Nonadiabatic tunneling in circularly polarized laser fields. II. Derivation of formulas

We provide detailed analysis of strong field ionization of degenerate valence $p$ orbitals by circularly polarized fields. Our analytical approach is conceptually equivalent to the Perelomov, Popov, and Terent'ev (PPT) theory and is virtually exact for short-range potentials. After benchmarking our results against the PPT theory for $s$ orbitals, we obtain the results for $p$ orbitals. We also show that, as long as the dipole approximation is valid, both the PPT method and our results are gauge invariant, in contrast with widely used strong field approximation (SFA). Our main result, which has already been briefly outlined in [I. Barth and O. Smirnova, Phys. Rev. A 84, 063415 (2011)], is that strong field ionization preferentially removes electrons counter-rotating to the circularly polarized laser field. The result is illustrated using the example of Kr atom. Strong, up to one order of magnitude, sensitivity of strong field ionization to the sense of electron rotation in the initial state is one of the key signatures of nonadiabatic regime of strong field ionization.

Figure 1

Time-averaged ionization rates $w_{+}(\mathcal{E},\omega )$ for $s$ (orange), $p_0$ (green), $p_{+}$ (blue), $p_{-}$ (red) orbitals, and right circular polarization versus laser frequency $\omega$ for $I_p=0.5$a.u. and laser amplitude $\mathcal{E}=0.06$a.u. The solid, dashed, and dotted curves correspond to the accurate [Eqs. (76, 77, 78)], approximate [Eqs. (88, 89, 90)], and simple [Eqs. (100, 101, 102)] results, respectively. Note that $|C_{\kappa l}{|}^2=1$ are used.

Figure 2

Time-averaged ionization rates $w_{+}(\mathcal{E},\omega )$ for $s$ (orange), $p_0$ (green), $p_{+}$ (blue), $p_{-}$ (red) orbitals, and right circular polarization versus laser intensity $I=c^2{\varepsilon }_0^2{\mathcal{E}}^2$ for $I_p=0.5$a.u. and laser frequency $\omega =0.057$a.u. (800 nm) in logarithmic scale. The solid curves corresponds to the accurate results [Eqs. (76, 77, 78)]. Because of the logarithmic scale, the accurate [Eqs. (76, 77, 78)], approximate [Eqs. (88, 89, 90)], and simple [Eqs. (100, 101, 102)] results coincide within graphical resolution. Note that $|C_{\kappa l}{|}^2=1$ are used.

Figure 3

Time-averaged $n$-photon ionization rates $w_{n+}(\mathcal{E},\omega )$ [Eq. (36)] or equivalently photoelectron energy distribution at the detector for $s$ (orange), $p_0$ (green), $p_{+}$ (blue), $p_{-}$ (red), and total $p$ (brown) orbitals and right circular polarization versus final electronic kinetic energy $k_n^2/2$ [Eq. (37)] for $I_p=0.5$a.u., laser amplitude $\mathcal{E}=0.06$a.u., and laser frequency $\omega =0.057$a.u. (800 nm). The solid and dashed curves correspond to the accurate [Eqs. (76, 77, 78)] and approximate [Eqs. (88, 89, 90)] results, respectively. The spectra for total $p$ orbitals are calculated according to $w_{n+}^p(\mathcal{E},\omega )=w_{n+}^{p_0}(\mathcal{E},\omega )+w_{n+}^{p_{+}}(\mathcal{E},\omega )+w_{n+}^{p_{-}}(\mathcal{E},\omega )$. Note that $|C_{\kappa l}{|}^2=1$ are used. The approximate results of the ionization rates for $p_{+}$ and $p_{-}$ orbitals are equal at the final kinetic energy $2U_p+I_p\approx 1.05$a.u; see text for discussion. In the adiabatic limit $\gamma \ll 1$, all photoelectron distributions are peaked at $2U_p\approx 0.55$a.u. and are the same for $p_{+}$ and $p_{-}$ orbitals.