Electron Wave Packet Research Articles

Factorization of high-order-harmonic-generation yields in impurity-doped materials

We present a theoretical investigation of high-harmonic generation (HHG) from impurity-doped materials using the time-dependent Schr\"odinger equation (TDSE) approach. We demonstrate the factorization of HHG yields as a product of an electron wave packet and the recombination cross section, in analogy to HHG from atoms and molecules in the gas phase. Furthermore, we show that the quantitative rescattering model based on this factorization accurately reproduces the TDSE results. This opens up new possibilities to study impurities in materials using the available techniques from strong-field physics.

On the preservation of coherence in the electronic wavepacket of a neutral and rigid polyatomic molecule

We present various types of reduced models including five vibrational modes and three electronic states for the pyrazine molecule in order to investigate the lifetime of electronic coherence in a rigid and neutral system. Using ultrafast optical pumping in the ground state (11Ag), we prepare a coherent superposition of two bright excited states, 11B2u and 11B1u, and reveal the effect of the nuclear motion on the preservation of the electronic coherence induced by the laser pulse. More specifically, two aspects are considered: the anharmonicity of the potential energy surfaces and the dependence of the transition dipole moments (TDMs) with respect to the nuclear coordinates. To this end, we define an ‘ideal model’ by making three approximations: (i) only the five totally symmetric modes move, (ii) which correspond to uncoupled harmonic oscillators, and (iii) the TDMs from the ground electronic state to the two bright states are constant (Franck–Condon approximation). We then lift the second and third approximations by considering, first, the effect of anharmonicity, second, the effect of coordinate-dependence of the TDMs (first-order Herzberg–Teller contribution), third, both. Our detailed numerical study with quantum dynamics is meant to be realistic for pyrazine over about 20 femtoseconds, and was further extended so as to probe the effect of such approximations on a model system. We show that long-term revivals of the electronic coherence persist up to the picosecond time range even for the most realistic model.

Spin–orbit-resolved spectral phase measurements around a Fano resonance

We apply a spectrally-resolved electron interferometry technique to the measurement of the spectral phase in the vicinity of the 3s13p64p Fano resonance of argon. We show that it allows disentangling the phases of the two nearly-overlapping electron wavepackets corresponding to different spin–orbit final states. Using simple assumptions, it is possible to process the experimental data and numerically isolate each component in a self-consistent manner. This in turn allows reconstructing the autoionization dynamics of the dominant channel.

Atomic photoionization dynamics in ultrashort cycloidal laser fields

We present numerical simulations of ultrafast multiphoton ionization dynamics in a two-dimensional atomic model driven by co- and counterrotating circularly polarized single-color and bichromatic carrier envelope phase (CEP) stable ultrashort laser pulse sequences. Taking into account phase variations due to CEP fluctuations and the Gouy phase, our results accurately reproduce recently measured photoelectron momentum distributions [Pengel et al., Phys. Rev. Lett. 118, 053003 (2017), Phys. Rev. A 96, 043426 (2017); Kerbstadt et al., Nat. Comm. 10, 685 (2019), Adv. Phys. X 4, 1672583 (2019)]. The time evolution of the complex-valued electron wave function in coordinate and momentum space is calculated to study the bound state- and the vortex formation dynamics. The non-vanishing azimuthal probability current density proves the vortex nature of electron wave packets with odd-numbered rotational symmetry. Their angular momentum expectation value assumes half-integer values of 3.5 (corotating) and 0.5 (counterrotating). Knowledge of the wave function allows us to analyze the photoionization dynamics and to validate the physical pictures proposed in previous experimental studies. As an outlook, we investigate how electron vortices develop from the multiphoton- to the tunnel regime with increasing laser intensity.

Zitterbewegung-mediated RKKY coupling in topological insulator thin films

The dynamics of itinerant electrons in topological insulator (TI) thin films is investigated using a multi-band decomposition approach. We show that the electron trajectory in the 2D film is anisotropic and confined within a characteristic region. Remarkably, the confinement and anisotropy of the electron trajectory are associated with the topological phase transition of the TI system, which can be controlled by tuning the film thickness and/or applying an in-plane magnetic field. Moreover, persistent electron wavepacket oscillation can be achieved in the TI thin film system at the phase transition point, which may assist in the experimental detection of the jitter motion (Zitterbewegung). The implications of the microscopic picture of electron motion in explaining other transport-related effects, e.g., electron-mediated RKKY coupling in the TI thin film system, are also discussed.

