PHYSICAL REVIEW A 95, 063408 (2017) Spin-dependent quantum theory of high-order above-threshold ionization D. Zille, 1, 2 , * D. Seipt, 3, 4 M. Möller, 1, 2 S. Fritzsche, 2, 5 G. G. Paulus, 1, 2 and D. B. Miloševi´ c 6, 7, 8 1 Institute of Optics and Quantum Electronics, Friedrich Schiller University Jena, Max-Wien-Platz 1, 07743 Jena, Germany 2 Helmholtz Institut Jena, Fröbelstieg 3, 07743 Jena, Germany 3 Physics Department, Lancaster University, Lancaster LA1 4YB, United Kingdom 4 The Cockcroft Institute Daresbury Laboratory, Warrington WA4 4AD, United Kingdom 5 Theoretisch-Physikalisches Institut, Friedrich Schiller University Jena, Max-Wien-Platz 1, 07743 Jena, Germany 6 Faculty of Science, University of Sarajevo, Zmaja od Bosne 35, 71000 Sarajevo, Bosnia and Herzegovina 7 Academy of Sciences and Arts of Bosnia and Herzegovina, Bistrik 7, 71000 Sarajevo, Bosnia and Herzegovina 8 Max-Born-Institut, Max-Born-Strasse 2a, 12489 Berlin, Germany (Received 21 February 2017; published 12 June 2017) The strong-field-approximation theory of high-order above-threshold ionization of atoms is generalized to include the electron spin. The obtained rescattering amplitude consists of a direct and exchange part. On the examples of excited He atoms as well as Li + and Be ++ ions, it is shown that the interference of these two amplitudes leads to an observable difference between the photoelectron momentum distributions corresponding to different initial spin states: Pronounced minima appear for singlet states, which are absent for triplet states. DOI: 10.1103/PhysRevA.95.063408 I. INTRODUCTION The ionization dynamics of atoms by a strong laser field is usually described by using the nonrelativistic Schrödinger equation and neglecting the influence of the electron spin. It has been assumed that this is justified for optical frequencies and laser intensities much lower than 3.5 × 10 16 W/cm 2 . For higher intensities, or longer wavelengths, the magnetic field component becomes important and one should use the Pauli or even the Dirac equation [1,2]. High-order atomic processes in strong fields are commonly described using the three-step model (see Refs. [3,4] and references therein): The electron, liberated in the first step, moves in the laser field and may be driven back to the atomic core in the second step. Finally, in the third step, the electron may elastically scatter off the core and leave towards the detector with a much higher energy than it could acquire in the first step alone. This is the so-called high-order above-threshold ionization (HATI) process [5,6]. The electron may also recombine to the ground state, emitting a high-energy photon in the high-order harmonic generation (HHG) process [7]. One may expect that, with increasing laser intensity, the energy of the emitted HATI electrons or HHG photons increases. However, due to the magnetic component of the Lorentz force, the returning electron has a drift momentum in the direction of propagation of the laser field, which drastically decreases the significance of the rescattering process [3,8]. Therefore, rescattering is only important for nonrelativistic electron energies. In this case, the probability of spin flips is small [9]. If the spin is not important, the rescattering effect can be treated using the simpler relativistic Klein-Gordon equation instead of the Dirac equation [10]. On the other hand, spin-polarized electrons are important in many areas of physics [11,12]. Spin dynamics in relativistic strong-field ionization has been analyzed in Ref. [13] for circularly and in Ref. [14] for linearly polarized laser fields. * danilo.zille@uni-jena.de It has been proposed to use ionization with strong (but still nonrelativistic) circularly polarized laser fields for the creation of spin-polarized electrons [15]. This has been realized in a recent experiment [16]. However, since the probability of the electron returning to the core decreases with increasing ellipticity of the laser polarization, there is no rescattering for circularly polarized fields. This problem can be solved by using a so-called bicircular field (consisting of two coplanar counter-rotating circularly polarized fields with different fre- quencies), which enables rescattering [17]. In this case, the spin-dependent effects are due to the spin-orbit interaction. Coulombic forces do not act directly on the spins of electrons. In addition to the mentioned spin-orbit coupling, spin-dependent effects in collisions may be facilitated via the “Pauli force,” i.e., the requirement that wave functions of identical fermions are antisymmetric. In this paper, we will show that spin effects in strong-field ionization can be important if the Pauli force is taken into account, even in the nonrelativistic regime. Using this, we will extend the semiclassical results from Ref. [18]. It is known that for the scattering of identical particles the scattering amplitude consists of the direct and exchange term [19]. The relative sign in the sum of these two terms depends on the spin state of the scattering particles (e.g., singlet or triplet). In calculations of the scattering cross sections for laser-assisted electron-atom scattering, the influence of the exchange effect was taken into account in Refs. [20–22]. Here, we consider spin-dependent rescattering in the HATI process. To this end, we reformulate and generalize the strong-field approximation (SFA) theory of HATI to include the spin-wave functions. In the obtained result, the direct and exchange rescattering amplitudes are explicitly separated. We illustrate our theory with examples of strong-field ionization of excited states of He, Li + , as well as Be ++ and find that considerable differences in the rescattering process are to be expected, depending on the spin state of the returning electron and residual ion. Specifically, ionization from singlet states leads to distinct minima in the photoelectron momentum distributions (PEMDs), which are absent for triplet states. The investigated effect could potentially be exploited 2469-9926/2017/95(6)/063408(6) 063408-1 ©2017 American Physical Society