Generation of quantum vortices in photodetachment: The role of the ground-state wave function

Formation of quantum vortices in laser-induced photodetachment from negative ions is analyzed. The driving laser field consists of a single ultrashort pulse of circular polarization and the unperturbed ground-state wave function of the anion is found in either the $s$ or $p$ state. In particular, numerical illustrations for the photodetachment from ${\mathrm{H}}^{-},$ ${\mathrm{O}}^{-},$ ${\mathrm{K}}^{-},$ and a model ${\mathrm{A}}^{-}$ anion are presented. Special attention is paid to the symmetry of the ground-state wave function and ionization potential over the final vortex pattern. It is shown that the two-dimensional spectra of photoelectrons in momentum space comprise three well-defined regions: The low-energy (central) region, multiphotonlike zone, and supercontinuum. While the supercontinuum does not contribute to vorticity and the multiphoton zone depends only on the laser field characteristics, vortices in the low-energy region strongly depend on the bound-state wave function and its ionization potential.

Figure 1

Temporal evolution of the tips of the vector potential $e\mathit{A}\left(\varphi \right)$ (left panel), electric field $e\mathcal{E}\left(\varphi \right)$ (second panel), and the two-dimensional spiral of ionization $U_p\left(\varphi \right)$ (third panel) as parametric plots. Those curves start and end at (0,0) and evolve counterclockwise while changing $\varphi$ from zero to $2\pi$. The laser pulse is defined by Eqs. (42, 43, 44, 45, 46, 47) and comprises $N_{\mathrm{osc}}=3$ field oscillations. The polarization is circular $(\delta =\pi /4)$ and the wavelength and intensity are fixed to ${\lambda }_{\mathrm{L}}=4000$ nm and $I=2.5\times {10}^{11}$ W/${\mathrm{cm}}^2$, respectively. In the right panel we present the frequency distribution of the pulse [see Eqs. (49) and (50)] indicating the position at which its maximum is achieved (red vertical line) and the FWHM (red horizontal line).

Figure 2

In the upper panels we show the triply differential probability of detachment $\mathcal{P}\left(\mathit{p}\right)$ [Eq. (40)] in the $p_x{p}_y$ plane (${\theta }_{\mathit{p}}=\pi /2$) as a function of the scaled kinetic energy $E\left(\mathit{p}\right)/{\omega }_{\mathrm{L}}$ and the azimuthal angle ${\phi }_{\mathit{p}}$. While the distribution for the (1,0) state is calculated at $p_z=0.05$ a.u. (we show its projection over the $p_x{p}_y$ plane), the remaining distributions are calculated at $p_z=0$. In the second row we show the triply differential probability of detachment as a function of $E\left(\mathit{p}\right)/{\omega }_{\mathrm{L}}$ at constant polar and azimuthal angles of electron detection, ${\theta }_{\mathit{p}}=\pi /2$ and ${\phi }_{\mathit{p}}=\pi$. There, broad multiphotonlike peaks can be observed with deep minima between them. The vertical red line, at $E\left(\mathit{p}\right)\approx 0.7{\omega }_{\mathrm{L}}$, locates the transition from the central region to the multiphoton regime, and its position is similar for all distributions. In the third row we present the topological charges $m(p_r,p_z)$ calculated from Eq. (41) as a function of the energy $E\left(p_r\right)/{\omega }_{\mathrm{L}}=p_r^2/\left(2m_{\mathrm{e}}{\omega }_{\mathrm{L}}\right)$. The contour $\mathcal{C}$ is a circle of radius $p_r$, parallel to the $p_x{p}_y$ plane and at a distance $p_z$ from the origin of coordinates. Note that, in all cases, $m(0,p_z)=0$. The laser pulse is as defined in Fig. 1 and the target anion is ${\mathrm{H}}^{-}$ for the $s$ state [20] and a model anion ${\mathrm{A}}^{-}$ for $p$ states.

Figure 3

Magnitude (upper row) and phase (lower row) of the probability amplitude of detachment in the two-dimensional momentum space $\mathcal{A}\left({\mathit{p}}_{\perp }\right)$ [Eq. (20)]. The magnitude $|\mathcal{A}\left({\mathit{p}}_{\perp }\right){|}^{\nu }$ is raised to the power $\nu =0.5$ for visual purposes and all plots are obtained by setting $p_z=0$ (i.e., in the $p_x{p}_y$ plane), except for the case (1,0). There $p_z=0.05$ a.u. The driving laser pulse is described in Fig. 1. Each column corresponds to a different ground-state wave function. While the left column shows the results for the ${\mathrm{H}}^{-}$ ion in the $s$ state [20, 48], the remaining columns are for the model ${\mathrm{A}}^{-}$ anion in the $p$ states.

