁed by in-trap mass spectrometry,revealing the branching ratio between radiative and nonradiative charge exchange processes.DOI:10.1103/PhysRevLett.105.133201PACS numbers:34.50.Cx,34.70.+e,37.10.TyCold collisions are characterized by the de Brogliewavelength of the colliding particles becoming comparableto the length scale of the molecular interactions.Quantummechanics then dominates the elastic and inelastic scatter-ing phenomena and cross sections.Understanding elasticcollisions in this regime is fundamental for achieving andoptimizing sympathetic cooling of species which do notallow for direct laser cooling,for example,complex mole-cules.On the other hand,inelastic collisions of translation-ally cold reaction partners provide a unique opportunity toobserve quantum mechanical details of chemical processeswhich typically are averaged out at higher temperatures[1].It is expected that the detailed understanding of coldcollisions will eventually pave the way towards coherentchemistry,in which full quantum control of the internalstates and reaction pathways will be possible.Over the past few years,impressive advances have beenmade in exploring cold collisions of neutral atoms.Thishas lead,for example,to the controlled creation of weaklybound dimers[2]and trimers[3]and the production ofmolecular gases in the rovibrational ground state[4].Incomparison,cold ion-neutral collisions are much less ex-plored.Ion-neutral reactions offer the outstanding experi-mental possibility that reactants and reaction products canbe observed and manipulated at the single particle leveland can be trapped for very long times allowing for precisemeasurements[5–9].Collisions of cold(millikelvin)trapped ions with hot($102K)neutral atoms have pro-vided insights into chemical reactions[6,7,10]and play animportant role in understanding the molecular compositionof the interstellar gas[11].Only very recently,the regimein which both collision partners are translationally cold hasbeen accessed by using trapped ions interacting with coldmolecular beams[7]or trapped neutral atom samples[12,13].Nevertheless,elastic scattering in the quantumregime of many partial waves with its decisive energy dependence of the cross section( /EÀ1=3)was not yet observed.Regarding charge exchange,experimental observations[12]were well explained by the Langevinmodel( L/EÀ1=2)that is based on classical mechanics to describe the mobility of ions in gases[14].In this Letter,we explore heteronuclear collision pro-cesses between trapped atoms and ions in which both collision partners are translationally cold.We study elastic and reactive collisions using a single ion only in order to better control the kinetic energy than in larger ion crystals. This enables us to investigate the energy dependence of elastic and inelastic scattering processes from20to 450 eV(equivalent to thermal energies in the range of 0.2–5K).Weﬁnd the elastic collision rate signiﬁcantly larger than predicted by Langevin theory but in full agree-ment with a quantum mechanical calculation.Moreover, we investigate inelastic collisions and measure the energy dependence of the charge exchange cross section and the branching ratio for radiative vs nonradiative charge ex-change.This provides a comprehensive characterization of binary atom-ion collisions in this energy range.Atom-ion scattering is dominated by the polarizationpotential,which asymptotically behaves as VðrÞ¼ÀC42r4, and a hard-core repulsion at the length scale of the Bohrradius.Here r is the internuclear separation,C4¼ q2ð4 0Þ2,q denotes the charge of the ion, is the dc polarizability of the neutral particle,and 0is the vacuum permittivity. Atom-ion scattering was theoretically investigated as early as1905when Langevin calculated the drift velocity of ions in a buffer gas.The Langevin model is based on classical mechanics and the predicted cross section is L¼ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ2C4=Ep,leading to an energy-independent collision rate ngevin-type collisions occur in close en-counters between an atom and an ion when the impact parameter is below a critical value b c¼ð2C4=EÞ1=4[15] and exhibit a large momentum transfer between the atom and the ion.Going beyond the Langevin model,the quan-tum mechanical description of scattering yields the angular dependence of the differential cross section in more detail.In the energy range under consideration,elastic scattering from the polarization potential involves several partial waves since the s -wave scattering regime of atom-ion collisions is only at submicrokelvin temperatures [16,17].The semiclassical approximation of the full quantum mechanical elastic scattering cross section for a collisionenergy E is ðE Þ¼ ð1þ 2=16Þð C 24@2E Þ1=3[16,18]( de-notes the reduced mass).This includes,in particular,a large contribution towards forward scattering under very small angles [19],mainly from the centrifugal barrier.The forward scattering peak results from interference of partial waves similar to the bright ‘‘Poisson spot’’emerging behind a beam block in wave optics.In these collisions the momentum transfer between the collision partners is extremely small and very difﬁcult to detect.Until now,neither cold ion-neutral collisions nor ion-mobility spec-trometry could access this regime.In our experiment we study collisions between ultracold 87Rb atoms and a single Yb þion.We prepare 2:2Â106neutral atoms in the j F ¼2;m F ¼2i hyperﬁne ground state at T %300nK in a harmonic magnetic trap of char-acteristic frequencies ð!x ;!y ;!z Þ¼2 Âð8;28;28ÞHz [13,20].We spatially overlap the neutral atoms with a single Yb þion trapped and laser cooled in an rf-Paul trap.The ion in the Paul trap experiences a pseudopotential which is derived from time-averaging a very rapidly oscil-lating electric quadrupole ﬁeld E ðx;y Þ¼V rf cos ð t ÞR2Âðx;Ày;0Þ.V rf denotes the applied voltage,R is the distance of the ion to the electrodes,and ¼2 Â42:7MHz is the drive frequency.The time-averaged potential provides a harmonic conﬁnement for the ion with characteristic trap frequencies !?