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 ionization by circularly polarized fields, for atoms with arbitrary binding potentials. Through the ARM method, the effect of the core potential can now be included consistently both during and after ionization. We find that Coulomb effects modify the ionization dynamics in several ways, including modification of (i) the ionization times, (ii) the initial conditions for the electron continuum dynamics, (iii) the “tunneling angle” at which the electron “enters” the barrier, and (iv) the electron drift momentum. We derive analytical expressions for the Coulomb-corrected ionization times, initial velocities, momentum shifts, and ionization rates in circularly polarized fields, for arbitrary angular momentum of the initial state. We also analyze how nonadiabatic Coulomb effects modify (i) the calibration of the attoclock in the angular streaking method and (ii) the ratio of ionization rates from $p^{-}$ and $p^{+}$ orbitals, predicted by Barth and Smirnova [Phys. Rev. A 84, 063415 (2011)] for short-range potentials.

Figure 1

Kinematics of electron tunneling through the rotating barrier. The right circularly polarized laser field $E$ creates a tunneling barrier rotating counterclockwise. (a) Short-range potential: The electron observed at the detector placed along the $x$ axis exits the barrier along the negative direction of the $y$ axis at angle ${\alpha }^{\left(0\right)}=-\pi /2$. (b) Long-range potential: The electron observed at the detector placed along the $x$ axis exits the barrier at the angle ${\alpha }^{\left(1\right)}=-\pi /2-\Delta \alpha$, $\Delta \alpha =|\omega \Delta t_i^{\prime \left(0\right)}|$, and $\Delta t_i^{\prime \left(0\right)}<0$.

Figure 2

Initial velocity corresponding to the center of the velocity distribution: $v_{\perp }$ [dashed (green) line] [Eq. (84)] and $v_{\parallel }$ [dot-dashed (blue) line] [Eq. (85)] vs frequency for $E_0=3\times {10}^{10}$ V/m ($E_0=0.06$ a.u. and $I=2.6\times {10}^{14}$ W/cm${}^2$) and $I_p=14$ eV. $v_{\perp }^{\text{SFA}}$ [solid (red) line] shows the result arising in the nonadiabatic short-range theory (the PPT theory; see [8, 14, 21]) and in the length-gauge SFA.

Figure 3

Calibration of the attoclock for an Ar atom with $I_p=15.7$ eV. (a) Initial velocities $v_x$ [solid (red) line] and $v_y$ [dashed (green) line] resulting from the ARM theory and $v_x$ [dot-dashed (blue) line] from the nonadiabatic short-range theories [8, 14, 21] for the geometry specified in Fig. 1. (b) Initial coordinate (exit point) in the current implementation of the ARM and the PPT [8, 14, 21] theories [solid (red) line] and $I_p/E_0$ in the adiabatic theory [dashed (blue) line]. (c) Angular offset $\Delta \alpha$ corresponding to the ARM [solid (red) line], nonadiabatic short-range [dashed (green) line], and adiabatic [blue (dot-dashed) line] theories. (d) Uncertainty in the calibration of time in the attoclock corresponding to (i) the nonadiabatic two-step model [solid (red) line] and (ii) the adiabatic two-step model [dashed (blue) line].

Figure 4

Calibration of the attoclock for a He atom with $I_p=24.6$ eV. (a) Initial velocities $v_x$ [solid (red) line] and $v_y$ [dashed (green) line] resulting from the ARM theory and $v_x$ [dot-dashed (blue) line] from the nonadiabatic shortrange theories [8, 14, 21] for the geometry specified in Fig. 1. (b) Initial coordinate (exit point) in the current implementation of the ARM and the PPT [8, 14, 21] theories [solid (red) line], and $I_p/E_0$ in the adiabatic theory [dashed (blue) line]. (c) Angular offset $\Delta \alpha$ corresponding to the ARM [solid (red) line], nonadiabatic short-range [dashed (green) line], and adiabatic [dot-dashed (blue) line] theories. (d) Uncertainty in time reconstruction associated with the nonadiabatic two-step model [solid (red) line] and the adiabatic two-step model [dashed (blue) line].

Figure 5

Ratio of ionization rates from $p^{-}$ and $p^{+}$ orbitals for a Ne atom ($I_p=21.5645$ eV) and $E_0=7.7\times {10}^{10}$ V/m ($E_0=0.15$ a.u. and $I=1.6\times {10}^{15}$ W/cm${}^2$), with $w_{p^{-}}/w_{p^{+}}$ for a right circularly polarized field: short-range potential [solid (red) line] [8] and long-range potential [dashed (blue) line].