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arXiv:quant-ph/0206020 · Physical Review A

Time scale of forerunners in quantum tunneling

Gastón García‐Calderón, Jorge Villavicencio, F.S. Delgado, Juan Gonzalo Muga

Published Oct 30, 2002• 13 citations
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Abstract

The forerunners preceding the main tunneling signal of the wave created by a source with a sharp onset or by a quantum shutter have been generally associated with over-the-barrier (nontunneling) components. We demonstrate that, while this association is true for distances that are larger than the penetration length, for smaller distances the forerunner is dominated by under-the-barrier components. We find that its characteristic arrival time is inversely proportional to the difference between the barrier energy and the incidence energy, a tunneling time scale different from both the phase time and the B\"uttiker-Landauer time.

DOI
10.1103/physreva.66.042119

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The forerunners preceding the main tunneling signal of the wave created by a source with a sharp onset or by a quantum shutter have been generally associated with over-the-barrier (nontunneling) components. We demonstrate that, while this association is true for distances that are larger than the penetration length, for smaller distances the forerunner is dominated by under-the-barrier components. We find that its characteristic arrival time is inversely proportional to the difference between the barrier energy and the incidence energy, a tunneling time scale different from both the phase time and the B"uttiker-Landauer time.
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We obtain the solution of a relativistic wave equation and compare it with the solution of the Schr\"odinger equation for a source with a sharp onset and excitation frequencies below cutoff. A scaling of position and time reduces to a single case all the (below cutoff) nonrelativistic solutions, but no such simplification holds for the relativistic equation, so that qualitatively different ``shallow'' and ``deep'' tunneling regimes may be identified relativistically. The nonrelativistic forerunner at a position beyond the penetration length of the asymptotic stationary wave does not tunnel; nevertheless, it arrives at the traversal (semiclassical or B\"uttiker-Landauer) time $\ensuremath{\tau}.$ The corresponding relativistic forerunner is more complex: it oscillates due to the interference between two saddle-point contributions and may be characterized by two times for the arrival of the maxima of lower and upper envelopes. There is in addition an earlier relativistic forerunner, right after the causal front, which does tunnel. Within the penetration length, tunneling is more robust for the precursors of the relativistic equation.
Time-dependent analytical solutions to Schrödinger's equation with quantum shutter initial conditions are used to investigate the issue of the tunneling time of forerunners in rectangular potential barriers. By using a time-frequency analysis, we find the existence of a regime characterized by the opacity of the barrier, where the maximum peak of a forerunner measured at the barrier transmission edge x=Lx=Lx=L corresponds to a genuine tunneling process. The corresponding time scale represents the tunneling time of the forerunner through the classically forbidden region.
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.
Motivated by recent time-domain experiments on ultrafast atom ionization, we analyze the transients and time scales that characterize, aside from the relatively long lifetime, the decay of a localized state by tunneling. While the tunneling starts immediately, some time is required for the outgoing flux to develop. This short-term behavior depends strongly on the initial state. For the initial state, tightly localized so that the initial transients are dominated by over-the-barrier motion, the time scale for flux propagation through the barrier is close to the B\"uttiker-Landauer traversal time. Then a quasistationary, slow-decay process follows, which sets ideal conditions for observing diffraction in time at longer times and distances. To define operationally a tunneling time at the barrier edge, we extrapolate backward the propagation of the wave packet that escaped from the potential. This extrapolated time is considerably longer than the time scale of the flux and density buildup at the barrier edge.
Transient time-domain resonances found recently in time-dependent solutions to Schr\"odinger's equation are used to investigate the issue of the tunneling time in rectangular potential barriers. In general, a time-frequency analysis shows that these transients have frequencies above the cutoff frequency associated with the barrier height, and hence correspond to nontunneling processes. We find, however, a regime characterized by the barrier opacity, where the peak maximum tmax{t}_{\mathrm{max}}tmax​ of the time-domain resonance corresponds to under-the-barrier tunneling. We argue that tmax{t}_{\mathrm{max}}tmax​ represents the relevant tunneling time scale through the classically forbidden region.