**1. Introduction**

The simultaneous removal of two electrons from an atom following absorption of a single photon is an archetypal process driven entirely by many-electron correlation. Such a non-sequential single-photon two-electron ionization (double photoionization or DPI in short) has been the focus of experimental and theoretical activities for several decades [1–3]. The correlation mechanisms of DPI are well understood and can be described as the shakeoff (SO) and knock-out (KO) processes [4–7]. Shake-off is driven by a sudden change of the atomic potential after a fast removal of the primary photoelectron. Conversely, knock-out is a slow process in which the departing electron impinges on the ionic core and ejects the secondary photoelectron. A complementary quasi-free mechanism (QFM) of PDI, in which the nucleus remains a spectator, was predicted theoretically by Amusia and co-workers [8]. Recently, the QFM was studied experimentally [9–11], and it was ascribed to a combination of the SO and KO processes.

The first theoretical description of DPI in atoms invoked the lowest order perturbation theory [12–15]. With a growing computational power, more sophisticated non-perturbative methods were developed. The convergen<sup>t</sup> close-coupling (CCC) [16,17] and the timedependent close-coupling (TDCC) [18] are among the many predictive and accurate numerical techniques.

A resurged interest in DPI was stimulated by a newly acquired experimental capability to resolve atomic photoionization in time. Laser pulses are shaped in such a way that they can probe atomic ionization on the attosecond (1 as = 10−<sup>18</sup> s) time scale. Firstly, single photoionization was time resolved [19,20]. Then a DPI process was traced in time [21]. The accompanying theoretical studies have also appeared [22,23].

An alternative theoretical approach to DPI can be provided by the single-hole Green's function (SHGF) formalism introduced to photoionization [24,25]. Originally, this method was utilized to calculate shake-up satellites in atomic photoionization spectra [26–29].

However, by construction, the SHGF contains the DPI continuum, which can be attributed to the SO process. This capability of the SHGF method has been overlooked so far. Here, we rectify this omission.

The present work is structured in the following way. We start with a brief introduction of the SHGF technique, using a diagrammatic expansion of the ionization amplitude. We identify the double ionized continuum in this amplitude and link it with the imaginary part of the SHGF self-energy. This allows us to derive the analytic expressions for the DPI cross section resolved with the photoelectron energy as well as the time delay associated with the SO process. Next, we test our energy resolved DPI cross sections against experiments as well as the earlier CCC and TDCC calculations. This way, we identify the photoelectron energy range where the SO process makes the dominant contribution to DPI. Finally, we evaluate the time that it takes to shake off a bound electron. As expected, the SO process is fast with only a few attoseconds needed to shake off the secondary photoelectron into the two-electron continuum. However, there are few notable exceptions when the SO process takes much longer time to complete. We find this situation in strongly correlated targets such as the negative H− ion as well as the subvalent shells of the Ar and Xe atoms and the Cl− ion. The binding of the H− ion is wholly owed to many-electron correlation, and the photoionization of subvalent *ns* shells in Ar, Xe and Cl− is affected very strongly by correlation satellites. As the result, the SO process in H− takes as much as 30 as, whereas the similar process in Ar 3*s* and Xe 5*s* requires nearly 50 as to complete. The same process in Cl− 3*s* takes in excess of 100 as. We conclude by evaluating other components of the measurable time delay in DPI and thus making the case for the experimental resolution of the SO process in time.
