Observation of nonspreading wave packets in an imaginary potential

We propose and experimentally demonstrate a method to prepare a nonspreading atomic wave packet. Our technique relies on a spatially modulated absorption constantly chiseling away from an initially broad de Broglie wave. The resulting contraction is balanced by dispersion due to Heisenberg's uncertainty principle. This quantum evolution results in the formation of a nonspreading wave packet of Gaussian form with a spatially quadratic phase. Experimentally, we confirm these predictions by observing the evolution of the momentum distribution. Moreover, by employing interferometric techniques, we measure the predicted quadratic phase across the wave packet. Nonspreading wave packets of this kind also exist in two space dimensions and we can control their amplitude and phase using optical elements.     

Minimal position-velocity uncertainty wave packets in relativistic and non-relativistic quantum mechanics

We consider wave packets of free particles with a general energy-momentum dispersion relation E(p). The spreading of the wave packet is determined by the velocity v=∂_pE. The position-velocity uncertainty relation is saturated by minimal uncertainty wave packets Φ(p)=Aexp(-αE(p)+βp). In addition to the standard minimal Gaussian wave packets corresponding to the non-relativistic dispersion relation E(p)=p²/2m, analytic calculations are presented for the spreading of wave packets with minimal position-velocity uncertainty product for the lattice dispersion relation E(p)=-cos(pa)/ma² as well as for the relativistic dispersion relation . The boost properties of moving relativistic wave packets as well as the propagation of wave packets in an expanding Universe are also discussed.     

Motion of an electronic wave packet in an electromagnetic field

An equation is derived for the matrix of the parameters of a small Gaussian wave packet moving in arbitrary fixed electromagnetic fields. The equation can be used to describe the evolution of wave packets in a wide class of vacuum devices. A simple example of the evolution of a packet in a constant magnetic field is studied. Pis’ma Zh. éksp. Teor. Fiz. 64, No. 8, 515–520 (25 October 1996)     

Tomographic Description of a Quantum Wave Packet in an Accelerated Frame

The tomography of a single quantum particle (i.e., a quantum wave packet) in an accelerated frame is studied. We write the Schrödinger equation in a moving reference frame in which acceleration is uniform in space and an arbitrary function of time. Then, we reduce such a problem to the study of spatiotemporal evolution of the wave packet in an inertial frame in the presence of a homogeneous force field but with an arbitrary time dependence. We demonstrate the existence of a Gaussian wave packet solution, for which the position and momentum uncertainties are unaffected by the uniform force field. This implies that, similar to in the case of a force-free motion, the uncertainty product is unaffected by acceleration. In addition, according to the Ehrenfest theorem, the wave packet centroid moves according to classic Newton’s law of a particle experiencing the effects of uniform acceleration. Furthermore, as in free motion, the wave packet exhibits a diffraction spread in the configuration space but not in momentum space. Then, using Radon transform, we determine the quantum tomogram of the Gaussian state evolution in the accelerated frame. Finally, we characterize the wave packet evolution in the accelerated frame in terms of optical and simplectic tomogram evolution in the related tomographic space.     

Semiclassical trajectory-coherent states of the nonlinear Schrödinger equation with unitary nonlinearity

Semiclassically concentrated states of the nonlinear Schrödinger equation (NLSE) with unitary nonlinearity, representing multidimensional localized wave packets, are constructed on the basis of the Maslov complex germ theory. A system of ordinary differential equations of Hamilton-Ehrenfest (HE) type, describing the motion of the wave packet centroid, is derived. The structure of the HE system is strongly influenced by the initial conditions of the Cauchy problem for the NLSE. Wave packets of Gaussian type are constructed in an explicit form. Possible use of the solutions constructed in the problem of optical pulse propagation in a nonlinear medium with nonstationary dispersion is discussed.     

On the kinematics of broadband multipath scintillation and the approach to saturation

Utilizing a simple model in which the acoustic wave function is a sum of independent Gaussian wave packets, the relative intensity variance or scintillation index (SI) is analytically calculated. The model has an unspecified probability density function (PDF) for wave packet amplitudes and Gaussian PDFs for travel-time-induced and non-travel-time-induced phase shifts; amplitudes and both phase shifts are assumed to be mutually uncorrelated. It is shown that a proper treatment of the mean field is required to obtain the saturation value, SI = 1, in the limit of a large number of interfering wave packets. The analytic formulas for SI allow identification of important wave packet parameters in the approach to saturation. Criteria are identified for both broadband and narrow-band cases for which the approach to saturation is from above and below 1. It is demonstrated that the broadband approach to saturation is much slower than the narrow-band cases, since wave packets separated in time by more than an inverse bandwidth do not strongly contribute to interference. This effect is quantified by the time-bandwidth product. The model is also used to obtain an analytic expression for pulse time spread; it is shown that multipath conditions which favor a rapid approach to saturation do not favor large pulse spread.     

