Causality and quantum interference in time-delayed laser-induced nonsequential double ionization

We perform a detailed analysis of the importance of causality within the strong-field approximation and the steepest-descent framework for the recollision-excitation with subsequent tunneling ionization (RESI) pathway in laser-induced nonsequential double ionization (NSDI). In this time-delayed pathway, an electron returns to its parent ion and, by recolliding with the core, gives part of its kinetic energy to excite a second electron at a time $t^{\prime }$. The second electron then reaches the continuum at a later time $t$ by tunneling ionization. We show that, if $t^{\prime }$ and $t$ are complex, the condition that recollision of the first electron occurs before tunnel ionization of the second electron translates into boundary conditions for the steepest-descent contours and thus puts constraints on the saddles to be taken when computing the RESI transition amplitudes. We also show that this generalized causality condition has a dramatic effect on the shapes of the RESI electron momentum distributions for few-cycle laser pulses. Physically, causality determines how the dominant sets of orbits of an electron returning to its parent ion can be combined with the dominant orbits of a second electron tunneling from an excited state. All features encountered are analyzed in terms of such orbits and their quantum interference.

Figure 1

Partial RESI transition probabilities for a monochromatic driving field, as function of the momentum components $p_{1,2\parallel }$ parallel and $p_{1,2\perp }$ perpendicular to the laser field polarization. Panels (a) and (b) depict the transition probability $|M^{\left(1\right)}\left({\mathbf{p}}_1\right){|}^2$ for the first electron, Eq. (17), while panels (c) and (d) give the transition probability $|M^{\left(2\right)}\left({\mathbf{p}}_2\right){|}^2$ for the second electron, Eq. (18). Panel (a) shows the contribution of the dominant trajectories of the first electron (one per half cycle), while panel (b) shows its interference with the subdominant partner trajectories in the same half cycle, treated collectively in uniform approximation. The trajectory pairs from the two half cycles result in identical distributions for opposite parallel momenta; interference between these pairs of trajectories is negligible. Panel (c) shows the contribution of an individual orbit of the second electron, while panel (d) shows its interference with the second orbit in the same field cycle. The distributions have been normalized with regard to the largest value in each panel. Parameters are for helium ($E_1^{\left(\text{g}\right)}=0.97$ a.u., $E_2^{\left(\text{g}\right)}=2$ a.u., and $E_2^{\left(\text{e}\right)}=0.5$ a.u.), and the driving field is monochromatic with intensity $I=3\times {10}^{14}$ W/cm${}^2$ and frequency $\omega =0.057$ a.u.

Figure 2

Schematic representation of the electric field $\mathbf{E}\left(t\right)$ and the corresponding vector potential $\mathbf{A}\left(t\right)$ for a few-cycle pulse of five cycles. The arrows in panel (a) indicate the approximate times around which the first electron leaves, in case it returns at a crossing. The complex return and start times for the indicated pairs of orbits will have real parts in the vicinity of such times. The rectangles in part (b) mark the approximate ionization times for the second electron associated with the orbits utilized in this work (Orbits 1 to 5). The fields have been normalized to $E\left(t\right)/E_0$ and $A\left(t\right)/A_0$, where $E_0$, $A_0$ denote the field amplitudes.

Figure 3

Contributions from specific sets of orbits to the momentum-resolved RESI transition probabilities $|M^{\left(1\right)}\left({\mathbf{p}}_1\right){|}^2$ of the first electron, Eq. (17), for the few-cycle pulse of Fig. 2. Panels (a), (b), (c), and (d) correspond to Pairs 2, 3, 4, and 5$\left(e_1\right)$, respectively (to eliminate distorting interference effects, only the dominant orbit, which remains physical beyond the classical boundary, has been taken into account). The pulse has peak intensity $I=2.5\times {10}^{14}$ W/cm${}^2$, frequency $\omega =0.057$ a.u. and carrier-envelope phase $\phi =0$; the other parameters are the same as in Fig. 1.

Figure 4

Contributions of individual orbits to the momentum-resolved RESI transition probabilities $|M^{\left(2\right)}\left({\mathbf{p}}_2\right){|}^2$ for the second electron, Eq. (18), for the same few-cycle pulse and atomic parameters as in Fig. 3. Panels (a), (b), (c), and (d) depict the contributions from the orbits 1, 2, 3, and 4$\left(e_2\right)$, respectively (see Fig. 2).

Figure 5

Partial transition probabilities for a few-cycle pulse as function of the momentum components $p_{n\parallel }$ and $p_{n\perp }$. Panels (a) and (b) depict the partial transition probability $|M^{\left(1\right)}\left({\mathbf{p}}_1\right){|}^2$ for the first electron, while panels (c) and (d) depict $|M^{\left(2\right)}\left({\mathbf{p}}_2\right){|}^2$ for the second electron. These are obtained by coherently adding the contributions of orbits addressed in Figs. 3 and 4 (the field and atomic parameters are the same as in these figures). In panels (a) and (c), this sum is restricted to the dominant orbits of pairs $3\left(e_1\right)$ and $4\left(e_1\right)$, as well as $1\left(e_2\right)$ and $2\left(e_2\right)$, respectively. The distributions have been normalized with regard to the maximum probability density in each panel.