¼2 Â150kHz .On top of the time-averaged motion,the ion undergoes very rapid oscillations in position (micromotion)of an amplitude a mm which is proportional to the local electric ﬁeld strength.Axial conﬁnement of the ion with a characteristic frequency of !ax ¼2 Â42kHz is provided by electrostatic ﬁelds.A collision with an ultracold atom changes the instan-taneous velocity of the ion and disrupts its coherent micro-motion oscillation.As result,energy can be transferred from the ion to the neutral atoms,leading to cooling of the secular motion of the ion [13],but also energy of up to$m ion2 2a 2mm can be transferred from the driving ﬁeld to the ion’s secular motion [21–23].Depending on the exact parameters,the ion will equilibrate at a mean energy determined by the two processes.We tune the energy of the atom-ion collisions by applying a homogeneous offset electric ﬁeld E 0transverse to the symmetry axis of the ion trap.This displaces the ion from the geometric center of the Paul trap and introduces excess micromotion of amplitude a mm;ex ¼ﬃﬃﬃ2p qE 0=ðm ion !? Þ.m ion is the mass of the ion.We center the neutral atomic cloud at the new equilibrium position by applying homogeneous magnetic ﬁelds of $100mG .We investigate the effect of the excess micromotion a mm;ex on the equilibrated mean energy of the ion after ithas interacted with a neutral atom cloud for t 0¼8s .During the interaction with the neutral atoms,the ion is in its electronic ground state.For the energy measurement we release the neutral atoms from the trap and set the offset electric ﬁeld to zero.Then,we illuminate the ion with laser light red-detuned by Á¼À0:25Àfrom the S 1=2!P 1=2transition of Yb þat 370nm (À¼2 Â20MHz denotes the linewidth of the atomic transition)and monitor the temporal increase of the ﬂuorescence rate [see Fig.1(a)].We ﬁt the data with the solution of the time-dependent optical Bloch equations for the motion of the particle in the pseudopotential and perform a thermal ensemble average [24,25].In Fig.1(b),we display the measurement of the mean energy of the ion vs the applied offset electric ﬁeld E 0,which exhibits the expected quadratic dependence.We use this energy tuning mechanism to study the energy dependence of the collision cross sections in the next paragraphs.The minimum energy is determined by resid-ual micromotion.We ﬁnd that the ion equilibrates after less than 100ms interaction time to its average kinetic energy.The kinematics of a binary collision between an ion of energy E ion and a neutral atom of mass m n ,which we assume at rest (E n ¼0),results in a relative energy transfer ÁE=E ion ¼4 =ð þ1Þ2sin 2ð =2Þwith ¼m ion =m n .Collisions in which the scattering angle exceeds cut ¼2arcsin ½ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃE cut ð þ1Þ2=ð4 E ion Þp lead to atom loss from the trap.E cut is the effective trap depth of the atom trap.A neutral atom after a collision will be detected as lost in absorption imaging if it moves at a very large orbit in the trap and does not thermalize with the neutral atom cloud within the time t 0.Numerical calculations show that ther-malization can be achieved only for energy transfers less than 0.8neV .In Fig.2(a),we show the measured atom loss as a function of the ion energy.We ﬁnd that the atom loss exceeds the classical Langevin prediction ÁN ¼n 0 L vt 0and,moreover,shows a pronounced en-ergy dependence,also not explained by the Langevin model.Here n 0is the local neutral atom density,and v is the relative velocity of the atom and the ion.To modeloura)b)FIG.1(color online).(a)Time-resolved ﬂuorescence of a single 172Yb ion.The signal is averaged over 69repetitions,and the error bars are determined by the photon shot noise.The line is a ﬁt [24]resulting in a mean energy of the ion of ð230Æ9Þ eV .(b)Measured mean ion energy vs offset electric ﬁeld used for tuning the excess micromotion.The line is a linear ﬁt to the data.data,we numerically calculate the quantum mechanical differential elastic scattering cross section I ð ;E Þ.We calculate the scattering phase shifts in a semiclassical approximation for large values of the angular momentum for a ÀC 4=2r 4potential and perform a random-phase approximation for small angular momenta for the unknown short-range repulsion [16].Using I ð ;E Þwe calculate the expected neutral atom losses in a Monte Carlo simulation.The solid line in Fig.2(a)shows the predicted loss for large energy transfer (ÁE >E cut ).In the case of small energy transfer ( < cut )the atom remains in the trap and thermalizes with the remaining atom cloud.The very small amount of transferred energy and momentum ( cut is in the millirad range)makes these collisions very challenging to observe.However,using ultracold atoms at nanokelvin temperatures as reaction partners allows us to resolve a relative energy transfer on the order of $10À7,inconceivable in previous experi-ments.The observed temperature increase [Fig.2(b)]is explained by two different contributions.Small energy transfers (ÁE <E cut )are dominant.The other effect re-sults from selectively removing atoms from the center of the cloud where their potential energy is below the thermal average,leading to ‘‘evaporative heating.’’The dashed line shows the maximally possible contribution from evapora-tive heating,assuming all observed losses of Fig.2(a)being from the center of the trap.The solid line is the predicted temperature increase taking into account both contributions.The simulations in both Figs.2(a)and 2(b)use the same value of E cut as the only free parameter,and we obtain E cut ¼0:7neV ,in agreement with the experi-mentally expected value.Reactive chemical processes in atom-ion collisions,like charge exchange or molecule formation,can occur only when an atom and an ion approach each other to distancesat which the electronic wave functions overlap,which is the case in close encounters.In the regime of many partial waves,the rate constant for chemical processes is thus predicted to be proportional to the Langevin rate constant L v and to be independent of the collision energy [15,16,26].For equal chemical elements,strong loss due to near-resonant charge exchange was observed [12]in agreement with the Langevin estimate.In our case,the Rb (5s 2S 1=2)atom in j F ¼2;m F ¼2i and the Yb þ(6s 2S 1=2)ion in j J ¼1=2;m J ¼Æ1=2i col-lide in the excited A 1Æþsinglet or the a 3Æþtriplet state of the ðRbYb Þþmolecular potential [see Fig.3(a)].The electronic ground state X 1Æþ,which asymptotically cor-responds to Rb þ(4p 61S 0)and neutral Yb ð6s 21S 0Þ,lies 2.08eV below.The most striking experimental observation of a chemical process is the disappearance of the Yb þﬂuorescence which we detect after the interaction with the neutral ing the same ion energy tuning mechanism as described above,we have investigated the energy dependence of the reaction rate constant.For two different isotopes,172Yb and 174Yb ,we ﬁnd the rate con-stant to be independent of the collision energy,as predicted [14,15][see Fig.3(b)].The average rate constants of R 172¼ð2:8Æ0:3ÞÂ10À20m 3=s and R 174¼ð4:0Æ0:3ÞÂ10À20m 3=s are 5orders of magnitude smaller than in the homonuclear case [12]and of the same order of magnitude as predicted for the similarly asymmetric system ðNaCa Þþ[26].There is a systematic error between the measurements for the two isotopes of 15%due to uncertainty of the density determination of the thermal cloud.Additionally,we have measured the density dependence of the inelastic atom-ion collisions by positioning the ionatFIG.2(color online).