Delay time of electron wave packet through a two-dimensional semiconductor heterostructure

In this work, we systematically investigate the group delay time of an electron wave packet through a two-dimensional semiconductor heterostructure. It is shown that the lateral displacement, resulting from the angular spread of the electron wave packet, plays an important role in total delay time. In the propagating case, the group delay time can be negative due to the effect of lateral displacement, and is greatly enhanced by transmission resonances. In the evanescent case, the delay time saturates to a constant in the opaque limit, which is simply the Hartman effect observed for a two-dimensional situation.     

Pulse dynamics in an optical fiber with linear intermode coupling and Kerr nonlinearity dispersion

The effect of Kerr nonlinearity dispersion on the envelope duration and the velocity of the envelope maximum for a wave packet propagating in an optical fiber and formed by two coupled copropagating waves is studied. The critical (threshold) energy of wave packet collapse can substantially be diminished and the supraluminal mode of propagation of the pulse envelope maximum in a nonamplifying medium can be realized owing to a significantly higher nonlinearity dispersion in such systems in comparison with “single-wave” ones.     

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     

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.     

Three dimensional diffusion delay time tomography

Stable numerical convolution of the diffusion impulse time response with a Gaussian wave packet produces a wave packet whose delay time is governed by an eikonal equation where the diffusivity distribution plays the role of the square of the velocity distribution. Diffusion delay time tomography data can approximately image the diffusivity distribution by solving an inverse problem for the eikonal equation with multiple rays that traverse low diffusivity regions not traversed by earlier related methods, and consequently the resolution in such regions is improved. This is important for medical imaging.     

Quantum dynamics of charge transfer on the one-dimensional lattice:Wave packet spreading and recurrence

The wave function temporal evolution on the one-dimensional(1D) lattice is considered in the tight-binding approximation. The lattice consists of N equal sites and one impurity site(donor). The donor differs from other lattice sites by the on-site electron energy E and the intersite coupling C. The moving wave packet is formed from the wave function initially localized on the donor. The exact solution for the wave packet velocity and the shape is derived at different values E and C. The velocity has the maximal possible group velocity v = 2. The wave packet width grows with time ～ t1/3and its amplitude decreases ～ t-1/3. The wave packet reflects multiply from the lattice ends. Analytical expressions for the wave packet front propagation and recurrence are in good agreement with numeric simulations.     

三态K2分子飞秒含时光电子能谱的理论研究

利用三态模型和含时波包法, 研究了K₂分子在强飞秒抽运-探测激光场中延时、脉宽以及抽运波长对光电子能谱和波包动力学过程的影响. 研究结果表明, 激光场强较弱或者脉宽较短都可能不发生Autler-Townes分裂, 光电子能谱呈现出单峰结构; 延时和抽运波长的改变影响能峰结构、位置和相对峰高; 对于不同的抽运波长, 波包的振动周期是相同的, 波包振荡幅度随脉宽增大而减小; 光电子能谱反映了波包动力学信息. 研究结果可以为实验上实现分子的光控制以及量子调控过程提供一定的参考, 并为进一步研究K₂分子的动力学性质提供有用的信息.     

Acoustic emission source location in plates using wavelet analysis and cross time frequency spectrum

In this study, the theories of wavelet transform and cross-time frequency spectrum (CTFS) are used to locate AE source with frequency-varying wave velocity in plate-type structures. A rectangular array of four sensors is installed on the plate. When an impact is generated by an artificial AE source such as Hsu–Nielsen method of pencil lead breaking (PLB) at any position of the plate, the AE signals will be detected by four sensors at different times. By wavelet packet decomposition, a packet of signals with frequency range of 0.125–0.25 MHz is selected. The CTFS is calculated by the short-time Fourier transform of the cross-correlation between considered packets captured by AE sensors. The time delay is calculated when the CTFS reaches the maximum value and the corresponding frequency is extracted per this maximum value. The resulting frequency is used to calculate the group velocity of wave velocity in combination with dispersive curve. The resulted locating error shows the high precision of proposed algorithm.     

Time development of a wave packet and the time delay

A one-dimensional scattering problem off a δ-shaped potential is solved analytically and the time development of a wave packet is derived from the time-dependent Schrödinger equation. The exact and explicit expression of the scattered wave packet supplies us with interesting information about the “time delay” by potential scattering in the asymptotic region. It is demonstrated that a wave packet scattered by a spin-flipping potential can give us quite a different value for the delay times from that obtained without spin-degrees of freedom.     

