Role of reflections in the generation of a time delay in strong-field ionization

The problem of time delay in tunneling ionization is revisited. The origin of time delay at the tunnel exit is analyzed, underlining the two faces of the concept of the tunneling time delay: the time delay around the tunnel exit and the asymptotic time delay at a detector. We show that the former time delay, in the sense of a delay in the peak of the wave function, exists as a matter of principle and arises due to the sub-barrier interference of the reflected and transmitted components of the tunneling electronic wave packet. We exemplify this by describing the tunneling ionization of an electron bound by a short-range potential within the strong-field approximation in a “deep tunneling” regime. If sub-barrier reflections are extracted from this wave function, then the time delay of the peak is shown to vanish. Thus, we assert that the disturbance of the tunneling wave packet by the reflection from the surface of the barrier causes a time delay in the neighborhood of the tunnel exit.

Figure 1

(a) Ionization amplitude $|{\psi }_i{(x,t)|}^2$ as defined in Eq. (17). At every cross section in $x$, the temporal peak of the wave function was determined (plotted in blue) which can be interpreted as the trajectory of the peak. (b) Comparison of trajectories in the region near the tunnel exit, $x_e$. The displayed curves are the peak trajectory [blue, given by the maximum of $|{\psi }_i{(x,t)|}^2$], the classical trajectory (orange, given by the Newton equation starting at the tunnel exit), and the Wigner trajectory [green, given by Eq. (25)]. Under the barrier ($x<10$, shaded blue), there is an increasing delay of the peak with respect to the laser peak. Outside the barrier, the peak trajectory rapidly converges with the classical and Wigner electron trajectory. Near the atomic core ($x<2$), the bound state, $|{\psi }_0\left(x\right)|\gg |{\psi }_i\left(x\right)|$, dominates the full wave function $\psi \left(x\right)={\psi }_0\left(x\right)+{\psi }_i\left(x\right)$ so the meaning of a trajectory for ionization is lost.

Figure 2

Probability distribution $|{\psi }_i(x_e,t){|}^2$ vs time at the classical tunnel exit $x_e=I_p/E_0$. This distribution is peaked at a greater time, $t_m\approx 7.6$ a.u. $\approx 183$ as, marked by the dot, than the peak of the laser pulse.

Figure 3

Log-log plots of the scaling with respect to field strength, $E_0$, of the (a) Wigner time delay and (b) group velocity at the classical tunnel exit, $x_e=I_p/E_0$, for the time evolved SFA wave function (red dots) and the adiabatic constant field wave function (blue lines). For the time evolved wave function, the maxima in time, $t_m$, of the probability distribution $|\psi (x_e,t){|}^2$ and its derivative were calculated; for the adiabatic case, results follow from Eqs. (26) and (27). The agreement in the trends is good, deteriorating as one approaches the OTBI threshold $E_{\mathrm{th}}\approx 0.25$ a.u.

Figure 4

The solution to the electron in a constant field problem is given by a superposition of Airy functions of the first and second kind, $\mathrm{Ai}\left(\tilde{x}\right)$ and $\mathrm{Bi}\left(\tilde{x}\right)$, plotted in blue and orange, respectively. These functions have very accurate asymptotic expansions, shown dashed, which can be derived by considering the saddle points of the Airy integrals (28) and (29). Under the barrier, $x<10$ (shaded blue), these expansions show that the wave-function components $\mathrm{Ai}\left(\tilde{x}\right)$ and $\mathrm{Bi}\left(\tilde{x}\right)$, respectively, correspond to the reflected and transmitted components of the wave function.

Figure 5

The Airy integral ${\int }_{\gamma }dsexp(\tilde{x}s-s^3/3)$ is defined on the complex $s$ plane; it converges when the end points of its infinite contour, $\gamma$, lie in the shaded areas. The canonical contours defining the Airy function of the first (${\gamma }_1$) and second (${\gamma }_2-{\gamma }_3$) kind are shown in (a). Contributions to the Airy integral arise principally from portions of the contour near the saddle points of the integrand function $exp(\tilde{x}s-s^3/3)$. The configuration of the saddles $s_{\pm }\left(x\right)=\pm \sqrt{\tilde{x}}$ (red dots) in the complex plane is shown for positions (b) inside the barrier ($\tilde{x}>0$) and (c) outside the barrier ($\tilde{x}<0$). Through each saddle point pass two perpendicular level curves, the path of steepest descents (solid blue) and the path of steepest ascents (dashed gray). Deforming the defining contours in (a) through the steepest descents contours in (b) and (c) provides asymptotic expansions for the Airy function, as in Eqs. (31) and (32).

