Stationary phase method and delay times for relativistic and non-relativistic tunneling particles

The stationary phase method is frequently adopted for calculating tunneling phase times of analytically-continuous Gaussian or infinite-bandwidth step pulses which collide with a potential barrier. This report deals with the basic concepts on deducing transit times for quantum scattering: the stationary phase method and its relation with delay times for relativistic and non-relativistic tunneling particles. After reexamining the above-barrier diffusion problem, we notice that the applicability of this method is constrained by several subtleties in deriving the phase time that describes the localization of scattered wave packets. Using a recently developed procedure - multiple wave packet decomposition - for some specifical colliding configurations, we demonstrate that the analytical difficulties arising when the stationary phase method is applied for obtaining phase (traversal) times are all overcome. In this case, we also investigate the general relation between phase times and dwell times for quantum tunneling/scattering. Considering a symmetrical collision of two identical wave packets with an one-dimensional barrier, we demonstrate that these two distinct transit time definitions are explicitly connected. The traversal times are obtained for a symmetrized (two identical bosons) and an antisymmetrized (two identical fermions) quantum colliding configuration. Multiple wave packet decomposition shows us that the phase time (group delay) describes the exact position of the scattered particles and, in addition to the exact relation with the dwell time, leads to correct conceptual understanding of both transit time definitions. At last, we extend the non-relativistic formalism to the solutions for the tunneling zone of a one-dimensional electrostatic potential in the relativistic (Dirac to Klein-Gordon) wave equation where the incoming wave packet exhibits the possibility of being almost totally transmitted through the potential barrier. The conditions for the occurrence of accelerated and, eventually, superluminal tunneling transmission probabilities are all quantified and the problematic superluminal interpretation based on the non-relativistic tunneling dynamics is revisited. Lessons concerning the dynamics of relativistic tunneling and the mathematical structure of its solutions suggest revealing insights into mathematically analogous condensed-matter experiments using electrostatic barriers in single- and bi-layer graphene, for which the accelerated tunneling effect deserves a more careful investigation.     

The exact correspondence between phase times and dwell times in a symmetrical quantum tunneling configuration

The general and explicit relation between phase times and dwell times for quantum tunneling or scattering is investigated. Considering two identical propagating wave packets symmetrically impinging a one-dimensional barrier, here we demonstrate that these two distinct transit time definitions give connected results where, for such a colliding configuration, the phase time (group delay) accurately describes the exact position of the scattered particles. The analytical difficulties that arise when the stationary phase method is employed for obtaining the phase (traversal) times are all overcome, since the multiple wave packet decomposition allows us to recover the exact position of the reflected and transmitted waves. In addition to the exact relation between the phase time and the dwell time, this leads to the right interpretation for both of them. PACS 03.65.Xp     

Spectral line narrowing in a gas by atoms trapped in a standing light wave

This paper presents results of a theoretical analysis of a new method for eliminating the Doppler broadening of spectral lines and the broadening by the transit time of atoms through a light beam. The atomic motion in a one-dimensional standing wave is studied and the conditions for translational-to-vibrational motion transformation are found. The variation in the Doppler contour by the trapping effect is investigated. It is illustrated, in particular, that the width of the narrow peak at the line centre depends mainly on the finite transit time of the atoms through the light beam. Next it is shown that, by accumulating slow atoms in a three-dimensional standing wave, it is possible, in principle, to observe narrow peaks with their widths determined only by the natural line width. The possibility of experimentally detecting of the phenomenon is discussed.     

Misinterpretation yields supervelocities during transmission of wave packets through a barrier

This paper is concerned with the transmission time of an incident Gaussian wave packet through a symmetric rectangular barrier. Following Hartman (J. Appl. Phys. 33, 3427 (1962)), the transmission time is usually taken as the difference between the time at which the peak of the transmitted packet leaves the barrier of thickness and the time at which the peak of the incident Gaussian wave packet arrives at the barrier. This yields a corresponding transmission velocity which appears under certain conditions as a supervelocity, i.e. becomes larger than the corresponding propagation velocity in free space which is the group velocity for electrons or the velocity of light for photons, respectively. By analysing the propagation of a broadband wave packet (which leads in free space to an extremely concentrated wave packet at a certain time) we obtain the pulse response function of the barrier and show that the insertion of the barrier is physically unable to produce a supervelocity. Therefore, the peak of an incident Gaussian wave packet and the peak of the transmitted wave packet are in no causal relationship. The shape of the transmitted wave packet is produced from the incident wave by convolution with the pulse response of the barrier. This yields a distortion of the shape of the wave packet which includes also the observed negative time shift of the peak. We demonstrate further that the phenomenon of Hartman's supervelocities is not restricted to barriers with their exponentially decaying fields but occurs for instance also in transmission lines with an inserted LCR circuit. Received 7 January 1999 and Received in final form 22 April 1999     