(a)Atom loss from the magnetic trap vs mean energy of a single 172Yb þion.The experimental data are averaged over $70realizations each,and the standard error is given.The solid line is a numerical simulation based on the full elastic scattering cross section.The dashed line shows the prediction of the Langevin model.(b)Temperature increase of the neutral atom cloud vs mean energy of a single 172Yb þion.The solid line shows the numerical results using the full elastic scattering cross section.The dashed line shows the maximally possible evaporative heating effect.The mean energy of the ion is derived from Fig.1.a)d)c)FIG.3(color online).(a)Schematic of the molecular poten-tials.(b)Measured ion loss probability as a function of offset electric ﬁeld.The ion’s mean energy is derived from Fig.1.Each data point is averaged over approximately 100repetitions of the experiment.The dashed lines show the average of the data.(c)Position dependence of the ion loss probability for 174Yb þas the ion is scanned through the cloud.The red line is the theoretical density proﬁle of the thermal cloud.(d)The linear dependence of the ion loss rate of the density signals binary collisions leading to charge exchange reactions.Each data point is averaged over approximately 350repetitions of the experiment.different locations inside the neutral atom cloud [Fig.3(c)].We achieve this without increasing the ion’s micromotion by applying a homogeneous magnetic bias ﬁeld to displace the position of the neutral atomic cloud.Monitoring the ion loss rate as a function of the position,we ﬁnd that the inelastic atom-ion collisions scale linearly with the local atomic density,indicating that the charge exchange reactions are binary atom-ion collisions [see Fig.3(d)].This also demon-strates the principal capability of the single ion as a local probe for density measurements in a neutral atom cloud [27].Charge exchange can occur by emission of a photon (radiative charge exchange)or as a nonadiabatic transition between molecular levels.For low temperatures,radiative charge exchange has been predicted to be the dominating process for ðNaCa Þþ[26].In order to investigate the reac-tion products of the charge exchange process,we perform mass spectrometry.To this end,we load two Yb þions into the ion trap and overlap them with the neutral cloud.In the cases in which only one of the two ions undergoes a reaction,the other ion serves to identify the reaction prod-uct by measuring a common vibrational mode in the ion trap [5,7].We excite the axial secular motion of Yb þby applying intensity-modulated light at 370nm at an angle of 60 relative to the symmetry axis of the Paul trap and monitor the ﬂuorescence.The frequency of the intensity modulation is swept linearly from 30to 55kHz in 2.5s.When the intensity modulation coincides with a collective resonance of the secular motion,the ions are heated and the ﬂuorescence reduces [see Fig.4(a)].The collective mode for a Rb þand an Yb þion in the trap is 13%above the bare Yb þmode at 42kHz,whereas the mode of ðRbYb Þþand Yb þwould be 12%lower in frequency.Figure 4(b)shows the histogram for the observed charge exchange processes.In the cases in which one Yb þion is lost,we ﬁnd approxi-mately 30%probability for the production of cold Rb þand 70%for a complete loss.If the reaction products take up 2eV as kinetic energy in a nonradiative charge exchange,they would be lost due to the ﬁnite depth ($150meV )of our ion trap.We have not observed the formation of a charged molecule in this process.In conclusion,we have provided a comprehensive sur-vey of cold heteronuclear atom-ion collisions.We have observed that Langevin theory is insufﬁcient to describe cold atom-ion collisions but the full quantum mechanical cross section is required.Moreover,we have observed charge exchange reactions between a single ion and ultra-cold atoms and analyzed their products.Our results pro-vide an excellent starting point for future experiments targeting the full quantum control of chemical reactions at the single particle level.Photoassociation or Feshbach resonances [17]could be used to create single trapped cold molecules in speciﬁc rovibrational states.We thank D.Meschede,G.Shlyapnikov,and V .Vuletic for discussions and EPSRC (EP/F016379/1,EP/H005676/1),ERC (Grant No.240335),and the Herchel Smith Fund (C.S.)for support.*cs540@[1]R.V .Krems,Phys.Chem.Chem.Phys.10,4079(2008).[2] E.Donley et al.,Nature (London)417,529(2002).[3]T.Kraemer et al.,Nature (London)440,315(2006).[4]K.-K.Ni et al.,Science 322,231(2008).[5]M.Drewsen et al.,Phys.Rev.Lett.93,243201(2004).[6]P.F.Staanum,K.Højbjerre,R.Wester,and M.Drewsen,Phys.Rev.Lett.100,243003(2008).[7]S.Willitsch et al.,Phys.Rev.Lett.100,043203(2008).[8]P.F.Staanum et al.,Nature Phys.6,271(2010).[9]T.Schneider et al.,Nature Phys.6,275(2010).[10] B.Roth et al.,Phys.Rev.A 73,042712(2006).[11] D.Smith,Chem.Rev.92,1473(1992).[12] A.T.Grier,M.Cetina,F.Orucˇevic ´,and V .Vuletic ´,Phys.Rev.Lett.102,223201(2009).[13] C.Zipkes,S.Palzer, C.Sias,and M.Ko¨hl,Nature (London)464,388(2010).[14]ngevin,Ann.Chim.Phys.5,245(1905).[15] E.V ogt and G.H.Wannier,Phys.Rev.95,1190(1954).[16]R.Coˆte ´and A.Dalgarno,Phys.Rev.A 62,012709(2000).[17]Z.Idziaszek,T.Calarco,P.S.Julienne,and A.Simoni,Phys.Rev.A 79,010702(2009).[18]H.Massey,Contemp.Phys.12,537(1971).[19]P.Zhang,A.Dalgarno,and R.Coˆte ´,Phys.Rev.A 80,030703(2009).[20]S.Palzer,C.Zipkes,C.Sias,and M.Ko¨hl,Phys.Rev.Lett.103,150601(2009).[21] F.G.Major and H.G.Dehmelt,Phys.Rev.170,91(1968).[22]R.G.DeV oe,Phys.Rev.Lett.102,063001(2009).[23] E.R.Hudson,Phys.Rev.A 79,032716(2009).[24]J.H.Wesenberg,Phys.Rev.A 76,053416(2007).[25]R.J.Epstein et al.,Phys.Rev.A 76,033411(2007).[26]O.P.Makarov,R.Coˆte ´,H.Michels,and W.W.Smith,Phys.Rev.A 67,042705(2003).[27] C.Kollath,M.Ko¨hl,and T.Giamarchi,Phys.Rev.A 76,063602(2007).Common mode excitation frequency [kHz]F l u o r e s c e n c e c o u n t s [a .u .]a)FIG.4(color online).(a)Mass spectrometry signal together with dashed lines indicating the expected frequencies.The black trace corresponds to two 174Yb þions.For one Rb þion and one Yb þion (red curve)the collective mode frequency is higher.No signals at the resonance for Yb þand ðRbYb Þþwere obtained.(b)Observed distribution of the reaction products for two Yb þions after 8s interaction time with the neutral Rb atoms (blue bars,486events).The red bars (543events)show a comparative measurement without neutral atoms.。

