Delay time and persistent oscillations for a shifted quantum shutter

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 ΔX. The dynamical advance-time (negative delay-time), ΔT, 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, ΔT exhibits a monotonic behaviour as a function of ΔX, 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, ΔX _c, the monotonic behaviour of ΔT 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. (paper)