Wigner Time Delay Research Articles

Probing Wigner time delays with photoelectron interferometry: Anisotropic long-range imprint of the short-range centrifugal potential

Probing Wigner time delays with photoelectron interferometry: Anisotropic long-range imprint of the short-range centrifugal potential

Wigner time delay revisited

The concept of Wigner time delay was introduced in scattering theory to quantify the delay or advance of an incoming particle in its interaction with the scattering potential. The scattered-wave part of the wave packet, which describes the interacting particle, is delayed/advanced with respect to the non-interacting plane-wave packet. It has been widely assumed that this concept can be transferred to ionization considering it as a half-scattering process. In the present work we show, by analyzing the corresponding wave packets, that this assumption is incorrect. For ionization the outgoing wave packet is a superposition of the continuum states ψk(−)(r,t), which satisfy the incoming-wave boundary condition at large distances r from the origin and which for large time t→∞ form a plane-wave packet. The expansion coefficients in this wave packet are the ionization probability amplitudes. If these amplitudes depend on the scattering phase shift, the plane-wave part of the ionized wave packet may be modified, leading to an ionization analog of the Wigner time delay. However, in most cases such a Wigner time delay is absent for ionization. We illustrate this with two examples: (i) photoionization by a weak field of an atom with the electron bound by a zero-range potential and (ii) strong-field ionization.

Wigner time delay and Hartman effect in quantum motion along deformed Riemannian manifolds

Elastic scattering of a wave can be quantified by a shift in the phase with respect to the phase of the incoming wave. A qualitative measure of the time during which the effect occurs is given by the Wigner time delay. The tunneling time is known to saturate with increasing tunneling barrier width (Hartman effect). Here, we analyze the elastic quantum mechanical scattering in a deformed one-dimensional Riemannian manifold, particularly with respect to the Wigner time delay and conclude on the Hartman effect. It is shown that scattering due to local curvature variations results in imperfect conduction and leads to a Wigner time delay that at low energies, is in variance with the classical time delay that is inferred from the arc length. At moderate and high energies, however, classical and quantum time delays coincide.

Measurability of Wigner time delay in a photoionization experiment

We visit significant historical steps in quantum collision physics that make photoionization time delay measurable, especially considering the time-reversal symmetry in related processes. The works of A. Ravi P. Rau and Ugo Fano inspire key ideas presented in this article.

Time delays in anisotropic systems

Scattering properties and time delays for general (nonsymmetric) potentials in terms of the respective S-matrices are discussed paradigmatically in one dimension and in comparison to symmetric potentials. Only for the latter the Wigner and Smith time delays coincide. Considering asymmetric potentials also reveals that only one version of S-matrices used in the literature (the one with reflection coefficients on the diagonal) generalizes to the asymmetric case. Finally, we give a criterion how to identify a potential with intrinsic symmetry that behaves like an asymmetric one if it is merely offset from the scattering center.

Resonant scattering by a loop: the Wigner delay time and Poisson’s kernel


 
 
 The resonances of a loop pierced by a magnetic field are analized in terms of the scattering matrix phase, the Wigner delay time and its relation to Poisson’s kernel. Except for specific values of the magnetic flux, the resonances appear overlapped by pairs due to the broken degeneracy. Although it is well known that the Poisson kernel describes how the phase is distributed in the Argand plane, we demonstrate that Poisson’s kernel coincides with the reciprocal of the Wigner delay time, thus providing a novel interpretation of this quantity. The distribution of the Wigner delay time is also determined, it exhibits explicitly the effect of the magnetic flux, contrary to what happens to the distribution of the phase.
 
 

Wigner time delay in photoionization: a 1D model study

In scattering theory, the Wigner–Smith time delay, calculated through a phaseshift derivative or its multichannel generalization, has been demonstrated to measure the amount of delay or advance experienced by colliding particles during their interaction with the scattering potential. Fetic, Becker, and Milosevic argue that this concept cannot be extended to include photoionization, viewed as a half-scattering experiment. Their argument is based on the lack of information about scattering phaseshifts in the part of the wavefunction (satisfying the ingoing-wave boundary condition) going to the detector. This article aims to test this claim by examining a photodetachment process in a simple 1D model with a short-range symmetrical potential. Using time-dependent perturbation theory with a dipole interaction, the relevant wavepacket of the outgoing particle is analyzed and compared to the free wavepacket as a reference. Our findings confirm that, indeed, a time delay arises in the liberated fragmentation wavepacket, which is expressed as an energy derivative of the scattering phaseshift. Our study highlights that the source of the phaseshift content in the wavepacket arriving at the detector is the dipole matrix element, which is a direct consequence of imposing the ingoing-wave boundary condition. We illustrate our results through numerical simulations of both the non-free and free wavepackets. The amount of the observed time delay is found to be half of that appearing in a typical scattering experiment.