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Quasivacuum solar neutrino oscillations

a rXiv:h ep-ph/5261v219Se p2Quasi-vacuum solar neutrino oscillations G.L.Fogli a ,E.Lisi a ,D.Montanino b ,and A.Palazzo a a Dipartimento di Fisica and Sezione INFN di Bari,Via Amendola 173,I-70126Bari,Italy b Dipartimento di Scienza dei Materiali dell’Universit`a di Lecce,Via Arnesano,I-73100Lecce,Italy Abstract We discuss in detail solar neutrino oscillations with δm 2/E in the range [10−10,10−7]eV 2/MeV.In this range,which interpolates smoothly be-tween the so-called “just-so”and “Mikheyev-Smirnov-Wolfenstein”oscillation regimes,neutrino ﬂavor transitions are increasingly aﬀected by matter eﬀects as δm 2/E increases.As a consequence,the usual vacuum approximation has to be improved through the matter-induced corrections,leading to a “quasi-vacuum”oscillation regime.We perform accurate numerical calculations of such corrections,using both the true solar density proﬁle and its exponen-tial approximation.Matter eﬀects are shown to be somewhat overestimated in the latter case.We also discuss the role of Earth crossing and of energy smearing.Prescriptions are given to implement the leading corrections in the quasi-vacuum oscillation range.Finally,the results are applied to a global analysis of solar νdata in a three-ﬂavor framework.PACS number(s):26.65.+t,14.60.PqTypeset using REVT E XI.INTRODUCTIONA well-known explanation of the solarνeﬂux deﬁcit[1]is provided byﬂavor oscillations[2]of neutrinos along their way from the Sun(⊙)to the Earth(⊕).For two active neutrino states[say,(νe,νµ)in theﬂavor basis and(ν1,ν2)in the mass basis],the physics of solarνoscillations is governed,at any given energy E,by the mass-mixing parametersδm2andωin vacuum,1as well as by the electron density proﬁle N e(x)in matter[3].Diﬀerent oscillation regimes can be identiﬁed in terms of three characteristics lengths, namely,the astronomical unitL=1.496×108km,(1) the oscillation length in vacuumL osc=4πEeV2/MeV−1km,(2)and the refraction length in matterL mat=2π2G F N e=1.62×104 N e(cos2ω−L osc/L mat)2+sin22ω.(4)Typical solutions to the solar neutrino problem(see,e.g.,[4])involve values of L osc either in the so-called“just-so”(JS)oscillation regime[5],characterized byL JS osc∼L≫L mat,(5) or in the“Mykheyev-Smirnov-Wolfenstein”(MSW)oscillation regime[3],characterized byL MSWosc∼L mat≪L.(6) The two regimes correspond roughly toδm2/E∼O(10−11)eV2/MeV and toδm2/E>∼10−7 eV2/MeV,respectively.For just-so oscillations,since L mat/L osc→0,the eﬀect of matter is basically to suppress the oscillation amplitude both in the Sun and in the Earth(sin22ωm→0),so that(coherent)ﬂavor oscillations take place only in vacuum,starting from the Sun surface[6].Conversely, for MSW oscillations,L osc∼L mat andﬂavor transitions are dominated by the detailed matter density proﬁle,while the many oscillation cycles in vacuum(L osc≪L)are responsible for completeνdecoherence at the Earth,once smearing eﬀects are taken into account[7].2].Therefore,it is intuitively clear that in the intermediate range between(5)and(6), corresponding approximately to10−10<∼δm2/E<∼10−7eV2/MeV,the simple vacuum oscillation picture of the JS regime becomes increasingly decoherent and aﬀected by matter eﬀects for increasing values ofδm2/E,leading to a hybrid regime that might be called of “quasi-vacuum”(QV)oscillations,characterized byL mat<∼L QV osc<∼L,(7) In the past,quasi-vacuum eﬀects on the oscillation amplitude and phase have been ex-plicitly considered only in relatively few papers(see,e.g.,[8–14])as compared with the vast literature on solar neutrino oscillations,essentially because typicalﬁts to solarνrates al-lowed only marginal solutions in the range where QV eﬀects are relevant.However,more recent analyses appear to extend the former ranges of the JS solutions upwards[15]and of the MSW solutions downwards[16]inδm2/E,making them eventually merge in the QV range[17],especially under generous assumptions about the experimental or theoreticalνﬂux uncertainties.2Therefore,a fresh look at QV corrections seems warranted.Recently, a semianalytical approximation improving the familiar just-so formula in the QV regime was discussed in[18]and,in more detail,in[19],where additional numerical checks were performed.In this work we revisit the whole topic,by performing accurate numerical cal-culations which include the exact density proﬁle in the Sun and in the Earth,within the reference mass-mixing rangesδm2/E∈[10−10,10−7]eV2/MeV and tan2ω∈[10−3,10].3We also discuss some approximations that can simplify the computing task in present applica-tions.We then apply such calculations to a global analysis of solar neutrino data in the rangeδm2≤10−8eV2.Our paper is structured as follows.The basic notation and the numerical techniques used in the calculations are introduced in Sec.II and III,respectively.The eﬀects of solar matter in the quasi-vacuum oscillation regime are discussed in Sec.IV,where the results for true and exponential density proﬁles are compared.Earth matter eﬀects are described in Sec.V.The decoherence of oscillations induced by energy(and time)integration is discussed in Sec.VI.The basic results are summarized and organized in Sec.VII,and then applied to a three-ﬂavor oscillation analysis in Sec.VIII.Section IX concludes our work.II.NOTATIONTheνpropagation from the Sun core to the detector at the Earth can be interpreted as a“double slit experiment,”where the originalνe can take two paths,corresponding to the intermediate transitionsνe→ν1and toνe→ν2.The globalνe survival amplitude is then the sum of the amplitudes along the two paths,A(νe→νe)=A⊙(νe→ν1)·A vac(ν1→ν1)·A⊕(ν1→νe)+A⊙(νe→ν2)·A vac(ν2→ν2)·A⊕(ν2→νe),(8) where the transition amplitudes from the Sun production point to its surface(A⊙),from the Sun surface to the Earth surface(A vac)and from the Earth surface to the detector(A⊕) have been explicitly factorized.Theνe survival probability P ee is then given byP ee=|A(νe→νe)|2.(9) In general,the above amplitudes can be written asA⊙(νe→ν1)=P⊕exp(iξ⊕),(10c) for theﬁrst path and asA⊙(νe→ν2)=1−P⊕,(11c) for the second path,where R⊙is the Sun radius.4In the above equations,P⊙and P⊕are real numbers(∈[0,1])representing the transition probability P(νe↔ν1)along the two partial paths inside the Sun(up to its surface)and inside the Earth(up to the detector). The corresponding phase diﬀerences between the two paths,ξ⊙andξ⊕(∈[0,2π]),have been associated to theﬁrst path without loss of generality.Theνe survival probability P ee from Eq.