Closed-orbit theory for photodetachment in a time-dependent electric field

The standard closed-orbit theory is extended for the photodetachment of negative ions in a time-dependent electric field. The time-dependent photodetachment rate is specifically studied in the presence of a single-cycle terahertz pulse, based on exact quantum simulations and semiclassical analysis. We find that the photodetachment rate is unaffected by a weak terahertz field, but oscillates complicatedly when the terahertz pulse gets strong enough. Three types of closed classical orbits are identified for the photoelectron motion in a strong single-cycle terahertz pulse, and their connections with the oscillatory photodetachment rate are established quantitatively by generalizing the standard closed-orbit theory to a time-dependent form. By comparing the negative hydrogen and fluorine ions, both the in-phase and antiphase oscillations can be observed, depending on a simple geometry of the contributed closed classical orbits. On account of its generality, the presented theory provides an intuitive understanding from a time-dependent viewpoint for the photodetachment dynamics driven by an external electric field oscillating at low frequency.

Figure 1

(a) Field configurations reproduced from Ref. [23] with a slight modification. The gray curve and the solid red line represent the weak laser field and the single-cycle THz pulse, respectively, divided by their corresponding field amplitudes. The laser-field oscillation cannot be resolved due to its high frequency. (b) Time-dependent photodetachment rate obtained from exact quantum simulations for ${\mathrm{H}}^{-}$. The THz pulse strengths $F_m$ are given in the legend for different lines, respectively. All of the curves are normalized using the value of the photodetachment rate at $t=0$ ps without the single-cycle pulse applied ($F_m=0$ kV/cm).

Figure 2

Graphic demonstration of the possible closed classical orbits in a strong single-cycle THz pulse. The specific case in Fig. 1 for $F_m=40$ kV/cm is used as an example. Panels (i) and (ii) are two possible cases for ${\theta }_i=0$, and (iii) and (iv) are two possible cases for ${\theta }_i=\pi$. In each panel, the top subplot is a geometric expression of Eq. (8) and the bottom one shows the corresponding trajectory. For each trajectory, the position of the vertical bold solid (blue) line indicates the starting time, while the vertical bold dashed (red) lines locate its returning time. Both the starting and returning time instants are marked accordingly along each closed orbit (solid curve) in (b), (d), (f), and (h), where the horizontal bold dashed line shows the ion-center location, with the dotted line representing the continuation of each trajectory.

Figure 3

(a) Returning time plot for the relevant closed orbits in Figs. 1 and 2, giving the initial, outgoing time $t_i$ for each possible closed orbit returning back to the atom center at time $t$. The three types of closed orbits are identified in order by the solid blue curve, the dotted lines, and the dashed lines. (b) Time-dependent photodetachment rate for ${\mathrm{H}}^{-}$. The bold dashed curve is given by Eqs. (35, 36, 37) from closed-orbit theory (COT), while the solid blue curve is from exact quantum simulations by directly solving the time-dependent Schrödinger equation (TDSE) in Eq. (4). (c) Comparison between the photodetachment rates for ${\mathrm{H}}^{-}$ (thin blue curve) and ${\mathrm{F}}^{-}$ (bold red line) obtained from quantum simulations. All of the quantum results have been normalized using the value of the photodetachment rate at $t=0$ ps without any external fields applied.

Figure 4

Phase dependence on a simple geometry of the closed orbit. (a) Returning time plot for the relevant closed orbits. The solid blue curve and the dotted and the dashed lines correspond to the closed orbits with (${\theta }_i=0, {\theta }_{\mathrm{ret}}=\pi$), ${\theta }_i={\theta }_{\mathrm{ret}}=0$, and (${\theta }_i=\pi , {\theta }_{\mathrm{ret}}=0$), respectively. (b),(c) The time-dependent photodetachment rate for ${\mathrm{H}}^{-}$ and ${\mathrm{F}}^{-}$, respectively. The bold dashed and the solid blue curves are given by closed-orbit theory and quantum simulations, respectively. (d) Comparison between the photodetachment rates for ${\mathrm{H}}^{-}$ (thin red curve) and ${\mathrm{F}}^{-}$ (bold gray line) obtained from quantum simulations. All of the quantum results have been normalized using the value of the photodetachment rate at $t=0$ ps without any external fields applied.

Figure 5

The same as Fig. 4, but for higher photon energy and larger field strength as listed on the top. Here, $t_L$ in Eq. (1) was set to be $50T_0$ as in Ref. [23].

Figure 6

Amplitude dependence on the returning electron momentum for ${\mathrm{H}}^{-}$. The thin solid (red) curves in (a)–(c) are reproduced from the bold dashed lines in Figs. 3, 4, and 5, respectively, for comparison. The bold gray lines are calculated from Eq. (36) without the term $C\left(k_{\mathrm{ret}}\right)/C\left(k_0\right)$ included.

Figure 7

(a)–(c) Comparison between the time-dependent calculations from Eq. (35) (heavy thin curves) and a static-field approximation after Eq. (38) (light bold lines) by increasing the single-cycle pulse duration. (d) The returning time plot for the relevant closed orbits in (c). The solid blue curve and the dotted and dashed lines correspond to the closed orbits with (${\theta }_i=0, {\theta }_{\mathrm{ret}}=\pi$), ${\theta }_i={\theta }_{\mathrm{ret}}=0$, and (${\theta }_i=\pi , {\theta }_{\mathrm{ret}}=0$), respectively.

Figure 8

Comparison between the time-dependent calculations from Eq. (35) (heavy thin curves) and a static-field approximation after Eq. (38) (light bold lines) by increasing the single-cycle pulse strength.