Photoelectron Momentum Distributions Research Articles

Importance of bulk excitations and coherent electron-photon-phonon scattering in photoemission from PbTe(111): Ab initio theory with experimental comparisons

This paper presents a fully ab initio many-body photoemission framework that includes coherent three-body electron-photon-phonon scattering to predict the transverse momentum distributions and the mean transverse energies (MTEs) of bulk photoelectrons from single-crystal photocathodes. The need to develop such a theory stems from the lack of studies that provide complete understanding of the underlying fundamental processes governing the transverse momentum distribution of photoelectrons emitted from single crystals. For example, initial predictions based on density-functional theory calculations of effective electron masses suggested that the (111) surface of PbTe would produce very small MTEs $(\ensuremath{\le}15 \mathrm{meV})$, whereas our experiments yielded MTEs 10 to 20 times larger than these predictions and also exhibited a lower photoemission threshold than predicted. The ab initio framework presented in this paper correctly reproduces both the magnitude of the MTEs from our measurements in PbTe(111) and the observed photoemission below the predicted threshold. Our results show that both photoexcitations into states that propagate in the bulk of the material and coherent many-body electron-photon-phonon scattering processes, which initial predictions ignored, play surprisingly important roles in photoemission from PbTe(111). Finally, from the lessons learned, we recommend a procedure for rapid computational screening of potential single-crystal photocathodes for applications in next-generation ultrafast electron diffraction and x-ray free-electron lasers, which will enable significant advances in condensed matter research.

Nondipole modification of the ac Stark effect in above-threshold ionization

Above-threshold ionization (ATI) of atoms in intense laser pulses produces photoelectron momentum distributions that exhibit discrete concentric circles related to the absorption of more photons than required to exceed the ionization threshold. Using numerical solutions of the one-electron time-dependent Schr\"odinger equation beyond the electric dipole approximation for circularly polarized laser pulses, we show that the geometric centers of ATI rings are counterintuitively shifted against the propagation direction of the light, i.e., in the direction opposite to the on-average transferred photon momentum. From a comparison with a simple model based on energy considerations, we identify a nondipole modification of the ac Stark effect for electron continuum states as the origin of our observation.

Atomic Siegert states in a rotating electric field

We show that the time-dependent Schr\"odinger equation describing in the dipole approximation the interaction of a one-electron atom with a monochromatic circularly polarized electromagnetic field can be reduced to a stationary Schr\"odinger equation in the form which allows to separate variables in the asymptotic region and explicitly formulate outgoing-wave boundary conditions. The solutions to this equation are called atomic Siegert states (SSs) in a rotating electric field. We develop the theory of such states and propose an efficient method to construct them using powerful techniques of stationary scattering theory. The method yields not only the SS eigenvalue defining the Stark-shifted energy and total ionization rate of the state, but also the SS eigenfunction defining partial ionization amplitudes and the photoelectron momentum distribution. The theory is illustrated by calculations for a model potential.

Symmetries and Selection Rules of the Spectra of Photoelectrons and High-Order Harmonics Generated by Field-Driven Atoms and Molecules

Using the strong-field approximation we systematically investigate the selection rules for high-order harmonic generation and the symmetry properties of the angle-resolved photoelectron spectra for various atomic and molecular targets exposed to one-component and two-component laser fields. These include bicircular fields and orthogonally polarized two-color fields. The selection rules are derived directly from the dynamical symmetries of the driving field. Alternatively, we demonstrate that they can be obtained using the conservation of the projection of the total angular momentum on the quantization axis. We discuss how the harmonic spectra of atomic targets depend on the type of the ground state or, for molecular targets, on the pertinent molecular orbital. In addition, we briefly discuss some properties of the high-order harmonic spectra generated by a few-cycle laser field. The symmetry properties of the angle-resolved photoelectron momentum distribution are also determined by the dynamical symmetry of the driving field. We consider the first two terms in a Born series expansion of the T matrix, which describe the direct and the rescattered electrons. Dynamical symmetries involving time translation generate rotational symmetries obeyed by both terms. However, those that involve time reversal generate reflection symmetries that are only observed by the direct electrons. Finally, we explain how the symmetry properties, imposed by the dynamical symmetry of the driving field, are altered for molecular targets.