Attosecond Time Delay Trends across the Isoelectronic Noble Gas Sequence

The analysis and measurement of Wigner time delays can provide detailed information about the electronic environment within and around atomic and molecular systems, with one the key differences being the lack of a long-range potential after a halogen ion undergoes photoionization. In this work, we use relativistic random-phase approximation to calculate the average Wigner delay from the highest occupied subshells of the atomic pairings (2p, 2s in Fluorine, Neon), (3p, 3s in Chlorine, Argon), (4p, 4s, 3d, in Bromine, Krypton), and (5p, 5s, 4d in Iodine, Xenon). The qualitative behaviors of the Wigner delays between the isoelectronic pairings were found to be similar in nature, with the only large differences occurring at photoelectron energies less than 20 eV and around Cooper minima. Interestingly, the relative shift in Wigner time delays between negatively charged halogens and noble gases decreases as atomic mass increases. All atomic pairings show large differences at low energies, with noble gas atoms showing large positive Wigner delays, while negatively charged halogen ions show negative delays. The implications for photoionization studies in halide-containing molecules is also discussed.

Making sense of transmission resonances and Smith lifetimes in one-dimensional scattering: The extended phase space quantum trajectory picture

Resonances are ubiquitous in a wide range of physical and chemical phenomena. Their impact on quantum scattering processes renders their study as important as it can be puzzling. In this paper, we illustrate the accuracy of a fully quantum, purely trajectory based reformulation of quantum mechanics proposed by one of the authors (Poirier) to acquire insights on shape resonances through direct and accurate computation of the diagonal elements of Smith’s lifetime matrix. This study also generalizes the relationship between the quantum trajectory propagation time and the Eisenbud–Wigner time delay – introduced in our previous publication (Dupuy et al., 2022) for symmetric potentials – to the general case of asymmetric potential profiles. In addition, we show how the complex amplitudes of the scattering matrix can be extracted from left- and right-incident quantum trajectories. Finally, we demonstrate that extended phase space quantum trajectories not only recover S-matrix and quantum time quantities, but they also provide their own picture of resonant phenomena, as dynamically distinct events characterized by an integer number of closed orbits in the quantum phase space.

Shape resonances in photoionization cross sections and time delay

Shape resonances in photoionization of atoms and molecules arise from a particular geometry of the ionic potential which traps the receding photoelectron in a quasi-bound state in a particular partial wave. This mechanism allows us to connect the photoionization cross section in the resonant region with the photoelectron scattering phase in this partial wave by a simple formula $\sigma \propto \sin^2\delta_\ell$. Due to this relation, the phase $\delta_\ell$ can be extracted from an experimentally known cross section and then converted to the photoelectron group delay (Wigner time delay) $\tau_{\rm W} = \partial \delta_\ell/\partial E$ which is measurable by recently developed laser interferometric techniques. Such a direct connection of the photoionization cross section and the time delay is a fundamental property of shape resonances which provides a comprehensive test of novel measurements against a large body of older synchrotron data.

Direct and accurate calculation of dwell times and time delays using quantum trajectories

Among the numerous concepts of time in quantum scattering, Smith's dwell time (Smith, 1960 [7]) and Eisenbud & Wigner's time delay (Wigner, 1955 [12]) are the most well established. The dwell time represents the amount of time spent by the particle inside a given coordinate range (typically a potential barrier interaction region), while the time delay measures the excess time spent in the interaction region because of the potential. In this paper, we use the exact trajectory-ensemble reformulation of quantum mechanics, recently proposed by one of the authors (Poirier), to study how tunneling and reflection unfold over time, in a one-dimensional rectangular potential barrier. Among other dynamical details, the quantum trajectory approach provides an extremely robust, accurate, and straightforward method for directly computing the dwell time and time delay, from a single quantum trajectory. The resultant numerical method is highly efficient, and in the case of the time delay, completely obviates the traditional need to energy-differentiate the scattering phase shift. In particular, the trajectory variables provide a simple expression for the time delay that disentangles the contribution of the self-interference delay. More generally, quantum trajectories provide interesting physical insight into the tunneling process.

