Atomic Siegert states in a rotating electric field

We show that the time-dependent Schrödinger equation describing in the dipole approximation the interaction of a one-electron atom with a monochromatic circularly polarized electromagnetic field can be reduced to a stationary Schrödinger equation in the form which allows to separate variables in the asymptotic region and explicitly formulate outgoing-wave boundary conditions. The solutions to this equation are called atomic Siegert states (SSs) in a rotating electric field. We develop the theory of such states and propose an efficient method to construct them using powerful techniques of stationary scattering theory. The method yields not only the SS eigenvalue defining the Stark-shifted energy and total ionization rate of the state, but also the SS eigenfunction defining partial ionization amplitudes and the photoelectron momentum distribution. The theory is illustrated by calculations for a model potential.

Figure 1

Adiabatic potentials defined by Eq. (23) for $\omega =0.1$ at three representative field strengths indicated in the figure. $r_0=1$, 3, and 10 for the three values of $F$, respectively. The three lowest potentials are shown by solid colored (blue, green, and orange) lines, the higher potentials are shown by solid gray lines. The dashed (red) line shows the lowest envelope function defined by Eq. (66) with $n=0$. The horizontal dashed (black) line shows the real part $\mathcal{E}$ of the SS eigenvalue (40).

Figure 2

Similar to Fig. 1, but for $\omega =0.3$. In this case $r_0\approx 0.333$, 1.11, and 3.33 for the three values of $F$, respectively.

Figure 3

Trajectory traced by the energy eigenvalue $E$ of the dominant SS as $F$ grows from 0 to 0.1 at $\omega =0.1$. Some selected values of $F$ are indicated by the circles. The vertical dashed lines show branch cuts beginning at the threshold energies $-m\omega$. The color of the line showing the trajectory changes to a lighter nuance each time the trajectory jumps to another branch of $E$ crossing a branch cut. The inset shows a comparison of $E_{\text{L}}$ obtained from Eq. (51a) (solid black line) with the energy eigenvalue of the corresponding SS in a static electric field [31] (dashed red line) in the same interval of $F$.

Figure 4

Similar to Fig. 3, but for $F$ growing from 0 to 0.3 at $\omega =0.3$. The inset is a closeup of a tiny region shown by the red square near the point where the trajectory crosses the branch cut beginning at $E=-0.9$.

Figure 5

Maps of the flux (45) in the $(x,y)$ plane of the RKH frame for $\omega =0.1$ at three field strengths $F$ indicated in the figure. Color shows (on a base 10 logarithmic scale) the magnitude of $\mathbf{j}\left(\mathbf{r}\right)$ while arrows indicate its local direction. The white line shows a classical trajectory defined by Eq. (67) which describes the motion of an electron after tunneling in the adiabatic regime.

Figure 6

Similar to Fig. 5, but for $\omega =0.3$.

Figure 7

Partial ionization rates ${\mathrm{\Gamma }}_m$ as a function of $m$ (top panels) and ${\mathrm{\Gamma }}_{lm}$ as a function of $m$ and $l$ (bottom panels) for $\omega =0.1$ at three field strengths indicated in the figure. Color of the circles in the bottom panels shows (on a logarithmic scale) the value of ${\mathrm{\Gamma }}_{lm}$. $m_{\text{min}}=6$, 6, and 11 for the three values of $F$, respectively.

Figure 8

Similar to Fig. 7, but for $\omega =0.3$. In this case, $m_{\text{min}}=2$, 2, and 4 for the three values of $F$, respectively.

Figure 9

Angular distributions $p_m\left({\theta }_k\right)$ of photoelectrons which have absorbed $m$ photons for $\omega =0.1$ at three field strengths indicated in the figure. The values of $p_m\left({\theta }_k\right)$ as a function of $m$ and ${\theta }_k$ are shown by color (on a logarithmic scale) along arcs of radius $k=\mathrm{Re}{k}_m$ in the plane with coordinates $k_{\rho }=ksin{\theta }_k$ and $k_z=kcos{\theta }_k$. The numbers indicate several lowest values of $m$ for open channels.

Figure 10

Similar to Fig. 9, but for $\omega =0.3$.

Figure 11

The radius $k_{\text{max}}$ of the circle in the $(k_x,k_y)$ plane along which the PEMD (61) is localized in the adiabatic regime as a function of $F$ at $\omega =0.1$ (top panel) and as a function of $\omega$ at $F=0.1$ (bottom panel). Small solid circles with error bars show the values obtained by fitting the present results for the PEMD by Eq. (68). Solid (black) and dashed (red) lines show the predictions of the adiabatic theory in the leading-order [14, 89, 90] and first-order (for the present system $\varkappa =1$) [34] approximations, respectively.