Electron Wavepacket Dynamics Research Articles

Recollision dynamics of electron wave packets in high-order harmonic generation

We numerically investigate the dynamics of recollision of an electron in high-order harmonic generation (HHG) for an H atom and a molecular ion $\text{H}_{2}{}^{+}$ using a short (ten optical cycles), and intense $({I}_{0}\ensuremath{\ge}{10}^{14}\text{ }\text{W}/{\text{cm}}^{2})$, $z$-polarized linear laser pulse with wavelength 800 nm by accurately solving the three-dimensional time-dependent Schr\odinger equation. A time-frequency analysis obtained via Gabor transforms is employed to identify electron recollision and recombination times responsible for the generation of harmonics. We find that the HHG spectra are mainly attributed to the recollision of an inner electron wave packet with the parent ion in agreement with the classical recollision model. A time delay of the electron recollision occurs between wave packets in inner and outer regions, near to and far from the parent ion, due to different phase of the acceleration (as well as dipole velocity) of the electron. Inner wave packets at recollision contain mainly short and long trajectories whereas outer wave packets contain only single trajectories. Lower-order harmonics are generated mainly by single recollisions near field extrema, i.e., in strong electric fields whereas higher-order harmonics are generated by double trajectories with different intensities. In the case of $\text{H}_{2}{}^{+}$ at a critical nuclear distance for charge resonance enhanced ionization, we also find that HHG mainly comes from contributions of the inner electron wave packet, but with more complex recollision trajectories due to the presence of more than one Coulomb center. Triple recollision trajectories are shown to occur generally for the latter.

Electron-Wavepacket Reaction Dynamics in Proton Transfer of Formamide

We apply the semiclassical Ehrenfest theory, which provides electron wavepacket dynamics coupled to nuclear motion, to a study of water-assisted proton relay in formamide compared with a forced proton transfer in gas phase, both of which are associated with the tautomerization. We start with the enol (imidic acid) form HO-CH=NH and track its proton transfer process to the keto (amide) form O=CH-NH(2). Identifying the fact that this is indeed a "proton transfer process" rather than hydrogen-atom migration in terms of radical character on the proton, we show a collective quantum flux of electrons, which flows backward against the proton motion. This backward flux compensates the electrons tightly covering the proton, as represented in the Mulliken charge. The enol form formamide is one of the simplest species in the group O=CR(1)-NHR(2), which is a unit of polypeptide. In the gas phase, the nitrogen atom may have a pyramidal structure as in the ammonium molecule; therefore, the C-N bond may allow low barrier rotation along it. This rotation is strongly prohibited by the formation of the double bond C horizontal lineN induced by the proton transfer. Not only the dynamical process of proton transfer itself but also the electronic structures left behind are greatly affected by the presence of water molecule(s) and polar solvents. In discussing the relative stability of the formamide after the proton transfer, the following resonance structures are frequently mentioned, O(-)-CH=N(+)H(2) <--> O=CH-NH(2). Here we address the dynamical manifestation of the resonance structures in terms of our dynamical electron theory.

Wave packet splitting and zitterbewegung in two-dimensional semiconductor structures with spin-orbit coupling

Dynamics of electron wave packets in an asymmetric quantum well in the presence of Rashba spinorbit coupling was analytically and numerically studied. Electron Green’s functions were introduced and the evolution of 1D and 2D wave packets was studied. The effect of packet splitting caused by the presence of two branches with different chiralities in the Rashba Hamiltonian spectrum and zitterbewegung, i.e., packet center’s jitter, was studied. Spatial components of the spin density were calculated. It was shown that the component of the spin density Sy in split parts of the wave packet has opposite signs, and two other spin density components oscillate in space between scattering packets.

