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Physica Scripta

Delay time and persistent oscillations for a shifted quantum shutter

Alberto Hernández-Maldonado, Jorge Villavicencio, Andrea Hernández-Aviña

Published Feb 19, 2021• 2 citations
Publisher

Abstract

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.

DOI
10.1088/1402-4896/abe82f

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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.
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 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.
Abstract We explore the dynamics of matter wave scattering in finite potential wells using analytical solutions of the Schrödinger equation within the framework of a quantum shutter model. We find that the incident wave interferes with the bound states of the quantum well, resulting in time-domain oscillations. These oscillations exhibit Rabi-type frequencies, characterised by the energy differences between the incident wave and the bound states of the quantum well. We show that in systems with double-bound states, the interference pattern is characterised by quantum beats in the time-dependent probability density. The period of these beatings depends on the energy difference between the bound states, which can be tuned by controlling the potential parameters. In the general case where bound, anti-bound, and resonant states coexist in the system spectrum, complex oscillations in the probability density arise from the interactions of the incident wave with different quantum states. We demonstrate that the bound states sector can effectively describe this complex behaviour, providing a simple and reliable analytical expression for the probability density in multilevel systems. This formula highlights the significant role of bound states, whose interaction with the incident wave dominates the transient probability density. This contrasts with conventional systems with potential barriers and wells, where resonances govern the wave dynamics.
Transient phenomena of phase-modulated cutoff wave packets are explored by deriving an exact general solution to Schr\"odinger's equation for finite-range potentials involving arbitrary initial quantum states. We show that the dynamical features of the probability density are governed by a virtual self-induced two-level system with energies E+{E}_{+}E+​ and ${E}_{\ensuremath{-}}$ due to the phase modulation of the initial state. The asymptotic probability density exhibits Rabi oscillations characterized by the frequency $\mathrm{\ensuremath{\Omega}}=({E}_{+}\ensuremath{-}{E}_{\ensuremath{-}})/\ensuremath{\hbar}$, which are independent of the potential profile. It is also found that for a system with a bound state, the interplay between the virtual levels with the latter causes a quantum beat effect with a beating frequency, $\mathrm{\ensuremath{\Omega}}$. We also find a regime characterized by a time-diffraction phenomenon that allows us to measure unambiguously the delay time, which can be described by an exact analytical formula. It is found that the delay time agrees with the phase time only for the case of strictly monochromatic waves.
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.