Attosecond control of spin polarization in electron-ion recollision driven by intense tailored fields

We show that electrons recolliding with the ionic core upon tunnel ionization of noble gas atoms driven by a strong circularly polarized laser field in combination with a counter-rotating second harmonic are spin polarized and that their degree of polarization depends strongly on the recollision time. Spin polarization arises as a consequence of (1) entanglement between the recolliding electron and the ion, and (2) sensitivity of ionization to the sense of electron rotation in the initial state. We demonstrate that one can engineer the degree of spin polarization as a function of time by tuning the relative intensities of the counter-rotating fields, opening the door for attosecond control of spin-resolved dynamics.


I. INTRODUCTION
The Stern-Gerlach experiment [1][2][3] revealed, in 1922, that an electron possesses an intrinsic angular momentum that is quantized and that is independent of its orbital angular momentum: the spin. Electron spin governs the behavior of matter, arranging the electronic shells of the elements in the periodic table through the Pauli exclusion principle [4] and giving rise to magnetism [5]. Ever since its discovery, finding ways of producing spin polarized electrons has attracted the interest of physicists [6]. In 1969, Fano demonstrated that one-photon ionization of atoms with circularly polarized light in the energy region of Cooper minima can lead to the generation of electrons with a high degree of spin polarization [7]. Another way of producing polarized currents is via ionization from a selected state of an atom or a molecule presenting fine structure splitting [8]. This investigation has been extended to the multiphoton case in the perturbative regime [9][10][11]. However, despite its importance, spin polarization with strong laser fields has received no attention until very recently [12][13][14][15]. The first theoretical predictions of spin polarization in noble gases upon strong field ionization with circularly polarized light [12] have just been experimentally confirmed [14].
Spin polarization in the strong field regime is a consequence of electron-ion entanglement and the sensitivity of the ionization yield to the sense of electron rotation in the initial state [12]: electrons that counter-rotate with the field ionize more easily than the co-rotating electrons, yielding different ionization rates for p − and p + electrons in noble gases [16][17][18][19][20][21] and diatomic molecules [15]. The possibility of inducing recollision of spin-polarized electrons with the parent ion can open new directions in attosecond spectroscopy [13,14].
Not surprisingly, the degree of spin polarization is higher for higher ellipticity of the ionizing field. The flip side of the coin, however, is that high ellipticity of the ionizing field reduces the chance of electron return to the parent ion. In this context, the use of an intense circularly polarized laser field in combination with its counter-rotating second harmonic, known as a bi-circular field, constitutes a powerful tool for introducing the spin degree of freedom into attosecond science, due to the opportunity to combine circular polarization with the efficient recollision offered by these fields [13,[22][23][24][25][26]. The application of bi-circular fields can lead to the production of ultrashort circularly and elliptically polarized laser pulses in the XUV domain [25][26][27][28][29][30][31]. Their chiral nature offers unique possibilities for probing molecular chirality [32] or symmetry breaking [33] at their natural time scales via high harmonic generation spectroscopy. Recent theoretical work [13] has indicated that electrons produced upon strong field ionization with bi-circular fields are spin polarized.
Here we present a detailed theoretical study of spin polarization in electron-core recollision driven by bi-circular fields, emphasizing the possibilities for, and the physical mechanisms of controlling the degree of spin-polarization by changing the parameters of the bi-circular field. The paper is organized as follows. Section II describes the theoretical approach, which is based on the strong field approximation (SFA). Section III describes our results, focusing on the analytical analysis of how the properties of the quantum electron trajectories define the spin polarization. This allows us to establish the origin of spin polarization in bi-circular fields (section III A) and show how to achieve its attosecond control by tailoring the laser fields (section III B). Section IV concludes the paper.

