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Physical Review A

Role of top and under-the-barrier complex transmission poles for the delay time in single barrier tunneling

Jorge Villavicencio, Gastón García‐Calderón

Published Sep 22, 2004• 3 citations
Publisher

Abstract

The transition from time advance to time delay in one-dimensional single barrier tunneling is investigated by considering the trajectories in the complex $k$ plane of the top-barrier transmission poles as a function of the opacity of the barrier. We show that such a transition is linked to the particular event where the first top-barrier transmission pole migrates under the barrier as the opacity of the system diminishes. It is also shown that the above situation has an effect on tunneling dynamics since it governs the presence of transient structures in the time evolution of the probability density.

DOI
10.1103/physreva.70.032107

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The transition from time advance to time delay in one-dimensional single barrier tunneling is investigated by considering the trajectories in the complex kkk plane of the top-barrier transmission poles as a function of the opacity of the barrier. We show that such a transition is linked to the particular event where the first top-barrier transmission pole migrates under the barrier as the opacity of the system diminishes. It is also shown that the above situation has an effect on tunneling dynamics since it governs the presence of transient structures in the time evolution of the probability density.
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.
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.
An exact analytical solution to the time-dependent Schr\"odinger equation with cutoff wave initial conditions is used to investigate the fast tunneling response of a rectangular potential barrier. We find that just across the tunneling region, the probability density exhibits at short times a transient behavior that may be characterized by a peak tp{t}_{p}tp​ and a width $\ensuremath{\Delta}t.$ We show that tp{t}_{p}tp​ provides the earliest tunneling response of the system and that the top-barrier S-matrix poles play an important role in the process. As a function of the barrier width, tp{t}_{p}tp​ exhibits two regimes. Along the first regime, tp{t}_{p}tp​ remains almost a constant; as the barrier width increases, a second regime appears where tp{t}_{p}tp​ grows linearly with the barrier width.
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.
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.
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.