Simple Man's Model Research Articles

Strong-field processes induced by an ultrashort linearly polarized pulse with two carrier frequencies

We perform a detailed semianalytical study of the characteristics of strong-field processes induced by an ultrashort linearly polarized pulse with two carrier frequencies. We show that the photoelectron spectra depend to a great extent on the absolute phases of the field components and on the duration of the pulse. By using the saddle-point method, we show that this dependence can be exploited to control the electron dynamics in the laser field. Besides the photoelectron spectra, we also investigate the spectra of high-order harmonics. Again, we find a strong dependence on the absolute phases and the duration of the pulse. We present the spectra using a false color scale in the absolute phase-harmonic energy plane, and show that particular regions in this plane can be assessed using the simple man's model founded on the classical solution of the Newton equation of motion. Finally, we investigate the symmetry properties of the photoelectron and high-order harmonic spectra with respect to the transformation which includes the change of the absolute phase, and compare them with those valid for a long driving pulse with a flat envelope. Published by the American Physical Society 2024

Influence of initial tunneling step on the return energy of high-order harmonic generation

To investigate high-order harmonic generation in a monochromatic laser field, we derive an analytical expression for the return energy of an electron as a function of the time interval between ionization and return. We then expand the expression for kinetic energy to second order with respect to the Keldysh parameter $\ensuremath{\gamma}$. In this expansion, the zero-order term is the return energy in the simple man model and the second-order term corresponds to corrections to this model. The origin of this additional kinetic energy is frequently attributed to the nonzero exit of the initial tunneling step. Here, we show that this commonly used picture is incomplete. We present a framework to fully understand the additional kinetic energy as resulting from additive contributions of zero-order and second-order velocities. Our results show that the nonzero velocity of the initial tunneling step has a quantifiable effect on the cutoff energy measured in high harmonic generation (HHG). This opens the door to experimentally addressing the question of the initial electron velocity at the tunnel exit, with important implications for the correct calibration of the attoclock, as well as our interpretation of the strong field-ionization process more broadly.

High-order harmonic generation by two linearly polarized laser fields with an arbitrary angle between their polarization axes

High-order harmonic generation by two linearly polarized fields that enclose an arbitrary crossing angle $\ensuremath{\theta}$ is investigated using the strong-field approximation. We investigate the combinations with frequency ratios $s/r=2$ and 3 with integer $r$ and $s$. In the former case, the emitted harmonics are elliptically polarized unless $\ensuremath{\theta}={0}^{\ensuremath{\circ}}$ or ${90}^{\ensuremath{\circ}}$, while in the latter they are always elliptically polarized except for $\ensuremath{\theta}={0}^{\ensuremath{\circ}}$. The possibility to control the intensity and ellipticity of the emitted harmonics by variation of the relative phase $\ensuremath{\varphi}$ between the field components and the crossing angle $\ensuremath{\theta}$ is explored. There are regions with large harmonic ellipticity and appreciable intensity in the relative-phase--harmonic-order plane and they are more extended for values of the crossing angle closer to ${90}^{\ensuremath{\circ}}$. The overall shape of the spectra (smooth, oscillatory, or erratic) for different values of $\ensuremath{\theta}$ and $\ensuremath{\varphi}$ is explained using the saddle-point method and quantum-orbit theory. The simple-man model is utilized to assess the regions with large harmonic intensity and to predict the position of the cutoff. For the frequency ratio $s/r=2$, the harmonic ellipticity is significant even for extremely small deviations of the crossing angle $\ensuremath{\theta}$ from ${90}^{\ensuremath{\circ}}$.

Subcycle time-resolved nondipole dynamics in tunneling ionization

The electron nondipole dynamics in tunneling ionization in an elliptically polarized laser field is investigated theoretically using a relativistic Coulomb-corrected strong-field approximation (SFA) based on the eikonal approximation of the Klein-Gordon equation. We calculate attoclock angle-resolved light-front momentum distributions at different ellipticities of the laser field in quasistatic and nonadiabatic regimes and analyze them with an improved Simpleman model. The nondipole correlations between longitudinal and transverse momentum components are examined. Deviations of the photoelectron momentum distribution calculated via SFA with respect to the available experimental results as well as with the improved Simpleman model are discussed and interpreted in terms of nonadiabatic as well as Coulomb effects in the continuum and under-the-barrier. The favorable prospects of an experimental observation are discussed.