Figure 4

The same as in Fig. 2 but for the ${\mathrm{O}}^{-}$ anion. This time, only the probability distributions and topological charges for $p$ states are displayed. The ground-state wave function is defined by Eq. (29) with the parameters $\kappa =0.328$ a.u. and $A=0.42$ a.u. This corresponds to an ionization potential $I_p=1.46$ eV. As before, we set $p_z=0$ for the (1,1) and $(1,-1)$ cases, and $p_z=0.05$ a.u. for (1,0). In all cases, $m(0,p_z)=0$ in the topological charge plots.

Figure 5

The same as in Fig. 3 but for the ${\mathrm{O}}^{-}$ anion. Here we only present the results for $p$ states. The parameters determining the ground-state wave function are $\kappa =0.328$ a.u. and $A=0.42$ a.u. such that $I_p=1.46$ eV [see Eq. (29)]. All distributions are calculated for $p_z=0$, except for the case (1,0) for which $p_z=0.05$ a.u.

Figure 6

Phase of the probability amplitude, $\arg \left[\mathcal{A}\left({\mathit{p}}_{\perp }\right)\right]$ (upper panel), and topological charges, $m(p_r,0)$ (lower panel), for the photodetachment from ${\mathrm{O}}^{-}$ in the (1,1) state. The laser field parameters are the same as in Fig. 1. In contrast to Figs. 4 and 5, we show here our results as a function of the azimuthal angle ${\phi }_{\mathit{p}}\in ]-\pi ,\pi ]$ and the radius $p_r=\left|{\mathit{p}}_{\perp }\right|$, with $p_z=0$. The range in momentum $0.03⩽p_r/p_{\mathrm{at}}⩽0.06$ coincides with the central polygon of low probability shown in the lower-right panel of Fig. 5.

Figure 7

The same as in Fig. 6 but for the three states $m_{\ell }=-1,0,1$. This time, the range of momentum is chosen such that the single vortices in the central region are observed. The red vertical line at $p_0=0.1035p_{\mathrm{at}}$ indicates the approximate transition to the multiphotonlike zone (note that $p_0^2/2m_{\mathrm{e}}=0.47{\omega }_{\mathrm{L}}$).

Figure 8

Magnitude of the probability amplitude of detachment, $|\mathcal{A}\left({\mathit{p}}_{\perp }\right){|}^{\nu }$, with $\nu =0.5$ [see Eq. (20)], in the $p_x{p}_y$ plane (i.e., $p_z=0$). The target anion is ${\mathrm{O}}^{-}$ in the ground state $(\ell ,m_{\ell })=(1,1)$. While each panel corresponds to a different intensity, the remaining laser field and ion parameters are the same as in Fig. 5. Here, each intensity corresponds to $I_1=5.0\times {10}^{10}$ W/${\mathrm{cm}}^2$, $I_2=2I_1$, $I_3=3I_1,...,I_{12}=12I_1=6.0\times {10}^{11}$ W/${\mathrm{cm}}^2$.

Figure 9

The same as in Fig. 8 but for the phase of the probability amplitude of detachment, $\arg \left[\mathcal{A}\left({\mathit{p}}_{\perp }\right)\right]$.

Figure 10

The same as in Figs. 2 and 4 but for the ${\mathrm{K}}^{-}$ anion. We show only the results for $s$ states [see Eq. (28)] with the parameters $\kappa =0.192$ a.u. and $A=1.24$ a.u. such that $I_p=0.502$ eV. Again, we observe that $m(0,0)=0$, as expected.

Figure 11

The same as in Figs. 3 and 5 but for the ${\mathrm{K}}^{-}$ anion. Due to the electron configuration of the ion, only the results for $s$ states are displayed. The probability amplitude of detachment is calculated for the ground-state wave function in Eq. (28) with the parameters $\kappa =0.192$ a.u. and $A=1.24$ a.u. ($I_p=0.502$ eV).

Figure 12

The same as in Fig. 11 but for a weaker driving field. While the intensity is now $I=1.0\times {10}^{11}$ W/${\mathrm{cm}}^2$, the remaining field parameters are the same as in Fig. 1.