Azimuthal asymmetry of the extracted electron in field ionization of a hydrogen atom with orbital angular momentum

The tunneling ionization of an excited hydrogen atom by a static electric field $\mathbf{E}$ is investigated for the case where the initial electron has an orbital angular momentum $\mathbf{L}$ nonparallel to $\mathbf{E}$. The outgoing electron has a nonzero mean transverse velocity $\langle {\mathbf{v}}_{\mathrm{T}}\rangle$ in the direction of $\mathbf{E}\times \langle \mathbf{L}\rangle$. During this process the linear Stark effect makes $\langle \mathbf{L}\rangle$ and $\langle {\mathbf{v}}_{\mathrm{T}}\rangle$ oscillate or rotate about $\mathbf{E}$. Measures of the asymmetry are calculated at leading order in $\mathbf{E}$ for an initial state $2P$ state. The generalization to coherent elliptic Rydberg states is outlined. A subset of these states whose classical Kepler ellipses rotate rigidly about $\mathbf{E}$ is particularly interesting for the observation of the asymmetry. The preparation of states with $\mathbf{L}$ nonparallel to $\mathbf{E}$ and the conditions to get a sizable ${\mathbf{v}}_{\mathrm{T}}$ asymmetry are discussed.

Figure 1

Semiclassical motion of the electron extracted from the hydrogen atom by a strong field $\mathbf{E}$ when the electron is initially in a $L_y=+1$ state.

Figure 2

The $\{h,k\}$ lattices of Stark states for $n=5$ and 2. The top right (bottom left) corner represents the long-lived blue (short-lived red) state. The $n=2$ states are those of Eq. (9).

Figure 3

Classical picture of the Stark oscillations of $L_y$ and $A_x$. Orbits 0 to 7 correspond to the eight panels of Fig. 4.

Figure 4

Evolution of the electron density ${\left|\mathrm{\Psi }(t,\mathbf{r})\right|}^2$ of Eqs. (24) and (25) ($y=0$ slice) during one Stark period $T_{\mathrm{S}}$. At $4t/T_{\mathrm{S}}=0,1,2,3,4,\mathrm{\Psi }\left(t\right)$ takes the values ${\mathrm{\Psi }}_{Ly+},i{\mathrm{\Psi }}_{x+},-{\mathrm{\Psi }}_{Ly-},-i{\mathrm{\Psi }}_{x-}$, and ${\mathrm{\Psi }}_{Ly+}$ again, where ${\mathrm{\Psi }}_{x\pm }$ is the $A_x=\pm 1$ eigenstate. $\mathrm{\Psi }$ vanishes along the current vortex $z=0,2\pm x/sin\left(3Ft\right)=r$ (whirling as indicated by arrows). For $t=T_{\mathrm{S}}/4$ or $3T_{\mathrm{S}}/4,\mathrm{\Psi }$ vanishes on the surface $2\pm x=r$. The $x$ and $z$ windows are $[-7,+7]$.

Figure 5

Classical orbit of a ROE state. The orbit plane rotates as indicated by the curved arrow. The overbar of $\overline{\mathbf{j}},\overline{\mathbf{L}}$, and $\overline{\mathbf{A}}$ has been omitted.

Figure 6

Construction of a ROE state. At $t<0$ the external field $\mathbf{E}={\mathbf{E}}_0$ is static, and the atom is in the blue Stark state: ${\overline{\mathbf{j}}}_1\left(0\right)=-{\overline{\mathbf{j}}}_2\left(0\right)=j{\widehat{\mathbf{\omega }}}_{\mathrm{S}}\left(0\right)$, where ${\mathbf{\omega }}_{\mathrm{S}}\left(t\right)\equiv -(3n/2)\mathbf{E}\left(t\right)$. During the interval $[0,T]$ one makes $\mathbf{E}$ rotate about $\mathbf{\Omega }$. The figure shows what happens in the rotating frame: ${\mathbf{\omega }}_{\mathrm{S}}$ is fixed, and the Coriolis force makes ${\overline{\mathbf{j}}}_1$ and ${\overline{\mathbf{j}}}_2$ rotate about ${\mathbf{\omega }}_1^{*}=-\mathbf{\Omega }+{\mathbf{\omega }}_{\mathrm{S}}$ and ${\mathbf{\omega }}_2^{*}=-\mathbf{\Omega }-{\mathbf{\omega }}_{\mathrm{S}}$, respectively. We choose $T=2\pi /{\omega }_1^{*}$ and ${\omega }_2^{*}<{\omega }_1^{*}$, so that ${\overline{\mathbf{j}}}_1\left(T\right)$ has rejoined $j{\widehat{\mathbf{\omega }}}_{\mathrm{S}}\left(T\right)$ while ${\overline{\mathbf{j}}}_2\left(T\right)$ had not yet rejoined $-j{\widehat{\mathbf{\omega }}}_{\mathrm{S}}\left(T\right)$. At $t>T$ the field is static again: $\mathbf{E}=\mathbf{E}\left(T\right)$. The atom is then in a ROE state where ${\overline{\mathbf{j}}}_1\left(t\right)=j{\widehat{\mathbf{\omega }}}_{\mathrm{S}}\left(T\right)$ and ${\overline{\mathbf{j}}}_2\left(t\right)$ starts rotating about $-{\mathbf{\omega }}_{\mathrm{S}}\left(T\right)$. The overbar of ${\overline{\mathbf{j}}}_1$ and ${\overline{\mathbf{j}}}_2$ has been omitted.

Figure 7

Motion of the probability density ${\left|\mathrm{\Psi }(t,\mathbf{r})\right|}^2$ of the outgoing electron. The curve represents $\langle x(t,z)\rangle \simeq 2\langle \check{x}\left(t\right)\rangle \sqrt{z}$ versus $z$ at fixed $t$. With time, the undulations move to the right. It looks like a crawling snake.

Figure 8

Intensity $I\left(t^{*}\right)$ of the outgoing electron flux from an $n=2$ CLy state [Eq. (55a)] and amplitude $a^{\mathrm{o}}\left(t^{*}\right)$ of the oscillations of the asymmetry (49). The time of exit $t^{*}$ is in picoseconds, and $I\left(t^{*}\right)$ is in ${\mathrm{ps}}^{-1}$. Field intensity $F=0.004$ a.u. The oscillations, of period $T_{\mathrm{S}}=0.0127$ ps, are too dense to show. Note the change of “eras” and local maxima at $t_{1,2}^{*}$ and $t_{2,3}^{*}$. The sloping straight lines under the $I\left(t^{*}\right)$ curve represent the successive dominant contributions to (55a).