Wigner Time Delay Research Articles

Statistics of Complex Wigner Time Delays as a Counter of S-Matrix Poles: Theory and Experiment.

We study the statistical properties of the complex generalization of Wigner time delay τ_{W} for subunitary wave-chaotic scattering systems. We first demonstrate theoretically that the mean value of the Re[τ_{W}] distribution function for a system with uniform absorption strength η is equal to the fraction of scattering matrix poles with imaginary parts exceeding η. The theory is tested experimentally with an ensemble of microwave graphs with either one or two scattering channels and showing broken time-reversal invariance and variable uniform attenuation. The experimental results are in excellent agreement with the developed theory. The tails of the distributions of both real and imaginary time delay are measured and are also found to agree with theory. The results are applicable to any practical realization of a wave-chaotic scattering system in the short-wavelength limit, including quantum wires and dots, acoustic and electromagnetic resonators, and quantum graphs.

Probing tunneling dynamics of dissociative H2 molecules using two-color bicircularly polarized fields

Probing and manipulating the electronic motion in the ultrafast laser molecular interaction provides the pathways for quantum imaging and controlling chemical reactions. Recently, the emerging application of attosecond metrology of ultrafast electron dynamics has accessed the time scale of the most fundamental processes in molecular chemical reactions. Here, we probe the tunneling dynamics of internuclear-dependent dissociative reaction of ${\mathrm{H}}_{2}$ with angular streaking using two-color bicircularly polarized femtosecond laser pulses. By measuring high-resolution photoelectron spectroscopy, we disentangle the orientation and internuclear-distance dependent effect of the long-range Coulomb potential and the initial phase on molecular-frame photoelectron momentum distributions, and stereo extract the phase gradient of the tunneling electron wave packets and Wigner time delay during the dissociative ionization using two-color bicircular fields. The work has an insight into the clocking of ultrafast spatiotemporal photoelectron dynamics and the quantum control of molecular chemical processes via sculptured circular fields, which can be applied to polyatomic molecules.

Transverse shifts and time delays of spatiotemporal vortex pulses reflected and refracted at a planar interface

Abstract Transverse (Hall-effect) and Goos–Hänchen shifts of light beams reflected/refracted at planar interfaces are important wave phenomena, which can be significantly modified and enhanced by the presence of intrinsic orbital angular momentum (OAM) in the beam. Recently, optical spatiotemporal vortex pulses (STVPs) carrying a purely transverse intrinsic OAM were predicted theoretically and generated experimentally. Here we consider the reflection and refraction of such pulses at a planar isotropic interface. We find theoretically and confirm numerically novel types of OAM-dependent transverse and longitudinal pulse shifts. Remarkably, the longitudinal shifts can be regarded as time delays, which appear, in contrast to the well-known Wigner time delay, without temporal dispersion of the reflection/refraction coefficients. Such time delays allow one to realize OAM-controlled slow (subluminal) and fast (superluminal) pulse propagation without medium dispersion. These results can have important implications in various problems involving scattering of localized vortex states carrying transverse OAM.

Two-center interference and stereo Wigner time delay in photoionization of asymmetric molecules

We present numerical simulations of the time delay in photoemission of asymmetric diatomic molecules using the technique of reconstruction of attosecond beating by interference of two-photon transitions (RABITT) by solving the time-dependent Schr\"odinger equation. Our results show an obvious time delay between photoelectrons emitted to the left and right and this relative time delay oscillates as the photoelectron energy changes. More interestingly, the amplitude of this oscillation increases when the asymmetry degree of diatomic molecules decreases. With the method of the selected continuum wave functions, we calculate the Wigner time delay in photoionization. The obtained stereo Wigner time delay also oscillates with photoelectron energy. This oscillation is traced back to two-center interferences and it could explain the relative time delay in the RABITT measurement. Furthermore, our results indicate that the continuum-continuum time delay in photoemission of heteronuclear molecules is asymmetric.

