Analytical R-matrix Research Articles

Relativistic analytical R -matrix theory for strong-field ionization

The analytical R-matrix (ARM) theory has been known for an efficient description of the Coulomb effects of the atomic core in strong-field ionization in the nonrelativistic regime. We generalize the ARM theory into the relativistic domain aiming at the application to strong-field ionization of highly charged ions in ultrastrong laser fields. Comparison with the relativistic Coulomb-corrected strong-field approximations (SFA) is provided, highlighting the advantages and disadvantages. The weakly relativistic asymptotics and its accordance with the nondipole Coulomb-corrected SFA are examined. As an example of a physical application of the relativistic ARM, the Coulomb enhancement of tunneling ionization probability for highly charged ions at the cutoff of the direct channel is discussed.

Reconciling Conflicting Approaches for the Tunneling Time Delay in Strong Field Ionization.

Several recent attoclock experiments have investigated the fundamental question of a quantum mechanically induced time delay in tunneling ionization via extremely precise photoelectron momentum spectroscopy. The interpretations of those attoclock experimental results were controversially discussed, because the entanglement of the laser and Coulomb field did not allow for theoretical treatments without undisputed approximations. The method of semiclassical propagation matched with the tunneled wave function, the quasistatic Wigner theory, the analytical R-matrix theory, the backpropagation method, and the under-the-barrier recollision theory are the leading conceptual approaches put forward to treat this problem, however, with seemingly conflicting conclusions on the existence of a tunneling time delay. To resolve the contradicting conclusions of the different approaches, we consider a very simple tunneling scenario which is not plagued with complications stemming from the Coulomb potential of the atomic core, avoids consequent controversial approximations and, therefore, allows us to unequivocally identify the origin of the tunneling time delay.

Looking inside the tunnelling barrier: I. Strong field ionisation from orbitals with high angular momentum in circularly polarised fields

Here and in the two companion papers we use time-dependent analytical R-Matrix (ARM) theory to consider the impact of electron-core interaction on (i) sensitivity of strong field ionisation to the sense of electron rotation in the initial state and (ii) electron spin polarisation. ARM is a rigorous approach to strong field ionisation that takes advantage of the semiclassical nature of electron dynamics in strong laser fields and naturally incorporates complex-valued classical trajectories in the time-propagation. While the appearance of complex time is associated with optical tunnelling, complex coordinates reflect non-adiabaticity of the electron response to the strong laser field. We focus on non-adiabatic electron-core interaction in the sub-barrier region and show how it affects ionisation from orbitals with high angular momentum ℓ = 2, 3 in multiply charged atomic ions with strong electron-core interaction. In this paper we discuss sub-cycle ionisation yields, while in the companion papers we focus on the effects of the long-range core potential on the ‘initial conditions’ (from the ARM perspective, companion paper II) for the photo electron as it emerges from the classically forbidden region and on the spin polarisation of the ejected electron (companion paper III).

Looking inside the tunnelling barrier III: spin polarisation in strong field ionisation from orbitals with high angular momentum

Strong field ionisation is known as optical tunnelling of an electron through the barrier created by the laser field and core potential. Time-dependent analytical R-matrix takes advantage of the semiclassical nature of electron dynamics in strong laser fields and naturally incorporates complex-valued trajectories in time propagation. The language of trajectories enables straightforward ‘kinematic’ interpretation of several new phenomena such as ‘inversion’ of the conventional propensity rules in photo-ionisation (Barth and Smirnova 2011 Phys. Rev. A 84 063415) and generation of spin-polarised electrons (Barth and Smirnova 2013 Phys. Rev. A 88 013401) during ionisation in strong laser fields. Both phenomena are due to the kinematics of the electron motion under the barrier and were predicted for short-range potentials, i.e. neglecting the electron–core interaction. However, the electron–core interaction can and does affect the kinematics of tunnelling defining the ionisation propensity rules (companion paper I) and spin-polarisation. The latter is the focus of this paper. We consider orbitals with arbitrary angular momentum and extend our results to positively charged ions.

