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Physical Review A

Delay time in a single barrier for a movable quantum shutter

Alberto Hernández-Maldonado

Published May 28, 2010• 4 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.81.054101

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The transient solution and delay time for a \ensuremathδ\ensuremath{\delta}\ensuremathδ 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 \ensuremathΔtsat\ensuremath{\Delta}{t}_{\mathrm{sat}}\ensuremathΔtsat​ in the limit of very short distances, which represents the maximum delay produced by the potential barrier near the interaction region. The phase time \ensuremathτ\ensuremathθ{\ensuremath{\tau}}_{\ensuremath{\theta}}\ensuremathτ\ensuremathθ​, 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.
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.
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.
Abstract We derive a time-dependent solution for the Schrödinger-Pauli equation within a quantum shutter model to analyse cut-off plane waves scattered by a finite-range potential under a constant magnetic field. Exact analytical solutions for the time-dependent spinor components are derived for the low-energy plane-wave case, enabling the exploration of dynamical effects within the transmission region of a potential barrier. We demonstrate that the spinor probability density exhibits a diffraction in time pattern, with peaks shifted by the presence of the magnetic field relative to the field-free case. This shift introduces a delay time, which we analyse as a function of the Larmor frequency, with tunable peak positions controlled by the field intensity. We further explore the dynamical features of the spin expectation values 〈S k 〉, which exhibit unique diffraction in time behaviour and converge to stationary values in the long-time asymptotic regime. Additionally, we show that measuring the average total spin of the system offers an alternative approach for calculating the dynamical delay time. For infinitesimally small magnetic fields, we define a characteristic dynamical Larmor timescale associated with the evolution of 〈S y 〉, enabling the recovery of the Büttiker Larmor time, τ y , at low energies in the long-time limit. These findings demonstrate that magnetic fields can precisely manipulate tunnelling timescales in quantum systems, providing a means to control matter wave dynamics.
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 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.