Coulomb-corrected Strong-field Approximation Research Articles

Feynman path-integral strong-field dynamics calculation method

The emergence and development of ultrafast intense lasers and attosecond measurement techniques have made it possible to observe and control the motions of electrons on a timescale of attoseconds and a spatial scale of atoms. With the improvement of experimental measurement accuracy, higher requirements are put forward for the accuracy of theoretical calculation methods. Extracting temporal and spatial information about ultrafast dynamics from experimental results through using theoretical models presents a significant challenge. Compared with the exact solutions of the time-dependent Schrödinger equation, the Feynman path-integral method for strong-field dynamics calculations offers a simpler model and higher computational efficiency. The electronic wave packet is regarded as a particle with different initial states, and by analyzing the motion of the particle, the causes of various nonlinear physical phenomena in strong fields can be clarified. This work introduces the saddle point approximation into strong field dynamics calculations based on the strong field approximation theory. Furthermore, the Coulomb-corrected strong field approximation method, trajectory-based Coulomb-corrected strong field approximation method, and Coulomb quantum trajectory strong field approximation method are presented in detail. This review aims to provide relevant methods and literature references for studying strong field dynamics theoretical calculations and also to present some ideas for developing new algorithms.

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.

Coulomb-corrected strong-field approximation based on a genetic algorithm

Coulomb-corrected strong-field approximation (CCSFA), which integrates Coulomb potential and the interference effects, is a semiclassical method successful in the study of atomic strong-field ionization. However, it is difficult to numerically solve the saddle-point equation describing tunnelling in the CCSFA, especially for complex laser fields such as elliptically or orthogonally polarized two-color laser fields. In this work, we propose an efficient method based on the genetic algorithm (CCSFA-GA) to overcome this difficulty. The accuracy of our method is verified by comparing our result with the calculation of the standard CCSFA on a hydrogen atom, subjected to an intense laser field. Moreover, we show that the result of the numerical solution of the time-dependent Schrödinger equation with an elliptically polarized laser field can be well reproduced by the CCSFA-GA.

Inclusion of Coulomb effects in laser-atom interactions

We investigate the role of the Coulomb interaction in strong field processes. We find that the Coulomb field of the ion makes its presence known even in highly intense laser fields, in contrast to the assumptions of the strong field approximation. The dynamics of the electron after ionization is analyzed with four models for an arbitrary laser polarization: the Hamiltonian model in the dipole approximation, the strong field approximation, the Coulomb-corrected strong field approximation and the guiding center. These models illustrate clearly the Coulomb effects, in particular Coulomb focusing and Coulomb asymmetry. We show that the Coulomb-corrected strong field approximation and the guiding center are complementary, in the sense that the Coulomb-corrected strong field approximation describes well short time scale phenomena (shorter than a laser cycle) while the guiding center is well suited for describing long time scale phenomena (longer than a laser cycle) like Coulomb-driven recollisions and Rydberg state creation.

Coulomb-Corrected Strong Field Approximation without Singularities and Branching Points

The domain of validity of the Coulomb-Corrected Strong-Field Approximation (CCSFA) is going to be analyzed in relation to the semi-classical dynamics of electrons during ionization of hydrogen-like targets. Our analysis is limited to ionization driven by Ti-sapphire laser pulses with intensities up to roughly 1014 Wcm−2. For such parameters, the effects related to radiation pressure are small and the laser field can be described in the dipole approximation. By applying the Magnus expansion for the exact retarded electron propagator we obtain an effective action which is free of Coulomb singularities and branch points when the complex-time trajectories are used. Furthermore, we show that the classical action is exactly recovered as the asymptotic limit of its effective counterpart. The applicability of such limit is also discussed.

