Transmitted Wave Packet Research Articles

Wave packet tunneling and imaginary wave vector dispersion

Dispersion associated with the complex band structure of a periodic potential can be used to explain single-electron wave packet tunneling through a finite-sized tunnel barrier. The imaginary k-space dispersion in a tunnel barrier may be viewed as a filter that controls the arrival time, shape, and group velocity of a transmitted wave packet. This k-space filter can be designed to increase or decrease the group velocity of the transmitted pulse and a transmitted peak-pulse probability arrival time can be engineered to occur before or after that of an electron wave packet propagating in the absence of the tunnel barrier.

Non-invasive analysis of macro-crack on the characteristics of ultrasonic guided wave

The effects of macro-crack in plates on the characteristic of ultrasonic guided wave were studied. Using the Finite Element Model and numerical calculation method, the finite element mesh models of various macro-crack in aluminum plate were established. The single S0 or A0 mode Lamb wave was excited in plate by the wave structure loading method. The propagation time and amplitude of wave packet in the received signal were analyzed to study the phenomenon of mode transformation and the relationship between macro-crack size and signal amplitude. The simulation results show that there is mode conversion when the single mode Lamb wave interacts with an asymmetric macro-crack, and no mode conversion with a symmetric macro-crack. The energy of reflected wave increases with the increase of the depth and width of macro-crack, and the energy of transmitted wave decreases correspondingly. The size and inclined angle of macro-crack can be quantitatively estimated by the fitting function from the energies of reflected and transmitted waves. The variation of crack depth, width and angle is approximately linear with the amplitudes of reflected and transmitted wave packets. A fitting function is constructed by using the amplitude ratio of reflected wave packet or transmitted wave packet to total wave packet, which can be used to estimate the macro-crack depth, width and angle. This conclusion is helpful to the detection of macro-crack in the actual production and service of plate structures.

Speed-up and slow-down of a quantum particle

We study non-relativistic propagation of Gaussian wave packets in one-dimensional Eckart potential, a barrier, or a well. In the picture used, the transmitted wave packet results from interference between the copies of the freely propagating state with different spatial shifts (delays), x', induced by the scattering potential. The Uncertainty Principle precludes relating the particle’s final position to the delay experienced in the potential, except in the classical limit. Beyond this limit, even defining an effective range of the delay is shown to be an impracticable task, owing to the oscillatory nature of the corresponding amplitude distribution. Our examples include the classically allowed case, semiclassical tunnelling, delays induced in the presence of a virtual state, and scattering by a low barrier. The properties of the amplitude distribution of the delays, and its pole representation are studied in detail.

Phase-space consideration on barrier transmission in a time-dependent variational approach with superposed wave packets

A known limitation of time-dependent mean-field approaches is a lack of quantum tunneling for collective motions such as in sub-barrier fusion reactions. As a first step toward a solution, a time-dependent model is considered using a superposition of Gaussian wave packets, to describe the relative motion between two colliding nuclei, which may be simplified to a problem for one particle in one dimension. In this article, how the model describes the potential-barrier transmission is investigated by paying attention to the time evolution of the phase space distribution, which in particular reveals that the behavior of the free propagation of the incoming state is not trivial, depending on the number of superposed wave packets. Passage over the barrier can occur due to the high-momentum components in the incoming state corresponding to energies above the barrier height, which is, however, of classical nature and needs to be distinguished from the true quantum tunneling. Although a transmitted wave packet in some case may end up with an energy lower than the barrier, a difficulty is noticed in guaranteeing the energy conservation when the energies of different exit channels, e.g. of transmission and reflection, are individually measured. To overcome these issues for a description of quantum tunneling is still a challenging problem. This article mainly treats the same system with the same model as in the paper Hasegawa, Hagino and Tanimura (2020) [1]. However, we reach different conclusions.