Looking inside the tunnelling barrier: II. Co- and counter-rotating electrons at the ‘tunnelling exit’

The initial conditions for electron trajectories at the exit from the tunnelling barrier are often used in strong field models, for example to bridge the first and the second steps of the three-step model celebrated in this issue. Since the analytical R-matrix theory does not rely on the three-step model or the concept of the tunnelling barrier in coordinate space, obtaining the initial conditions for electron trajectories at the barrier exit is, strictly speaking, not necessary to calculate standard observables. Not necessary, but possible—especially when motivated by the occasion of this issue. The opportunity to evaluate such initial conditions emerges as a corollary of analysing sub-barrier kinematics, which includes the interplay of laser and Coulomb fields on the sub-cycle scale (see the companion paper I). We apply our results to discuss the difference in such initial conditions for co- and counter-rotating electrons liberated during strong field ionisation. We derive quantum orbits and classical trajectories describing ionization dynamics of co- and counter-rotating electrons in long-range potentials.

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

Coulomb time delays in high harmonic generation

Measuring the time it takes to remove an electron from an atom or molecule during photoionization has been the focus of a number of recent experiments using newly developed attosecond spectroscopies. The interpretation of such measurements, however, depends critically on the measurement protocol and the specific observables available in each experiment. One such protocol relies on high harmonic generation. In this paper, we derive rigorous and general expressions for ionisation and recombination times in high harmonic generation experiments. We show that these times are different from, but related to, ionisation times measured in photoelectron spectroscopy: that is, those obtained using the attosecond streak camera, RABBITT and attoclock methods. We then proceed to use the analytical R-matrix theory to calculate these times and compare them with experimental values.

Slalom in complex time: Emergence of low-energy structures in tunnel ionization via complex-time contours

The ionization of atoms by strong, low-frequency fields can generally be described well by assuming that the photoelectron is, after the ionization step, completely at the mercy of the laser field. However, certain phenomena, like the recent discovery of low-energy structures in the long-wavelength regime, require the inclusion of the Coulomb interaction with the ion once the electron is in the continuum. We explore the first-principles inclusion of this interaction, known as analytical R-matrix theory, and its consequences on the corresponding quantum orbits. We show that the trajectory must have an imaginary component, and that this causes branch cuts in the complex time plane when the real trajectory revisits the neighbourhood of the ionic core. We provide a framework for consistently navigating these branch cuts based on closest-approach times, which satisfy the equation $\mathbf{r}(t) \cdot \mathbf{v}(t) = 0$ in the complex plane. We explore the geometry of these roots and describe the geometrical structures underlying the emergence of LES in both the classical and quantum domains.

Progress in study of low-energy photoelectron in ultra-fast strong fields-analytical R-matrix theory based semiclassical trajectory method

The semi-classical method based on the recently developed analytical R-matrix theory is reviewed in this work. The method is described with the application to ultra-fast strong-field direct ionization of atoms with one active electron in a linearly polarized field[Torlina L, Smirnova O 2012 Phys. Rev. A 86 043408]. The analytical R-matrix theory separates the space into inner and outer regions, naturally allowing the possibility of an analytical or semi-analytical description of wave function in the outer region, which can be approximated by Eikonal-Volkov solutions while the inner region provides well-defined boundary conditions. Applying the stationary phase method, the calculation of the ionization amplitude is cast into a superposition of components from trajectories and their associated phase factors. The shape of the tunneling wave packets associated with different instants of ionization is presented. It shows the exponential cost of deviating from the optimal tunneling trajectory renders the tunneling wave packet a Gaussian shape surrounding the semi-classical trajectory. The intrinsically non-adiabatic corrections to the sub-cycle ionization amplitude in the presence of both the Coulomb potential and the laser field is shown to have different influences on the probability of ionization. As a specific study case, soft recollisions of the released electron near the ionic core is investigated by using pure light-driven trajectories with Coulomb-corrected phase factor[Pisanty E, Ivanov M 2016 Phys. Rev. A 93 043408]. Incorporating the Coulomb potential, it is found problematic to use the conventional integration contour as chosen in other methods with trajectory-based Coulomb corrections, because the integration contour may run into the Coulomb-induced branch cuts and hence the analyticity of the integrand fails. In order to overcome the problem, the evolution time of the post-tunneling electron is extended into the complex domain which allows a trajectory to have an imaginary component. As the soft recollision occurs, the calculation of the ionization amplitude requires navigating the branch cuts cautiously. The navigating scheme is found based on closest-approach times which are the roots of closest-approach times equations. The appropriately selected closest-approach times that always present in the middle of branch-cut gate may serve to circumvent these branch cuts. The distribution of the closest-approach times presents rich geometrical structures in both the classical and quantum domains, and intriguing features of complex trajectories emerge as the electron returns near the core. Soft recollisions responsible for the low-energy structures are embedded in the geometry, and the underlying emergence of near-zero energy structures is discussed with the prediction of possible observations in experiments.

