Theory of attosecond absorption spectroscopy in krypton

A theory for time-domain attosecond pump–attosecond probe photoabsorption spectroscopy is formulated and related to the atomic response. The theory is illustrated through a study of attosecond absorption spectroscopy in krypton. The atomic parameters entering the formulation such as energies and Auger widths, as well as wave functions and dipole coupling matrix elements, are determined by accurate many-body structure calculations. We create a hole in a valence shell by an attosecond pump, couple an inner-shell electron to the hole by an attosecond probe, and then monitor the formation of the hole in this manner. In a second example, a hole is created in an inner shell by the first pulse, and the second probe pulse couples an even more tightly bound state to that hole. The hole decays in this example by Auger electron emission, and the absorption spectroscopy follows the decay of the hole and the associated coherences in real time.

Figure 1

Binding energies for the relevant states in the krypton atom. The energies are obtained from our calculations as discussed in Sec. 4.

Figure 2

The radial dipole matrix element, Eq. (30), between the 4$p$ states in krypton and the $s$-wave continuum (full, blue) or the $d$-wave continuum (dotted, red). The overlaps are calculated for the 4$p_{j=1/2}$ and 4$p_{j=3/2}$ states. The radial difference in these latter states is very small and the difference in the radial overlaps cannot be seen on the scale of the figure.

Figure 3

The radial dipole matrix element, Eq. (30), between the 4$s$ state in krypton and the $p$-wave continuum (full, blue) and the overlap between the 3$d$ states and the $p$-wave continuum (dotted, red) or the $f$-wave continuum (dashed, black). As in Fig. 2, the difference in the radial overlaps for the different $j$ states cannot be see in the figure.

Figure 4

Spectral profiles of the two pulses (in arbitrary units). The photon energy of the pump pulse, centered at 50 eV, is insufficient to drive the transitions from the 3$d$ core levels to the 4$p$ valence holes, which are just above 80 eV. The intensity of the probe pulse, centered at 81 eV, is insufficient to cause ionization of the system.

Figure 5

Normalized absorption spectrum for holes in the 4$p$ valence shell. The spectrum shows three lines at 81.3, 81.9, and 82.6 eV corresponding to the $3d_{5/2}\to {4p}_{3/2}$, $3d_{3/2}\to {4p}_{1/2}$, and $3d_{3/2}\to {4p}_{3/2}$ transitions.

Figure 6

The absorption signal versus the delay between the pulses. The curves are (from strongest signal to weakest) 3$d_{5/2}\to {4p}_{3/2}$(lowest curve, red), 3$d_{3/2}\to {4p}_{1/2}$(middle curve, blue), and 3$d_{3/2}\to {4p}_{3/2}$(highest curve, green).

Figure 7

Focusing only on the times when the two pulses overlap, we may resolve the buildup of the holes on an attosecond time scale. The upper panel shows the absorption spectrum and the lower panel shows the absorption for the three lines, normalized at $t_0=0$. The curves are 3$d_{5/2}\to {4p}_{3/2}$ (lowest curve at a delay of 0.2 fs, red), 3$d_{3/2}\to {4p}_{1/2}$ (middle curve at a delay of 0.2 fs, blue), and 3$d_{3/2}\to {4p}_{3/2}$ (highest curve at a delay of 0.2 fs, green).

Figure 8

The absorption in Fig. 7 stabilizes at about a delay of 0.2 fs. This corresponds to the situation shown above.

Figure 9

Absorption spectra calculated for the set of pulses tuned to drive transitions from the 3$p$ level. The spectra in the first and the second panels show the lines corresponding to the 3$d$ levels and the third and the fourth panels show the spectra for the 4$s$ level. The first and third panels show the full spectra. The second and the fourth panels show the intersections at the center of the lines. In the first and second panels the lines are 126.6 eV, 3$p_{1/2}$ -3$d_{3/2}$ (middle curve, blue); 120.3 eV, 3$p_{3/2}$-3$d_{5/2}$ (lowest curve, red); 119 eV, 3$p_{3/2}$-3$d_{3/2}$ (highest curve, green). In the third and fourth panels the lines are 189.e eV, 3$p_{1/2}$-4$s_{1/2}$ (upper curve, blue), 3$p_{3/2}$-4$s_{1/2}$ (lower curve, blue). Notice that the set of parameters used implies that the absorption at the 3$p$ to 4$s$ lines is two orders of magnitude weaker than that at the 3$p$ to 3$d$ lines.