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New Journal of Physics

Effects of time on the evolution of a wave packet in the tunneling dynamics

Lijuan Jia, Long Xu, Peng Zhang, Libin Fu

Published Nov 1, 2021• 4 citations
Publisher

Abstract

Abstract We investigate the time-dependent electron wave packet in a one-dimensional geometry with the potential bent by a homogeneous external field. Based on the behaviors of the wave packet over time, we observe a crossover time. After this crossover time, the temporal evolution of the wave packet comes into a new regime, where the wave packet evolves in a self-similar structure. To establish the time scale of this crossover quantitatively, we utilize the Loschmidt echo function, through which the time at which the crossover occurs can be extracted. We also find the time of the maximum ionization velocity can be comparable with the semi-classical tunneling delay time.

DOI
10.1088/1367-2630/ac39b3

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Abstract We investigate the time-dependent electron wave packet in a one-dimensional geometry with the potential bent by a homogeneous external field. Based on the behaviors of the wave packet over time, we observe a crossover time. After this crossover time, the temporal evolution of the wave packet comes into a new regime, where the wave packet evolves in a self-similar structure. To establish the time scale of this crossover quantitatively, we utilize the Loschmidt echo function, through which the time at which the crossover occurs can be extracted. We also find the time of the maximum ionization velocity can be comparable with the semi-classical tunneling delay time.
In order to better understand the trapping and response time of resonant electron wave packets in a resonant tunneling structure, we examined the time-dependent transport of a spherically symmetric Gaussian wave packet through a model double-barrier potential. Within the effective mass picture, the wave packet is projected onto the complete eigenstates of the model potential. The wave packet is initially localized to the left of the barriers and allowed to propagate through the structure. At wave packet energies corresponding to the resonant energies, the probability transmitted to the right of the barriers exhibits a steplike behavior as a function of time. A computation is made of the quantum mechanical delay time, to~tclass, where to is the packet transit time and tclass is the transit time for a classical particle. An increase in this delay time is observed when the packet energy corresponds to the energy of the resonant levels, and a comparison is made to the decay time of a particle in the first resonant state of the well. The delay time increases with increasing barrier area and decreasing packet energy width.
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.
We consider the evolution of a wave packet that is made up of a group of the wave functions describing the stationary scattering process and tunnels through a one-dimensional potential of arbitrary form. As the main characteristics of the time difference of the tunnelling process, use is made of the propagation speed of the wave-packet maximum. We show that the known Hartman formula for the tunnelling time corresponds to the wave packet with a wavenumber-uniform spectral composition in the case, when the phase and transmission coefficient modulus dispersions are taken into account only in the linear approximation. The amplitude of the main peak of the transmitted wave intensity is proven to be independent of the tunnelling time and is determined by the transmission coefficient of the spectral component at the carrier frequency and the spectral width of the wave packet. In the limit of an infinitely wide potential barrier the amplitude of the wave-packet maximum is shown to tend to zero slower than the tunnelling time tends to its asymptotic value, i.e., indeed we deal with the paradox of an infinitely large propagation speed of a wave disturbance through the barrier.
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.
Abstract We theoretically study the tunneling time by investigating a wave packet of Bose-condensed atoms passing through a square barrier. We find that the tunneling time exhibits different scaling laws in different energy regimes. For negative incident energy of the wave packet, counterintuitively, the tunneling time decreases very rapidly with decreasing incident velocity. In contrast, for positive incident energy smaller than the barrier height, the tunneling time increases slowly and then reaches a maximum, which is in agreement with the Larmor clock experiments. The effect of the barrier width related to the uncertainty principle on the maximum tunneling time is also addressed. Our work provides a general scenario of tunneling time that can be used to understand and explain the controversy over tunneling time.
Localized wave-packet tunneling through a one-dimensional potential barrier is investigated with respect to the distribution of tunneling times. Specifically, we construct a quantity--- termed the ``tunneled flux''---which has properties of a tunneling-time probability distribution. The tunneled flux is determined completely in terms of the time evolution of an initially localized wave packet. The tunneled flux and the corresponding tunneling-time distributions are investigated analytically via a semiclassical approximation. Additionally, numerical studies are performed with both the semiclassical and quantal versions of the tunneled flux. The semiclassical calculation qualitatively reproduces numerical tunneled fluxes, converging quantitatively as \ensuremath{\Elzxh}\ensuremath{\rightarrow}0. It also provides a wealth of insight into the nature of tunneling processes. Our principle conclusions are as follows: (i) There is an essential disparity between the time- and energy-domain pictures of wave-packet tunneling; (ii) the tunneling-time distribution can be modeled as an exponential of a skewed Lorentzian function, with a width governed primarily by the phase-space asymmetry of the initial wave packet; and (iii) tunneling is faster than simple classical motion from one side of the barrier to another, even assuming an instantaneous transit between the turning points. Effectively, the barrier acts as a filter for the high-momentum components of the initial wave packet.