Applications in random matrix theory of a PIII′ τ-function sequence from Okamoto’s Hamiltonian formulation

We consider the singular linear statistic of the Laguerre unitary ensemble (LUE) consisting of the sum of the reciprocal of the eigenvalues. It is observed that the exponential generating function for this statistic can be written as a Toeplitz determinant with entries given in terms of particular [Formula: see text] Bessel functions. Earlier studies have identified the same determinant, but with the [Formula: see text] Bessel functions replaced by [Formula: see text] Bessel functions, as relating to the hard edge scaling limit of a generalized gap probability for the LUE, in the case of non-negative integer Laguerre parameter. We show that the Toeplitz determinant formed from an arbitrary linear combination of these two Bessel functions occurs as a [Formula: see text]-function sequence in Okamoto’s Hamiltonian formulation of Painlevé III[Formula: see text], and consequently the logarithmic derivative of both Toeplitz determinants satisfies the same [Formula: see text]-form Painlevé III[Formula: see text] differential equation, giving an explanation of a fact which can be observed from earlier results. In addition, some insights into the relationship between this characterization of the generating function, and its characterization in the [Formula: see text] limit, both with the Laguerre parameter [Formula: see text] fixed, and with [Formula: see text] (this latter circumstance being relevant to an application to the distribution of the Wigner time delay statistic), are given.

Generalization of Wigner time delay to subunitary scattering systems.

We introduce a complex generalization of the Wigner time delay τ for subunitary scattering systems. Theoretical expressions for complex time delays as a function of excitation energy, uniform and nonuniform loss, and coupling are given. We find very good agreement between theory and experimental data taken on microwave graphs containing an electronically variable lumped-loss element. We find that the time delay and the determinant of the scattering matrix share a common feature in that the resonant behavior in Re[τ] and Im[τ] serves as a reliable indicator of the condition for coherent perfect absorption (CPA). By reinforcing the concept of time delays in lossy systems this work provides a means to identify the poles and zeros of the scattering matrix from experimental data. The results also enable an approach to achieving CPA at an arbitrary frequency in complex scattering systems.

Angular dependence of the Wigner time delay upon tunnel ionization of H2

When a very strong light field is applied to a molecule an electron can be ejected by tunneling. In order to quantify the time-resolved dynamics of this ionization process, the concept of the Wigner time delay can be used. The properties of this process can depend on the tunneling direction relative to the molecular axis. Here, we show experimental and theoretical data on the Wigner time delay for tunnel ionization of H2 molecules and demonstrate its dependence on the emission direction of the electron with respect to the molecular axis. We find, that the observed changes in the Wigner time delay can be quantitatively explained by elongated/shortened travel paths of the emitted electrons, which occur due to spatial shifts of the electrons’ birth positions after tunneling. Our work provides therefore an intuitive perspective towards the Wigner time delay in strong-field ionization.

RABBITT phase transition across the ionization threshold

The authors present a theoretical treatment of attosecond phase shifts in photoionization due to the interference of pathways involving two-photon transitions with adjacent harmonics for the special case where one of the involved states is a bound state, rather than the usual situation where both states are in the continuum. They show that this process provides an alternative way of assessing the continuum-continuum phase, which is usually closely linked to the Wigner time delay and hard to access experimentally.

Photoionization of Xe 5s: angular distribution and Wigner time delay in the vicinity of the second Cooper minimum

The angular distribution and photoionization Wigner time delay of Xe 5s photoelectrons are studied in the region of the second Cooper minimum (SCM) using (i) the relativistic multiconfiguration Tamm–Dancoff approximation, (ii) the relativistic-random-phase approximation (RRPA) and (iii) the RRPA-with-relaxation to demonstrate how differing treatments of correlation, and the relativistic interactions, affect the results. The results of the three methods are compared with each other and with available experimental data. The comparison reveals the importance of electron correlations for which a multiconfiguration description of the initial state is essential. The spin-resolved and spin-averaged photoionization time delay results show important signatures in the region of the SCM in the Xe 5s photoionization cross-section.