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

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.

Massive and massless two-dimensional Dirac particles in electric quantum dots

In this work we investigate the confining properties of charged particles of a Dirac material in the plane subject to an electrostatic potential well, that is, in an electric quantum dot. Our study focuses on the effect of mass and angular momenta on such confining properties. To have a global picture of confinement, both bound and resonance states are considered. The resonances will be examined by means of the Wigner time delay of the scattering states, as well as through the complex eigenvalues of outgoing states in order to show that they are physically meaningful. By tuning the potential intensity of the well, electron captures and atomic collapses are observed for critical values. In these processes, the bound states of the discrete spectrum become resonances of the continuous spectrum or vice versa. For massive charges, the atomic collapse phenomenon keeps the number of bound levels in the quantum dot below a maximum value. In the massless case, the bound states have zero energy and occur only for some discrete values of the potential depth, as is known. We also show that although the intensity of the resonances for massive particles is not significantly influenced by angular momenta, on the contrary, for massless particles they are quite sensitive to angular momenta, as it is the case of graphene.

Time delay statistics for finite number of channels in all symmetry classes

Within a random matrix theory approach, we obtain spectral statistics of the Wigner time delay matrix Q, for arbitrary channels number M and for all symmetry classes, in fact for the general Dyson parameter β. We also put forth two conjectures: one is related to the large-M expansion of joint cumulants of traces of powers of Q, which generalizes and implies a previous conjecture of Cunden, Mezzadri, Vivo and Simm; the other concerns the tail of the distribution of traces of powers of Q.

Use of transmission and reflection complex time delays to reveal scattering matrix poles and zeros: Example of the ring graph.

We identify the poles and zeros of the scattering matrix of a simple quantum graph by means of systematic measurement and analysis of Wigner, transmission, and reflection complex time delays. We examine the ring graph because it displays both shape and Feshbach resonances, the latter of which arises from an embedded eigenstate on the real frequency axis. Our analysis provides a unified understanding of the so-called shape, Feshbach, electromagnetically induced transparency, and Fano resonances on the basis of the distribution of poles and zeros of the scattering matrix in the complex frequency plane. It also provides a first-principles understanding of sharp resonant scattering features and associated large time delay in a variety of practical devices, including photonic microring resonators, microwave ring resonators, and mesoscopic ring-shaped conductor devices. Our analysis involves use of the reflection time difference, as well as a comprehensive use of complex time delay, to analyze experimental scattering data.

Cluster approach to scattering in MoS2 photoemission

We examine the photoemission cross sections and Wigner time delays from MoS 2 core-level orbitals Mo4s, Mo4p, S3s. We employ a cluster approach and study the photoemission process for progressively larger model clusters, in order to assess the scattering effects by neighboring atoms. First we explore how the fully differential cross section (FDCS) from the localized orbitals evolves as we increase the cluster size. In order to establish the underlying scattering mechanics that are mapped into the FDCS and Wigner time delays, we jointly analyze them in terms of the polar emission angle and photoemission energy for fixed azimuthal angles. The Wigner-time-delay scale goes from a few tens up into the hundreds of attoseconds when increasingly large clusters are considered, indicating the presence of high order scattering processes and interference. We find that, for the largest clusters, some internal propagation pathways start taking shape, which in an infinite system would amount to band structure positive energy states. We conclude that second- or higher-order neighbors around the main emitter strongly affect the photoelectron propagation and yield, therefore supporting their importance for models that aim for quantitative descriptions.

Theoretical study of the molecular-frame photoemission time delay for K -shell photoionization of N2

Photoelectron emission Wigner time delay for $K$-shell ionization of a nitrogen molecule is studied theoretically in the frame of a molecular reference for parallel and perpendicular orientations of the molecule with respect to the electric field vector of linearly polarized ionizing light. Electronic structure calculations are performed by the single center method for photoelectron kinetic energies from 5 to 20 eV and at different levels of approximation. Thereby, the influence of core relaxation effects and interchannel coupling on the $1s$-photoemission time delays, computed across the ${\ensuremath{\sigma}}^{*}$-shape resonance, is investigated in detail. The present theoretical results can be considered as reliable predictions for future experiments.