Multichromatic Polarization-Controlled Pulse Sequences for Coherent Control of Multiphoton Ionization

In this review, we report on recent progress in the generation and application of multichromatic polarization-tailored pulse sequences for the coherent control of multiphoton ionization (MPI) dynamics and present unpublished experimental results that complement our previous findings. Specifically, we utilize single-color, bichromatic, and trichromatic polarization-controlled pulse sequences generated by spectral amplitude, phase and polarization modulation of a carrier-envelope phase (CEP)-stable white light supercontinuum for MPI. The analysis of the number of ionization pathways and the number of distinct final free electron states shows that both increase significantly, but scale differently with the number of absorbed photons and the number of pulses in the sequence. In our experiments, ultrafast polarization shaping is combined with high-resolution photoelectron tomography to generate, control, and reconstruct three-dimensional photoelectron momentum distributions from atomic and molecular MPI. We discuss the use of polarization-controlled single-color and bichromatic pulse sequences in perturbative and non-perturbative coherent control of coupled electron-nuclear dynamics in molecules, atomic spin-orbit wave packet dynamics and the directional photoemission from atoms and chiral molecules. We compare the coherent control of CEP-insensitive intraband multipath interference in the MPI with a fixed number of photons with CEP-sensitive interband multipath interference in the ionization with a different number of photons. The generation and control of free electron vortices with even-numbered rotational symmetry by MPI with single-color pulse sequences is contrasted with the bichromatic control of CEP-sensitive electron vortices with odd-numbered rotational symmetry. To illustrate the potential of multichromatic pulse sequences for coherent control, we present a trichromatic scheme for shaper-based quantum state holography.

Attoclock with bicircular laser fields as a probe of velocity-dependent tunnel-exit positions

Strong-field ionization of atoms can be investigated on the attosecond time scale by using the attoclock method, i.e. by observing the peak of the photoelectron momentum distribution (PMD) after applying a laser pulse with a two-dimensional polarization form. Examples for such laser fields are close-to-circular or bicircular fields. Here, we report numerical solutions of the time-dependent Schrödinger equation for bicircular fields and a comparison with a compact classical model to demonstrate that the tunnel-exit position, i.e. the position where the electron emerges after tunnel ionization, is encoded in the PMD. We find that the tunnel-exit position depends on the transverse velocity of the tunneling electron. This gives rise to a momentum-dependent attoclock shift, meaning that the momentum shift due to the Coulomb force on the outgoing electron depends on which slice of the momentum distribution is analysed. Our finding is supported by a momentum-space-based implementation of the classical backpropagation method.

Coherent control mechanisms in bichromatic multiphoton ionization

Free electron vortices (FEVs) generated by multiphoton ionization (MPI) with ultrashort laser pulses have attracted significant attention due to their varied symmetries and unusual topological properties. We study two physical mechanisms of coherent control in atomic MPI with bichromatic polarization-shaped femtosecond laser pulses which give rise to the rich variety of FEVs. In the experiments, we combine pulse shaping of a carrier-envelope phase-stable supercontinuum with photoelectron tomography to generate and reconstruct three-dimensional photoelectron momentum distributions (PMDs). Simultaneous measurements of energetically separated photoelectrons from intraband and interband interference in a single PMD allow us to compare phase and polarization control of the angular distributions by both mechanisms. We investigate phase control in three scenarios: first, counterrotating circularly polarized pulses are employed to contrast the phase-insensitive angular momentum eigenstate created by intraband interference via frequency mixing with the phase-sensitive c 7 rotationally symmetric FEV from pure interband interference of two single-color ionization pathways. In the second scenario, we use orthogonal linearly polarized pulses to compare the phase control properties of a six-lobed angular momentum wave packet from intraband interference to those of a complex shaped interband PMD in the presence of phase fluctuations. Finally, we demonstrate phase control of a photoelectron hologram from mixed interband interference. In a (3 + 1) resonance enhanced MPI scheme, the red pump pulse induces a bound electron wave packet probed by the time-delayed blue pulse. The latter simultaneously creates a reference wave packet by three-photon ionization to form the photoelectron hologram. Rotation of the hologram with c 1 or c 5 rotational symmetry maps the time evolution of the bound wave packet. To analyze our results, we develop analytical expressions for the wave functions of intraband and interband interference in perturbative non-resonant MPI. The experiments are complemented with two-dimensional TDSE simulations to follow the FEV formation dynamics and to validate the physical pictures.