II. METHOD
Consider ionization, followed by electron-parent ion recollision, of xenon atoms driven by a strong right circularly polarized (RCP) field in combination with the counter-rotating second harmonic. The resulting electric field can be written, in the dipole approximation, as: where F 0,ω and F 0,2ω are the amplitudes of the right and left circularly polarized fields, respectively, with frequencies ω and 2ω. Within the strong-field approximation (SFA), the continuum electron wave function at time t is given by [34]: where IP is the ionization potential, p is the drift (canonical) momentum, related to the the kinetic momentum k(t) by k(t) = p + A(t), d(p + A(t)) = p + A(t)|d|Ψ 0 is the transition dipole matrix element from the initial ground state |Ψ 0 (the system is assumed to be in the ground state at t = t 0 ) to a Volkov state |p + A(t) V , given by where S V (t, t , p) is the Volkov phase: Eq. 2 can be used to calculate different observables, such as photoelectron yields, induced polarization and harmonic spectra [34]. Here we are interested in analyzing the degree of spin-polarization of the electrons that are driven back to the ionic core. This requires a measure of the recollision probability, resolved on the state of the ion and on the spin of the returning electron. The latter is determined by the initial magnetic quantum number of the state from which the electron tunnels and the state of the ion that has been created upon ionization, as described in [12]. As for the recollision probability, given that the size of the returning wave packet far exceeds the size of the atom, an excellent measure of the recollision amplitude is the projection of the continuum wave function (eq. 2) |Ψ(t) on any compact object at the origin; the recollision current will scale with the object area. To obtain the required recollision probability density at the origin, we simply project |Ψ(t) on the delta-function at the origin, yielding The degree of spin polarization of the recolliding electrons as a function of the recollision time t is given by the normalized difference between the recollision probability densities for electrons recolliding with spin up (w ↑ (t) = |a ↑ (t)| 2 ) and spin down (w ↓ (t) = |a ↓ (t)| 2 ) [12]: The densities w ↑ (t) and w ↓ (t) are obtained from the recollision densities w 2 correlated to ionization from the p + and p − orbitals, resolved on the ionic states 2 P 3/2 and 2 P 1/2 , and the corresponding Clebsch-Gordan coefficients [12]: The contribution of the p 0 orbital is negligible [16,18]. The key quantities in these expressions are the recollision densities resolved on the initial orbital and the final ionic state, 2 , etc. Application of the saddle-point method (see e.g. [34]) to the integral eq. 5 allows us to perform the semi-classical analysis of this expression in terms of electron trajectories, getting insight into the physical origin of spin polarization during recollision. The saddle points are calculated by solving the following set of equations [34]: where IP is the ionization potential, t i and t r are the complex ionization and recollision times, respectively. Eq. 9 describes tunneling and eq. 10 requires that the electron returns to the core. The recollision densities correlated to ionization from p + and p − orbitals are proportional to: In this expression, the first key quantity that determines the magnitude of w pm IP is the imaginary part of action. It is mostly accumulated between the times t i = t i + it i and t i , i.e. in the classically forbidden region. The second key quantity, which depends on the projection m of the angular momentum, is the complex-valued ionization angle φ k(t i ) . It is given by the following expression: with k x (t i ) = k x (t i ) + ik x (t i ) and k y (t i ) = k y (t i ) + ik y (t i ) being the complex velocities along x and y directions, respectively. Note that the difference between the recollision densities from p + and p − orbitals depends solely on the imaginary part of the ionization angle.
Finally, the electron recollision energy is calculated as neglecting small imaginary contribution when keeping t r on the real time axis.
Real time Imaginary time

III. RESULTS
The Lissajous curves of the electric field considered here (see eq. 1) and of the corre-   1). Results are shown for the ionic states of xenon 2 P 3/2 (red lines) and 2 P 1/2 (blue lines), with ionization potentials IP 2 P 3/2 = 12.13 eV and IP 2 P 1/2 = 13.43 eV.
We have evaluated the degree of spin polarization in recollision (eq. 6) using the saddle point solutions shown in fig. 3. Total spin polarization is shown in fig. 4 as a function of the recollision time, together with the degree of polarization resolved in the 2 P 1/2 and 2 P 3/2 states of the core. It is clear from the figure that recolliding electrons are spin-polarized and that their degree of polarization depends strongly on the recollision time. Electrons that return to the core at earlier (later) times are more likely to have spin up (down). Note also that spin polarization resolved in the ionic states 2 P 1/2 and 2 P 3/2 has opposite sign. Both spin polarization resolved on the states of the ion and the total spin polarization change sign at the recollision phase (time) of 0.7π rad (1.11 fsec). Each return time is associated with a given recollision energy, which is the well-known time-energy mapping [34] (see fig.   3d). Fig. 4 shows spin polarization as a function of the recollision energy for short and long trajectories. Whereas for the short trajectories spin polarization changes dramatically as a function of the recollision energy, for the long trajectories the variation is rather smooth.   fig. 3.