Wave excitation and dynamics in non-Hermitian disordered systems

Dynamic and steady-state aspects of wave propagation are deeply connected in lossless open systems in which the scattering matrix is unitary. There is then an equivalence among the energy excited within the medium through all channels, the Wigner time delay, which is the sum of dwell times in all channels coupled to the medium, and the density of states. But these equivalences fall away in the presence of material loss or gain. In this paper, we use microwave measurements, numerical simulations, and theoretical analysis to discover the changing relationships among fundamental wave properties with loss and gain, and their dependence upon dimensionality and spectral overlap. We begin with the demonstrations that the transmission time in random 1D media is equal to the density of states even in the presence of ultrastrong absorption and that its ensemble average is independent of the strengths of scattering and absorption. In contrast, the Wigner time becomes imaginary in the presence of loss, with real and imaginary parts that fall with absorption. In multichannel media, the transmission time remains equal to the density of states and is independent of the scattering strength in unitary systems but falls with absorption to a degree that increases with the strengths of absorption and scattering, and the number of channels coupled to the medium. We show that the relationships between key propagation variables in non-Hermitian systems can be understood in terms of the singularities of the phase of the determinant of the transmission matrix. The poles of the transmission matrix are the same as those of the scattering matrix, but the transmission zeros are fundamentally different. Whereas the zeros of the scattering matrix are the complex conjugates of the poles, the transmission zeros are topological: in unitary systems they occur only singly on the real axis or as conjugate pairs. We follow the evolution and statistics of zeros in the complex plane as random samples are deformed. The sensitivity of the spacing of zeros in the complex plane with deformation of the sample has a square-root singularity at a zero point at which two single zeros and a complex pair interconvert. The transmission time is a sum of Lorentzian functions associated with poles and zeros. The sum over poles is the density of states with an average that is independent of scattering and dissipation. But the sum over zeros changes with loss, gain, scattering strength and the number of channels in ways that make it possible to control ultranarrow spectral features in transmission and transmission time. We show that the field, including the contribution of the still coherent incident wave, is a sum over modal partial fractions with amplitudes that are independent of loss and gain. The energy excited may be expressed in terms of the resonances of the medium and is equal to the dwell time even in the presence of loss or gain.

Wigner time delay of particles elastically scattered by a cluster of zero-range potentials

The Wigner time delay of slow particles in the process of their elastic scattering by complex targets formed by several zero-range potentials is investigated. It is shown that at asymptotically large distances from the target, the Huygens-Fresnel interference pattern formed by spherical waves emitted by each of the potentials is transformed into a system of spherical waves generated by the geometric center of the target. These functions determine flows of particles in and out through the surface of the sphere surrounding the target. The energy derivatives of phase shifts of these functions are the partial Wigner time delay. General formulas that connect the s-phase shifts of particle scattering by each of the zero-range potentials with the phases of particle scattering by the potential cluster forming the target are obtained. Model targets consisting of two-, three- and 4-centers are considered. It is assumed that these targets are built from identical delta-potentials with equal distances between their centers. The partial Wigner time delay of slow particles by considered targets are obtained. We apply the derived general formulas to consideration of electron scattering by atomic clusters that trap electron near the target, and by calculating the times delay of mesons scattered by few-nucleons systems.

Probing Molecular Frame Wigner Time Delay and Electron Wavepacket Phase Structure of CO Molecule

The time delay of photoelectron emission serves as a fundamental building block to understand the ultrafast electron emission dynamics in strong-field physics. Here, we study the photoelectron angular streaking of CO molecules by using two-color ( 400 + 800 nm) corotating circularly polarized fields. By coincidently measuring photoelectrons with the dissociative ions, we present molecular frame photoelectron angular distributions with respect to the instantaneous driving electric field signatures. We develop a semiclassical nonadiabatic molecular quantum-trajectory Monte Carlo (MO-QTMC) model that fully captures the experimental observations and further ab initio simulations. We disentangle the orientation-resolved contribution of the anisotropic ionic potential and the molecular orbital structure on the measured photoelectron angular distributions. Furthermore, by analyzing the photoelectron interference patterns, we extract the sub-Coulomb-barrier phase distribution of the photoelectron wavepacket and reconstruct the orientation- and energy-resolved Wigner time delay in the molecular frame. Holographic angular streaking with bicircular fields can be used for probing polyatomic molecules in the future.