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.

Rescattering time-energy analysis of high-order above-threshold ionization in few-cycle laser fields

A Wigner distributionlike function based on the improved strong-field approximation theory is proposed to calculate the rescattering time-energy distribution (RTED) of high-energy photoelectrons of atomic above-threshold ionization process in few-cycle laser fields with different frequencies. The RTED shows bell-like stripes and the outermost stripe is compared with semiclassical results given by the simple-man model with consideration of different positions of tunnel exit and different initial longitudinal momenta. Analysis indicates the existence of the tunnel exit. However, though it shifts farther away from the core with decreasing frequency, the position of the tunnel exit is significantly less than the prediction by adiabatic theory even for the low-frequency case which is well in the tunneling regime. Our results also imply that the effect of the tunnel exit is more important than that of the initial longitudinal momentum at the tunnel exit for the backward-scattering electrons. Moreover, the inner stripe structures in the RTED are attributed to interference between electrons with the same final energy emitted at different ionization times.

Simple man model in the Heisenberg picture

Describing the ionization of an atom exposed to a strong laser field entails computationally expensive quantum simulations based on the numerical solutions of the time-dependent Shrödinger equation. The well-known Simple Man Model provides a qualitatively accurate description of the ionization process. Here, we propose a quantum generalization of the physical picture given by the Simple Man Model. We describe an approximate solution to the Heisenberg operator equations of motion for an atom in a laser field. We provide justification of this generalization and test its validity by applying it to calculate the coordinate and velocity autocorrelation functions. Both our model and results of the ab initio numerical calculations show distinct types of correlations due to different types of electron’s motion providing insight into the strong field ionization dynamics.

Wavelength scaling of atomic nonsequential double ionization in intense laser fields

Experimental and theoretical investigations of the wavelength dependence of the nonsequential double ionization (NSDI) process of the xenon atom in intense laser fields are reported. The observed wavelength dependence of the ratio Xe2+/Xe+ deviates significantly from the prediction of the three-step model of ionization (sometimes called the simple-man model), both in the slope of the overall decrease with increasing wavelength and in the existence of some pronounced humps. A semiclassical model calculation reproduces the changing slope of the overall decrease, which is due to the interplay of wave-packet diffusion, Coulomb focusing, and a closely related Coulomb defocusing effect of the liberated electron. The hump is beyond the scope of the semiclassical model.

Nonadiabatic Effect on the Rescattering Trajectories of Electrons in Strong Laser Field Ionization Process**Supported by the National Natural Science Foundation of China under Grant Nos 11425414 and 11504215, and the Scientific Research Training Program of Shanxi University.

The important features of the rescattering trajectories in strong field ionization process such as the cutoff of the return energy at 3.17UP and that of the final energy at 10UP are obtained, based on the adiabatic approximation in which the initial momentum of the electron is assumed to be zero. We theoretically study the nonadiabatic effect by assuming a nonzero initial momentum on the rescattering trajectories based on the semiclassical simpleman model. We show that the nonzero initial momentum will modify both the maximal return energy at collision and the final energy after backward scattering, but in different ways for odd and even number of return trajectories. The energies are increased for even number of returns but are decreased for odd number of returns when the nonzero (positive or negative) initial momentum is applied.

Coulomb effect on the left–right asymmetry in photoelectron emission with few-cycle laser pulses

We theoretically study the strong-field ionization of hydrogen atom in few-cycle laser pulses with the Coulomb–Volkov distorted-wave approximation (CVA) theory and focus on the role of the Coulomb potential in the left–right asymmetry of the photoelectron yields along the laser polarization direction, by comparing the CVA results with strong-field approximation (SFA) simulations. Our simulations show that the carrier-envelope phase (CEP) dependent asymmetry in CVA deviates from the SFA simulation and more interestingly, there is a phase shift of the asymmetry curve as a function of CEP when the laser intensity increases, contrary to what is expected in the SFA simulations. In terms of the simple man's model, the deviation of the asymmetry curves in CVA from the SFA simulations is attributed to the significant influence of the Coulomb potential on the forward rescattering electron which will get close to the core again after tunneling ionization. Furthermore, the laser-intensity dependence of the phase shift of the asymmetry curves in CVA is elucidated.