Tunnelling of Hermite-Gaussian wavepackets

We examine the tunnelling process of incident Hermite-Gaussian wavepackets which feature several nodes and peaks. An analytic formula for the transmitted wave function is developed which applies to arbitrary incident wave functions and potential barriers with known transmission amplitudes. This formula permits the accurate numerical calculation of the transmitted wave packet, even when the transmission probability is extremely small. For square potential barriers, the shape of the transmitted packet, its relationship to the shape of the incident packet, the momentum of the transmitted packet, and the duration of the tunnelling process are investigated. The shape of the reconstructed transmitted wave packet is also studied for the smooth repulsive Pöschl-Teller potential.

Approximate quantum trajectory approach to the Schrödinger-Langevin equation for barrier transmission

The Schrödinger-Langevin equation is approximately solved by propagating individual quantum trajectories for barrier transmission problems. Equations of motion are derived through use of the derivative propagation method, which leads to a hierarchy of coupled differential equations for the amplitude of the wave function and the spatial derivatives of the complex action along each trajectory. Computational results are presented for a one-dimensional Eckart barrier and a two-dimensional system involving either a thick or thin Eckart barrier along the reaction coordinate coupled to a harmonic oscillator. Frictional effects on the trajectory, the transmitted wave packet, and the transmission probability are analyzed.

Wave packet propagation across barriers by semiclassical initial value methods.

Semiclassical initial value representation (IVR) formulas for the propagator have difficulty describing tunneling through barriers. A key reason is that these formulas do not automatically reduce, in the classical limit, to the version of the Van Vleck-Gutzwiller (VVG) propagator required to treat barrier tunneling, which involves trajectories that have complex initial conditions and that follow paths in complex time. In this work, a simple IVR expression, that has the correct tunneling form in the classical limit, is derived for the propagator in the case of one-dimensional barrier transmission. Similarly, an IVR formula, that reduces to the Generalized Gaussian Wave Packet Dynamics (GGWPD) expression [D. Huber, E. J. Heller, and R. Littlejohn, J. Chem. Phys. 89, 2003 (1988)] in the classical limit, is derived for the transmitted wave packet. Uniform semiclassical versions of the IVR formulas are presented and simplified expressions in terms of real trajectories and WKB penetration factors are described. Numerical tests show that the uniform IVR treatment gives good results for wave packet transmission through the Eckart and Gaussian barriers in all cases examined. In contrast, even when applied with the proper complex trajectories, the VVG and GGWPD treatments are inaccurate when the mean energy of the wave packet is near the classical transmission threshold. The IVR expressions for the propagator and wave packet are cast as contour integrals in the complex space of initial conditions and these are generalized to potentially allow treatment of a larger variety of systems. A steepest descent analysis of the contour integral formula for the wave packet in the present cases confirms its relationship to the GGWPD method, verifies its semiclassical validity, and explains results of numerical calculations.

Particle Transmission Through a Barrier of Time-Dependent Opacity

We discuss the one-dimensional scattering of an arbitrary incident wave packet by a localized potential subjected to a time-dependent perturbation. It is shown that an exact solution can be obtained for an ample class of time dependencies provided that they are smaller than the unperturbed potential. Simpler solutions can also be obtained (under particular conditions) by looking for the incident wave which produces a specified transmitted wave packet.

Transmission time in the reflectionless complex potential

A phase time definition directly obtained from the Schrödinger equation is used to investigate the time delay of a particle scattered by complex reflectionless potential. The artifacts introduced by truncating in the numerical simulation are clarified. The time delay of the transmitted wave packet is found to be equal to the reflection time of the truncated potential. Both time delays are the same as the traversal time in the free space, but shorter than the time taken by a classical particle to pass the same potential.

Complex time paths for semiclassical wave packet propagation with complex trajectories.

The use of complex-valued trajectories in semiclassical wave packet methods can lead to problems that prevent calculation of the wave function in certain regions of the configuration space. We investigate this so-called bald spot problem in the context of generalized Gaussian wave packet dynamics. The analysis shows that the bald spot phenomenon is essentially due to the complex nature of the initial conditions for the trajectories. It is, therefore, expected to be a general feature of several semiclassical methods that rely on trajectories with such initial conditions. A bald region is created when a trajectory, needed to calculate the wave function at a given time, reaches a singularity of the potential energy function in the complex plane at an earlier, real time. This corresponds to passage of a branch point singularity across the real axis of the complex time plane. The missing portions of the wave function can be obtained by deforming the time path for the integration of the equations of motion into the complex plane so that the singularity is circumvented. We present examples of bald spots, singularity times, and suitable complex time paths for one-dimensional barrier transmission in the Eckart and Gaussian systems. Although the bald regions for the Eckart system are often localized, they are found to be semi-infinite for the Gaussian system. For the case of deep tunneling, the bald regions for both systems may encompass the entire portion of space occupied by the transmitted wave packet. Thus, the use of complex time paths becomes essential for a treatment of barrier tunneling.

