Field-ionization threshold and its induced ionization-window phenomenon for Rydberg atoms in a short single-cycle pulse

We study the field-ionization threshold behavior when a Rydberg atom is ionized by a short single-cycle pulse field. Both hydrogen and sodium atoms are considered. The required threshold field amplitude is found to scale inversely with the binding energy when the pulse duration becomes shorter than the classical Rydberg period, and, thus, more weakly bound electrons require larger fields for ionization. This threshold scaling behavior is confirmed by both three-dimensional classical trajectory Monte Carlo simulations and numerically solving the time-dependent Schrödinger equation. More surprisingly, the same scaling behavior in the short pulse limit is also followed by the ionization thresholds for much lower bound states, including the hydrogen ground state. An empirical formula is obtained from a simple model, and the dominant ionization mechanism is identified as a nonzero spatial displacement of the electron. This displacement ionization should be another important mechanism beyond the tunneling ionization and the multiphoton ionization. In addition, an “ionization window” is shown to exist for the ionization of Rydberg states, which may have potential applications to selectively modify and control the Rydberg-state population of atoms and molecules.

Figure 1

(a) Three typical single-cycle pulses. The solid, the dashed, and the dotted curves represent, respectively, “symmetric,” “asymmetric,” and “inverted” field profiles given by Eq. (1). (b) Threshold behaviors for sodium with $t_w=0.1$ ps. The red solid points, the open squares, and the points indicated by “$+$” correspond respectively to symmetric, asymmetric, and inverted pulses. The bold dot-dashed line is from Eq. (4). The solid line indicates the $n^{-3}$ scaling behavior. (c) Threshold behaviors for both hydrogen and sodium with different $t_w$ values, where a symmetric pulse is applied. The blue circles, the gray triangles, and the red squares display the thresholds for sodium with $t_w=1$ ps, $t_w=0.5$ ps, and $t_w=0.1$ ps, respectively. The corresponding thresholds for hydrogen are denoted by $+$. The bold dashed lines are from Eq. (5). In both (b) and (c), the thin dashed and the thin dot-dashed lines (black online) represent the thresholds for hydrogen and sodium in the long pulse limit, respectively.

Figure 2

Other similar results as those in Fig. 1. (a) The threshold behaviors for hydrogen with different pulse profiles. The red solid points, the open squares, and the points indicated by “$+$” correspond respectively to “symmetric,” “asymmetric,” and “inverted” pulses. (b) The threshold values for hydrogen Rydberg states with different angular-momentum quantum numbers, where a symmetric field profile is used. The red solid points, the open squares, and the points labeled by $+$ correspond respectively to the required threshold amplitudes for the Rydberg atom initially at an $s$ state, a $p$ state, and a $d$ state. (c) is the same as (b) but for sodium, where an effective quantum number $n^{*}$ is used by including the quantum defects. In all the subplots (a)–(c), $t_w=0.1$ ps. The dashed and the dot-dashed lines (black online) represent the thresholds for hydrogen and sodium in the long pulse limit, respectively.

Figure 3

Threshold scaling behaviors with (a) for Rydberg atoms and (b) for much lower bound states. $\omega$ is defined as $\pi /\left(2t_w\right)$. A “symmetric” pulse is applied. In (a), the open and the solid circles are from Fig. 1 for $t_w=1$ ps. The quantum results for hydrogen $15d$ state at different scaled frequencies are shown by the red open squares. The dashed and the dot-dashed lines represent the thresholds for hydrogen and sodium in the low-frequency limit, respectively. In (b), the open and the solid circles correspond to the thresholds for hydrogen atoms initially in $1s$ state and $3d$ state, respectively. In both (a) and (b), the bold solid curve (green online) is from Eq. (6). The thin solid curve in (b) shows the variation of Keldysh parameter $\gamma$ associated with the bold solid line. The unity value of $\gamma$ is indicated by the dotted line.

Figure 4

“Ionization window” for hydrogen Rydberg atoms initially in a $d$ state. The “symmetric” pulse in Fig. 1 is used. The applied field amplitudes for the squares, the triangles, the open, and the solid circles are $2.5\text{kV}/\mathrm{cm},5\text{kV}/\mathrm{cm},50\text{kV}/\mathrm{cm}$, and $100\text{kV}/\mathrm{cm}$, respectively. $t_w=0.1$ ps for the open circles, and $t_w=1$ ps for the others.