Theory of attosecond transient absorption spectroscopy of strong-field-generated ions

Strong-field ionization generally produces ions in a superposition of ionic eigenstates. This superposition is generally not fully coherent and must be described in terms of a density matrix. A recent experiment [E. Goulielmakis et al., Nature (London) 466, 739 (2010)] employed attosecond transient absorption spectroscopy to determine the density matrix of strong-field-generated Kr${}^{+}$ ions. The experimentally observed degree of coherence of the strong-field-generated Kr${}^{+}$ ions is well reproduced by a recently developed multichannel strong-field-ionization theory, but there is significant disagreement between experiment and theory with respect to the degree of alignment of the Kr${}^{+}$ ions. In the present paper, the theory underlying attosecond transient absorption spectroscopy of strong-field-generated ions is developed. The theory is formulated in such a way that the nonperturbative nature of the strong-field-ionization process is systematically taken into account. The impact of attosecond pulse propagation effects on the interpretation of experimental data is investigated both analytically and numerically. It is shown that attosecond pulse propagation effects cannot explain why the experimentally determined degree of alignment of strong-field-generated Kr${}^{+}$ ions is much smaller than predicted by existing theory.

Figure 1

Atomic levels of Kr${}^{+}$ associated with resonant absorption at photon energies near $80$ eV. The notation ${\mathit{nl}}_j^{-1}$ indicates that relative to the closed-shell ground state of the neutral atom, an electron is missing (a hole is present) in the ${\mathit{nl}}_j$ subshell.

Figure 2

Attosecond transient absorption cross section (in Mb) of strong-field-generated Kr${}^{+}$, plotted as a function of the photon energy and the time delay. The cross section was calculated using Eq. (35). Panels (a) and (b) show the attosecond transient absorption cross section for the theoretical ion density matrix elements in Table . In panels (c) and (d), the attosecond transient absorption cross section is shown for the experimental ion density matrix elements in Table .

Figure 3

Apparent attosecond transient absorption cross section (in Mb) of strong-field-generated Kr${}^{+}$, plotted as a function of the photon energy and the time delay. The cross section was calculated by applying Beer’s law to the numerically propagated EUV field. Panels (a) and (b) show the apparent attosecond transient absorption cross section for the theoretical ion density matrix elements in Table . In panels (c) and (d), the apparent attosecond transient absorption cross section is shown for the experimental ion density matrix elements in Table .

Figure 4

The spectrum of the EUV pulse before (dash-dotted line) and after propagation for $t_{\mathrm{EUV}}=0$. The dashed line, representing a simulation assuming the validity of Beer’s law, corresponds to panels (c) and (d) in Fig. 2. The solid line shows the result of numerical propagation, corresponding to panels (c) and (d) in Fig. 3.