Temporal ghost imaging of a time object,dispersion cancelation,and nonlocal time lens with bi-photon state

We present a theory of temporal diffraction, temporal imaging of a bi-photon state, and temporal ghost imaging of a time object. By applying factional Fourier transform to the bi-photon wave packet propagating in space, we could obtain a theory that shows the physical origin of dispersion cancelation, temporal imaging, nonlocal effects of time lenses, and temporal ghost imaging. We introduce the temporal diffraction distance for bi-photon wave packet and show that the bi-photon wave packet behaves like a single wave packet whose temporal diffraction distance is determined by the coherent sum of the temporal diffraction distances for the signal and the idler beams. This property yields the well known dispersion cancelation, the recovery of original bi-photon wave packet in temporal imaging, and the nonlocal combination of two time lenses placed in different arms. We also propose a method for ghost imaging of an arbitrary time object.     

矩形弹子球中的量子波包分析(英文)

利用波包分析量子力学体系的动力学行为在研究经典和量子的对应关系方面越来越成为一个非常重要的方法.利用高斯波包分析方法,我们计算了矩形弹子球体系的自关联函数,自关联函数的峰和经典周期轨道的周期符合的很好,这表明经典周期轨道的周期可以通过含时的量子波包方法产生.我们还讨论了矩形弹子球的波包回归和波包的部分回归,计算结果表明在每一个回归时间,波包出现精确的回归.对于动量为零的波包,初始位置在弹子球内部的特殊对称点处,出现一些时间比较短的附加的回归.     

H-在金属面附近光剥离的波包动力学研究

利用波包演化和自关联函数方法对H^-在金属面附近光剥离的波包动力学进行了研究.结果表明,金属面附近光剥离电子的波包演化和回归结构与H^-到金属面的距离、激光脉冲的脉冲宽度和初始动量都有一定的关系.因此,可以通过改变离子表面距离和激光脉冲的参数对光剥离电子的动力学性质进行调控研究.除此之外,光剥离电子的镜像态寿命对波包的演化和自关联函数也会产生一定的影响:考虑镜像态寿命的影响时,随着时间的演化,波包概率密度的振幅逐渐减小,波包整体上有明显的衰减,寿命对波包演化过程中的干涉有削弱的作用;通过对电子波包的自关联函数研究,发现无限长寿命的电子波包有很好的量子回归现象,而当考虑寿命因素后光剥离电子波包随着时间的演化会发生周期性的坍塌和扩散,经过一段时间后,该回归现象消失.本文的理论研究可以为表面附近电子波包动力学的实验研究提供一定的参考价值. 关键词： 波包 演化和回归 自关联函数 金属面     

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     

Cauchy matrix structure of the Mel'nikov model of long-short wave interaction

We propose a systematic method to construct the Mel'nikov model of long-short wave interactions, which is a special case of the Kadomtsev-Petviashvili (KP) equation with self-consistent sources (KPSCS). We show details how the Cauchy matrix approach applies to Mel'nikov's model which is derived as a complex reduction of the KPSCS. As a new result we find that in the dispersion relation of a 1-soliton there is an arbitrary time-dependent function that has previously not reported in the literature about the Mel'nikov model. This function brings time variant velocity for the long wave and also governs the short-wave packet. The variety of interactions of waves resulting from the time-freedom in the dispersion relation is illustrated.     

Semiclassical propagation of Gaussian wave packets

We analyze the semiclassical evolution of Gaussian wave packets in chaotic systems. We show that after some short time a Gaussian wave packet becomes a primitive WKB state. From then on, the state can be propagated using the standard time-dependent WKB scheme. Complex trajectories are not necessary to account for the long-time propagation. The Wigner function of the evolving state develops the structure of a classical filament plus quantum oscillations, with phase and amplitude being determined by geometric properties of a classical manifold.     

General quasi-nonspreading linear three-dimensional wave packets

We introduce a general approach for generation of sets of three-dimensional quasi-nonspreading wave packets propagating in linear media, also referred to as linear light bullets. The spectrum of rigorously nonspreading wave packets in media with anomalous group velocity dispersion is localized on the surface of a sphere, thus drastically restricting the possible wave packet shapes. However, broadening slightly the spectrum affords the generation of a large variety of quasi-nonspreading distributions featuring complex topologies and shapes in space and time that are of interest in different areas, such as biophysics or nanosurgery. Here we discuss the method and show several illustrative examples of its potential.     