Characterization of electron-deficient chemical bonding of diborane with attosecond electron wavepacket dynamics and laser response

We report a theoretical study of non-adiabatic electrons–nuclei coupled dynamics of diborane H 2BH 2BH 2 under several types of short pulse lasers. This molecule is known to have particularly interesting geometrical and electronic structures, which originate from the electron-deficient chemical bondings. We revisit the chemical bonding of diborane from the view point of electron wavepacket dynamics coupled with nuclear motions, and attempt to probe the characteristics of it by examining its response to intense laser fields. We study in the following three aspects, (i) bond formation of diborane by collision between two monoboranes, (ii) attosecond electron wavepacket dynamics in the ground state and first excited state by circularly polarized laser pulse, and (iii) induced fragmentation back to monoborane molecules by linearly polarized laser. The wave lengths of two types of laser field employed are 200 nm (in UV range) and 800 nm (in IR range), and we track the dynamics from hundreds of attoseconds up to few tens of femtoseconds. To this end, we apply the ab initio semiclassical Ehrenfest theory, into which the classical vector potential of a laser field is introduced. Basic features of the non-adiabatic response of electrons to the laser fields is elucidated in this scheme. To analyze the electronic wavepackets thus obtained, we figure out bond order density that is a spatial distribution of the bond order and bond order flux density arising only from the bonding regions, and so on. Main findings in this work are: (i) dimerization of monoboranes to diborane is so efficient that even intense laser is hard to prevent it; (ii) collective motions of electron flux emerge in the central BHHB bonding area in response to the circularly polarized laser fields; (iii) laser polarization with the direction of central two BH bonding vector is efficient for the cleavage of BH 3–BH 3; and (iv) nuclear derivative coupling plays a critical role in the field induced fragmentation dynamics.

Electron flux in molecules induced by nuclear motion

As a general tool for analysis of chemical reactions from the view point of electron wavepacket dynamics, electron flux within a molecule is numerically realized in terms of physically time-dependent electronic wavefunctions given by the semiclassical Ehrenfest theory. These wavefunctions are synchronized with real time motion of molecular nuclei through the nuclear kinematic coupling (nonadiabatic elements). Since the standard quantum flux gives only a null field for a real-valued electronic eigenfunction, we extend the definition of flux such that the essential information of dynamical flow of electrons can be retrieved even from adiabatic electronic wavefunctions calculated in the scheme of the so-called ab initio molecular dynamics.

Non-Born–Oppenheimer electronic and nuclear wavepacket dynamics

A practical quantum theory for unifying electronic and nuclear dynamics, which were separated by the Born-Oppenheimer approximation, is proposed. The theory consists of two processes. Nonadiabatic (quantum) electron wavepacket dynamics on branching (non-Born-Oppenheimer) nuclear paths are first constructed. Since these paths are not the classical trajectories, most of the existing semiclassical theories to generate quantum wavepacket do not work. Therefore, we apply our own developed semiclassical wavepacket theory to these generated non-Born-Oppenheimer paths. This wavepacket is generated based on what we call the action decomposed function, which does not require the information of the so-called stability matrix. Thus, the motion of nuclei is also quantized, and consequently the total wave function is represented as a series of entanglement between the electronic and nuclear wavepackets. In the last half of the article, we show the practice to demonstrate how these independent theories can be unified to give electron-nuclear wavepackets in a two-state model. The wavepackets up to the phases and resultant transition probabilities are compared to the full quantum-mechanical counterparts. It turns out that the lowest level approximation to the wavepacket approach already shows a good agreement with the full quantum quantities. Thus, the present theoretical framework gives a basic method with which to study non-Born-Oppenheimer electronic and nuclear wavepacket states relevant to ultrafast chemical events.