Principle of stationary phase for propagating wave packets in the unidimensional scattering problem

We point out some incompatibilities which appear when one applies the stationary phase method for deriving phase times to obtain the spatial localization of wave packets scattered by a unidimensional potential barrier. We concentrate on the above barrier diffusion problem where the wave packet collision implies the possibility of multiple reflected and transmitted wave packets, which, depending on the boundary conditions, can overlap or stand in relative separation in space. We demonstrate that the indiscriminate use of the method for such a particular configuration leads to paradoxical results for which the correct interpretation, confirmed by analytical/numerical calculations, imposes the necessity of the appearance of multiple peaks as a consequence of multiple reflections by the barrier steps. Also at Instituto de Física Gleb Wataghin, UNICAMP, PO Box 6165, 13083-970 Campinas, SP, Brazil.     

Spin-polarizing properties of heterostructures with magnetic nano elements

The problem of electron resonant and non-resonant scatterings on two magnetized barriers is studied in the one-dimension. The transfer-matrix is built up to exactly calculate the coefficient of the electron transmittance through the system of two magnetic barriers with non-collinear magnetizations. The polarization of the transmitted electron wave for resonance and non-resonance transmittances is calculated. The transmittance coefficient and spin polarization can be drastically enhanced and controlled by the angle between the barrier magnetizations.     

Equivalent circuit of the barrier-conductor structures

Novel heterostructure devices are comprised of potential barriers connected by short conductors. In this paper we present a simple theory for the transport properties of the barrier-conductor chain. The analysis is based on the solution of the Boltzmann equation supplemented by the boundary conditions provided by the barrier reflection and transmission probabilities. As an application of the theory the small signal equivalent circuit is constructed for the single and double barrier cases and for the infinite periodic chain. The high frequency properties of these structures are discussed. In general, the multibarrier structures show transit time resonances associated with multiple reflections.     

超声检测中常用激励波形的高精度相控发射实现

超声无损检测根据不同的检测需求使用不同的激励波形,而超声相控阵技术的应用则要求各通道的激励波形能够相控发射。分析了不同检测对象、不同检测方法对激励信号的波形类型要求,研究了基于FPGA主控的数字式超声检测平台下实现单通道方波脉冲、尖脉冲、正弦波形和任意波形等的发射方法,以及以上各种波形在多通道高精度相控发射时的实现方法。实验表明,各种波形的激励信号都实现了高精度相控发射,最高可达1ns的延时精度。     

Interaction of a shear wave with a moving domain wall in a ferromagnetic crystal with allowance for the external magnetic field

The boundary-value problem of the magnetoelastic wave interaction with a moving domain wall in a ferromagnetic crystal is solved in the nonexchange magnetostatic approximation with allowance for the external magnetic field. It is shown that the difference introduced by magnetic field between the ferromagnetic resonance frequencies of the domains does not cause any noticeably departure of the refraction characteristics of reflected and transmitted waves from those observed at zero frequency mismatch. By contrast, the magnitudes of the transmission and reflection coefficients strongly depend on the external magnetic field and on the mobility of the domain wall. The dependence of the magnitude of the reflection coefficient on the external magnetic field at a fixed angle of shear wave incidence is found to possess two ferromagnetic resonance peaks. The positions and heights of the peaks may vary depending on the mobility of the domain wall.     