Coherent control for qubit state readout

Short pulses from mode-locked lasers can produce background-free atomic fluorescence by allowing temporal separation of the prompt incidental scatter from the subsequent atomic emission. We use this to improve our quantum state detection of optical-frequency and electron-shelved trapped ion qubits by more than two orders of magnitude. For direct detection of qubits defined on atomic hyperfine structure, however, the large bandwidth of short pulses is greater than the hyperfine splitting, and repeated excitation is not qubit state selective. Here, we show that the state resolution needed for projective quantum measurement of hyperfine qubits can be recovered by applying techniques from coherent control to the orbiting valence electron of the queried ion. We demonstrate electron wavepacket interference to allow readout of the original qubit state using broadband pulses, even in the presence of large amounts of background laser scatter.

Collisional interferometry of levitons in quantum Hall edge channels at ν=2

We consider a Hong-Ou-Mandel interferometer for Lorentzian voltage pulses applied to Quantum Hall edge channels at filling factor $\nu=2$. Due to inter-edge interactions, the injected electronic wave-packets fractionalize before partitioning at a quantum point contact. Remarkably enough, differently from what theoretically predicted and experimentally observed by using other injection techniques, we demonstrate that, when the injection occurs through time-dependent voltage pulses (arbitrarily shaped), the Hong-Ou-Mandel noise signal always vanishes for a symmetric device, and that a mismatch in the distances between the injectors and the point of collision is needed in order to reduce the visibility of the dip. We also show that, by properly tuning these distances or by applying different voltages on the two edge channels in each arm of the interferometer, it is possible to estimate the intensity of the inter-edge interaction. The voltage pulses are chosen of the Lorentzian type because of their experimental relevance.

Photoelectron holography of the H2+ molecule

We investigate the photoelectron spectrum of the H2+ target induced by few-cycle XUV laser pulses using first principle calculations. In the photoelectron spectrum, by performing calculations for different internuclear separations, we investigate how the structure of the target is influencing the spatial interference pattern. This interference pattern is created by the coherent superposition of electronic wave packets emitted at the same time, but following different paths. We find that the location of the interference minima in the spectra is dominantly determined by the target’s ionization energy, however, by comparing the H2+ results with model calculations with spherically symmetric potentials, clear differences were observed for the molecular potential relative to the central potentials. Next to the main feature (spatial interference) we have also identified the traces of the two-center interference in the photoelectron spectrum, however, these were mainly washed out due to the complex electronic wave packet dynamics that occurs during the interaction with the considered laser field.Graphical abstract

Principle and application of two-color attoclock

Time-resolving ultrafast electron dynamics driven by lights in atoms and molecules is one of the central topics in both science and technology, as a prerequisite for controlling the electron dynamics and creating unprecedented functions to serve for human. Photoionization by intense lasers is the fundamental interaction between lights and atoms. For strong-field ionization, the amplitude and the phase of the ionized electron wave packet could give direct insights of electron dynamics in the classically forbidden, sub-Coulomb-barrier region. Electron tunneling through the suppressed Coulomb potential and traveling in continuum leave fingerprints not only on the amplitude of the electron wave packet, but also on its phase distribution. Recently, benefiting from that the laser pulse duration approaches to the natural time scale of intra-atomic electron dynamics, one was allowed to image the amplitude and the phase of an electron wave function on the attosecond scale using new metrologies combined with attosecond light pulses, such as attosecond streaking camera and the technique of reconstruction of attosecond beating by interference of two-photon transition. Alternatively, attoclock is based on the angle-time mapping principle in one-color circularly polarized femtosecond light fields, without resorting to the attosecond light pulses. It is originally introduced as a powerful method to address the issue of tunneling time delays with the attosecond time resolution in strong-field physics. In recent years, due to the flexible manipuility and the rich diversity of the two-color femtosecond light fields, the two-color fields have gradually become an important method to control the ultrafast electron dynamics driven by intense lasers. The two-color attoclock combined with attoclock and two-color fields, then attracts the increasing attention, and hence becomes the research front of strong-field physics. In this review, we will introduce two kinds of the two-color attoclock. First, based on the co-rotating two-color light fields, we demonstrate a novel time-resolved photoelectron interferometry, i.e., double-hand attoclock. Using this technique, one can extract the phase information and the amplitude information of the ionized electron wavepackets by strong-field ionization with intense light pulses. From the extracted time-resolved distributions of electron phase and amplitude, we reveal the electron sub-barrier dynamics with attosecond time resolution. As to the second two-color attoclock, we study the old and controversial question, i.e., tunneling time delay. Introducing a linearly polarized second harmonic field to calibrate the attoclock constructed by a circularly polarized fundamental-frequency light field, we demonstrate an improved attoclock. The improved attoclock can measure the photoelectron offset angle more accurately, which corresponds to tunneling time delay. Theoretically, we also demonstrate that the instantaneous tunneling picture and the Wigner delay picture for tunneling, which seem contradictory with each other, can be unified within the same theoretical framework, i.e., strong-field approximation theory. For the result of the improved attoclock, the two tunneling pictures can both agree with the experiment. To conclude, the two-color attoclock possesses not only very high time resolution but also good space resolution. Therefore, the two-color attclock has a very exciting potential in molecular orbital imaging, tunneling ionization of molecules and so on.

