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arXiv:2204.01220 · Journal of Physics A Mathematical and Theoretical

Wigner time delays and Goos–Hänchen shifts of 2D quantum vortices scattered by potential barriers

Maxim Mazanov, Konstantin Yu Bliokh

Published Aug 23, 2022• 2 citations
arXivHTMLPDF

Abstract

Abstract We consider reflection and transmission of 2D quantum wavepackets with phase vortices (also known in optics as spatiotemporal vortex pulses) at potential step-like, delta-function, and rectangular barriers. The presence of a vortex significantly modifies the Wigner time delays and Goos–Hänchen shifts, previously studied for Gaussian-like wavepackets. In particular, the scattered wavepackets undergo non-zero time delays and lateral shifts even for purely real scattering coefficients, when the standard Wigner and Artmann formulae vanish. We derive analytical expressions for the vortex-induced times delays and spatial shifts and verify these with numerical calculations of the Schrödinger equation. The time delays and shifts are resonantly enhanced in the vicinity of the critical-angle incidence for a step-like potential and near transmission resonances for a rectangular barrier.

DOI
10.1088/1751-8121/ac8c09

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The Journal of Chemical Physics•Dec 15, 1974•196 citations
Abstract We consider reflection and transmission of 2D quantum wavepackets with phase vortices (also known in optics as spatiotemporal vortex pulses) at potential step-like, delta-function, and rectangular barriers. The presence of a vortex significantly modifies the Wigner time delays and Goos–Hänchen shifts, previously studied for Gaussian-like wavepackets. In particular, the scattered wavepackets undergo non-zero time delays and lateral shifts even for purely real scattering coefficients, when the standard Wigner and Artmann formulae vanish. We derive analytical expressions for the vortex-induced times delays and spatial shifts and verify these with numerical calculations of the Schrödinger equation. The time delays and shifts are resonantly enhanced in the vicinity of the critical-angle incidence for a step-like potential and near transmission resonances for a rectangular barrier.
Quantum weak measurements, wavepacket shifts and optical vortices are universal wave phenomena, which originate from fine interference of multiple plane waves. These effects have attracted considerable attention in both classical and quantum wave systems. Here we report on a phenomenon that brings together all the above topics in a simple one-dimensional scalar wave system. We consider inelastic scattering of Gaussian wave packets with parameters close to a zero of the complex scattering coefficient. We demonstrate that the scattered wave packets experience anomalously large time and frequency shifts in such near-zero scattering. These shifts reveal close analogies with the Goos-Hänchen beam shifts and quantum weak measurements of the momentum in a vortex wavefunction. We verify our general theory by an optical experiment using the near-zero transmission (near-critical coupling) of Gaussian pulses propagating through a nano-fibre with a side-coupled toroidal micro-resonator. Measurements demonstrate the amplification of the time delays from the typical inverse-resonator-linewidth scale to the pulse-duration scale.
We explore the analogy between following the motion of a reflected wave packet and a quantum measurement of the spatial delay imposed on the particle by the scattering potential. It is shown that converting such delays into temporal durations can lead to “negative times,” and best be avoided. It is also demonstrated that a “soft” potential can be replaced by a superposition of hard walls. This representation is used for calculating the Goos-Hänchen shift of a Gaussian beam incident on a potential step or a barrier at an oblique angle.
We study the quantum Goos-H\"anchen (GH) shift and the tunneling transmission at a curved step potential by investigating the time evolution of a wave packet. An initial wave packet is expanded in terms of the eigenmodes of a circular step potential. Its time evolution is then given by the interference of their simple eigenmode oscillations. We show that the GH shift along the step boundary can be explained by the energy-dependent phase loss upon reflection, which is defined by modifying the one-dimensional (1D) effective potential derived from the two-dimensional (2D) circular system. We also demonstrate that the tunneling transmission of the wave packet is characterized by a free-space image distant from the boundary. The tunneling transmission exhibits a rather wide angle divergence and the direction of maximum tunneling is slightly rotated from the tangent at the incident point, which is consistent with the time delay of the tunneling wave packet computed in the 1D modified effective potential.