Ab initio verification of the analytical R-matrix theory for strong field ionization

We summarize the key aspects of the recently developed analytical R-matrix (ARM) theory for strong field ionization (Torlina and Smirnova 2012 Phys. Rev. A 86 043408; Kaushal and Smirnova 2013 Phys. Rev. A 88 013421), and present tests of this theory using ab initio numerical simulations for hydrogen and helium atoms in long circularly polarized laser pulses. We find excellent agreement between the predictions of ARM and the numerical calculations.

Hole dynamics and spin currents after ionization in strong circularly polarized laser fields

We apply the time-dependent analytical R-matrix theory to develop a movie of hole motion in a Kr atom upon ionization by strong circularly polarized field. We find rich hole dynamics, ranging from rotation to swinging motion. The motion of the hole depends on the final energy and the spin of the photoelectron and can be controlled by the laser frequency and intensity. Crucially, hole rotation is a purely non-adiabatic effect, completely missing in the framework of quasistatic (adiabatic) tunneling theories. We explore the possibility to use hole rotation as a clock for measuring ionization time. Analyzing the relationship between the relative phases in different ionization channels we show that in the case of short-range electron-core interaction the hole is always initially aligned along the instantaneous direction of the laser field, signifying zero delays in ionization. Finally, we show that strong-field ionization in circular fields creates spin currents (i.e. different flow of spin-up and spin-down density in space) in the ions. This phenomenon is intimately related to the production of spin-polarized electrons in strong laser fields Barth and Smirnova (2013 Phys. Rev. A 88 013401). We demonstrate that rich spin dynamics of electrons and holes produced during strong field ionization can occur in typical experimental conditions and does not require relativistic intensities or strong magnetic fields.

Time-resolving electron-core dynamics during strong-field ionization in circularly polarized fields

Electron-core interactions play a key role in strong-field ionization and the formation of photoelectron spectra. We analyse the temporal dynamics of strong field ionization associated with these interactions using the time-dependent analytical R-matrix (ARM) method, developed in our previous work [J. Kaushal and O. Smirnova, Phys. Rev. A 88, 013421 (2013)]. The approach is fully quantum but includes the concept of trajectories. However, the trajectories are not classical in the sense that they have both real and imaginary components all the way to the detector. We show that the imaginary parts of these trajectories, which are usually ignored, have a clear physical meaning and are crucial for the correct description of electron-core interactions after ionization. In particular, they give rise to electron deceleration, as well as dynamics associated with electron recapture and release. Our approach is analytical and time-dependent, and allows one to gain access to the electron energy distribution and ionization yield as a function of time. Thus we can also rigorously answer the question: when is ionization completed?

Nonadiabatic Coulomb effects in strong-field ionization in circularly polarized laser fields

We develop the recently proposed analytical R-matrix (ARM) method to encompass strong field ionisation by circularly polarised fields, for atoms with arbitrary binding potentials. Through ARM, the effect of the potential can now be included consistently both during and after ionisation, providing a complete picture for the effects of the long-range potential. We find that the Coulomb effects modify the ionisation dynamics in several ways, including modification of (i) the ionisation (exit) times, (ii) the initial conditions for the electron continuum dynamics, (iii) the "tunnelling angle", at which the electron "enters" the barrier, and (iv) the electron drift momentum. We derive analytical expressions for the Coulomb-corrected ionisation times, initial velocities, momentum shifts and ionisation rates in circularly polarised fields, for arbitrary angular momentum of the initial state. We also analyse how the non-adiabatic Coulomb effects modify (i) the calibration of the attoclock in the angular streaking method, and (ii) the ratio of ionisation rates from $p^{-}$ and $p^{+}$ orbitals.