Photoionization phase shift and Wigner time delay of endohedrally confined atoms using transient phase methods

In contrast to the conventional finite difference methods, two transient phase methods have been effectively used in the present work to directly compute the photoionization phase shift and Wigner time delay of confined atoms (A@C60) in the single-active electron (SAE) approximation. The different phase methods: (A) employing logarithmic derivatives at shell boundaries, and (B) Born approximation are verified with the help of well-established finite difference methods in SAE approximation and sophisticated many-electron techniques. In this work, confinement oscillations on the dipole phase and photoelectron group delay following ionization from 1s subshell of H@C60, 3p subshell of Ar@C60 and 5p subshell of Xe@C60 are analyzed. The comparison with many-body calculation shows that the features in the time delay of a confined system are governed mainly by the effects of screening apart from that due to the external potential. A systematic study and comparison of the results from phase methods and many-electron techniques indicate that these techniques can be effectively used in the analysis of photoionization phase shift and time delay in confined atoms.

Wigner Time-Delay and Distribution for Polarization Interaction in Strongly Coupled Semiclassical Plasmas

The quantum effect on the Wigner time-delay and distribution for the polarization scattering in a semiclassical dense plasma is explored. The partial wave analysis is applied for a partially ionized dense plasma to derive the phase shift for the polarization interaction. The Wigner time-delay and the Wigner distribution are derived for the electron–atom polarization interaction including the effects of quantum-mechanical characteristic and plasma screening. In this work, we show that the Wigner time-delay and the Wigner distribution for the polarization interaction can be suppressed by the quantum effect. The Wigner time-delay and the Wigner distribution are also significantly suppressed by the increase of plasma shielding. The variation of the Wigner time-delay and the Wigner distribution function due to quantum screening is discussed.

Holographic angular streaking of electrons and the Wigner time delay

For a circularly polarized single-color field at a central frequency of $2\omega$ the final electron momentum distribution upon strong field ionization does not carry any information about the phase of the initial momentum distribution. Adding a weak, co-rotating, circularly polarized field at a central frequency of $\omega$ gives rise to a sub-cycle interference pattern (holographic angular streaking of electrons (HASE)). This interference pattern allows for the retrieval of the derivative of the phase of the initial momentum distribution after tunneling $\phi^{\prime}_{\mathrm{off}}(p_i)$. A trajectory-based semi-classical model (HASE model) is introduced which links the experimentally accessible quantities to $\phi^{\prime}_{\mathrm{off}}(p_i)$. It is shown that a change in $\phi^{\prime}_{\mathrm{off}}$ is equivalent to a displacement in position space $\Delta x$ of the initial wave packet after tunneling. This offset in position space allows for an intuitive interpretation of the Wigner time delay $\Delta \tau_W$ in strong field ionization for circularly polarized single-color fields. The influence of Coulomb interaction after tunneling is investigated quantitatively.

Chaotic scattering with localized losses: S-matrix zeros and reflection time difference for systems with broken time-reversal invariance.

Motivated by recent studies of the phenomenon of coherent perfect absorption, we develop the random matrix theory framework for understanding statistics of the zeros of the (subunitary) scattering matrices in the complex energy plane, as well as of the recently introduced reflection time difference (RTD). The latter plays the same role for S-matrix zeros as the Wigner time delay does for its poles. For systems with broken time-reversal invariance, we derive the n-point correlation functions of the zeros in a closed determinantal form, and we study various asymptotics and special cases of the associated kernel. The time-correlation function of the RTD is then evaluated and compared with numerical simulations. This allows us to identify a cubic tail in the distribution of RTD, which we conjecture to be a superuniversal characteristic valid for all symmetry classes. We also discuss two methods for possible extraction of S-matrix zeros from scattering data by harmonic inversion.

Angular dependent time delay near correlation induced Cooper minima

We analyze an angular dependence of the Wigner time delay near the Cooper minimum (CM) of the sub-valent ns shell in argon, krypton and xenon. Such an angular dependence is a result of interplay between the relativistic and correlation effects. The correlation with the outermost np valence shell induces a CM in the sub-valent ns shell which is otherwise CM free. A phase difference between the two spin–orbit split ionization continua Ep1/2 and Ep3/2 makes the Wigner time delay angular dependent. Both these effects are accounted for within relativistic formulations of the random phase approximation and the time-dependent density functional theory. Comparison between these two approaches illustrates a very strong sensitivity of the observed effect to the computation detail, especially the account of the ground state correlation.

