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Physical Review A

Quantum shutter transient solutions and the delay time for the δ potential

Alberto Hernández-Maldonado, Gastón García‐Calderón

Published Jul 24, 2003• 16 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.68.014104

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Physical Review A•Jun 22, 2010•4 citations
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 \ensuremathδ\ensuremath{\delta}\ensuremathδ 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.
Abstract We derive an exact analytical solution to the time-dependent Schrödinger equation based on a resonant state expansion, to explore the time-evolution of cutoff plane waves scattered by finite range potentials, within a shifted quantum shutter model. The latter allows to control both, the position of the shutter and an hypothetical detector, which are initially separated by a distance Δ X . The dynamical advance-time (negative delay-time), Δ T , is explored for the particular case of a delta potential well, by measuring the difference of the first maxima of the time-diffraction pattern of the probability density, corresponding to the free and delta potential cases. We show that in general, Δ T exhibits a monotonic behaviour as a function of Δ X , and derive a simple formula for the timescale by using the symmetry and rescaling properties of the dynamical solution. We demonstrate that below a critical value of the shutter-detector separation, Δ X c , the monotonic behaviour of Δ T is hindered due to an oscillatory phenomena of the probability density, known as persistent oscillations , that drastically distorts the maxima used in the measurement process. These persistent oscillations are periodic Rabi-type oscillations that arise from an interplay between the incidence energy of the initial quantum wave and the bound state of the system, and their frequency play an important role in the dynamics of the delay time.
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.
The transient solution and delay time for a $\ensuremath{\delta}$ potential scatterer with a movable quantum shutter is calculated by solving analytically the time-dependent Schr\"odinger equation. The delay time is analyzed as a function of the distance between the shutter and the potential barrier and also as a function of the distance between the potential barrier and the detector. In both cases, it is found that the delay time exhibits a dynamical behavior and that it tends to a saturation value $\ensuremath{\Delta}{t}_{\mathrm{sat}}$ in the limit of very short distances, which represents the maximum delay produced by the potential barrier near the interaction region. The phase time ${\ensuremath{\tau}}_{\ensuremath{\theta}}$, on the other hand, is not an appropriate time scale for measuring the time delay near the interaction region, except if the shutter is moved far away from the potential. The role played by the antibound state of the system on the behavior of the delay time is also discussed.
An exact analytical solution to the time-dependent Schr\"odinger equation for an attractive $\ensuremath{\delta}$ potential using cutoff plane-wave initial conditions is derived to investigate the effect of a bound state on the transmission transients. We find that at short distances from the potential the probability density behaves harmonically at all times. This unexpected behavior, which corresponds to the trapping of a fraction of the transmitted wave, decreases exponentially with distance, so that at large distances the solution goes into the stationary solution. We find also that this behavior has no effect on the phase time.