Figure 6

Correlated RESI two-electron momentum distributions $F(p_{1\parallel },p_{2\parallel })$ as functions of the electron momentum components $p_{n\parallel }$ ($n=1,2$) parallel to the laser-field polarization. The atomic parameters correspond to helium (see Fig. 1). Panels (a) to (c) have been computed for a monochromatic field of intensity $I=3\times {10}^{14}$ W/cm${}^2$ and frequency $\omega =0.057$ a.u., while in panels (d) to (i) a five-cycle pulse of intensity $I=2.5\times {10}^{14}$ W/cm${}^2$, frequency $\omega =0.057$ a.u., and carrier-envelope phase $\phi =0$ has been used. The middle row of panels (d)–(f) show results for the few-cycle pulse if causality is discarded (these results are therefore unphysical), while the bottom panels (g)–(i) show the results for the few-cycle pulse including the causality restrictions. From the left to the right column of panels, we integrated over a transverse momentum range centered around low, medium, and large transverse momenta, respectively.

Figure 7

Construction of steepest-descent contours for the evaluation of $M^{\left(2\right)}({\mathbf{p}}_2;t^{\prime })$, Eq. (21), for monochromatic driving, with complex rescattering times $t^{\prime }$ corresponding to $p_{1\perp }=0$, $p_{1\parallel }=2\sqrt{U_{\text{p}}}$. In panel (a), these rescattering times, obtained by solving the saddle-point Eqs. (11, 12, 13), are indicated by the star symbols. For orientation, the curves show the trace of times $t^{\prime }$ when $p_{1\parallel }$ is varied from $-5\sqrt{U_{\text{p}}}$ to $+5\sqrt{U_{\text{p}}}$ while $p_{1\perp }=0$ remains fixed. In panels (b) and (d), the circles indicate the ionization times $t$ of the second electron for $(p_{2\parallel },p_{2\perp })=(0,0)$. These times are obtained by solving the saddle-point Eq. (15). Panel (b) displays the steepest-descent and steepest-ascent contours that pass through the saddles obtained for the second electron, without taking into account causality (solid and dashed lines, respectively). The solid circles indicate saddles that are physically allowed (with positive $\mathrm{Im}{S}_2$), while the open circles are unphysical saddles. Panel (c) displays the boundary curves; that is, the steepest-descent lines passing through the rescattering times $t^{\prime }$ (star symbols). These lines are all determined from the condition $\mathrm{Re}{S}_2=\mathrm{const}$. Panel (d) illustrates a steepest-descent contour passing through the relevant saddles and incorporating causality. A full steepest-descent contour must make use of solid lines in order to connect a lower integration limit $t=t^{\prime }$ (stars) to the upper integration limit (at $t=+\infty$). The thick lines show the full contour for one selected starting value, indicated by the solid star. This contour only visits a selection of all physical saddles (dark dots), and the saddles that violate causality (light dots) are left out.

Figure 8

Steepest descent contours incorporating causality for the same field and momentum components of the first electron as in Fig. 7, but for $(p_{2\parallel },p_{2\perp })=(1.5\sqrt{U_{\text{p}}},0)$ and $(p_{2\parallel },p_{2\perp })=(3\sqrt{U_{\text{p}}},0)$ [panels (a) and (b), respectively]. As in the previous figure, the thick lines show a sample contour for one selected starting value, indicated by the solid star. Between panels (a) and (b) a Stokes transition occurs, and one saddle is lost. The remaining notation is the same as in the previous figure.

Figure 9

Construction of steepest-descent contours for the few-cycle pulse employed in Sec. 3b. Labels refer to the orbit pairs (first electron) and orbits (second electron) indicated in Fig. 2. Panel (a) shows the effect of the finite pulse length on the rescattering times $t^{\prime }$ of the first electron. The stars again denote the rescattering times for $p_{1\perp }=0$, $p_{1\parallel }=2\sqrt{U_{\text{p}}}$. In the tails of the pulse the driving is weak, resulting in rescattering times that are shifted away from the real axis. Therefore, some of these trajectories of the first electron become negligible. Panels (b)–(d) shows the ionization times $t$ of the second electron, together with the steepest-descent contours incorporating causality, for momentum components $(p_{2\parallel },p_{2\perp })=(0,0)$, $(p_{2\parallel },p_{2\perp })=(1.5\sqrt{U_{\text{p}}},0)$, and $(p_{2\parallel },p_{2\perp })=(3\sqrt{U_{\text{p}}},0)$, respectively. The solid star indicates a rescattering time in the center of the pulse, which gives the largest semiclassical contribution for the momentum combination $(p_{1\parallel },p_{1\perp })$ chosen. The value of this time is close to that in the monochromatic case. A Stokes transition in which the saddle corresponding to Orbit $3\left(e_2\right)$ is lost occurs between panels (c) and (d). The remaining parameters and the notation employed are the same as in Figs. 7 and 8.