On the Traversal Time of Barriers

Fifty years ago Hartman studied the barrier transmission time of wave packets (J Appl Phys 33:3427–3433, 1962). He was inspired by the tunneling experiments across thin insulating layers at that time. For opaque barriers he calculated faster than light propagation and a transmission time independent of barrier length, which is called the Hartman effect. A faster than light (FTL or superluminal) wave packet velocity was deduced in analog tunneling experiments with microwaves and with infrared light thirty years later. Recently, the conjectured zero time of electron tunneling was claimed to have been observed in ionizing helium inside the barrier. The calculated and measured short tunneling time arises at the barrier front. This tunneling time was found to be universal for elastic fields as well as for electromagnetic fields. Remarkable is that the delay time is the same for the reflected and the transmitted waves in the case of symmetric barriers. Several theoretical physicists predicted this strange nature of the tunneling process. However, even with this background many members of the physics community do not accept a FTL signal velocity interpretation of the experimental tunneling results. Instead a luminal front velocity was calculated to explain the FTL experimental results frequently. However, Brillouin stated in his book on wave propagation and group velocity that the front velocity is given by the group velocity of wave packets in the case of physical signals, which have only finite frequency bandwidths. Some studies assumed barriers to be cavities and the observed tunneling time does represent the cavity lifetime. We are going to discus these continuing misleading interpretations, which are found in journals and in textbooks till today.     

Scattering by a plane-parallel layer with high concentration of optically soft particles

A method describing light propagation in a plane-parallel light-scattering layer with large concentration of homogeneous particles is developed. It is based on the radiative transfer equation and the doubling method. The interference approximation is used to take into account collective scattering effects. Spectral dependence of transmitted light for a layer of nonabsorbing optically soft particles with subwavelength-sized particles is investigated. At small volume concentration of the particles the weak spectral dependences of wave exponents for coherently transmitted and diffuse light are observed. It is shown that in a layer with large volume concentration of the subwavelength-sized particles the wave exponent can exceed considerably the value of four, which takes place for the Rayleigh particles. The dependence of wave exponents for coherently transmitted and diffuse light on the refractive index and concentration of particles is investigated in detail. Multiple scattering of light results in the reduction of the exponent. The quantitative results are presented and discussed. It is shown that there is a range of wavelengths where the negative values of the wave exponent at the regime of multiple scattering are implemented.     

Transit time of chirped pulses through one-dimensional, nonabsorbing barriers

Experiments show that the transit times of chirped, narrow-band pulses that move across nonabsorbing, one-dimensional barriers are modified dramatically by the interplay between the chirp and the transmission function of the sample. In an experiment we monitored 0.9-ns chirped, nearly Gaussian pulses as they traversed a 450-mum GaAs etalon. At certain wavelengths pulse transit times can be superluminal or even negative. To explain these phenomena we have proposed a generalization of the transit time for chirped pulses that is still meaningful even when the transit times are superluminal or negative. Our predictions agree well with the experimental results.     

A Simple Method of the Flow Rate Determination of Fluids on the Basis of Diagrams obtained by a Radioactive Tracer Measurement

The simpliest and most convenient in a practical use method of the flow rate measurement, based on the tracer technique, is a such one, in which the passing time of the “tracer wave” is determined. The results are then obtained as a diagram of radiation intensity in the chosen points of the pipeline. The passing of the tracer over these points is marked on a recording tape in a form of the characteristic peaks. Depending on the kind of flow, the recorded peaks take the different shapes. In the first order they depend on the velocity profile in the pipe cross section. This differentiation of velocities causes that, when the distance between the detecting probe and the injecting unit increases, the tracer wave becomes more diluted, involving consequently a corresponding lengthening and flattening of peaks.     

Diffraction of light by an acoustic wave with three frequency components

Collinear light diffraction by three-frequency sound is investigated theoretically. The amplitude distributions of transmitted and diffracted light waves along the cell are calculated for different amplitudes of sound signals. The dependence of the intensity of principal diffraction peaks on the frequency difference between acoustic signal components is studied for different amplitude ratios of these components. It is shown that the character of this dependence for a wave being in synchronism differs substantially from that for two other waves characterized by detuning. The dependence of the amplitudes of principal and parasitic diffraction peaks on the efficiency of acoustooptical interaction is investigated. It is demonstrated that parasitic sideband components in diffracted light can play a considerable role if the diffraction efficiency is sufficiently high and exceeds 80%.     

Investigations on AlGaN‐Based Deep‐Ultraviolet Light‐Emitting Diodes With Si‐Doped Quantum Barriers of Different Doping Concentrations

In this work, we investigate the impact of Si doped AlGaN quantum barriers on the optical powers for [0001] oriented III‐nitride based deep‐ultraviolet light‐emitting diodes (DUV LEDs). The polarization‐induced electric field in the active region is screened as the result of Si‐doped quantum barriers, which gives rise to the improved spatial overlap between electron and hole wave functions. The polarization screening effect within the quantum wells is further proven by the observation of the blue shift for the wavelength. However, the hole distribution across the active region can be significantly retarded if the Si dosage in the quantum barriers is too high. Therefore, the improved radiative recombination within the active region can be realized provided that the Si dosage in the quantum barriers is moderately adjusted to guarantee both the better hole injection efficiency and the screened polarization effect in the multiple quantum wells.     