Attosecond-resolved photoelectron holography for triatomic molecule

Photoelectron holography in strong field tunneling ionization has subatomic spatial and attosecond temporal resolutions and thus it is a promising method to probe the attosecond electronic dynamics in atoms and molecules. The proof-of-concept studies have been performed in atoms and diatomic molecules. Here, we illustrate the application of the strong-field photoelectron holography in probing the attosecond electron dynamics in more complex system. As a demonstrated, by solving the time-dependent Schrödinger equation, we shown that the ultrafast charge migration in the triatomic molecule H3 could be retrieved from the holographic fringes in the photoelectron momentum distribution. Our study indicates that the concept of photoelectron holography can be extended to probe the attosecond electron dynamics in more complex molecules which is of essential significance for the newly emerging attosecond chemistry.

Local most-probable routes and classical-quantum correspondence in strong-field two-dimensional tunneling ionization

We study ionization of atoms in strong two-dimensional (2D) laser fields with various forms, numerically and analytically. We focus on the local most-probable tunneling routes (some specific electron trajectories) which are corresponding to the local maxima of photoelectron momentum distributions (PMDs). By making classic-quantum correspondence, we obtain a condition for these routes characterized by the electron position at the tunnel exit. With comparing the identified routes with the classical limit and the partial-decoupling approximation where it is assumed that tunneling is dominated by the main component of the 2D field, some semiclassical properties of 2D tunneling are addressed. The local maxima of PMD related to the local most-probable routes can be used as one of the preferred observables in ultrafast measurements.

Probing the tunneling electron wave packet using the counter-rotating bi-circular fields

We experimentally measure the photoelectron momentum distributions of Ar atoms using two-color 800 nm (ω) + 400 nm (2ω) counter-rotating circularly polarized fields. By employing a weak fundamental circularly polarized field at 800 nm, the atomic potential barrier formed by the strong second harmonic field at 400 nm can be modulated. This introduces a spatially rotating temporal two-slit interferometry between the photoelectron wave packets tunneling from the modified barriers in the consecutive cycles. The modulation of the barrier would leave its imprint on the photoelectron interference pattern. In analogy to the analysis of the traditional optical interferometry, we perform the Fourier transform on the photoelectron interference. The modulation on the amplitude and phase of the electron wave packets induced by the weak fundamental field is retrieved.

Exploration of interference structure of a hydrogen atom with different wavelengths in a linearly polarized laser field

We investigate the ionization process of an atom with different wavelengths in a linearly polarized laser field. With the change of wavelength, the interference structure in photoelectron momentum distributions (PMDs) changes obviously. The temporal double-slit fringes can be clearly observed in the relatively high-energy region below the energy of 2U p for the laser wavelength , but we only observe the temporal double-slit fringes in the relatively low-energy region below the energy of 2U p for the laser wavelength . By analyzing the number of relative contribution of directly ionized electrons, we find a sensitive dependence of the direct electron on the laser wavelength. In addition, the temporal double-slit fringes can be also observed in the photoelectron momentum distributions in the deep tunneling regime ().

Probing the Spin-Orbit Time Delay of Multiphoton Ionization of Kr by Bicircular Fields.

We study multiphoton ionization of Kr atoms by circular 400-nm laser fields and probe its photoelectron circular dichroism with the weak corotating and counterrotating circular fields at 800nm. The unusual momentum- and energy-resolved photoelectron circular dichroisms from the ^{2}P_{1/2} ionic state are observed as compared with those from ^{2}P_{3/2} ionic state. We identify an anomalous ionization enhancement at sidebands related to the ^{2}P_{1/2} ionic state on photoelectron momentum distribution when switching the relative helicity of the two fields from corotating to counterrotating. By performing the two-color intensity-continuously-varying experiments and the pump-probe experiment, we find a specific mixed-photon populated resonant transition channel in counterrotating fields that contributes to the ionization enhancement. We then probe the time delay between the two spin-orbit coupled ionic states (^{2}P_{1/2} and ^{2}P_{3/2}) using bicircular fields and reveal that the resonant transition has an insignificant effect on the relative spin-orbit time delay.