Multichannel coherence in strong-field ionization

Atomic and molecular ions generated by a strong optical laser pulse are not in general in the electronic ground state. The density matrix for such ions is characterized by the electronic quantum-state populations and by the coherences among the electronic quantum states. Nonvanishing coherences signal the presence of coherent electronic wave-packet dynamics in the laser-generated ions. For noble-gas atoms heavier than helium, the most important channels populated via strong-field ionization are the outer-valence single-hole states with a total angular momentum of $j=3/2$ or $j=1/2$. For this case, we develop a time-dependent multichannel theory of strong-field ionization. We derive the ion density matrix and express the hole density in terms of the elements of the ion density matrix. Our wave-packet calculations demonstrate that neon ions generated in a strong optical field (800 nm) are almost perfectly coherent. In strong-field-generated xenon ions, however, the coherence is substantially suppressed.

Dynamics of 2D electron wave packets with different initial spin polarization in 2D structures with spin-orbit coupling

The effect of splitting and zitterbewegung of 2D-electrons wave packets in the presence of Rashba spin-orbit coupling has been investigated. It is shown that nonstandard dynamics of wave packets occurs in the systems where the complete system of eigenfunctions is formed by states with different chirality. The time evolution of wave packets depends on the initial electron spin orientation. It is established that the oscillations of packet centers decay with time. Similar effects were studied by us previously for packets with initial spin orientation perpendicular to the 2D electron gas plane.

Coherent Terahertz Echo of Tunnel Ionization in Gases

We study tunnel ionized electron wave packet dynamics during the initial transition from a gas to a plasma by detecting the terahertz radiation emitted in the process. Experimental and theoretical results show that much of the observed radiation is due to coherent buildup of bremsstrahlung released during the first electron-atom collision. Coherent control of the tunnel ionization process combined with ab initio modeling provides a real-time view of the initial stages of the formation of a laser-induced plasma and allows us to fully understand this important source of terahertz radiation.

Toward non‐Born–Oppenheimer quantum chemistry

AbstractA practical quantum theory for unifying electronic and nuclear dynamics, which were separated by the Born–Oppenheimer approximation, is proposed. The theory consists of two processes. Nonadiabatic (quantum) electron wavepacket dynamics on branching (non‐Born–Oppenheimer) nuclear paths are first constructed. Since these paths are not the classical trajectories, most of the existing semiclassical theories to generate quantum wavepacket do not work. Therefore, we apply our own developed semiclassical wavepacket theory to these generated non‐Born–Oppenheimer paths. This wavepacket is generated based on what we call the action decomposed function, which does not require the information of the so‐called stability matrix. Thus, the motion of nuclei is also quantized, and consequently the total wavefunction is represented as a series of entanglement between the electronic and nuclear wavepackets. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2009

Wave packet dynamics in a monolayer graphene

The dynamics of charge particles described by Gaussian wave packet in monolayer graphene is studied analytically and numerically. We demonstrate that the shape of wave packet at arbitrary time depends on correlation between the initial electron amplitudes ${\ensuremath{\psi}}_{1}(\stackrel{P\vec}{r},0)$ and ${\ensuremath{\psi}}_{2}(\stackrel{P\vec}{r},0)$ on the sublattices $A$ and $B$ correspondingly (i.e., pseudospin polarization). For the transverse pseudospin polarization the motion of the center of wave packet occurs in the direction perpendicular to the average momentum ${\stackrel{P\vec}{p}}_{0}=\ensuremath{\hbar}{\stackrel{P\vec}{k}}_{0}$. Moreover, in this case the initial wave packet splits into two parts moving with opposite velocities along ${\stackrel{P\vec}{p}}_{0}$. If the initial direction of pseudospin coincides with average momentum the splitting is absent and the center of wave packet is displaced at $tg0$ along the same direction. The results of our calculations show that all types of motion experience Zitterbewegung. Besides, depending on initial polarization the velocity of the packet center may have the constant component ${v}_{c}=uf(a)$, where $u\ensuremath{\approx}{10}^{8}\text{ }\text{cm}/\text{s}$ is the Fermi velocity and $f(a)$ is a function of the parameter $a={k}_{0}d$ ($d$ is the initial width of wave packet). As a result, the direction of the packet motion is determined not only by the orientation of the average momentum, but mainly by the phase difference between the up and low components of the wave functions. Similar peculiarities of the dynamics of two-dimensional electron wave packet connected with initial spin polarization should take place in the semiconductor quantum well under the influence of the Rashba spin-orbit coupling.

