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Physical Review A

Tunneling and delay time of cutoff Gaussian wave packets

Jorge Villavicencio, Roberto Romo, Elmer Cruz

Published Jan 23, 2007• 17 citations
Publisher

Abstract

A formal analytic solution of the time-dependent Schr\"odinger equation is obtained for cutoff Gaussian wavepackets incident on finite range potentials of arbitrary shape. We derive closed analytical expressions for transmitted wavepackets with a small cutoff and the solutions are applied to investigate the dynamics of wavepacket tunneling. We showed that the analytic solutions satisfy a useful rescaling property that was applied to study and characterize the delay time in delta potential barriers.

DOI
10.1103/physreva.75.012111

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A formal analytic solution of the time-dependent Schr"odinger equation is obtained for cutoff Gaussian wavepackets incident on finite range potentials of arbitrary shape. We derive closed analytical expressions for transmitted wavepackets with a small cutoff and the solutions are applied to investigate the dynamics of wavepacket tunneling. We showed that the analytic solutions satisfy a useful rescaling property that was applied to study and characterize the delay time in delta potential barriers.
Abstract 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.
Analytic solutions to the time-dependent Schr\"odinger equation for cutoff wave initial conditions are used to investigate the time evolution of the transmitted probability density for tunneling. For a broad range of values of the potential barrier opacity $\ensuremath{\alpha},$ we find that the probability density exhibits two evolving structures. One refers to the propagation of a forerunner related to a time domain resonance [Phys. Rev. A 64, 0121907 (2001)], while the other consists of a semiclassical propagating wave front. We find a regime where the forerunners are absent, corresponding to positive time delays, and show that this regime is characterized by opacities $\ensuremath{\alpha}<{\ensuremath{\alpha}}_{c}.$ The critical opacity ${\ensuremath{\alpha}}_{c}$ is derived from the analytical expression for the delay time, which reflects a link between transient effects in tunneling and the delay time.
Analytic expressions are given for the time spent by a particle tunneling through a potential barrier. The expressions are derived for an incident wave packet which is initially Gaussian, centered about a point an arbitrary distance away from a rectangular potential barrier and moving toward the barrier with constant average velocity. Upon collision with the barrier, the packet splits into a transmitted and a reflected packet. The resultant transmission time is positive, nonzero and in principle measurable. Although the transmission time becomes quite large as the incident kinetic energy becomes very small, in general, for nonzero incident momentum and finite potential barriers which are neither very thick nor very thin, the transmission times are less than the time that would be required for the incident particle to travel a distance equal to the barrier thickness. The transmission times for metal-insulator-metal thin film sandwiches, given approximately by δt3=ℏ/(Efφ)12, where Ef is the Fermi energy of the metal, and φ the vacuum work function, are of the order of 10−16 sec, compared to RC time constants of about 10−13 sec.
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.
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.
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.