Nonlinear evolution of a three-dimensional transverse wave packet in a hot plasma including the effect of its interaction with an ion-acoustic wave

By a perturbation method two coupled nonlinear partial differential equations are obtained for the nonlinear evolution of a three dimensional transverse wave packet in a hot plasma including the effect of its interaction with a long wavelength ion-acoustic wave. From these two equations a nonlinear dispersion relation is obtained, from which the instability condition of a uniform transverse wave train including the effect of its interaction, both at resonance and at nonresonance with a long wavelength ion-acoustic wave, are deduced. Resonance occurs when the component of group velocity of the longitudinal wave along the direction of propagation of the ion-acoustic wave is equal to the phase velocity of the wave. Assuming the usual type of dependence of amplitude on space and time the coupled equations are transformed into two other coupled equations, which reduced to a single nonliear Schrödingsr equation when three dimensionality is disregarded. It is found that these three transformed equations cannot give instability condition at resonance.On leave fromThe Department of Mathematics, University of Kalyani, West Bengal, India.     

一般波包在均匀场中的运动

求出了一般波包的中心和宽度在一维均匀场中的变化规律，波包的中心遵循经典粒子的运动规律，宽度的平方则以时间的二次函数增长，但动量空间的相应波包却保持其宽度不变，特别是平面波在运动过程中仍保持为平面波，对Gauss波包，求出了波函数随时间变化的显式，验证了一般结论。     

On the velocity of propagation of a frequency-modulated wave packet in a dispersive absorbing medium

It is shown that the velocity of propagation of a frequency-modulated wave packet through a strongly dispersive absorbing medium can be significantly different from (either higher or lower than) that of a non-frequency-modulated wave packet. This difference is attributed to the absorption dispersion of the medium. The easiest way to take the absorption dispersion into account is to use the formalism of the complex group velocity of a wave packet. This paper considers the propagation of a linear frequency-modulated wave packet, whose carrier frequency is close to the frequency of a spectral absorption line of the medium.     

De Sitter spacetime as a momentum measuring apparatus

We discuss the evolution of a quantum wave packet in the expanding de Sitter spacetime using the plane wave solutions of the Dirac equation. We concentrate on the case of large negative times when the packet approaches the event horizon and confirm that the evolution accords with that expected from the classical trajectories. We point out that in certain conditions the packet can split into two components that become localized at different parts of the horizon and that this effect can be seen, in an idealized sense, as a measuring process for the momentum of the particle, in direct analogy with the measurement of spin in a Stern-Gerlach experiment.     

Li2分子含时波包动力学的理论研究

利用含时量子波包方法计算得到了Li2分子的光电子能谱,并运用波包动力学理论对含有不同参量的光电子能谱现象给出了合理的解释.通过分析文中的直观图像,研究了波包的动力学过程.结果表明,泵浦-探测脉冲的延迟时间对光电子能谱的形状有重要的影响;在较短延迟时间下,能谱独特的四峰现象是由光诱导势的产生引起的.     

Li2分子含时波包动力学的理论研究

利用含时量子波包方法计算得到了Li2分子的光电子能谱,并运用波包动力学理论对含有不同参量的光电子能谱现象给出了合理的解释。通过分析文中的直观图像,研究了波包的动力学过程。结果表明,泵浦-探测脉冲的延迟时间对光电子能谱的形状有重要的影响;在较短延迟时间下,能谱独特的四峰现象是由光诱导势的产生引起的。     

Direct observation of the space-time transformation of a femtosecond laser pulse in fused quartz

The transformation of a femtosecond laser pulse propagating in fused quartz before and after its transition to a filamentation regime has been investigated by femtosecond time-resolved optical polarigraphy. The spatial periodicity of a light field along and across the propagation axis has been detected and its nature has been attributed to the interference of the conical and plane components of the wave packet of the filament. The “supraluminal” motions of the observed filament intensity maximum are due to the longitudinal transformation of the pulse profile.     

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.     

Velocity selective trapping of atoms in a frequency-modulated laser field

The wavefunction of a moderately cold atom in a stationary near-resonant standing light wave delocalizes very fast due to wave packet splitting. However, we show that frequency modulation of the field may suppress packet splitting for some atoms having specific velocities in a narrow range. These atoms remain localized in a small space for a long time. We propose that in a real experiment with cold atomic gas this effect may decrease the velocity distribution of atoms (the field traps the atoms with such specific velocities while all other atoms leave the field).     

On the mass of a wave packet in plasma

The relativistic mass density which is transported by a quasi-monochromatic transverse wave packet is derived. The wave packet is assumed to propagate in isotropic cold collisionless plasma. The co-ordinate frame moving with the group velocity appears as the rest frame for both the mass density and the total mass of the wave packet. The equivalence of the energy-mass relation and the dispersion equation is demonstrated.