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arXiv:quant-ph/0411052 · Physical Review A

Negative group delay for Dirac particles traveling through a potential well

Xi Chen, Chun-Fang Li

Published Nov 17, 2003• 12 citations
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Abstract

The properties of group delay for Dirac particles traveling through a potential well are investigated. A necessary condition is put forward for the group delay to be negative. It is shown that this negative group delay is closely related to its anomalous dependence on the width of the potential well. In order to demonstrate the validity of stationary-phase approach, numerical simulations are made for a Gaussian-shaped temporal wave packet. A restriction to the potential-well's width is obtained that is necessary for the wave packet to remain distortionless in the traveling. Numerical comparison shows that the relativistic group delay is larger than its corresponding nonrelativistic one.

DOI
10.1103/physreva.68.052105

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Publisher’s Note: Negative group delay for Dirac particles traveling through a potential well [Phys. Rev. A68, 052105 (2003)]

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with a typographical error in Eq

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The properties of group delay for Dirac particles traveling through a potential well are investigated. A necessary condition is put forward for the group delay to be negative. It is shown that this negative group delay is closely related to its anomalous dependence on the width of the potential well. In order to demonstrate the validity of stationary-phase approach, numerical simulations are made for a Gaussian-shaped temporal wave packet. A restriction to the potential-well's width is obtained that is necessary for the wave packet to remain distortionless in the traveling. Numerical comparison shows that the relativistic group delay is larger than its corresponding nonrelativistic one.
Abstract We present a unified analytical study of the transmission phase time for Dirac particles scattered by delta-function potentials in one and two dimensions. For the one-dimensional case, we obtain a closed-form expression for the transmission relativistic phase time and show that it coincides with both the reflection phase time and the self-interference delay, in agreement with general predictions for symmetric, zero-width scatterers. The resulting delay alternates periodically between positive and negative values as the coupling strength increases, and converges to the known non-relativistic expression in the appropriate limit. This delay-advance pattern can be interpreted through the spectral structure of the Dirac Hamiltonian, where electron and positron bound-state sectors correlate with time advances and delays, respectively, reflecting the underlying nature of the delta interaction. In the two-dimensional case, we derive an exact expression for the transmission phase time in graphene and examine its angular and coupling dependence. The delay exhibits a periodic, butterfly-like interference pattern governed by resonant transmission and Klein tunnelling, vanishing at normal incidence and at integer values of the reduced coupling strength. These features reveal how angular interference and relativistic dynamics shape temporal signatures of quantum transport in Dirac materials. The closed-form analytical expressions for the relativistic phase time in one dimension and in two dimensions (graphene), which constitute the main results of this work. Our results provide analytic insight into phase-time phenomena in relativistic systems and provide compact benchmarks for time-resolved quantum transport in narrow-gap and Dirac-like media.
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.
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), [Formula: see text], 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.
A traversal time that has no problem of superluminality was advanced for particles to tunnel through potential barriers in the non-relativistic quantum theory in a previous paper by C.-F. Li and Q. Wang, Physica B 296 (2001) 356. This time is generalized in this paper to Dirac's relativistic quantum theory. Both evanescent and propagating cases are considered. It is shown that the traversal time in the evanescent case has much the same properties as in the non-relativistic quantum theory and thus has no problem of superluminality. It also gets rid of the problem of superluminality in the propagating case. Comparisons with the dwell time, the group delay, and the velocity of monochromatic front are also made.