Continuum electron giving birth to terahertz emission

The origin of terahertz (THz) generation in a gas-phase medium is still in controversy, although the THz sources have been applied across many disciplines. Herein, the THz generation in a dual-color field is investigated experimentally by precisely controlling the relative phase and polarization of dual-color lasers, where the accompanying third-harmonic generation is employed for in situ determination of the relative phase up to sub-wavelength accuracy. Joint studies with the strong approximation (SFA) theory reveal that the continuum-continuum (CC) transition within an escaped electron wave packet in the single atom gives birth to THz emission, without the necessity of considering the plasma effect. Meanwhile, we develop the analytic form from SFA-based CC description, which is able to reproduce and decompose the classical photocurrent model from the viewpoint of microscopic quantum theory, establishing the quantum-classical correspondence and bringing a novel insight into the mechanism of THz generation. Present studies leave open the possibility for probing the ultrafast dynamics of continuum electrons and a new dimension for the study of THz-related science and methodology.

Complete phase retrieval of photoelectron wavepackets

Coherent, broadband pulses of extreme ultraviolet light provide a new and exciting tool for exploring attosecond electron dynamics. Using photoelectron streaking, interferometric spectrograms can be generated that contain a wealth of information about the phase properties of the photoionization process. If properly retrieved, this phase information reveals attosecond dynamics during photoelectron emission such as multielectron dynamics and resonance processes. However, until now, the full retrieval of the continuous electron wavepacket phase from isolated attosecond pulses has remained challenging. Here, after elucidating key approximations and limitations that hinder one from extracting the coherent electron wavepacket dynamics using available retrieval algorithms, we present a new method called absolute complex dipole transition matrix element reconstruction (ACDC). We apply the ACDC method to experimental spectrograms to resolve the phase and group delay difference between photoelectrons emitted from Ne and Ar. Our results reveal subtle dynamics in this group delay difference of photoelectrons emitted form Ar. These group delay dynamics were not resolvable with prior methods that were only able to extract phase information at discrete energy levels, emphasizing the importance of a complete and continuous phase retrieval technique such as ACDC. Here we also make this new ACDC retrieval algorithm available with appropriate citation in return.

Revealing the effect of atomic orbitals on the phase distribution of an ionizing electron wave packet with circularly polarized two-color laser fields.

We theoretically study the interference of photoelectrons released from atomic p± orbitals in co-rotating and counter-rotating circularly polarized two-color laser pulses consisting of a strong 400-nm field and a weak 800-nm field. We find that in co-rotating fields the interference fringes in the photoelectron momentum distributions are nearly the same for p± orbitals, while in counter-rotating fields the interference fringes for p+ and p- orbitals oscillate out of phase with respect to the electron emission angle. The simulations based on the strong-field approximation show a good agreement with the numerical solutions of the time-dependent Schrödinger equation. We find that different phase distributions of the electron wave packets emitted from p+ and p- orbitals can be easily revealed by the counter-rotating circularly polarized two-color laser fields. We further show that the photoelectron interference patterns in the circularly polarized two-color laser fields record the time differences of the electron wave packets released within an optical cycle.

The comparison of high-order harmonic generation of H in a circularly polarized laser pulse combined with a terahertz field and with a static electric field

By solving a two-dimensional time-dependent Schrödinger equation we investigate high-order harmonic generation (HHG) for the molecular ion in a circularly polarized laser pulse (CPLP). A comparison is made between a HHG in a CPLP combined with a terahertz (THz) field and with a static electric field. When the CPLP is combined with a static electric field, it is observed that a double-plateau structure appears, and the harmonic spectrum presents large modulation. With the CPLP with a THz field, a continuum spectrum with a spectral width of about 137 eV from 102 eV to 239 eV can be achieved. The HHG process is studied using the semi-classical three-step model and time-frequency analysis. Our studies show that only one trajectory contributes to HHG when a THz field is added. Furthermore, we present the temporal evolution of the probability density of an electron wave packet, which shows the recombination of electrons at different times.

Trapping and Counting Ballistic Nonequilibrium Electrons.

We demonstrate the trapping of electrons propagating ballistically at far-above-equilibrium energies in GaAs/AlGaAs heterostructures in high magnetic field. We find low-loss transport along a gate-modified mesa edge in contrast to an effective decay of excess energy for the loop around a neighboring, mesa-confined node, enabling high-fidelity trapping. Measuring the full counting statistics via single-charge detection yields the trapping (and escape) probabilities of electrons scattered (and excited) within the node. Energetic and arrival-time distributions of captured electron wave packets are characterized by modulating tunnel barrier transmission.