Diabatic Mechanisms of Higher-Order Harmonic Generation in Solid-State Materials under High-Intensity Electric Fields.

We theoretically investigate mechanisms of higher-order harmonic generation in solid-state materials under a high-intensity ac electric field. A new theoretical framework presented in this Letter holds the legitimacy of Bloch's theorem even under the influence of the high-intensity electric field and provides an exact treatment of the diabatic processes of Bloch electrons. Utilizing this framework, we first discovered that the diabatic processes, namely, ac Zener tunneling and semimetallization of semiconductors, are key factors for nonperturbative mechanisms of HHG. These mechanisms are classified by the field intensity and could be understood by an extended simple man model based on an analogy between tunnel ionization in gaseous media and Zener tunneling in semiconductors. These conclusions would stimulate the universal understanding of HHG mechanisms in both atomic and solid cases.

Above-threshold ionization for very low electron energy

The rescattering term of the ionization amplitude in the strong-field approximation is analyzed for very low electron energy and emission in arbitrary direction, first in terms of the classical simple-man model and then in the quantum-mechanical quantum–orbit expansion of the strong-field-approximation amplitude. Particular orbits can be associated with particular patterns in the velocity map. The different roles of forward and backscattering are investigated. In addition to known features such as the LES and the fork, a characteristic and pronounced V structure in the velocity map is identified, which has been observed in recent experiments (2014 Phys. Rev. A 90 063424).

Tunneling dynamics in multiphoton ionization and attoclock calibration.

The intermediate domain of strong-field ionization between the tunneling and multiphoton regimes is investigated using the strong-field approximation and the imaginary-time method. An intuitive model for the dynamics is developed which describes the ionization process within a nonadiabatic tunneling picture with a coordinate dependent electron energy during the under-the-barrier motion. The nonadiabatic effects in the elliptically polarized laser field induce a transversal momentum shift of the tunneled electron wave packet at the tunnel exit and a delayed appearance in the continuum as well as a shift of the tunneling exit towards the ionic core. The latter significantly modifies the Coulomb focusing during the electron excursion in the laser field after exiting the ionization tunnel. We show that nonadiabatic effects are especially large when the Coulomb field of the ionic core is taken into account during the under-the-barrier motion. The simple man model modified with these nonadiabatic corrections provides an intuitive background for exact theories and has direct implications for the calibration of the attoclock technique.

Perturbation theory for quasienergy Floquet solutions in the low-frequency regime of the oscillating electric field

For a simple illustrative model Hamiltonian for xenon in a low-frequency linearly polarized laser field we obtain a remarkable agreement between the zero-order energy as well as the amplitude and phase of the zero-order Floquet states and the exact eigenvalues and eigenfunctions of the Floquet operator. Here we use as a zero-order Hamiltonian the adiabatic Hamiltonian where time is used as an instantaneous parameter. Moreover, for a variety of low laser frequencies, $\ensuremath{\omega}$, the deviation of the zero-order solutions from the exact quasienergy (QE) Floquet solutions approaches zero at the time the oscillating laser field is maximal. This remarkable result gives a further justification to the validity of the first step in the simple man model. It should be stressed that the numerical calculations of the exact QE Floquet solutions become extremely difficult when $\ensuremath{\omega}$ approaches zero and many Floquet channels are nested together and are coupled by the laser field. This is the main motivation for the development of perturbation theory for QE Floquet solutions when the laser frequency is small, to avoid the need to represent the Floquet operator by a matrix when the Fourier functions are used as a basis set. A way to calculate the radius of convergence of the perturbational expansion of the Floquet solutions in $\ensuremath{\omega}$ is given.