声表面波在圆弧处反射及透射的数值研究*

为了研究直达表面波在圆弧过渡面处传播性能的变化,采用有限元方法模拟了热弹机制下,线性脉冲激光辐照金属铝块表面时激发的表面波在近表面传播过程中,在不同曲率半径的圆弧处发生的反射及透射现象,建立了圆弧半径与反射表面波以及透射表面波时域信号特征之间的联系。计算结果表明:曲率半径与表面波中心波长的数值关系对表面波在圆弧处的传播有显著的影响;同时证明了根据透射表面波信号的到达时间可以反演圆弧半径的大小,为之后利用表面波信号定量检测材料表面圆弧凹痕的深度提供了理论依据。     

Quasi-bound states, resonance tunnelling, and tunnelling times generated by twin symmetric barriers

In analogy with the definition of resonant or quasi-bound states used in three-dimensional quantal scattering, we define the quasi-bound states that occur in one-dimensional transmission generated by twin symmetric potential barriers and evaluate their energies and widths using two typical examples: (i) twin rectangular barrier and (ii) twin Gaussian-type barrier. The energies at which reflectionless transmission occurs correspond to these states and the widths of the transmission peaks are also the same as those of quasi-bound states. We compare the behaviour of the magnitude of wave functions of quasi-bound states with those for bound states and with the above-barrier state wave function. We deduce a Breit-Wigner-type resonance formula which neatly describes the variation of transmission coefficient as a function of energy at below-barrier energies. Similar formula with additional empirical term explains approximately the peaks of transmission coefficients at above-barrier energies as well. Further, we study the variation of tunnelling time as a function of energy and compare the same with transmission, reflection time and Breit-Wigner delay time around a quasi-bound state energy. We also find that tunnelling time is of the same order of magnitude as lifetime of the quasi-bound state, but somewhat larger.     

Small corrections to the tunneling phase-time formulation

After reexamining the above-barrier diffusion problem where we notice that the wave packet collision implies the existence of multiple reflected and transmitted wave packets, we analyze the way of obtaining phase times for tunneling/reflecting particles in a particular colliding configuration where the idea of multiple peak decomposition is recovered. To partially overcome the analytical incongruities which frequently arise when the stationary phase method is adopted for computing the (tunneling) phase-time expressions, we present a theoretical exercise involving a symmetrical collision between two identical wave packets and a unidimensional squared potential barrier where the scattered wave packets can be recomposed by summing the amplitudes of simultaneously reflected and transmitted wave components so that the conditions for applying the stationary phase principle are totally recovered. Lessons concerning the use of the stationary phase method are drawn. PACS 02.30.Mv, 03.65.Xp     

微通道板动态特性的数值模拟

对皮秒高压脉冲驱动下微通道板中电子的渡越时间特性和增益特性进行了数值模拟,在电压脉冲波形分别为高斯形、三角形和梯形时,得到了电子渡越时间与电压脉冲宽度、幅度的关系曲线。在考虑入射电子为一高斯电子脉冲的情况下,获得了增益曲线的半峰全宽和峰值随脉冲电压幅度、宽度的变化规律。分析结果表明：当微通道板两端所加电压为梯形波时,微通道板中电子的渡越时间特性和增益特性较加三角波和高斯波要好。     

声表面波检测铝板表面微缺陷深度实验研究

为了探究声表面波与不同深度微裂纹缺陷相互作用的关系,将脉冲激光作用于一系列不同缺陷的试件铝板上进行线光源激励,激发激光超声波。用超声传感器接收在铝板中传播的激光超声信号,通过数字荧光示波器采集激光超声在铝板中的传播数据。对采集到的反射波数据进行分离谱分离过程得到的铝板中激光超声的时域分布和透射波数据进行频域分析。实验发现:缺陷深度影响着反射回波两峰值特征点到达时间差,两者之间近似线性关系,也影响着透射波的截止频率且二者呈现递减关系。