Group delay of electromagnetic pulses through multilayer dielectric mirrors combined with gravitational wave

Group delay of electromagnetic pulses through multilayer dielectric mirrors (MDM) combined with gravitational wave (GW) is investigated. Unlike in traditional quantum tunneling, the group delay of a transmitted wave packet irradiated by a GW increases linearly with MDM length. This peculiar tunneling effect can be attributed to electromagnetic wave leakage in a time-dependent photonic bandgap caused by the GW. In particular, we find that the group delay of the tunneling photons is sensitive to GW. Our study provides insight into the nature of the quantum tunnelling as well as a novel process by which to detect the GW.

WAVE PACKETS SCATTERED BY NON-PERIODIC BRAGG-TYPE LAYERED STRUCTURES

The time delay, space shift and widening of wave packet transmitted and re∞ected by structures with Bragg mirrors have been investigated. The speciflc structures such as Bragg mirrors, resonators, and structures with chirp variation of thickness of the \period have been considered. The calculation has been carried out under the conditions that carrier frequency, and incidence angle is in the vicinity of the Bragg resonance. Integral (mass center) and difierential (group) estimates of the delay time and space shift have been compared. The conditions for the appearance of anomalous (negative) mass center delay or mass center shift (Goos-Hanchen shift) of the re∞ected wave packet have been determined. The shape transformations of the wave packet illuminating periodic and quasiperiodic apodized Bragg re∞ectors have been under consideration. Spatial apodization of permittivity contrast yields much smaller shape deformation of the transmitted wave packet upon incidence at angles and carrier frequency near the edges of re∞ection band, as well in Bragg re∞ection band, in comparison with phenomena in similar periodic structures. The values of group delay for layered structures with a small chirp variation of optical (electrical) thickness of the period along longitudinal coordinates have been experimentally obtained in microwave range.

Quantum versus classical dynamics in a driven barrier: The role of kinematic effects

We study the dynamics of the classical and quantum mechanical scattering of a wave packet from an oscillating barrier. Our main focus is on the dependence of the transmission coefficient on the initial energy of the wave packet for a wide range of oscillation frequencies. The behavior of the quantum transmission coefficient is affected by tunneling phenomena, resonances, and kinematic effects emanating from the time dependence of the potential. We show that when kinematic effects dominate (mainly in intermediate frequencies), classical mechanics provides very good approximation of quantum results. In that frequency region, the classical and quantum transmission coefficients are in optimal agreement. Moreover, the transmission threshold (i.e., the energy above which the transmission coefficient becomes larger than a specific small threshold value) is found to exhibit a minimum. We also consider the form of the transmitted wave packet and we find that for low values of the frequency the incoming classical and quantum wave packet can be split into a train of well-separated coherent pulses, a phenomenon that admits purely classical kinematic interpretation.

Optical field modulation on the group delay of chiral tunneling in graphene

The influences of optical fields on the group delay of chiral tunneling in graphene are investigated in real time using the finite-difference time-domain method. The group delay of tunneling electrons irradiated by an optical field is significantly different from that observed in traditional quantum tunneling. We found that when the barrier width increases, the group delay becomes constant for the reflected wave packet, but increases linearly for the transmitted wave packet. This peculiar tunneling effect can be attributed to current leakage in a time-dependent barrier generated via the optical Stark effect.

