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Physical Review A

Transient effects and delay time in the dynamics of resonant tunneling

Gastón García‐Calderón, Alberto Rubio

Published May 1, 1997• 97 citations
Publisher

Abstract

We consider an analytic solution of the time-dependent Schr\"odinger equation with the initial condition \ensuremath{\psi}(x,0)=exp(ikx) along -\ensuremath{\infty}0 to investigate the time evolution for x>0 of the wave function in a double-barrier resonant structure at resonance. For typical parameters of the structure we find that the single-resonance approximation is valid from a few tenths of the corresponding lifetime onward. Very short times require the contribution of many far away resonances. The buildup time along the internal region takes a few lifetimes. At birth the transmitted wave front is blurred; however, for long times it becomes well defined and moves with classical velocity yielding a delay time \ensuremath{\tau}\ensuremath{\approx}2\ensuremath{\hbar}/\ensuremath{\Gamma} as in the stationary-phase treatment.

DOI
10.1103/physreva.55.3361

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We consider an analytic solution of the time-dependent Schr"odinger equation with the initial condition \ensuremath{\psi}(x,0)=exp(ikx) along -\ensuremath{\infty}0 to investigate the time evolution for x>0 of the wave function in a double-barrier resonant structure at resonance. For typical parameters of the structure we find that the single-resonance approximation is valid from a few tenths of the corresponding lifetime onward. Very short times require the contribution of many far away resonances. The buildup time along the internal region takes a few lifetimes. At birth the transmitted wave front is blurred; however, for long times it becomes well defined and moves with classical velocity yielding a delay time \ensuremath{\tau}\ensuremath{\approx}2\ensuremath{\hbar}/\ensuremath{\Gamma} as in the stationary-phase treatment.
An exact analytic solution of the time-dependent Schrödinger equation with cutoff wave initial conditions, that goes into the stationary solution at asymptotically long times, is used to investigate the transient behavior in the time evolution of the probability density along the internal and transmitted regions of a potential barrier. The approach involves the complex poles and residues (resonant states) of the outgoing Green's function of the problem. The solution exhibits absence of propagation along the classically forbidden internal region of the potential, and along the transmitted region it splits into two propagating structures. One of them is a transient effect describing the earliest response of the system to the initial condition whereas the other structure evolves towards the stationary solution.
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.
The time evolution of the buildup process inside a double-barrier system for off-resonance incidence energies is studied by considering the analytic solution of the time dependent Schr\\"{o}dinger equation with cutoff plane wave initial conditions. We show that the buildup process exhibits invariances under arbitrary changes on the system parameters, which can be successfully described by a simple and easy-to-use one-level formula. We find that the buildup of the off-resonant probability density is characterized by an oscillatory pattern modulated by the resonant case which governs the duration of the transient regime. This is evidence that off-resonant and resonant tunneling are two correlated processes, whose transient regime is characterized by the same transient time constant of two lifetimes.
The resonant tunneling of a wave packet is studied, for a one-dimensional double barrier, by solving the time-dependent Schrödinger equation. The transmission is determined as a function of the spatial width of the incident wave packet. It is shown that if the double barrier acts as an energy filter, then the spatial width of the transmitted wave packet is increased, due to the trapping of the particle between the two barriers for a time, ℏ/Γ, where Γ is the width of the resonance. The reflected wave packet consists of two components: one corresponding to reflection from the first barrier and the other due to the particle being trapped between the barriers for a time ℏ/Γ.
Transient tunneling effects in triple barrier systems are investigated by considering a time-dependent solution to the Schr\"odinger equation with a cutoff wave initial condition. We derive a two-level formula for incidence energies E near the first resonance doublet of the system. Based on that expression we find that the probability density along the internal region of the potential is governed by three oscillation frequencies: one of them refers to the well known Bohr frequency, given in terms of the first and second resonance energies of the doublet, and the two others represent a coupling with the incidence energy E. This allows us to manipulate the above frequencies to control the tunneling transient behavior of the probability density in the short-time regime.