Analytical solutions for strong field-driven atomic and molecular one- and two-electron continua and applications to strong-field problems

We develop the eikonal-Volkov approximation (EVA) to describe atomic and molecular strong-field dynamics. The main component of this approach is the approximate solution for the continuum states of one and two electrons in the presence of a long-range ionic potential and a strong laser field. These solutions include the laser field fully, the ionic potential, and the electron-electron interaction in the eikonal approximation, and describe the nonperturbative coupling between these interactions. Comparison with numerically evaluated continuum electron wave functions demonstrates quantitative accuracy of the approximate solutions. Their long-time limit yields the quasienergy (Floquet) states of the continuum electron. We also show how to extend the applicability of these solutions to deal with the singularities of the ionic potential, where straightforward eikonal approximation breaks down. The large-angle scattering (hard collision) is incorporated in the EVA formalism using an expansion in the number of hard collisions. Finally, we describe how the EVA formalism can be used to obtain simple analytical expressions for various strong-field problems.

Figure 1

(Color online) Trajectory given by Eq. (16) is specified by position $\mathbf{r}$ at time $t$ and the asymptotic velocity before the pulse is turned on at $t=T$.

Figure 2

(Color online) Comparison of the approximate solution for the phase correction induced by a 1D soft-core potential with the exact numerical solution. The coordinate- and time-dependent phase of the Volkov function is subtracted. The results are shown after three periods of propagation in a cw laser field starting with $A\left(T\right)=0$, for laser intensity $I=9.3\times {10}^{12}\mathrm{W}∕{\mathrm{cm}}^2$ and wavelength $\lambda =800\mathrm{nm}$. The initial asymptotic momentum is $k=1\mathrm{a.u.}$ and electron arrives from $-\infty$. The dashed red curve shows the initial phase.

Figure 3

(Color online) Comparison of different approximations for the phase correction induced by a 1D soft-core potential with the numerical solution (crosses). The coordinate- and time-dependent phase of Volkov function is subtracted. The results are shown after quarter, half, three quarters, and one period of propagation in cw laser field starting with $A\left(T\right)=0$, for laser intensity $I=1.5\times {10}^{14}\mathrm{W}∕{\mathrm{cm}}^2$ and wavelength $\lambda =800\mathrm{nm}$. The initial asymptotic momentum is $k=2\mathrm{a.u.}$ The dashed green curve shows the initial Coulomb phase. Blue solid line shows nonregularized phase $G_{k,T}(x,t)$; red dash-dotted curve shows regularized phase ${\sigma }_{k,T}(x,t)$.

Figure 4

(Color online) Comparison of the WKB amplitude Eq. (37) with the numerical solution (solid line). The results are shown after quarter, half, three quarters, and one period $T_L=2\pi ∕\omega$ of propagation in cw laser field starting with $A\left(T\right)=0$, for laser intensity $I={10}^{13}\mathrm{W}∕{\mathrm{cm}}^2$ and wavelength $\lambda =800\mathrm{nm}$. The initial asymptotic momentum is $k=1\mathrm{a.u.}$ Dashed (green) curve shows the initial WKB amplitude Eq. (55). Solid (black) curve shows the electron density evaluated numerically. Dash-dotted (red) curve shows the WKB amplitude.

Figure 5

(a) Numerically evaluated electron trajectory is plotted as a function of initial coordinate $x_0$ at $t=T_L$. The focal point is $x=-130\mathrm{a.u.}$ (b) To illustrate the appearance of caustics we plot the derivative of the trajectory with respect to the initial coordinate $x_0$. The caustic appears at $x_0\simeq 200\mathrm{a.u.}$, when the derivative is equal to zero.

Figure 6

(Color online) Comparison of different approximations for the amplitude with the numerical solution (solid line). The results are shown after quarter, half, three quarters, and one period of propagation in cw laser field starting with $A\left(T\right)=0$, for laser intensity $I={10}^{13}\mathrm{W}∕{\mathrm{cm}}^2$ and wavelength $\lambda =800\mathrm{nm}$. The initial asymptotic momentum is $k=1\mathrm{a.u.}$ Dashed (green) curve shows the initial WKB amplitude Eq. (55). Solid (black) curve shows the electron density evaluated numerically. Dash-dotted (magenta) curve shows the eikonal amplitude Eq. (41). Dotted (blue) curve shows the adiabatic approximation for the amplitude Eq. (51).

Figure 7

Subcycle ionization dynamics (a) as predicted by SFA and (b) calculated numerically. For details, see [17]. Thin dashed line shows the laser field, with the wavelength $\lambda =800\mathrm{nm}$. The laser field amplitude is ${\mathcal{E}}_0=0.065\mathrm{a.u.}$ (intensity $I=1.5\times {10}^{14}\mathrm{W}∕{\mathrm{cm}}^2$). The system is a 1D soft-core Coulomb potential with the ionization potential of Xe, $I_p=12.13\mathrm{eV}$.