Effects of intense femtosecond pumping on ultrafast electronic-vibrational dynamics in molecular systems with relaxation

We investigate the influence of strong femtosecond optical pulses on the ultrafast dynamics of molecular systems. The study is based on a series of generic molecular models of increasing complexity, which incorporate multiple and mutually coupled electronic states, electronic-vibrational interaction, and vibrational relaxation. The influence of vibrational relaxation is treated using multilevel Redfield theory. Comparisons to benchmark results of the multilayer multiconfiguration time-dependent Hartree (ML-MCTDH) method demonstrate the validity of the field-free implementation of Redfield theory employed in this work for weak system-bath interaction. The calculated electronic population and vibrational wave-packet dynamics demonstrate the intricate interplay of strong-field excitation, laser-induced Rabi oscillations, electronic interaction, vibronic coupling, and dissipation. In particular, we show that the interaction with a strong laser pulse may result in pronounced coherent vibrational motion in a dissipative system, even for laser pulses that are longer than the vibrational period. Furthermore, vibrational relaxation in combination with strong laser pulse excitation can lead to revivals of the electronic population after the excitation pulse is over.

Phase-space averaging and natural branching of nuclear paths for nonadiabatic electron wavepacket dynamics

We propose a simple and tractable method to treat quantum electron wavepacket dynamics that nonadiabatically couples with "classical" nuclear motions in mixed quantum-classical representation. The electron wavepacket is propagated inducing electronic-state mixing along our proposed paths. It has been shown in our previous studies that classical force working on nuclei in a nonadiabatic region is represented in a matrix form (called the force matrix), and the solutions of the Hamilton canonical equations of motion for nuclei based on this force matrix give rise to a cascade of infinitely many branching paths when solved simultaneously with electronic-state mixing. As a tractable approximation to these rigorous solutions, we here devise a method to provide much simpler nonadiabatic paths: (i) extract one or a few number of representative paths by taking an average over the paths in phase space (not averaging over the forces) that should be otherwise undergo the fine branching. (ii) After the nonadiabatic coupling becomes sufficiently small, let these paths naturally branch by running them with their own individual eigenforces (the eigenvalues of the force matrix). Since the eigenforces coincide with the forces of adiabatic potential energy surfaces in the limit of zero nonadiabatic coupling, these branching paths eventually run on one of possible adiabatic potential energy surfaces, converging to a classical path (Born-Oppenheimer path). The paths thus created are theoretically satisfactory in that they realize the coherent mixing of electronic states in the manner of quantum entanglement and yet eventually become consistent with the Born-Oppenheimer classical trajectories. We test the present method numerically with the use of two- and three-state systems that are extracted from ab initio calculations for the excited states of LiH molecule.

Modelling attosecond probing of electron wavepacket dynamics in non-aligned molecules

We propose and simulate an attosecond pump–probe scheme applied to an ensemble of non-aligned (or partially aligned) diatomic molecules. Non-dissociative dynamics initiated by an ultraviolet pump pulse are probed by photoionizing the molecule with an attosecond extreme-ultraviolet probe pulse. Photoelectron spectra recorded for different delays between the pulses exhibit signatures of streaking and sub-femtosecond quantum beating. If more than one vibrational state is excited, the nuclear motion modulates the visibility of quantum interference effects.