(9)reads thenP ee=P⊙P⊕+(1−P⊙)(1−P⊕)+2(1−δR−δ⊙−δ⊕),(13)2Ewith the deﬁnitionsR⊙δR=ξ⊙,(15)δm2L2Eδ⊕=4Although R⊙is relatively small(R⊙/L=4.7×10−3),it is explicitly kept for later purposes.The Earth radius R⊕can instead be safely neglected(R⊕/L=4.3×10−5).A relevant extension of the2νformula(12)is obtained for3νoscillations,required to accommodate solar and atmospheric neutrino data[21].Assuming a third mass eigenstate ν3with m2=|m23−m21,2|≫δm2,the3νsurvival probability can be written asP3νee=c4φP2νee+s4φ,(17)whereφis the(νe,ν3)mixing angle,and P2νee is given by Eq.(12),provided that the electron density N e is replaced everywhere by c2φN e(see[20]and refs.therein).Such replacement implies that the3νcase is not a simple mapping of the2νcase,and requires speciﬁc calculations for any given value ofφ.We conclude this section by recovering some familiar expressions for P ee,as special cases of Eq.(12).The JS limit(L mat/L osc→0,with complete suppression of oscillations inside matter)corresponds to P⊙≃c2ω≃P⊕and to negligibleδ⊙,δ⊕.Then,neglecting alsoδR, one gets from(12)the standard“vacuum oscillation formula,”P JS ee≃1−sin22ωsin2(πL/L osc).(18) In the MSW limit(L/L osc→∞),the global oscillation phaseξis very large and cosξ≃0on average.Furthermore,assuming for P⊙a well-known approximation in terms of the“crossing”probability P c between mass eigenstates in matter[in our nota-tion,P⊙≃sin2ω0m P c+cos2ω0m(1−P c),withω0m calculated at the production point],one gets from(12)and for daytime(P⊕=c2ω)the so-called Parke’s formula[22],P MSWee,day≃12−P c)cos2ωcos2ω0m.(19)III.NUMERICAL TECHNIQUESIn general,numerical calculations of theνtransition amplitudes must take into account the detailed N e proﬁle along the neutrino trajectory,both in the Sun and in the Earth.Concerning the Sun,we take N e from[23](“year2000”standard solar model).Figure1 shows such N e proﬁle as a function of the normalized radius r/R⊙,together with its expo-nential approximation[1]N e=N0e exp(−r/r0),with N0e=245mol/cm3and r0=R⊙/10.54. For the exponential density proﬁle,the neutrino evolution equations can be solved analyt-ically[8–11].In order to calculate the relevant probability P⊙and the phaseξ⊙,we have developed several computer programs which evolve numerically the familiar MSW neutrino evolution equations[3]along the Sun radius,for generic production points,and for any given value ofδm2/E∈[10−10,10−7]eV2/MeV and of tan2ω.We estimate a numerical(fractional) accuracy of our results better than10−4,as derived by several independent checks.As a ﬁrst test,we integrate numerically the MSW equations both in their usual complex form (2real+2imaginary components)and in their Bloch form involving three real amplitudes [24],obtaining the same results.We have then repeated the calculations with diﬀerent inte-gration routines taken from several computer libraries,and found no signiﬁcant diﬀerences among the outputs.We have optionally considered,besides the exact N e proﬁle,also the exponential proﬁle,which allows a further comparison of the numerical integration of theMSW equations with their analytical solutions,as worked out in[10,11]in terms of hyper-geometric functions(that we have implemented in an independent code).Also in this case, no diﬀerence is found between the output of the diﬀerent codes.Concerning the calculation of the quantities P⊕andξ⊕in the Earth,we evolve analyti-cally the MSW equations at any given nadir angleη,using the technique described in[25], which is based on aﬁve-step biquadratic approximation of the density proﬁle from the Pre-liminary Reference Earth Model(PREM)[26]and on aﬁrst-order perturbative expansion of the neutrino evolution operator.Such analytical technique provides results very close to a full numerical evolution of the neutrino amplitudes,the diﬀerences being smaller than those induced by uncertainties in N e[25].In particular,we have checked that,forδm2/E≤10−7 eV2/MeV,such diﬀerences are<∼10−3.In conclusion,we are conﬁdent in the accuracy of our results,which are discussed in the following sections.IV.MATTER EFFECTS IN THE SUNFigure2shows,in the mass-mixing plane and for standard solar model density,isolines of the diﬀerence c2ω−P⊙(solid curves),which becomes zero in the just-so oscillation limit of very smallδm2/E.The isolines shape reminds the“lower corner”of the more familiar MSW triangle[22].Also shown are isolines of constant resonance radius R res/R⊙(dotted curves),deﬁned by the MSW resonance condition L osc/L mat(R res)=cos2ω.The values of c2ω−P⊙are already sizable(a few percent)atδm2/E∼10−9,and increase for increasing δm2/E and for large mixing[tan2ω∼O(1)],especially in theﬁrst octant,where the MSW resonance can occur.The diﬀerence between matter eﬀects in theﬁrst and in the second octant can lead to observable modiﬁcations of the allowed regions inﬁts to the data[19],and to a possible discrimination between the casesω<π4[17,19].In the whole parameter range of Fig.2,it turns out that,within the region ofνproduction (r/R⊙<∼0.3),it is L mat(r)≪L osc(and thus sin22ω0m≃0).5As a consequence,all the curves of Fig.2do not depend on the speciﬁcνproduction point(as we have also checked numerically),and no smearing over theνsource distribution is needed in the quasi-vacuum regime.This is a considerable simpliﬁcation with respect to the MSW regime,which involves higher values ofδm2/E and thus shorter(resonance)radii,which are sensitive to the detailed νsource distribution.Figure3shows,in the same coordinates of Fig.2,the isolines of c2ω−P⊙corresponding to the exponential density proﬁle(dotted curves),for which we have used the fully analytical results of[10,11].(Identical results are obtained by numerical integration.)The solid lines in Fig.3refer to a well-known approximation(sometimes called“semianalytical”)to such results,which is obtained in the limit N e→0at the Sun surface(it is not exactly so for the exponential proﬁle,see Fig.1).More precisely,the zeroth order expansion of the hypergeometric functions[10,11]in terms of the small parameter z=i√5This is also indicated by the fact that R res/R⊙>∼0.55in theδm2/E range of Fig.2.and dotted curves in Fig.3are essentially due to the“solar border approximation”(N e→0) assumed in the semianalytical case;indeed,the diﬀerences would practically vanish if the exponential density proﬁle,and thus the“eﬀective”Sun radius,were unphysically continued for r≫R⊙(not shown).Such limitations of the semianalytical approximation have been qualitatively suggested by the authors of[27]but,contrary to their claim,our Fig.3shows explicitly that the semianalytical calculation of P⊙represents a reasonable approximation to the analytical one forωin both octants,as also veriﬁed in[28].A comparison of the results of Fig.2(true density)and of Fig.3(exponential density) shows that,in the latter case,the correction term c2ω−P⊙tends