High-energy direct photoelectron spectroscopy in strong-field ionization

Recently, in a strong Coulomb field regime of tunneling ionization an unexpected large enhancement of photoelectron spectra due to the Coulomb field of the atomic core has been identified by numerical solution of time-dependent Schr\"odinger equation [Phys. Rev. Lett. \textbf{117}, 243003 (2016)] in the upper energy range of the tunnel-ionized direct electrons. We investigate the origin of the enhancement employing a classical theory with Monte Carlo simulations of trajectories, and a quantum theory of Coulomb-corrected strong field approximation based on the generalized eikonal approximation for the continuum electron. Although the quantum effects at recollisions with a small impact parameter yield an overall enhancement of the spectrum relative to the classical prediction, the high energy enhancement itself is shown to have a classical nature and is due to momentum space bunching of photoelectrons released not far from the peak of the laser field. The bunching is caused by a large and nonuniform, with respect to the ionization time, Coulomb momentum transfer at the ionization tunnel exit.

Coulomb-corrected strong-field quantum trajectories beyond dipole approximation

Non-dipole effects in strong-field photoelectron momentum spectra have been revealed experimentally (Smeenk et al 2011 Phys. Rev. Lett. 106 193002; Ludwig et al 2014 Phys. Rev. Lett. 113 243001). For certain laser parameters and photoelectron momenta the spectra were found to be shifted against the laser propagation direction whereas one would naively assume that the radiation pressure due to the -force pushes electrons always in the propagation direction. Only the interplay between Lorentz and Coulomb forces may give rise to such counterintuitive dynamics. In this work, we calculate the momentum-dependent shift in and against the propagation direction by extending the quantum trajectory-based Coulomb-corrected strong-field approximation beyond the dipole approximation. A semi-analytical treatment where both magnetic and Coulomb forces are treated perturbatively but simultaneously reproduces the results from the full numerical solution of the equations of motion.

Strong-field ionization via a high-order Coulomb-corrected strong-field approximation

Signatures of the Coulomb corrections in the photoelectron momentum distribution during laser-induced ionization of atoms or ions in tunneling and multiphoton regimes are investigated analytically in the case of an one-dimensional problem. High-order Coulomb corrected strong-field approximation is applied, where the exact continuum state in the S-matrix is approximated by the eikonal Coulomb-Volkov state including the second-order corrections to the eikonal. Although, without high-order corrections our theory coincides with the known analytical R-matrix (ARM) theory, we propose a simplified procedure for the matrix element derivation. Rather than matching the eikonal Coulomb-Volkov wave function with the bound state as in the ARM-theory to remove the Coulomb singularity, we calculate the matrix element via the saddle-point integration method as by time as well as by coordinate, and in this way avoiding the Coulomb singularity. The momentum shift in the photoelectron momentum distribution with respect to the ARM-theory due to high-order corrections is analyzed for tunneling and multiphoton regimes. The relation of the quantum corrections to the tunneling delay time is discussed

Efficient time-sampling method in Coulomb-corrected strong-field approximation.

One of the main goals of strong-field physics is to understand the complex structures formed in the momentum plane of the photoelectron. For this purpose, different semiclassical methods have been developed to seek an intuitive picture of the underlying mechanism. The most popular ones are the quantum trajectory Monte Carlo (QTMC) method and the Coulomb-corrected strong-field approximation (CCSFA), both of which take the classical action into consideration and can describe the interference effect. The CCSFA is more widely applicable in a large range of laser parameters due to its nonadiabatic nature in treating the initial tunneling dynamics. However, the CCSFA is much more time consuming than the QTMC method because of the numerical solution to the saddle-point equations. In the present work, we present a time-sampling method to overcome this disadvantage. Our method is as efficient as the fast QTMC method and as accurate as the original treatment in CCSFA. The performance of our method is verified by comparing the results of these methods with that of the exact solution to the time-dependent Schrödinger equation.