Nonadiabatic electron wavepacket dynamics of molecules in an intense optical field: An ab initio electronic state study

A theory of quantum electron wavepacket dynamics that nonadiabatically couples with classical nuclear motions in intense optical fields is studied. The formalism is intended to track the laser-driven electron wavepackets in terms of the linear combination of configuration-state functions generated with ab initio molecular orbitals. Beginning with the total quantum Hamiltonian for electrons and nuclei in the vector potential of classical electromagnetic field, we reduce the Hamiltonian into a mixed quantum-classical representation by replacing the quantum nuclear momentum operators with the classical counterparts. This framework gives equations of motion for electron wavepackets in an intense laser field through the time dependent variational principle. On the other hand, a generalization of the Newtonian equations provides a matrix form of forces acting on the nuclei for nonadiabatic dynamics. A mean-field approximation to the force matrix reduces this higher order formalism to the semiclassical Ehrenfest theory in intense optical fields. To bring these theories into a practical quantum chemical package for general molecules, we have implemented the relevant ab initio algorithms in it. Some numerical results in the level of the semiclassical Ehrenfest-type theory with explicit use of the nuclear kinematic (derivative) coupling and the velocity form for the optical interaction are presented.

FFT-split-operator code for solving the Dirac equation in 2+1 dimensions

The main part of the code presented in this work represents an implementation of the split-operator method [J.A. Fleck, J.R. Morris, M.D. Feit, Appl. Phys. 10 (1976) 129–160; R. Heather, Comput. Phys. Comm. 63 (1991) 446] for calculating the time-evolution of Dirac wave functions. It allows to study the dynamics of electronic Dirac wave packets under the influence of any number of laser pulses and its interaction with any number of charged ion potentials. The initial wave function can be either a free Gaussian wave packet or an arbitrary discretized spinor function that is loaded from a file provided by the user. The latter option includes Dirac bound state wave functions. The code itself contains the necessary tools for constructing such wave functions for a single-electron ion. With the help of self-adaptive numerical grids, we are able to study the electron dynamics for various problems in 2+1 dimensions at high spatial and temporal resolutions that are otherwise unachievable. Along with the position and momentum space probability density distributions, various physical observables, such as the expectation values of position and momentum, can be recorded in a time-dependent way. The electromagnetic spectrum that is emitted by the evolving particle can also be calculated with this code. Finally, for planning and comparison purposes, both the time-evolution and the emission spectrum can also be treated in an entirely classical relativistic way. Besides the implementation of the above-mentioned algorithms, the program also contains a large C++ class library to model the geometric algebra representation of spinors that we use for representing the Dirac wave function. This is why the code is called “Dirac++”. Program summary Program title: Dirac++ or (abbreviated) d++ Catalogue identifier: AEAS_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AEAS_v1_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 474 937 No. of bytes in distributed program, including test data, etc.: 4 128 347 Distribution format: tar.gz Programming language: C++ Computer: Any, but SMP systems are preferred Operating system: Linux and MacOS X are actively supported by the current version. Earlier versions were also tested successfully on IRIX and AIX Number of processors used: Generally unlimited, but best scaling with 2–4 processors for typical problems RAM: 160 Megabytes minimum for the examples given here Classification: 2.7 External routines: FFTW Library [3,4], Gnu Scientific Library [5], bzip2, bunzip2 Nature of problem: The relativistic time evolution of wave functions according to the Dirac equation is a challenging numerical task. Especially for an electron in the presence of high intensity laser beams and/or highly charged ions, this type of problem is of considerable interest to atomic physicists. Solution method: The code employs the split-operator method [1,2], combined with fast Fourier transforms (FFT) for calculating any occurring spatial derivatives, to solve the given problem. An autocorrelation spectral method [6] is provided to generate a bound state for use as the initial wave function of further dynamical studies. Restrictions: The code in its current form is restricted to problems in two spatial dimensions. Otherwise it is only limited by CPU time and memory that one can afford to spend on a particular problem. Unusual features: The code features dynamically adapting position and momentum space grids to keep execution time and memory requirements as small as possible. It employs an object-oriented approach, and it relies on a Clifford algebra class library to represent the mathematical objects of the Dirac formalism which we employ. Besides that it includes a feature (typically called “checkpointing”) which allows the resumption of an interrupted calculation. Additional comments: Along with the program's source code, we provide several sample configuration files, a pre-calculated bound state wave function, and template files for the analysis of the results with both MatLab and Igor Pro. Running time: Running time ranges from a few minutes for simple tests up to several days, even weeks for real-world physical problems that require very large grids or very small time steps.