Experimental evidence for terahertz emission of continuum electrons in the dual-color laser field.

Terahertz (THz) wave generation (TWG) in a dual-color laser is investigated with joint measurements between THz and third-harmonic generation, where the relative phase delay of dual-color fields is determined in situ in sub-wavelength accuracy, allowing for the clarification of the TWG mechanism in a direct comparison with various theoretical predictions. The delay- and polarization-dependent experiment validates that the continuum-continuum transition within the escaped electron wavepacket in the single atom gives birth to THz emission, while the bound energetic level does not contribute to TWG. TWG from atoms and molecules would provide an all-optical, vacuum-free, and ultrafast tool to record the spatiotemporal evolution of tunneling electron wavepackets.

Revealing orbital dependent quantum interference of O2 underlying channel-resolved strong-field spectroscopies

We study the molecular orbital dependent tunneling ionization and quantum interference during laser-driven rescattering electron wavepacket of O2 by performing the channel-resolved photoion and photoelectron measurements. The shifted above threshold ionization electron spectra demonstrate that direct ionization of HOMO-2 to the excited cation state contributes significantly to the generation of fragment ions, while the parent ions are generated from tunneling ionization of highest occupied molecular orbital (HOMO). Furthermore, the destructive and constructive interferences between different rescattering orbits from strong-field ionization of HOMO and HOMO-2 are confirmed by the measured channel and angle-resolved photoelectron momentum distribution.

Looking into strong-field dynamics

The combination of high-order harmonic polarimetry and sub-cycle control of electronic trajectories gives insight into the birth of attosecond electronic wave packets in molecules.

Nonlinear THz radiation investigated by quantum simulations.

Terahertz (THz) waves can be generated by the nonlinear interaction between ultrashort laser pulses and air. The semiclassical photocurrent model is widely used. It is simple, but neglects the quantum effects. Some theoretical works are based on solving the time-dependent Schrödinger equation. However, it meets the difficulty of prohibitively large boxes in long-time evolution. Here we adopted the wave-function splitting algorithm to fully contain the information of photoelectrons. The contributions of the excited states and interference effects in electron wavepackets to THz radiation are studied numerically. We also theoretically investigated the THz generation from nitrogen molecules in a biased electric field. It is found that the THz yield enhancement as a function of the static field strength in experiments can be reproduced well by our method. In addition, the restriction of wavelength and phase difference in the two-color laser fields is less strict in the presence of the static field.

Single-electron quantum dynamics in high-harmonic generation spectrum from LiH molecule: Analysis of potential energy surfaces for electrons constructed from a model of localized Gaussian wave packets with valence-bond spin-coupling.

A high-harmonic generation (HHG) spectrum from a LiH molecule induced by an intense laser pulse is computed and analyzed with potential energy surfaces for electron motion (ePES) constructed from a model of localized electron wave packets with valence-bond spin-coupling. The molecule has two valence ePES with binding energies of 0.39 hartree and 1.1 hartree. The HHG spectrum from an electron dynamics on the weaker bound valence ePES, virtually assigned to Li 2s, exhibits a dominant peak at the first harmonic without plateau and cutoff. This compares with the free electron spectrum under an oscillating laser field and is comprehensive with the shape and depth of the ePES. The other valence ePES, assigned to H 1s, is deeper bound such that the overall profile of the wave function is well approximated by a Gaussian of the width comparable to the Li-H bond length. However, a small fraction, less than 10-3, of the probability density amplitude tunnels out from the bound potential with high wave number and spreads over tens of nm with parts recombining to the molecule due to the laser field oscillation. This minor portion of the electronic wave function is the major origin of the HHG extending up to 50 harmonic orders. Nonlinear dynamics within the potential well induced by the laser field oscillation also contributes to the HHG up to 30 harmonic orders.

Time delays from one-photon transitions in the continuum

Attosecond photoionization time delays reveal information about the potential energy landscape that an outgoing electron wavepacket probes upon ionization. In this study, we experimentally quantify the dependence of the time delay on the angular momentum of the liberated photoelectrons. For this purpose, we resolved electron quantum-path interference spectra in energy and angle using a two-color attosecond pump–probe photoionization experiment in helium. A fitting procedure of the angle-dependent interference pattern allows us to disentangle the relative phase of all four quantum pathways that are known to contribute to the final photoelectron signal. In particular, we resolve the dependence on angular momentum of the delay of one-photon transitions between continuum states, which is an essential and universal contribution to the total photoionization delay observed in attosecond pump–probe measurements. For such continuum–continuum transitions, we measure a delay between outgoing s and d electrons as large as 12 attoseconds, close to the ionization threshold in helium. Both single-active-electron and first-principles ab initio simulations confirm this observation for helium and hydrogen, demonstrating the universality of the observed delays.