Low-energy electron rescattering in laser-induced ionization

The low-energy structure (LES) in the energy spectrum of above-threshold ionization of rare-gas atoms is reinvestigated from three different points of view. First, the role of forward rescattering in the completely classical simple-man model (SMM) is considered. Then, the corresponding classical electronic trajectories are retrieved in the quantum-mechanical ionization amplitude derived in the strong-field approximation augmented to allow for rescattering. Third, classical trajectories in the presence of both the laser field and the Coulomb field are scrutinized in order to see how they are related to the LES. It is concluded that the LES is already rooted in the SMM. The Coulomb field enhances the structure so that it can successfully compete with other contributions and become visible in the total spectrum.

Electron rescattering at metal nanotips induced by ultrashort laser pulses

We theoretically investigate the interaction of moderate intensity near-infrared few cycle laser pulses with nano-scale metal tips. Local field enhancement in a nanometric region around the tip apex triggers coherent electron emission on the nanometer length and femtosecond time scale. The quantum dynamics at the surface are simulated with time-dependent density functional theory (TDDFT) and interpreted based on the simple man's model. We investigate the dependence of the emitted electron spectra on the laser wavelength.

Spin dynamics in relativistic ionization with highly charged ions in super-strong laser fields

Spin dynamics and induced spin effects in above-threshold ionization of hydrogenlike highly charged ions in super-strong laser fields are investigated. Spin-resolved ionization rates in the tunnelling regime are calculated by employing two versions of a relativistic Coulomb-corrected strong-field approximation (SFA). An intuitive simpleman model is developed which explains the derived scaling laws for spin flip and spin asymmetry effects. The intuitive model as well as our ab initio numerical simulations support the analytical results for the spin effects obtained in the dressed SFA where the impact of the laser field on the electron spin evolution in the bound state is taken into account. In contrast, the standard SFA is shown to fail in reproducing spin effects in ionization even at a qualitative level. The anticipated spin-effects are expected to be measurable with modern laser techniques combined with an ion storage facility.

Metastable states in microwave ionization

The 38-GHz microwave field required to effect 50$%$ ionization of Li atoms decreases from 60 V/cm at $n=37$, where the microwave frequency is one-third the Kepler frequency, to 45 V/cm at $n=56$, where the microwave frequency is equal to the Kepler frequency. For $ng56$ the 50$%$ ionization field is constant at $E\ensuremath{\approx}2.4{\ensuremath{\omega}}^{5/3}$ until $\ensuremath{\sim}80$ GHz below the ionization limit, where it begins to decline. In spite of the fact that the pulse is 8000 cycles long, we detect approximately 5$%$ of the initial population in very-high-lying ($ng215$) states subsequent to the microwave pulse. We suggest that these atoms are trapped in metastable atom-field states during the microwave pulse and relax to the high-lying states when the field is turned off. The existence of such states is predicted by the simple man's model of above-threshold ionization.

Strong Field Acceleration and Steering of Ultrafast Electron Pulses from a Sharp Metallic Nanotip

We report a strong, laser-field induced modification of the propagation direction of ultrashort electron pulses emitted from nanometer-sized gold tapers. Angle-resolved kinetic energy spectra of electrons emitted from such tips are recorded using ultrafast near-infrared light pulses of variable wavelength and intensity for excitation. For sufficiently long wavelengths, we observe a pronounced strong-field acceleration of electrons within the field gradient at the taper apex. We find a distinct narrowing of the emission cone angle of the fastest electrons. We ascribe this to the field-induced steering of subcycle electrons as opposed to the diverging emission of quiver electrons. Our findings are corroborated by simulations based on a modified Simpleman model incorporating the curved, vectorial field gradient in the vicinity of the tip. Our results indicate new pathways for designing highly directional nanometer-sized ultrafast electron sources.

Manipulating Photoelectron Distributions for Rare Gas Atoms Ionized with Polarized Femtosecond Laser Pulses

Above-threshold ionizations of rare gas atoms excited by polarized femtosecond laser pulses are investigated. The photoelectron momentum spectra are obtained applying the strong-field approximation (SFA) theory. It is found that, distribution of the emitted photoelectrons varies with different polarizations of laser pulses. We have interpreted the relationship between the observed distribution and the laser polarization taking advantage of tunneling ionization theory and simple-man model. The polarization sensitivity indicates that one can easily manipulate the photoelectron distribution by controlling the polarization of the exciting pulse.