Figure 6

Complex $t^{\prime }$ plane of the argument $\mathrm{\Phi }(x,t,t^{\prime })$ of the wave-function integral ${\psi }_i={\int }_{t_0}^{t}d{t}^{\prime }exp\left(i\mathrm{\Phi }\right)$, for parameters $x=9$ and $\omega t=\pi /40$. For these configurations there are two saddle points, denoted by a circle and triangle, the former of which, $t_{+}$, corresponds to reflections. The saddle-point method can be used to solve the wave-function integral, when the path is taken over both saddle points. We use a partial path given by (33) to remove the contribution of reflections by integrating over only one of the saddle points. Other possible configurations of the saddle points are shown in Fig. 11.

Figure 7

Log-log plot of the relative contributions to the wave-function amplitude from the portion of the integration path up to the saddle point (blue) and from the saddle point to the given real time $t$ (orange) for various positions $x$ under the classical barrier (i.e., far from the atomic core, but smaller than the tunnel exit). The behaviors are approximately decaying and growing exponentials, respectively, analogous to transmitted and reflected parts of the wave function.

Figure 8

(a) Temporal probability distributions at the position $x=8a.u.$ (under the barrier, near the tunnel exit), for the ionized wave function, ${\psi }_i$, and the pseudo-wave-function neglecting reflections, ${\psi }_{\mathrm{nr}}$. The physical wave function has accrued a Wigner time delay with respect to the laser field, whereas the maximum of the pseudo-wave-function is synchronous with the laser field peak. (b) Trajectories in the $(x,t)$ plane via the maximum of the wave function ${\psi }_i$ [given by $t_m\left(x\right)$], the maximum of the wave function neglecting reflections ${\psi }_{\mathrm{nr}}$, and the energy derivative of the constant field wave function ${\tau }_W\left(x\right)$. The classical electron trajectory, $x_{\mathrm{cl}}\left(t\right)$, starting at the tunnel exit at the peak of the laser field is also shown. Under the barrier, the probability distribution neglecting reflections $|{\psi }_{\mathrm{nr}}{|}^2$ shows zero time delay. In regions where Eq. (34) does not apply, mostly near the classical tunnel exit, we may not identify reflections or plot a subsequent trajectory.

Figure 9

Positions of the upper and lower saddle points of the integral (17) in the complex $t^{\prime }$ plane with varying $x$ and for fixed $\omega t=\pm 0.1\pi /2$ (blue and red, respectively). The saddles draw smooth curves with varying $x$, where the arrows indicate growing values of $x$. For values of $x\lesssim 7$, the lower saddle point disappears below the imaginary axis. Each pair of curves is centered around the line $Re\left(\omega t^{\prime }\right)=\omega t$, shown in dashed line. For the case $\omega t=0$, not shown, the two lines meet one point corresponding to $x=x_t$.

Figure 10

Pictorial representation of the square barrier potential for an incident monochromatic wave. The wave function is a piecewise solution of the Schrödinger equation for the three regions shown, given by Eqs. (A2) and (A3).

Figure 11

Configurations of the saddle points of the argument $\mathrm{\Phi }(x,t,t^{\prime })$ of the wave-function integral ${\psi }_i={\int }_{t_0}^{t}d{t}^{\prime }exp\left(i\mathrm{\Phi }\right)$ in the complex $t^{\prime }$ plane, for parameter ranges $x=7,10,14$ and $\omega t=-\pi /20,0,+\pi ,20$. The dashed line corresponds to $t^{\prime }=0$ and each plot is centered around the (scaled) observation time $\omega t$. The scale and color coding are identical to those of Fig. 6. As $x$ increases the two saddle points approach vertically and, after a closest approach, separate horizontally. For the peak of the pulse, $\omega t=0$, this closest approach is zero and the two saddle points merge at the point $x_t$. Otherwise, varying $t$ skews the relative orientation of the saddle points around the line $Re\left[t^{\prime }\right]=t$.