to be somewhat over-estimated,in particular when the semianalytical approximation is used.We have veriﬁed that such bias is dominantly due to the diﬀerence(up to a factor of∼2)between the true density proﬁle and its exponential approximation around r/R⊙∼0.8(see Fig.1)and, subdominantly,to the details of the density proﬁle shape at the border(r/R⊙→1).As a consequence,the“exponential proﬁle”calculation of P⊙(either semianalytic[17,19]or analytic)tends to shift systematically the onset of solar matter eﬀects to lower values of δm2/E.For instance,at tan2ω≃1(maximal mixing),the value c2ω−P⊙=0.05is reached atδm2/E≃8×10−10eV2/MeV for the true density,and atδm2/E a factor of∼2lower for the exponential density.In order to avoid artiﬁcially larger eﬀects at lowδm2/E in neutrino data analyses,one should numerically calculate P⊙with the true electron density proﬁle. The diﬀerence between the numerical calculation and the semianalytic approximation is also brieﬂy discussed in[19]forδm2/E≤10−8eV2/MeV(where P c≃P⊙).Concerning the phase factorδ⊙,we conﬁrm earlier indications[12,13]about its small-ness,in both cases of true and exponential density.In the latter case,the semianalytic approximation gives[10,12],forδm2/E→0,theω-independent resultδ⊙≃L−1 r0 ln(√6The interested reader can obtain numerical tables of P⊙,calculated for the standard solar model density,upon request from the authors.less than one permill,and thus it can be safely neglected in all current applications.V.MATTER EFFECTS IN THE EARTHStrong Earth matter eﬀects typically emerge in the range where L osc∼L mat within the mantle(N e∼2mol/cm3)or the core(N e∼5mol/cm3),as well as in other ranges of mantle-core oscillation interference[29],globally corresponding toδm2/E∼10−7–10−6 eV2/MeV.Therefore,only marginal eﬀects are expected in the parameter range considered in this work,as conﬁrmed by the results reported in Fig.4.Figure4shows isolines of the quantity c2ω−P⊕,which becomes zero in the just-so oscillation limit of very smallδm2/E.The solid curves corresponds to a nadir angleη=0◦(diametral crossing of neutrinos)and the dotted curves toη=45◦(crossing of mantle only). For other values ofη(not shown),the quantity c2ω−P⊕has a comparable magnitude.In the current neutrino jargon,the Earth eﬀect shown in Fig.4is operative in the lowermost part of the so-called“LOW”MSW solution[16]to the solar neutrino problem,or,from another point of view,to the uppermost part of the vacuum solutions[15].Concerning the phase correctionδ⊕(not shown),it is found to be smaller than1.5×10−5in the whole mass-mixing plane considered,and thus can be safely neglected.In practical applications,the correction term c2ω−P⊕must be time-averaged.This poses, in principle,a tedious integration problem,since such correction appears,in Eq.(12),both in the amplitude of the oscillating term(∝cosξ)and in the remaining,non-oscillating term. While the integration over time can be transformed,for the non-oscillating term,into a more manageable integration overη[25],this cannot be done for the oscillating term,which depends on time both through the prefactorC= cos δm2L2E ,(21)can be written,in the narrow-width approximation(∆E=E− E ≪ E ),in terms of the Fourier transform of the spectrum,˜s(τ)= dE s(E)e i∆Eτ.(22) More precisely,C≃ dE s(E)cos δm2L E (23)=D cos δm2Lδm2L2E E≃D·cos≃J0 ǫδm2L2E ,(26) 2π 2π0dt L22Ewhere J0is the Bessel function,acting as a further damping term for large values of its argument.Notice that the maximum fractional variation of the orbital radius,(L max−L min)/L= 2ǫ=3.34×10−2,is an order of magnitude larger thanδR=R⊙/L=4.7×10−3which,in turn,is larger than the phase correctionsδ⊙andδ⊕.Therefore,one can safely neglectδR,δ⊙andδ⊕in practical applications involving yearly(or even seasonal)averages,as we do in this work.However,for averages over shorter time intervals,such approximation might break down.In particular,δR(δ⊙)might be comparable to the monthly(weekly)variations of the solar neutrino signal.The observability of such short-time variations is beyond the present sensitivity of real-time solarνexperiments and would require,among other things, very high statistics and an extremely stable level of both the signal detection eﬃciency and of the background.If such diﬃcult experimental goals will be reached in the future,some of the approximations discussed so far(and recollected in the next section)should be revisited and possibly improved.VII.PRACTICAL RECIPESWe have seen in the previous sections that,asδm2/E increases,the deviations of P⊙(and subsequently of P⊕)from the vacuum value c2ωbecome increasingly important.We have also seen that the phase correctionsδ⊙andδ⊕are smaller thanδR=R⊙/L,which can in turn be neglected in present applications,so that one can practically take the usual vacuum value for the oscillation phase,ξ≃δm2L/2E.We think it useful to organize known and less known results through the following approximate expressions for the calculation of P ee,which are accurate to better than3%with respect to the exact,general formula(12) valid at anyδm2/E.Forδm2/E<∼5×10−10eV2/MeV,one can take P⊙≃P⊕≃c2ω,and obtain the just-so oscillation formulaP JS ee≃c4ω+s4ω+2s2ωc2ωcosξ,(27) withξ=δm2L/2E.For5×10−10<∼δm2/E<∼10−8eV2/MeV,one can still take P⊕≃c2ω, but since P⊙=c2ω(quasi-vacuum regime)one has thatP QVee≃c2ωP⊙+s2ω(1−P⊙)+2sωcωof P⊕can be transformed into a more manageable integration over the nadir angle,both for yearly[25]and for seasonal[34]averages.To summarize,the above sequence of equations describes the passage from the regime of just-so to that of MSW oscillations,via quasi-vacuum oscillations.In the JS regime,oscillations are basically coherent and do not depend on the electron density in the Earth or in the Sun(N e→∞).In the MSW regime,oscillations are basically incoherent(L→∞) and,in general,depend on the detailed electron density proﬁle of both the Sun and theEarth.In particular,in the MSW regime one has to take into account the interplay between the density proﬁle and the neutrino source distribution proﬁle.The intermediate QV regime is instead characterized by partially coherent oscillations(with increasing decoherence as δm2/E increases),and by a sensitivity to the electron density of the Sun(but not of the Earth).Such sensitivity is not as strong as in the MSW regime and,in particular,QV eﬀects are independent from the speciﬁcνproduction point,which can be eﬀectively taken at the Sun center.For the sake of completeness,we mention that,for high values ofδm2/E(≫10−4eV2/MeV),corresponding to L osc≪L mat in the Sun,the sensitivity to matter eﬀects is eventually lost both in the Sun and in the Earth(P⊙≃P⊕≃c2ω),and one reaches a fourth regime sometimes called of energy-averaged(EA)oscillations,which is totally incoherent and N e-independent:P EAee≃c4ω+s4ω.