Photodetachment of hydrogen negative ions in bichromatic oscillating electric fields

The photodetachment of a hydrogen negative ion in the bichromatic oscillating electric field is investigated using the time-dependent closed orbit theory. The photodetachment cross section is specifically calculated in the bichromatic oscillating electric field with a 1:2 frequency ratio and various phase differences. It is found that the oscillatory structure appears in the photodetachment cross section, which depends sensitively on the strength and the phase difference of the bichromatic oscillating electric field. The time-dependent photodetachment cross section is nearly unaffected by a weak bichromatic oscillating electric field. However, with the increase of the oscillating electric field strength, four different types of the detached electron's closed orbits are identified, which makes the photodetachment cross section oscillate in a much more complicated way. In addition, the influence of the phase difference in the bichromatic oscillating electric field on the photodetachment cross section is also investigated quantitatively. The formula we put forward for the time-dependent photodetachment cross section is universal and can be used to calculate the cross section for other frequency ratios in the bichromatic oscillating electric field. On account of its generality, our work provides a clear and intuitive picture for the photodetachment processes of a negative ion in the bichromatic oscillating electric field, and may guide future experimental research on the photodetachment dynamics in the oscillating electric field.

Figure 1

The $t-t_i$ curve for the detached electron's closed orbit in the bichromatic oscillating electric field. The photon energy is $E_p=1.0\mathrm{eV}$; the strength of the electric field is $F_0=200\mathrm{kV}/\mathrm{cm}$; the frequency of the bichromatic oscillating electric field is $2.42\times {10}^{-5}$ and $4.84\times {10}^{-5}\mathrm{a}.\mathrm{u}.$, respectively. Different types of closed orbits are distinguished by different lines.

Figure 2

Graphic demonstrations of four types of closed orbits for the detached electron in the bichromatic oscillating electric field. The photon energy is $E_p=1.0\mathrm{eV}$; the strength of the electric field is $F_0=200\mathrm{kV}/\mathrm{cm}$; the frequency of the bichromatic oscillating electric field is $2.42\times {10}^{-5}$ and $4.84\times {10}^{-5}\mathrm{a}.\mathrm{u}.$, respectively. (a) The up closed orbit; (b) the down closed orbit; (c) the up-down closed orbit; (d) the down-up closed orbit.

Figure 3

(a) The detached electron's trajectory moving along the $+z$ axis in the bichromatic oscillating electric field. The photon energy is $E_p=1.0\mathrm{eV}$; the strength of the electric field is $F_0=10\mathrm{kV}/\mathrm{cm}$; the frequency of the bichromatic oscillating electric field is $2.42\times {10}^{-5}$ and $4.84\times {10}^{-5}\mathrm{a}.\mathrm{u}.$, respectively. (b) The detached electron's trajectory moving along the $-z$ axis in the same bichromatic oscillating electric field. (c) The $t-t_i$ curve for the detached electron's up closed orbit in the bichromatic oscillating electric field. (d) The solid line denotes the corresponding time-dependent photodetachment cross section in the bichromatic oscillating electric field, while the red dashed line is the case without the time-dependent electric field.

Figure 4

The $t-t_i$ curve for the detached electron's up orbit and up-down orbits in the bichromatic oscillating electric field. The photon energy is $E_p=1.0\mathrm{eV}$; the strength of the electric field is $F_0=25\mathrm{kV}/\mathrm{cm}$; the frequency of the bichromatic oscillating electric field is $2.42\times {10}^{-5}$ and $4.84\times {10}^{-5}\mathrm{a}.\mathrm{u}.$, respectively. (b) The corresponding time-dependent photodetachment cross section in the bichromatic oscillating electric field.

Figure 5

(a) The $t-t_i$ curve for the detached electron's four different types of closed orbits in the bichromatic oscillating electric field. The photon energy is $E_p=1.0\mathrm{eV}$; the strength of the electric field is $F_0=50\mathrm{kV}/\mathrm{cm}$; the frequency of the bichromatic oscillating electric field is $2.42\times {10}^{-5}$ and $4.84\times {10}^{-5}\mathrm{a}.\mathrm{u}.$, respectively. (b) The corresponding time-dependent photodetachment cross section in the bichromatic oscillating electric field.

Figure 6

(a) The total time-dependent photodetachment cross section of the ${\mathrm{H}}^{-}\mathrm{ion}$ in the bichromatic oscillating electric field. The photon energy is $E_p=1.0\mathrm{eV}$; the strength of the electric field is $F_0=200\mathrm{kV}/\mathrm{cm}$; the frequency of the bichromatic oscillating electric field is $2.42\times {10}^{-5}$ and $4.84\times {10}^{-5}\mathrm{a}.\mathrm{u}.$, respectively. (b) The total oscillating cross section ${\sigma }_{\mathrm{osc}}(E,t)$ in the bichromatic oscillating electric field. (c–f) The oscillating cross section induced by the up closed orbit, down closed orbit, up-down closed orbit, and down-up closed orbit, respectively.

Figure 7

The influence of the phase difference in the bichromatic oscillating electric field on the time-dependent photodetachment cross section of the ${\mathrm{H}}^{-}\mathrm{ion}$. The photon energy is $E_p=1.0\mathrm{eV}$; the strength of the electric field is $F_0=300\mathrm{kV}/\mathrm{cm}$; the frequency of the bichromatic oscillating electric field is $2.42\times {10}^{-5}$ and $4.84\times {10}^{-5}\mathrm{a}.\mathrm{u}.$ The phase difference in the bichromatic oscillating electric field is as follows: (a) $\varphi =0$; (b) $\varphi =\pi /2$; (c) $\varphi =\pi$; (d) $\varphi =3\pi /2$.

Figure 8

Comparison of the time-dependent photodetachment cross section for the ${\mathrm{H}}^{-}\mathrm{ion}$ in the bichromatic oscillating electric field with the case of the single oscillating electric field. The photon energy is $E_p=1.0\mathrm{eV}$; the strength of the electric field is $F_0=500\mathrm{kV}/\mathrm{cm}$. The frequency of the single oscillating electric field is $2.42\times {10}^{-5}\mathrm{a}.\mathrm{u}.$ and the frequency in the bichromatic oscillating electric field is $2.42\times {10}^{-5}$ and $4.84\times {10}^{-5}\mathrm{a}.\mathrm{u}.$, respectively. (a) The $t-t_i$ curve for the detached electron's closed orbit in the single oscillating electric field. (b) The photodetachment cross section in the single oscillating electric field. (c) The $t-t_i$ curves for the detached electron's closed orbits in the bichromatic oscillating electric field. (d) The photodetachment cross section in the bichromatic oscillating electric field.