Semiclassical methods for strong field ionization of atoms

As the advances of laser technology, more and more nonlinear phenomena are observed in the atoms and molecules driven by strong laser pulses. Systematic investigations on these findings, such as above threshold ionization and high-order harmonic generation, will lead us to understanding the mechanisms in the microscopic world. The most exact way to simulate the experimental measurements is to solve the time-dependent Schrdinger equation (TDSE) numerically, in which the system is described by the wave function and thus one cannot have an intuitive insight into the underling process. Therefore, several semiclassical methods have been developed to understand the strong field ionization. In the classical point of view, the electrons tunnel out when the strong laser field suppresses the Coulomb potential. Then the electrons are driven by the laser electric field according to the Newtonian equations. Semiclassical methods take into account the tunnelling of the electron, the classical orbit of the electron, and the action as the phase of trajectory, which have successfully explained main structures in the ionization spectrum. Two of the most popular semiclassical methods are the quantum trajectory Monte Carlo method and the Coulomb-corrected strong field approximation method. In the present review, we will introduce these basic methods and show how they have been developed step by step, covering the most relevant and important works in the strong field physics. Finally we give two example of applications to show how these methods work. With the advantage of the classical picture, we can identify different kind of structures in the 2D photoelectron momentum distributions and tell how the structures are formed. Nonadiabatic effects can be studied by comparing the results of the two methods, together with accurate simulation from the numerical solution of TDSE. The current semiclassical methods can be further developed into advanced ones, which can be used in more complex molecular systems or multi-electron systems, and be widely used in the study of dynamics of molecule and atoms in strong laser fields.

Nonadiabatic Electron Dynamics in Orthogonal Two-Color Laser Fields with Comparable Intensities.

We theoretically investigate the nonadiabatic subcycle electron dynamics in orthogonally polarized two-color laser fields with comparable intensities. The photoelectron dynamics is simulated by exact solution to the 3D time-dependent Schrödinger equation, and also by two other semiclassical methods, i.e., the quantum trajectory MonteCarlo simulation and the Coulomb-corrected strong field approximation. Through these methods, we identify the underlying mechanisms of the subcycle electron dynamics and find that both the nonadiabatic effects and the Coulomb potential play very important roles. The contribution of the nonadiabatic effects manifest in two aspects, i.e., the nonadiabatic ionization rate and the nonzero initial velocities at the tunneling exit. The Coulomb potential has a different impact on the electrons' trajectories for different relative phases between the two pulses.

Influence of the Coulomb potential on above-threshold ionization: A quantum-orbit analysis beyond the strong-field approximation

We perform a detailed analysis of how the interplay between the residual binding potential and a strong laser field influences above-threshold ionization (ATI), employing a semi-analytical, Coulomb-corrected strong-field approximation (SFA) in which the Coulomb potential is incorporated in the electron propagation in the continuum. We find that the Coulomb interaction lifts the degeneracy of some SFA trajectories, and we identify a set of orbits which, for high enough photoelectron energies, may be associated with rescattering. Furthermore, by performing a direct comparison with the standard SFA, we show that several features in the ATI spectra can be traced back to the influence of the Coulomb potential on different electron trajectories. These features include a decrease in the contrast, a shift towards lower energies in the interference substructure, and an overall increase in the photoelectron yield. All features encountered exhibit a very good agreement with the \emph{ab initio} solution of the time-dependent Schr\"odinger equation.

Above-threshold ionization with highly charged ions in superstrong laser fields. III. Spin effects and their dependence on laser polarization

Spin effects in the tunneling regime of strong-field ionization of hydrogenlike highly charged ions in linearly as well as circularly polarized laser fields are investigated. The impact of the polarization of a laser field on the spin effects is analyzed. Spin-resolved differential ionization rates are calculated employing the relativistic Coulomb-corrected strong-field approximation developed in the previous paper of the series. Analytical expressions for spin asymmetries and the spin-flip probability, depending on the laser's polarization, are obtained for the photoelectron momentum corresponding to the maximum of the tunneling probability. A simple intuitive model is developed for the description of spin dynamics in tunnel ionization. The spin flip is shown to be experimentally observable by using moderate highly charged ions with a charge of the order of 20 and a laser field with an intensity of $I\ensuremath{\sim}{10}^{22}\phantom{\rule{0.28em}{0ex}}{\mathrm{W}/\mathrm{cm}}^{2}$.