The quantum phenomenon of above barrier reflection is investigated from a time-dependent perspective using Gaussian wavepackets. The transition path time distribution, which in principle is experimentally measurable, is used to study the mean flight times ⟨t⟩R and ⟨t⟩T associated with the reflection and the transmission over the barrier paying special attention to their dependence on the width of the barrier. Both flight times, and their difference Δt, exhibit two distinct regimes depending on the ratio of the spatial width of the incident wavepacket and the length of the barrier. When the ratio is larger than unity, the reflection and transmission dynamics are coherent and dominated by the resonances above the barrier. The flight times ⟨t⟩R/T and the flight time difference Δt oscillate as a function of the barrier width (almost in phase with the transmission probability). These oscillations reflect a momentum filtering effect related to the coherent superposition of the reflected and transmitted waves. For a ratio less than unity, the barrier reflection and transmission dynamics are incoherent and the oscillations are absent. The barrier width which separates the coherent and incoherent regimes is identified analytically. The oscillatory structure of the time difference Δt as a function of the barrier width in the coherent regime is absent when considered in terms of the Wigner phase time delays for reflection and transmission. We conclude that the Wigner phase time does not correctly describe the temporal properties of above barrier reflection. We also find that the structure of the reflected and transmitted wavepackets depends on the coherence of the process. In the coherent regime, the wavepackets can have an overlapping peak structure, but the peaks are not fully resolved. In the incoherent regime, the wavepackets split in time into distinct separated Gaussian like waves, each one reflecting the number of times the wavepacket crosses the barrier region before exiting. A classical Wigner approximation, using classical trajectories which upon reaching an edge of the barrier are reflected or transmitted as if the edge was a step potential, is quantitative in the incoherent regime. The implications of the coherence observed on resonance reactive scattering are discussed.
Magnetic control of Goos-Hänchen shifts and group delay time in monolayer WSe2_22​
We study the influence of an external magnetic field on the Goos-Hänchen (GH) shift and the group delay time (GDT) in monolayer WSe2_22​ in the presence of a magnetic barrier. The transport properties of Dirac-like carriers are obtained by solving the effective low-energy Hamiltonian and evaluating the corresponding transmission amplitudes. The GH shift and the GDT are subsequently extracted from the phase of the transmission coefficient. We systematically analyze their dependence on the magnetic field strength, incident energy, angle of incidence, and barrier width, with particular emphasis on the spin and valley degrees of freedom associated with the KKK and K′K'K′ valleys. Our results show that the magnetic barrier strongly modulates both the GH shift and the GDT, leading to oscillatory behavior and pronounced spin-valley-dependent transport characteristics. Remarkably, the magnetic field enables selective control of the lateral shift and traversal time of carriers for each spin and valley channel, allowing for tunable spatial and temporal separation of electronic wave packets. This provides a mechanism for manipulating fermionic trajectories after transmission through the barrier in a highly controllable manner. Such tunability opens promising avenues for designing nanoscale devices based on spin and valley filtering, as well as for potential applications in information storage and processing within spintronic and valleytronic platforms.
Exact numerical calculations are made for the scattering of quantum mechanical particles from a square two-dimensional potential barrier. This treatment is an exact analog of both frustrated total reflection of perpendicularly polarized light and the longitudinal Goos-Hänchen shift. Quantum mechanical streamlines (which are analogous to either classical mechanical trajectories or optical rays) are plotted. These streamlines are smooth, continuous, and have continuous first derivatives even through the classically forbidden region. The streamlines form quantized vortices surrounding each of the nodal points (which result from interference between the incident and reflected waves). Similar vortices occur in reactive collisions of H + H2 (McCullough and Wyatt; Kuppermann, Adams, and Truhlar) and undoubtedly play an important role in molecular collision dynamics. The theory for these vortices is given in a companion paper. A comparison is given between our numerical calculations and the stationary phase approximation (SPA). Although our incident wave packet has a half-width of more than one de Broglie wavelength in contrast to the SPA which replaces the diffuse beams by rays with delta function profiles, the agreement was surprisingly good for both the Goos-Hänchen shifts and for the reflection coefficients. However, we found that the Goos-Hänchen shift for the transmitted beam is significantly smaller than for the reflected beam, although in the stationary phase approximation these two shifts are equal. Furthermore, we found that the scattering from the potential barrier has very little effect on the shape of the wave packets. The power series expansion of the incident Debye-Picht wave packet ψI has an extremely small radius of convergence, whereas the power series for ψI*ψI has a radius of convergence of more than two de Broglie wavelengths. The imaginary velocity viis introduced into the Madelung-Landau-London hydrodynamical formulation of quantum mechanics. The corresponding imaginary streamlines will be considered in a forthcoming paper. The time-independent Schrödinger equation for real wavefunctions is reduced to solving the nonlinear first order partial differential equation: ℏ▿·vi = 2(E − V) + vi2. Here, vi is irrotational. This equation may lead to interesting new methods of solving the Schrödinger equation. It does lead to a generalization of the Prager-Hirschfelder perturbation scheme which invokes an electrostatic analogy. The philosophical implications of the hydrodynamical formulation of quantum mechanics are discussed. Penetration of a one-dimensional square potential barrier is used to demonstrate exactly how tunneling occurs by particles ``riding over the barrier'' à la Bohm. Cases are cited where quantum and classical mechanical motions are identical.