A. Origin of spin polarization
To better understand the physical origin of spin polarization in recollision, let us analyze the recollision densities for different ionic channels. These are presented in fig. 6 as a function of the recollision time, as well as the total recollision densities corresponding to electrons with spin up and spin down (eqs. 7 and 8). There are three important things worth noting here. First, the recollision densities correlated to the 2 P 3/2 state of the core (w p − IP 2 P 3/2 and w p + IP 2 P 3/2 ) are higher than those for the 2 P 1/2 state (w p − IP 2 P 1/2 and w p + IP 2 P 1/2 ) because the lower ionization potential of this ionic state leads to smaller imaginary ionization times (see fig.   3b) -the tunneling barrier is thinner. Second, all recollision densities exhibit a maximum value that arises at lower recollision times in the case of the p + orbital (w p + IP 2 P 3/2 and w p + IP 2 P 1/2 ). Third, the densities resolved on the 2 P 3/2 and 2 P 1/2 states of the core cross at φ r = 0.69π rad (t r = 1044 asec) and φ r = 0.70 rad (t r = 1061 asec), respectively, leading to changes of sign in spin polarization (see fig. 4).
In order to understand these features, we have examined the saddle point solutions at t = t i , when the electron enters the classically forbidden region. The ionization velocity and the ionization angle are shown in fig. 7 as a function of the recollision time. We can see that, for a recollision phase (time) of 0.7π rad (1.11 fsec), the real part of the ionization angle presents a jump of π and its imaginary component becomes zero. A purely real ionization angle leads to equal tunnelling probabilities for p + and p − orbitals (see eq. 11) and thus no spin polarization. The time-dependent sensitivity of the recollision densities to the sense of rotation of the electron in its initial state can be understood by examining different quantum trajectories.   In this section we discuss how modifying the parameters of the driving fields can affect the degree of spin polarization of the recolliding electrons. In particular, we analyze the effect of varying the relative intensities of the two counter-rotating fields. Fig. 10  Spin polarization is presented in fig. 10 (lower panels), also as a function of the recollision time. We can see that relatively small modifications of the fields intensities lead to dramatic changes in the degree of polarization, allowing to achieve a high degree of control. In particular, by tuning the relative intensities of the fields, it is possible to select the instant at which spin polarization changes it sign: increasing the intensity of the fundamental field shifts the change of sign towards earlier times, whereas increasing the intensity of its second harmonic has the opposite effect.   [35][36][37][38][39][40] or high harmonic generation [24,26,29,32,[41][42][43][44][45]. We have shown that the use of intense two-color counter-rotating bi-circular fields can drive electroncore recollision with a degree of spin polarization that depends on the recollision time and therefore on the recollision energy. Electron spin polarization upon tunnel ionization is intrinsically related to the generation of spin-polarized currents in the ionic core [46]. In this context, the potential of inducing recollision within one optical cycle of the driving field can allow for probing spin-polarized currents in atoms and molecules with sub-femtosecond and sub-Angstrom resolution. The time-dependence of spin polarization could be exploited to reconstruct information of the recollision process itself from spin-resolved measurements of diffracted electrons. Furthermore, our work shows that the degree of spin-polarization can be modified as desired by tailoring the driving fields. In particular, we have found that small variations in the relative intensities of the counter-rotating fields can change dramatically the level of polarization of the recolliding currents, opening the way for attosecond control of spin-resolved dynamics in atoms and molecules.
V. ACKNOWLEDGEMENTS