Photodetachment of ${\mathrm{H}}^{-}$ by a short laser pulse in crossed static electric and magnetic fields

We present a detailed quantum mechanical treatment of the photodetachment of ${\mathrm{H}}^{-}$ by a short laser pulse in the presence of crossed static electric and magnetic fields. An exact analytic formula is presented for the final state electron wave function (describing an electron in both static electric and magnetic fields and a short laser pulse of arbitrary intensity). In the limit of a weak laser pulse, final state electron wave packet motion is examined and related to the closed classical electron orbits in crossed static fields predicted by Peters and Delos [Phys. Rev. A 47, 3020 (1993)]. Owing to these closed orbit trajectories, we show that the detachment probability can be modulated, depending on the time delay between two laser pulses and their relative phase, thereby providing a means to partially control the photodetachment process. In the limit of a long, weak pulse (i.e., a monochromatic radiation field) our results reduce to those of others; however, for this case we analyze the photodetachment cross section numerically over a much larger range of electron kinetic energy (i.e., up to $500{\mathrm{cm}}^{-1}$) than in previous studies and relate the detailed structures both analytically and numerically to the above-mentioned, closed classical periodic orbits.

Figure 1

Geometrical arrangement of fields in photodetachment of ${\mathrm{H}}^{-}$ by a linearly polarized laser (with electric field ${\mathbf{E}}_{\mathbf{L}}$) in the presence of crossed static electric $\left({\mathbf{E}}_{\mathbf{S}}\right)$ and magnetic $\left(\mathbf{B}\right)$ fields. Both the laser and the static electric fields point along the $z$ axis. As indicated, the drift motion of the detached electron is along the $y$ axis.

Figure 2

Photodetachment cross section for ${\mathrm{H}}^{-}$ in the plane wave limit for a static magnetic field $B=1\mathrm{T}$ and three different values of the static electric field strength. Results are plotted versus detached electron kinetic energy above the zero field detachment threshold, $\omega +{\epsilon }_i$, up to $2.7{\mathrm{cm}}^{-1}$.

Figure 3

Same as Fig. 2 but for three higher static electric field strengths and detached electron kinetic energies up to $5{\mathrm{cm}}^{-1}$.

Figure 4

Photodetachment cross section for ${\mathrm{H}}^{-}$ in the plane wave limit for $B=1\mathrm{T}$, $E_S=60\mathrm{V}∕\mathrm{cm}$, and detached electron kinetic energies up to $30{\mathrm{cm}}^{-1}$.

Figure 5

The oscillatory part of the cross section, ${\sigma }_{\mathrm{osc}}$ [cf. Eq. (95)], for $B=1\mathrm{T}$ and $E_S=60\mathrm{V}∕\mathrm{cm}$ for $\omega +{\epsilon }_i$ ranging from (a) 0 to $60{\mathrm{cm}}^{-1}$; (b) 0 to $180{\mathrm{cm}}^{-1}$; and (c) 0 to $500{\mathrm{cm}}^{-1}$. Dashed lines indicate the boundary energies [cf. Eq. (90)] at which a new closed orbit appears [42] and the open triangles indicate the energies at which the amplitude of the oscillatory part of the quantum cross section is expected to have a local maximum [cf. Eq. (93)].

Figure 6

Time spectrum, $P_{osc}\left(t\right)$ [cf. Eq. (96)], of the oscillatory part of the cross section [cf. Eq. (95)] for different maximum energies, $E_f^{max}=\omega +{\epsilon }_i$, as indicated in each panel. See text for a detailed description.

Figure 7

Time spectrum, $P_{osc}\left(t\right)$ [cf. Eq. (96)], of the oscillatory part of the cross section [cf. Eq. (95)] given in Fig. 5c, calculated over two different energy ranges: (a) $0\le \omega +{\epsilon }_i\le 500{\mathrm{cm}}^{-1}$; and (b) $400{\mathrm{cm}}^{-1}\le \omega +{\epsilon }_i\le 500{\mathrm{cm}}^{-1}$. In (a) the open circles indicate the return times (i.e., orbit periods) of the 15 closed orbits for $\omega +{\epsilon }_i=500{\mathrm{cm}}^{-1}$ (see Table ). Also in (a), the open triangles indicate the return times of the circular closed orbits (in the drift frame) having ${\tilde{p}}_y^0=1$ [cf. Eqs. (82, 91)]; these return times are given by Eq. (94).

Figure 8

Effective photodetachment cross section of ${\mathrm{H}}^{-}$ [cf. Eq. (54)] by a single laser pulse of the form (6) with four different pulse durations [cf. Eq. (7)] in the presence of crossed static electric and magnetic fields, $E_S=15\mathrm{V}∕\mathrm{cm}$ and $B=1\mathrm{T}$. Results are plotted vs electron kinetic energy beginning from the zero-field threshold. Also shown (dotted line) is the photodetachment cross section for the case of a continuous (monochromatic) laser without any external static fields present.

Figure 9

(Color online). Contour plot snapshots of the probability density $P(x=0,y,z,t)$ in the $y\text{−}z$ plane for the case of zero initial momentum in the $x$ direction [cf. Eq. (97)]. The static electric and magnetic field strengths are $60\mathrm{V}∕\mathrm{cm}$ and $1\mathrm{T}$, respectively. The laser pulse duration is $T_p=2\mathrm{ps}$ and $t=0$ corresponds to the end of this pulse. The total electron energy is $E_f=8{\mathrm{cm}}^{-1}$.

Figure 10

(Color online). Same as Fig. 9 but for a total detached electron energy of $E_f=15.9{\mathrm{cm}}^{-1}$.

Figure 11

(Color online). Same as Figs. 10e, 10f, 10g, 10h except that the probability densities $P(x=0,y,z,t)$ are shown here in three-dimensions rather than as contour plots. Note the scale change in panel (b), which shows the refocusing of the wave packet amplitude along the drift axis (i.e., away from the atomic core at the origin).

Figure 12

The effective cross section for the double laser pulse case as modulated by: (a) the relative phase between the two pulses for several time delays, as indicated; and (b) the time delay between the two pulses for two fixed relative phases, 0 and $\pi$.