Time delay in slow electron elastic scattering by potentials with arising levels

SynopsisWe study Wigner time delay of slow electrons by spherically symmetric square potential well as a function of the well parameters. We show that the electron interaction with scattering center qualitatively depends on presence of discrete levels in the well. Electron retention mode by a potential well with no discrete level is replaced by the mode at which after level appears the incident electron hardly enters the scatterer. This behavior of time delay is universal since it is typical not only for the first but also for following arising s-levels.

Effects of relativistic interactions in photodetachment time delay of Br−

SynopsisWigner time delay in photodetachment of Br− has been studied. The absence of Coulomb phase in photodetachment process makes the Wigner time delay more sensitive to the centrifugal barrier shape resonance and threshold effects. Important relativistic effects have been found near the threshold region.

Perturbative method for resolving contact interactions in quantum mechanics

Long-range effective methods are ubiquitous in physics and in quantum theory, in particular. Furthermore, the reliability of such methods is higher when the nature of short-ranged interactions need not be modeled explicitly. This may be necessary for two reasons: (1) there are interactions that occur over a short range that cannot be accurately modeled with a potential function and/or (2) the entire Hamiltonian loses its reliability when applied at short distances. This work is an investigation of the utility and consequences of omitting a finite region of space from quantum mechanical analysis, accomplished by imposition of an artificial boundary behind which obscured short-ranged physical effects may operate. With this method, a free function of integration that depends on momentum is interpreted as a function encoding information needed to match a long-distance wavefunction to an appropriate state function on the other side of the boundary. Omitting part of the space from analysis implies that the strict unitarity requirement of quantum mechanics must be relaxed, since particles can actually propagate beyond the boundary. Strict orthogonality of eigenmodes and hermiticity of the Hamiltonian must also be relaxed in this method; however, all of these canonical relations are obeyed when averaged over sufficiently long times. What is achieved, therefore, appears to be an effective long-wavelength theory, at least for stationary systems. As examples, the quantum defect theory of the one-dimensional Coulomb interaction is recovered, as well as a new perspective of the inverse-square potential and the free particle, as well as the Wigner time delay associated with contact interactions. Potential applications of this method may include three-dimensional atomic systems and two-dimensional systems, such as graphene.

Distribution of the Wigner–Smith time-delay matrix for chaotic cavities with absorption and coupled Coulomb gases

Within the random matrix theory approach to quantum scattering, we derive the distribution of the Wigner–Smith time delay matrix for a chaotic cavity with uniform absorption, coupled via N perfect channels. In the unitary class we obtain a compact expression for the distribution of the full matrix in terms of a matrix integral. In the other symmetry classes we derive the joint distribution of the eigenvalues. We show how the large N properties of this distribution can be analysed in terms of two interacting Coulomb gases living on two different supports. As an application of our results, we study the statistical properties of the Wigner time delay in the presence of absorption.

Reflection Time Difference as a Probe of S-Matrix Zeroes in Chaotic Resonance Scattering

Motivated by recent interest in the zeroes of S-matrix entries in the complex energy plane, I use the Heidelberg model of resonance scattering to introduce the notion of Reflection Time Difference which is shown to play the same role for the {\it zeroes} as the Wigner time delay plays for the S- matrix {\it poles}.

Unifying Tunneling Pictures of Strong-Field Ionization with an Improved Attoclock.

We demonstrate a novel attoclock, in which we add a perturbative linearly polarized light field at 400nm to calibrate the attoclock constructed by an intense circularly polarized field at 800nm. This approach can be directly implemented to analyze the recent hot and controversial topics involving strong-field tunneling ionization. The generally accepted picture is that tunneling ionization is instantaneous and that the tunneling probability synchronizes with the laser electric field. Alternatively, recently it was described in the Wigner picture that tunneling ionization would occur with a certain of time delay. We unify the two seemingly opposite viewpoints within one theoretical framework, i.e., the strong-field approximation (SFA). We illustrate that both the instantaneous tunneling picture and the Wigner time delay picture that are derived from the SFA can interpret the measurement well. Our results show that the finite tunneling delay will accompany nonzero exit longitudinal momenta. This is not the case for the instantaneous tunneling picture, where the most probable exit longitudinal momentum would be zero.