(30) Such regime,which predicts an energy-independent suppression of the solar neutrinoﬂux, seems to be disfavored(but perhaps not yet ruled out)by current experimental data on total neutrino rates.In conclusion,forδm2/E going from extremely low values to inﬁnity, one can identify four rather diﬀerent oscillation regimes,JS→QV→MSW→EA,(31) each being characterized by speciﬁc properties and applicable approximations.Experiments still have to tell us unambiguously which of them truly applies to solar neutrinos.VIII.THREE-FLA VOR OSCILLATION ANALYSIS As discussed in[19],in the QV regime the2νsurvival probability(28)is non-symmetric with respect to the operationω→π2−ω.Therefore,while P JS3ν[obtained from Eqs.(17)and(27)]is symmetric with respect to theω=π8In the MSW regime,the mirror asymmetry of theﬁrst two octants was explicitly shown in[20].properties become evident in the triangular representation of the solar3νmixing parameter space discussed in[20,33],to which the reader is referred for further details.Figure6shows,in the triangular plot,isolines of P QV3ν(dotted lines)forδm2/E close to1.65×10−9eV2/MeV,corresponding to about100oscillation cycles.More precisely,the six panels correspond toξ=100×2π+∆ξ,with∆ξfrom0toπin steps ofπ/5.The dotted isolines are asymmetric with respect toω=π4also shows up in solar neutrino dataﬁts[19].Figure7reports the results of our global(rates+spectrum+day/night)three-ﬂavor analysis in the mass-mixing rangeδm2∈[10−11,10−8]eV2and tan2ω∈[10−2,102],for several representative values of tan2φ.We only show99%C.L.contours10(N DF=3)for the sake of clarity.The theoretical[35]and experimental[36–41]inputs,as well as theχ2statistical analysis[42],are the same as in Ref.[16](where MSW solutions were studied).Here, however,the range ofδm2is lower,in order to show the smooth transition from the MSW solutions to the QV andﬁnally the JS ones asδm2is decreased.In particular,the solutions shown in Fig.7represent the continuation,at lowδm2,of the LOW MSW solutions shown in Fig.10of[16](panel by panel).11As anticipated in the comments to Fig.6,the mirror asymmetry around tan2ω=1decreases for decreasingδm2(JS regime);a little asymmetry is still present even atδm2∼10−10eV2,where the gallium rates(sensitive to E as low as∼0.2MeV)start to feel QV eﬀects.In the region where QV eﬀects are important, the solutions are typically shifted in the second octant(ω>∼π/4),since the gallium rate is suppressed too much in theﬁrst octant(see also[19]).A similar drift was found for the LOWMSW solution [16].At any δm 2,the asymmetry decreases at large values of φ(tan 2φ>∼1.5)which,however,are excluded by the combination of accelerator and reactor data [21],unlessthe second mass square diﬀerence m 2turns out to be in the lower part of the sensitivity range of the CHOOZ experiment [43](m 2∼10−3eV 2).For φ=0,the standard two-ﬂavor case is recovered,and the results are comparable to those found in [15].The Super-Kamiokande spectrum plays only a marginal role in generating the mirror asymmetry of the solutions in Fig.7,since the modulation of QV eﬀects in the energy domain is much weaker than the one generated by the oscillation phase ξ.We ﬁnd that,at any given δm 2<∼10−8eV 2,the χ2diﬀerence at symmetric ωvalues is less than ∼1for the 18-bin spectrum data ﬁt.Therefore,QV eﬀects are mainly probed by total neutrino rates at present.We think it is not particularly useful to discuss more detailed features of the current QV solutions,such as combinations of spectral data or rates only,ﬁts with variations of hep neutrino ﬂux,etc.(which were instead given in [16]for MSW solutions).In fact,while the shapes of current MSW solutions are rather well-deﬁned,those of JS or QV solutions are still very sensitive to small changes in the theoretical or experimental input.Therefore,a detailed analysis of the “ﬁne structure”of the QV solutions in Fig.7seems unwarranted at present.Finally,in Fig.8we show sections of the allowed 3νsolutions (at 99%C.L.)in the triangle representation,for six selected (increasing)values of δm 2.Solutions are absent or shrunk at s 2φ∼0.5,where the theoretical νﬂux underestimates the gallium and water-Cherenkov data.The lowest value of δm 2(0.66×10−10eV 2)falls in the JS regime,so that the ring-like allowed region (which resembles the curves of iso-P ee in Fig.6)is symmetric with respect to the vertical axis at ω=π/4.However,as δm 2increases and QV eﬀects become operative,the solutions become more and more asymmetric,and shifted towards the second octant of ω[17,19].Figures 7and 8in this work,as well as Fig.10in [16],show that solar neutrino data,by themselves,put only a weak upper bound on the mixing angle φ.Much tighter constraints are set by reactor data [43],unless the second mass square diﬀerence m 2happens to be<∼10−3eV 2(which seems an unlikely possibility).In any case,QV eﬀects are operative also for a small (or zero)value of s 2φ.IX.CONCLUSIONSWe have presented a thorough analysis of solar neutrino oscillations in the “quasi-vacuum”oscillation regime,intermediate between the familiar just-so and MSW regimes.The QV regime is increasingly aﬀected by matter eﬀects for increasing values of δm 2.We have calculated such eﬀects both in the Sun and in the Earth,and discussed the accuracy of various possible approximations.We have implemented the QV oscillation probability in a full three-ﬂavor analysis of solar neutrino data,obtaining solutions which smoothly join (at δm 2∼10−8eV 2)the LOW MSW regions found in [16]for the same input data.The asymmetry of QV eﬀects makes such solutions diﬀerent for ω<π4,the two cases being symmetrized only in the just-so oscillation limit of small δm 2.ACKNOWLEDGMENTSWe thank J.N.Bahcall for providing us with updated standard solar model results.We thank A.Friedland and S.T.Petcov for useful discussions.REFERENCES[1]J.N.Bahcall,Neutrino Astrophysics(Cambridge University Press,Cambridge,England,1989).[2]B.Pontecorvo,Zh.Eksp.Teor.Fiz.53,1717(1967)[Sov.Phys.JETP26,984(1968)];Z.Maki,M.Nakagawa,and 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methods of theoretical physics