Resolving and weighing the quantum orbits in strong-field tunneling ionization

Tunneling ionization of atoms and molecules induced by intense laser pulses contains the contributions of numerous quantum orbits. Identifying the contributions of these orbits is crucial for exploring the application of tunneling and for understanding various tunneling-triggered strong-field phenomena. We perform a combined experimental and theoretical study to identify the relative contributions of the quantum orbits corresponding to the electrons tunneling ionized during the adjacent rising and falling quarter cycles of the electric field of the laser pulse. In our scheme, a perturbative second-harmonic field is added to the fundamental driving field. By analyzing the relative phase dependence of the signal in the photoelectron momentum distribution, the relative contributions of these two orbits are unambiguously determined. Our results show that their relative contributions sensitively depend on the longitudinal momentum and modulate with the transverse momentum of the photoelectron, which is attributed to the interference of the electron wave packets of the long orbit. The relative contributions of these orbits resolved here are important for the application of strong-field tunneling ionization as a photoelectron spectroscopy for attosecond time-resolved measurements.

Reconstruction of a single-active-electron potential from electron momentum distribution produced by strong-field ionization using optimization technique

We present a method for retrieving of single-active electron potential in an atom or molecule from a given momentum distribution of photoelectrons ionized by a strong laser field. In this method the potential varying within certain limits is found as the result of the optimization procedure aimed at reproducing the given momentum distribution. The optimization using numerical solution of the time-dependent Schrödinger equation for ionization of a model one-dimensional atom shows the good accuracy of the potential reconstruction method. This applies to different ways used for representing of the potential under reconstruction, including a parametrization and determination of the potential by specifying its values on a spatial grid.

Intensity-dependent angular distribution of low-energy electrons generated by intense high-frequency laser pulse.

By solving the three-dimensional time-dependent Schrödinger equation, we investigate the angular distributions of the low-energy electrons when an intense high-frequency laser pulse is applied to the hydrogen atom. Our numerical results show that the angular distributions of the low-energy electrons which generated by the nonadiabatic transitions sensitively depend on the laser intensity. The angular distributions evolve from a two-lobe to a four-lobe structure as the laser intensity increases. By analyzing nonadiabatic process in the Kramers-Henneberger frame, we illustrate that this phenomenon is attributed to the intensity-dependent adiabatic evolution of the ground state wavefunction. When the laser intensity further increases, the pathway of nonadiabatic transition from the ground state to the excited state and then to the continuum states is non-negligible, which results in the ring-like structure in the photoelectron momentum distribution. The angular distributions of the low-energy electrons provide a way to monitor the evolution of the electron wavefunction in the intense high frequency laser fields.

Effect of the Stark shift on the low-energy interference structure in strong-field ionization

An improved quantum trajectory Monte Carlo method including the Stark shift of the initial state, Coulomb potential, and multielectron polarization-induced dipole potential is adopted to revisit the origin of the low-energy interference structure in the photoelectron momentum distribution of the xenon atom subjected to an intense laser field, and resolve the different contributions of these three effects. We found that the Stark shift plays an essential role on the low-energy interference structure, which moves the ringlike constructive interference structure to the lower momentum region. The formation of the low-energy interference structure is a result of the combined effects of Stark shift, laser, and Coulomb fields, while the multielectron polarization mainly enhance the probability of the low energy photoelectron spectrum. Our finding provides insight into the electron dynamics of atoms and molecules when driven by the intense laser fields.