Figure 8

(Color online) Subcycle ionization dynamics around the barrier suppression field. Solid line shows results of numerical simulation and dashed line shows $W_T\left(t\right)$ (a) obtained with the eikonal-Volkov solutions and (b) obtained with the adiabatic approximation to the eikonal-Volkov solutions. Thin dashed line shows the laser pulse, with $\lambda =800\mathrm{nm}$. The laser field amplitude is ${\mathcal{E}}_0=0.065\mathrm{a.u.}$ $(I=1.5\times {10}^{14}\mathrm{W}∕{\mathrm{cm}}^2)$; the model system is the 1D soft-core Coulomb potential with $I_p=12.13\mathrm{eV}$.

Figure 9

(Color online) Second (lowest) order diagram for NSDI; see [38]. Two diagrams describe equivalent processes ensuring symmetry with respect to the two electrons. The approximations are schematically shown only in the left diagram. We use the Hamiltonian $H^{\mathrm{SF}}$ from Eq. (137) from the moment of ionization $t_1$ to the moment of recollision $t_2$ (red shaded area) and the regularized Hamiltonian equation (123) to describe the continuum motion of both liberated electrons from $t_2$ to the moment of observation $T$ (green shaded area).

Figure 10

(Color online) Comparison of different approximations for field-dressed continuum wave functions. Left and right panels show photoelectron spectra on the right and left detectors for the 200 asec XUV pulse arriving at the maximum of the vector potential of the IR field. Numerical results are shown with crosses, results obtained with Volkov functions are shown in red dashed-dotted line, results obtained with nonregularized eV functions are shown as blue dashed line, and results obtained with regularized eV functions with WKB-type amplitude are shown with solid magenta line.

Figure 11

(Color online) Checking different approximations for bound-free transition matrix elements into the field-free continuum wave functions for hydrogen atom. Panel (a) shows the ratio of various approximate matrix elements to the exact matrix element, as a function of final electron energy. Red solid curve marked as PW shows the ratio ${\mathbb{R}}_{\mathrm{pw}}$ for the plane wave continuum. The ratio ${\mathbb{R}}_e$ for the eikonal functions with amplitude $P\left(\mathit{r}\right)=1$ is shown as a solid green curve marked as EI(ph). The ratio ${\mathbb{R}}_{\mathrm{ep}}$ for the eikonal functions with amplitude Eq. (154) is shown as a solid blue curve marked as $\mathrm{EI}(\mathrm{ph},a)$. Panel (b) shows orthogonality of the eikonal continuum wave functions [see Eq. (148)] (EI) and the plane waves (PW) to the hydrogen ground state by plotting the overlap ${\mid \int d\mathit{r}{\Psi }_{\mathit{k}}^{*}\left(\mathit{r}\right)(e^{-r}∕\sqrt{\pi })\mid }^2$.

Figure 12

(Color online) Checking different approximations for bound-free transition matrix elements into the field-free continuum wave functions for hydrogen atom. Panels (a), (b) show the ratio of the approximate matrix element to the exact matrix element for the $3p$ and $3d$ states of the hydrogen atom correspondingly, as a function of the final electron energy. The curves marked as EI(ph) and $\mathrm{EI}(\mathrm{ph},a)$ are obtained using the eikonal continuum wave functions without and with the amplitude correction.

Figure 13

(Color online) Checking the accuracy of the iterative saddle-point procedure. The momentum shift $\Delta \mathbf{q}$ as a function of the arrival time of the ionizing 200 as XUV pulse. Time is in units of the IR cycle with $\lambda =800\mathrm{nm}$. Calculations for the soft-core potential Coulomb potential. The solid curve shows analytical results; dots are numerical results.

Figure 14

(Color online) (a) Ionization amplitude as a function of the moment of return $t_R$; $t_R$ is in the units of laser cycle. The dashed (blue) curve shows tunnel ionization amplitude $a_T\left[t_B\left(t_R\right)\right]$ for the moments of birth calculated neglecting the Coulomb effects in the continuum. The solid (red) curve shows the same tunneling amplitude, but for the moment of birth shifted due to the Coulomb effects. (b) $t_B$ vs $t_R$ with (solid red curve marked as “Coulomb”) and without (dashed blue curve marked as “SFA”) the Coulomb potential taken into account. Dashed-dotted (green) line shows “short-time” analytical approximation to the solid curve. In this calculation $I=2\times {10}^{14}\mathrm{W}∕{\mathrm{cm}}^2$, $\lambda =800\mathrm{nm}$, and $I_p=15.5\mathrm{eV}$.

Figure 15

(Color online) Nonadiabatic ionization amplitude $a_T$ as a function of the moment of return $t_R$, where $t_R$ is in the units of laser cycle. The dashed (blue) curve shows tunnel ionization amplitude $a_T\mathbf{\left(}{t}_B\left(t_R\right)\mathbf{\right)}$ for moments of birth calculated neglecting Coulomb effects in the continuum. The solid (red) curve shows the same tunneling amplitude, but for the moment of birth shifted due to the Coulomb effects. As in Fig. 14, $I=2\times {10}^{14}\mathrm{W}∕{\mathrm{cm}}^2$, $\lambda =800\mathrm{nm}$, and $I_p=15.5\mathrm{eV}$.