methods of theoretical physicsThe field of theoretical physics employs various methods and techniques to develop and test theories. Here are some common methods used in theoretical physics:1. Mathematical modeling: Theoretical physicists use mathematical equations and models to describe physical phenomena. They develop theoretical frameworks that predict and explain the behavior of the physical systems under study.2. Quantum field theory: This mathematical framework describes the behavior of particles and fields in quantum mechanics. It combines the principles of quantum mechanics with special relativity.3. Symmetry principles: Symmetries in nature play a crucial role in theoretical physics. Physicists often use symmetry principles to construct theories and make predictions. For example, the conservation of energy and momentum arises from the translational symmetry of physical laws.4. Perturbation theory: When the equations describing a physical system become mathematically complex, physicists often use perturbation theory. This method involves breaking down a problem into simpler parts and then solving it piece by piece.5. Computational simulations: Theoretical physicists employ computer simulations to study complex systems too difficult to analyze mathematically. By using numerical algorithms and computational power, simulations allow researchers to testtheoretical predictions, explore parameter spaces, and compare results with experiments.6. Effective field theory: This method involves constructing simplified models in which certain aspects of the physics are described at a lower energy scale. It is particularly useful when studying phenomena that occur at very high energy scales.7. Quantum mechanics: This fundamental theory provides a framework for understanding the behavior of matter and energy at the atomic and subatomic levels. Theoretical physicists often use quantum mechanics to develop theories and models that describe the behavior of particles and their interactions.8. General relativity: This theory describes gravity as the curvature of spacetime caused by mass and energy. Theoretical physicists use general relativity to study phenomena such as black holes, gravitational waves, and the large-scale structure of the universe.9. Statistical mechanics: This branch of physics characterizes the behavior of large ensembles of particles. Theoretical physicists use statistical mechanics to derive macroscopic properties from microscopic interactions, such as understanding the behavior of gases, magnets, and phase transitions.10. Symmetry breaking: In certain physical systems, symmetries can be spontaneously broken, leading to new emergent phenomena. Theoretical physicists study the effects of symmetry breaking to understand the behavior of quantum systems and the origin of mass.These are just a few of the methods employed by theoretical physicists. The field continues to evolve, and new techniques are constantly being developed as scientists seek to explore the frontiers of our understanding of the universe.。

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Introduction 1ical approximations in phase space using coherent states have been discussed extensively for several decades. This attractive topic, a favorite of many theoretical physicists and chemists, turns out to be very diﬃcult. In this contribution to its literature, we shall attempt to sort out and clarify the web of contradictions and inconsistencies that have characterized the recent state of the ﬁeld. We shall do so for the simplest possible case, one–dimensional coordinate space, i. e. two–dimensional phase space. This is the case where it is relatively easy to check the semiclassical approximations. We have done work in higher dimensions as well, but we do not include it here, as it would only obscure the basic relationships and further lengthen the paper. The conclusions we have reached are stated in section 7, and the reader who is already familiar with the subject may jump to them now to get an overall view. Because the pitfalls are numerous, however, we shall follow a slower approach, a historical one in this introduction, and then a systematic and detailed one in the body of the paper. The study of semiclassical methods has two basic motivations. First, it provides approximations to quantum mechanical quantities in terms of classical ingredients. These approximations should be very good if the typical classical actions are much larger than Planck’s constant. Interestingly, they are often fairly good even at very low quantum numbers. Second, semiclassical methods also help in understanding the quantum mechanical processes themselves, providing a more intuitive description. This description includes quantum mechanical interference, since both amplitudes and phases can be calculated semiclassically. The semiclassical approximation for the evolution operator, or propagator, in the coordinate representation has been known for more than 70 years and was ﬁrst written by Van Vleck [Van28]. It is a complex number with a modulus and a phase. The main part of the phase is the action of a classical trajectory joining a given initial coordinate to a given ﬁnal coordinate in a given time. Finding such trajectories is usually not a simple task. It is known as “the root search problem” and it gets more and more complicated as the number of dimensions increases. The modulus of the semiclassical propagator is related to the second derivative of the action with respect to these initial and ﬁnal points. It measures the dispersal of nearby trajectories. Gutzwiller, among others, revisited this problem around 1970 [Gut71, Gut90], focusing on non-integrable systems and giving birth to the ﬁeld of quantum chaos. Much progress has been made since then, particularly on the topological properties of Maslov indices [Rob91] and on the scars of periodic trajectories [Hel84, Bog88]. But there is another interesting representation for the evolution operator, which seems at ﬁrst sight to be more appropriate for comparisons with classical mechanics. This is the representation using the coherent states of a harmonic oscillator. They are gaussian states, localized in both coordinates and momenta, and therefore they can be thought of as quantum points in phase space. Although the exact coordinate propagator and the exact coherent state propagator are related by a simple change of representation,
3 Ambiguities in the Choice of Path Integral 3.1 Alternative Forms of the Path Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Ambiguities in Operator Ordering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Powers of . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 An 4.1 4.2 4.3 4.4 Initial Value Representation A mixed representation . . . . . . . . . . . . . Some properties of the mixed propagator . . . Comparison with Heller’s IVR . . . . . . . . . . Recovering Van Vleck’s Formula from the IVR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
arXiv:quant-ph/0105153v1 31 May 2001
Semiclassical Approximations in Phase Space with Coherent States
M. Baranger†, M. A. M. de Aguiar§†, F. Keck‡, H. J. Korsch‡, and B. Schellhaaß‡