Nonadiabatic tunneling ionization of atoms in elliptically polarized laser fields

We theoretically investigate the nonadiabatic effects in strong field tunneling ionization of atoms in elliptically polarized laser fields by solving the 3D time-dependent Schrödinger equation (TDSE). Comparing our TDSE results with those of two semi-classical methods, i.e., the quantum-trajectory Monte Carlo simulation (QTMC) and the Coulomb-corrected strong field approximation (CCSFA), we confirm the existence of the nonadiabatic effects with its fingerprint in the nonzero initial lateral velocity at the tunneling exit in the laser polarization plane. Our study shows that these nonadiabatic initial lateral momentum effects become significant in high ellipticity or circularly polarized laser field. These results indicate that the calibration of the experimental laser intensity in this situation should be performed nonadiabatically, which may strongly affect the results of the real tunneling time delay measurements.

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.

Above-threshold ionization with highly charged ions in superstrong laser fields. I. Coulomb-corrected strong-field approximation

Aiming at the investigation of above-threshold ionization in super-strong laser fields with highly charged ions, we develop a Coulomb-corrected strong field approximation (SFA). The influence of the Coulomb potential of the atomic core on the ionized electron dynamics in the continuum is taken into account via the eikonal approximation, treating the Coulomb potential perturbatively in the phase of the quasi-classical wave function of the continuum electron. In this paper the formalism of the Coulomb-corrected SFA for the nonrelativistic regime is discussed employing velocity and length gauge. Direct ionization of a hydrogen-like system in a strong linearly polarized laser field is considered. The relation of the results in the different gauges to the Perelomov-Popov-Terent'ev imaginary-time method is discussed.

Above-threshold ionization with highly charged ions in superstrong laser fields. II. Relativistic Coulomb-corrected strong-field approximation

We develop a relativistic Coulomb-corrected strong field approximation (SFA) for the investigation of spin effects at above-threshold ionization in relativistically strong laser fields with highly charged hydrogen-like ions. The Coulomb-corrected SFA is based on the relativistic eikonal-Volkov wave function describing the ionized electron laser-driven continuum dynamics disturbed by the Coulomb field of the ionic core. The SFA in different partitions of the total Hamiltonian is considered. The formalism is applied for direct ionization of a hydrogen-like system in a strong linearly polarized laser field. The differential and total ionization rates are calculated analytically. The relativistic analogue of the Perelomov-Popov-Terent'ev ionization rate is retrieved within the SFA technique. The physical relevance of the SFA in different partitions is discussed.

Interaction of ultra-intense laser pulses with relativistic ions

At high laser intensities, three step recollision processes such as high order harmonic generation and high-order ATI, are normally severely suppressed due to the magnetic field component of the laser pulse. However, if the laser pulse and relativistic ion beam are directed against each other, a significant increase in the frequency and the intensity of the pulse in the rest frame of the ions can occur. By performing calculations based on a Coulomb-corrected nondipole strong field approximation, we have shown that there is a range of intensities, Lorentz factors, and ion charges for which the suppression of the three step recollision processes is not severe, even for ponderomotive energies exceeding 10 keV. As an example, we consider parameters relevant to the accelerator that will be built at GSI-Darmstadt, capable of accelerating multicharged ions to Lorentz factors reaching 30.

Coulomb-corrected strong-field approximation for the two-electron atom

We present the model accounting for the Coulomb effects that occur in the direct double photoionization of the two-electron atom in an intense field of a circularly polarized plane wave. The model is the strong-field approximation (SFA) (the Reiss version of Keldysh-type models) applied to helium. We improve here our earlier study by inserting the electron-electron correlation in the initial state of the atom, and by inserting the Coulomb correction due to the influence of the nucleus on the final state of ionized electrons` pair. While the former improvement is of little significance, the latter extends the applicability range of the SFA (for typical laser wavelengths), and causes appreciable growth of the total ionization rate. {copyright} {ital 1997} {ital The American Physical Society}