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arXiv:quant-ph/0302096 · Physical Review A

Equivalence between the real-time Feynman histories and the quantum-shutter approaches for the “passage time” in tunneling

Gastón García‐Calderón, Jorge Villavicencio, Norifumi Yamada

Published May 27, 2003• 27 citations
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Abstract

We show the equivalence of the functions ${G}_{\mathrm{p}}(t)$ and $|\ensuremath{\Psi}(d,t){|}^{2}$ for the ``passage time'' in tunneling. The former, obtained within the framework of the real-time Feynman histories approach to the tunneling time problem, uses the Gell-Mann and Hartle's decoherence functional, and the latter involves an exact analytical solution to the time-dependent Schr\"odinger equation for cutoff initial waves.

DOI
10.1103/physreva.67.052106

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We show the equivalence of the functions Gp(t){G}_{\mathrm{p}}(t)Gp​(t) and ∣\ensuremathΨ(d,t)∣2|\ensuremath{\Psi}(d,t){|}^{2}∣\ensuremathΨ(d,t)∣2 for the ``passage time'' in tunneling. The former, obtained within the framework of the real-time Feynman histories approach to the tunneling time problem, uses the Gell-Mann and Hartle's decoherence functional, and the latter involves an exact analytical solution to the time-dependent Schr"odinger equation for cutoff initial waves.
A simple solution is given to the tunneling time problem from the viewpoint of interfering alternatives. Using Gell-Mann and Hartle's decoherence functional for real time Feynman histories, it is shown that (i) the probability distribution of tunneling times cannot be defined for ``resident time'' type but (ii) the range of values of the time is definable. The range turns out to be so broad for opaque barriers that a unique tunneling time cannot be considered even in a loose sense.
Exact analytical solutions of the time-dependent Schrödinger equation with the initial condition of an incident cutoff wave are used to investigate the traversal time for tunneling. The probability density starts from a vanishing value along the tunneling and transmitted regions of the potential. At the barrier width it exhibits, at early times, a distribution of traversal times that typically has a peak τpτ_pτp​ and a width ΔτΔτΔτ. Numerical results for other tunneling times, as the phase-delay time, fall within ΔτΔτΔτ. The Büttiker traversal time is the closest to τpτ_pτp​. Our results resemble calculations based on Feynman paths if its noisy behaviour is ignored.
Exact analytical solutions of the time-dependent Schrodinger equation with the initial condition of an incident cutoff wave are used to investigate the traversal time for tunneling. The probability density starts from a vanishing value along the tunneling and transmitted regions of the potential. At the barrier width it exhibits, at early times, a distribution of traversal times that typically has a peakp and a width ��. Numerical results for other tunneling times, as the phase-delay time, fall within ��. The Buttiker traversal time is the closest top. Our results resemble calculations based on Feynman paths if its noisy behaviour is ignored.
The quantum-shutter approach to tunneling time scales [G. Garc\'{\i}a-Calder\'on and A. Rubio, Phys. Rev. A 55, 3361 (1997)], which uses a cutoff plane wave as the initial condition, is extended to consider certain type of wave packet initial conditions. An analytical expression for the time-evolved wave function is derived. The time-domain resonance, the peaked structure of the probability density (as the function of time) at the exit of the barrier, originally found with the cutoff plane wave initial condition, is studied with the wave packet initial conditions. It is found that the time-domain resonance is not very sensitive to the width of the packet when the transmission process occurs in the tunneling regime.
The analytical solution to the time-dependent Schr\"odinger equation for tunneling using cutoff plane-wave initial conditions is in general given by the sum of two types of terms that exhibit a transient behavior. The time evolution of the probability density for the $\ensuremath{\delta}$ potential is compared with the free case to investigate in this case the role of these transient terms for the delay time. We find, by a dynamical calculation, that the delay time arises from the interference between these transient terms and we show that at very long times it goes into the phase delay time, given by the energy derivative of the phase of the transmission amplitude.