Quantum Mechanical Analysis of the Equilateral Triangle Billiard: Periodic Orbit Theory and Wave Packet Revivals

Using the fact that the energy eigenstates of the equilateral triangle infinite well (or billiard) are available in closed form, we examine the connections between the energy eigenvalue spectrum and the classical closed paths in this geometry, using both periodic orbit theory and the short-term semi-classical behavior of wave packets. We also discuss wave packet revivals and show that there are exact revivals, for all wave packets, at times given by T_rev=9μa²/4?π where a and μ are the length of one side and the mass of the point particle, respectively. We find additional cases of exact revivals with shorter revival times for zero-momentum wave packets initially located at special symmetry points inside the billiard. Finally, we discuss simple variations on the equilateral (60°-60°-60°) triangle, such as the half equilateral (30°-60°-90°) triangle and other “foldings,” which have related energy spectra and revival structures.     

Semiclassical dynamics of electron transfer at metal surfaces

In this Letter, we demonstrate that nonadiabatic dynamics of molecular scattering from metal surfaces can be efficiently simulated by semiclassical Gaussian wave packet propagation on a local complex potential. The method relies on the wideband limit decoupling of the nuclear equations of motion on different electronic states. If the continuum diabatic potential surfaces are assumed to be parallel, the number of Gaussian wave packets spawned scales at most linearly with propagation time, allowing efficient propagation of nuclear dynamics.     

Wave packets in graphene under external fields: Vortices formation

In this work we analyzed the time propagation of wave packets on a sheet of graphene under the action of external magnetic and electric fields in the Hall configuration. The treatment given in this work to the problem of particle propagation in graphene is based on the tight-binding model, not requiring to consider the linear approximation of the band structure around point K in the Brillouin zone. So, our calculation is able to describe the behavior of the particle in more general cases, not only the case of low lying excited states, the so-called massless Dirac electrons. Evaluating the time evolution of the wave function we assume as an initial state a Gaussian with a given velocity. We have considered the symmetric gauge for the vector potential. For specific cases one is able to show a very interesting effect such as the apparition of vortices, i.e., the initial wave is split into components each one of these forming vortices that remain stationaries as time goes. Moreover, for a packet with a wave vector near point K in the Brillouin zone, one is able to show the presence of the effect of zitterbewegung, that is, a trembling motion of the centroid of the wave packet. The inclusion of a dc electric field in the plane of the graphene lattice displaces the vortices in a direction perpendicular to the field.     

On the Value of Information in Status Update Systems

The age of information (AoI) is now well established as a metric that measures the freshness of information delivered to a receiver from a source that generates status updates. This paper is motivated by the inherent value of packets arising in many cyber-physical applications (e.g., due to precision of the information content or an alarm message). In contrast to AoI, which considers all packets are of equal importance or value, we consider status update systems with update packets carrying values as well as their generated time stamps. A status update packet has a random initial value at the source and a deterministic deadline after which its value vanishes (called ultimate staleness). In our model, the value of a packet either remains constant until the deadline or decreases in time (even after reception) starting from its generation to the deadline when it vanishes. We consider two metrics for the value of information (VoI) at the receiver: sum VoI is the sum of the current values of all packets held by the receiver, whereas packet VoI is the value of a packet at the instant it is delivered to the receiver. We investigate various queuing disciplines under potential dependence between value and service time and provide closed form expressions for both average sum VoI and packet VoI at the receiver. Numerical results illustrate the average VoI for different scenarios and relations between average sum VoI and average packet VoI.     

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     

Propagation of General Wave Packets in Some Classical and Quantum Systems

In quantum mechanics the center of a wave packet is precisely defined as the center of probability. The center-of-probability velocity describes the entire motion of the wave packet. In classical physics there is no precise counterpart to the center-of-probability velocity of quantum mechanics, in spite of the fact that there exist in the literature at least eight different velocities for the electromagnetic wave. We propose a center-of-energy velocity to describe the entire motion of general wave packets in classical physical systems. It is a measurable quantity, and is well defined for both continuous and discrete systems. For electromagnetic wave packets it is a generalization of the velocity of energy transport. General wave packets in several classical systems are studied and the center-of-energy velocity is calculated and expressed in terms of the dispersion relation and the Fourier coefficients. These systems include string subject to an external force, monatomic chain and diatomic chain in one dimension, and classical Heisenberg model in one dimension. In most cases the center-of-energy velocity reduces to the group velocity for quasi-monochromatic wave packets. Thus it also appears to be the generalization of the group velocity. Wave packets of the relativistic Dirac equation are discussed briefly.