Microscopic analysis of the optical and electronic properties of semiconductor photonic‐crystal structures

AbstractThe combination of dielectric photonic crystals and semiconductor nanostructures makes it possible to design many important aspects of the optoelectronic system response. A spatially‐varying dielectric environment induces modifications of the transversal and the longitudinal components of the electromagnetic field which have to be included in the self‐consistent microscopic analysis of the optical properties of hybrid semiconductor photonic‐crystal structures. In this paper, the development of a semiclassical microscopic theory is reviewed. Whereas the classical electromagnetic field is described at the level of Maxwell's equations, a full many‐body quantum theory is used for the interacting electronic excitations in the semiconductor material. Relevant examples of the numerical solutions of the resulting Maxwell semiconductor Bloch equations are presented showing, e.g., characteristic modifications of excitonic absorption spectra, the spatio‐temporal dynamics of electronic wave packets, as well as an increase of the optical gain in properly designed device geometries. (© 2007 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Efficient generation of an isolated single-cycle attosecond pulse

A method for efficiently generating an isolated single-cycle attosecond pulse is proposed. It is shown that the ultraviolet (UV) attosecond (as) pulse can be utilized as a robust tool to control the dynamics of electron wave packets (EWPs). By adding a UV attosecond pulse to an infrared (IR) few-cycle pulse at a proper time, only one return of the EWP to the parent ion is selected to effectively contribute to the harmonics; then, an isolated two-cycle $130\text{\ensuremath{-}}\mathrm{as}$ pulse with a bandwidth of $45\phantom{\rule{0.3em}{0ex}}\mathrm{eV}$ is obtained. After complementing the chirp, an isolated single-cycle attosecond pulse with a duration less than $100\phantom{\rule{0.3em}{0ex}}\mathrm{as}$ seems achievable. In addition, the contribution of the quantum trajectories can be selected by adjusting the delay between the IR and UV fields. Using this method, the harmonic and attosecond pulse yields are efficiently enhanced in contrast to the scheme [G. Sansone et al., Science 314, 443 (2006)] using a few-cycle IR pulse in combination with the polarization gating technique.

Probing valence electronic wave-packet dynamics by all x-ray stimulated Raman spectroscopy: A simulation study

The femtosecond impulsive stimulated x-ray Raman signal from quinolinol is simulated and analyzed using the doorway-window representation of pump-probe spectroscopy. A valence electronic wave packet prepared by the pump, tuned on resonance with a given core hole, is localized in the vicinity of a selected atom and probed by a window localized on a different atom, selected by the probe pulse. All valence electronic states within the pulse bandwidths can be observed with high spatial and temporal resolution by monitoring the variation of the signal with the delay between the pulses. Natural orbital representation of the reduced single-electron density matrix is used to visualize the dynamics of the valence wave packet, described as a linear combination of determinants made of occupied and unoccupied Kohn-Sham orbitals.

Electron wave-packet dynamics in a relativistic electromagnetic field: 3-D analytical approximation

A solution to the Klein Gordon equation for a laser-driven electron is constructed from a superposition of Volkov states. The time-and space-dependent three-dimensional superposition integral can be evaluated analytically for an initial Gaussian momentum distribution when the expression for relativistic energy is expanded in a Taylor series over the scaled initial momenta. The solution preserves many complicated wave-packet dynamics in a strong field, including so-called wave-packet shearing and the formation of multiple peaks when the wave packet spreads to the scale of the driving-field wavelength. The range of applicability of the solution applies to much of the parameter space accessible by current intense ultra-short laser systems.