Transient effects and reconstruction of the energy spectra in the time evolution of transmitted Gaussian wave packets

We derive an analytical solution to the time-dependent Schrödinger equation for the transmission of a Gaussian wave packet through an arbitrary potential of finite range. We consider the situation where the initial Gaussian wave packet is sufficiently broad in momentum space to include the full resonance structure of the system in the dynamical description. We find that the transmitted wave packet may be written as the free evolving Gaussian wave packet solution times a transient term. We demonstrate that both at very large distances and very long times the transient term tends to the transmission amplitude of the system and hence the time-evolving solution reproduces the resonance spectra of the system. We also prove that at a fixed distance and very long times the analytical solution is t−3/2 which extends previous analysis on this issue to arbitrary finite-range potentials. Our results are exemplified for a multibarrier system.

Tunnelling through two barriers

Recent theoretical studies of the tunnelling through two opaque barriers claim that the transit time is independent of the barrier widths and of the separation distance between the barriers. Such a result is based on the use of the stationary phase method and the hypothesis of a single transmitted wave packet. In this Letter, we propose an alternative treatment based on a multiple peaks decomposition of the transmitted wave. We observe, that if multiple reflections are allowed for correctly (infinite peaks) the transit time between the barriers appears exactly as expected.

Dynamics of electron tunneling in semiconductor nanostructures

We have modeled the transmission of an electron wave packet through a resonant tunneling semiconductor nanostructure by solving the time-dependent Schr\"odinger equation using the finite-difference method. We have found in all cases that the passage of the electron wave packet through the tunneling barrier is accompanied by a propagation delay relative to the propagation of an undisturbed wave packet. Tunneling transport is shown to be causal, and no evidence of superluminal behavior is seen, either for resonant or for off-resonant tunneling. In the case of off-resonant tunneling, the peak of the transmitted wave packet is observed to exit a double resonant tunneling barrier before the peak of the incident wave packet enters the structure. However, these two peaks are not directly related, and the appearance of a well-formed peak in the transmitted intensity is shown to be a result of the transient behavior of the tunneling event.

Bounds and enhancements for negative scattering time delays

The time of passage of the transmitted wave packet in a tunneling collision of a quantum particle with a square potential barrier becomes independent of the barrier width in a range of barrier thickness. This is the Hartman effect, which has been frequently associated with ``superluminality.'' A fundamental limitation on the effect is set by nonrelativistic ``causality conditions.'' We demonstrate first that the causality conditions impose more restrictive bounds on the negative time delays (time advancements) when no bound states are present. These restrictive bounds are in agreement with a naive, and generally false, causality argument based on the positivity of the ``extrapolated phase time,'' one of the quantities proposed to characterize the duration of the barrier's traversal. Nevertheless, square wells may in fact lead to much larger advancements than square barriers. We point out that close to the thresholds of new bound states, the time advancement increases considerably, while, at the same time, the transmission probability is large, which facilitates the possible observation of the enhanced time advancement.

Tunneling and traversal of ultracold atoms through vacuum-induced potentials

We examine the passage of ultra-cold two-level atoms through the potential produced by the vacuum of the cavity field. The peak of the transmitted wave packet generally occurs at the instant given by the expression for phase time even if that instant is earlier than the instant at which incident wave packet's peak enters the cavity and thus the phase time can be considered as the appropriate measure of the time required for the atom to traverse the cavity. We show that the phase tunneling time for ultra-cold atoms could be both super - and sub - classical time and we show how this behaviour can be understood in terms of the momentum dependence of the phase of transmission amplitude. The passage of the atom through the cavity is unique as it involves a coherent addition of the transition amplitudes corresponding to both barrier and well. New features such as the splitting of the wave packet arise from the entanglement between the center of mass motion and the electronic as well as field states in the cavity.

Effects of relativity on the time-resolved tunneling of electron wave packets

We solve numerically the time-dependent Dirac equation for a quantum wave packet tunneling through a potential barrier. We analyze the spatial probability distribution of the transmitted wave packet in the context of the possibility of effectively superluminal peak and front velocities of the electron during tunneling. Both the Dirac and Schr\"odinger theories predict superluminal tunneling speeds. However, in contrast to the Dirac theory the Schr\"odinger equation allows a possible violation of causality. Based on an analysis of the tunneling process in full temporal and spatial resolution, we introduce an instantaneous tunneling speed that can be computed inside the potential barrier.