Ionization in intense laser fields beyond the electric dipole approximation: concepts, methods, achievements and future directions

The electric dipole approximation is widely used in atomic, molecular and optical physics and is typically related to a regime for which the wavelength is much larger than the atomic structure. However, studies have shown that in strong laser fields another regime exists where the dipole approximation breaks down. During the ionization process in intense laser fields and at long wavelengths the photoelectrons can reach higher velocities such that the magnetic field component of the laser field becomes significant. The ionization dynamics and the final momentum of the electron is therefore modified by the entire Lorentz force. In contrast the magnetic field interaction is neglected in the dipole approximation. Rapid developments in laser technology and advancements in the accuracy of the measurements techniques have enabled the observation of the influence of such non-dipole effects on the final angular photoelectron momentum distributions. More recently the number of studies on ionization beyond the dipole approximation has increased significantly, providing more important insight into fundamental properties of ionization processes. For example we have shown that the final three dimensional photoelectron momentum spectra is significantly affected by the non-dipole drift with the parent–ion interaction, the linear multiphoton momentum transfer on a sub-cycle time scale and the sharing of the transferred linear photon momenta between the electron and the ion. In this article we present an overview of the underlying mechanisms and we review the experimental techniques and the achievements in this field. We focus on ionization in strong laser fields in the regime where the dipole approximation is not valid but a fully relativistic description is not required.

Strong-field ionization of heteronuclear diatomic molecules using an orthogonally polarized two-color laser field

We apply the improved molecular strong-field approximation to investigate high-order above-threshold ionization (HATI) of heteronuclear diatomic molecules by an orthogonally polarized two-color (OTC) laser field. The OTC field consists of two linearly polarized components with frequencies $r\ensuremath{\omega}$ and $s\ensuremath{\omega}$, where $r$ and $s$ are integers, and $\ensuremath{\omega}$ is the fundamental frequency. The molecule is aligned in the OTC laser field polarization plane. We show that the photoelectron momentum distribution obeys one reflection symmetry which is valid for arbitrary values of the relative phase between the OTC field components in the case when $r+s$ is odd. For molecules oriented along the polarization axis ${z}_{L}$ of the $r\ensuremath{\omega}$ component (${\ensuremath{\theta}}_{L}={0}^{\ensuremath{\circ}}$) and $r$ even and $s$ odd, the HATI spectrum exhibits the reflection symmetry with respect to the ${z}_{L}$ axis. When the molecular orientation is along the ${x}_{L}$ axis, which is perpendicular to the polarization axis of the $r\ensuremath{\omega}$ component (${\ensuremath{\theta}}_{L}={90}^{\ensuremath{\circ}}$), the spectrum exhibits the reflection symmetry with respect to the ${x}_{L}$ axis for $r$ odd and $s$ even. In addition, we analyze the asymmetry in the photoelectron spectra of heteronuclear molecules by comparing them with the photoelectron spectra obtained ionizing a homonuclear diatomic molecule. We also explore the influence of the shift of the relative phase by ${180}^{\ensuremath{\circ}}$ on the HATI spectra. We explain some characteristics of the obtained HATI spectra using a generalization of the classical two-dimensional simple man's model which includes ionization probabilities calculated using the imaginary-time method. Finally, we analyze the interference minima for different heteronuclear diatomic molecules and for particular values of the emission angle, laser-field parameters, and internuclear distance. These minima are well fitted with the curve obtained using the derived condition for the two-center destructive interference minima.

Observation of the quantum shift of a backward rescattering caustic by carrier-envelope phase mapping

We study carrier-envelope phase (CEP) -dependent photoelectron momentum distributions (PEMDs) generated by strong-field ionization of Xe atoms with linearly polarized sub-two-cycle near-infrared laser pulses. The PEMDs in the polarization direction end with a cutoff structure whose position and shape are analyzed using the adiabatic theory of rescattering [T. Morishita and O. I. Tolstikhin, Phys. Rev. A 96, 053416 (2017)]. This theory predicts that the cutoff, wherein the PEMD begins to exponentially decay, is located at a quantum caustic that is shifted towards higher energies with respect to the corresponding classical caustic at which long and short backward rescattering trajectories coalesce. The observed position of the cutoff in the PEMDs agrees well with the adiabatic theory, confirming the quantum shift of the caustic. This result paves the way to a more accurate procedure for extracting electron-ion elastic scattering differential cross sections from CEP-dependent cutoffs in strong-field PEMDs.

Disentangling the subcycle electron momentum spectrum in strong-field ionization

The authors use Fourier filtering techniques to eliminate multicycle interference patterns and reveal underlying subcycle structure in photoelectron momentum distributions produced by strong-field ionized atoms and molecules.