The role of shake-off for double ionization of atoms by a single photon with finite energy has become the subject of debate. In this letter, we attempt to clarify the meaning of shake-off at low photon energies by comparing different formulations appearing in the literature and by suggesting a working definition. Moreover, we elaborate on the foundation and justification of a mixed quantum-classical ansatz for the calculation of single-photon double ionization. PACS numbers: 32.80.Fb, 34.80.Kw Letter to the Editor 2 It is well known that the double-to-single cross section ratio for ionization of atoms by a single photon does not vanish at high photon energies. Rather, it approaches a finite constant which can be explained in the framework of a sudden approximation by the so-called shake-off mechanism. While shake-off is well defined in the asymptotic high-energy limit, its meaning at finite energies is less clear and has been the subject of debate recently. In fact, a number of definitions can be found in the literature, e.g., [1, 2, 3, 4, 5, 6, 7]. Some of these definitions are based on formal diagrammatic perturbation expansion techniques [3, 5], others on more general physical arguments [2, 4], others on a simple extension of the sudden approximation idea to finite energies [1, 6, 7]. Most of them (with the exception of [3], see below) have in common that they approach the well-known asymptotic expression at high energies. At low and intermediate energies, however, they may differ markedly (e.g., some show a monotonic dependence on energy while others have a maximum at some finite energy, some even exceed the total double-to-single cross section ratio measured experimentally). Thus, so far no unique definition for shake-off at finite energies exists, and it is not obvious what “the best” definition might be. On the other hand, in particular in connection with the interpretation of experimental data, attention has been given to the question what physical mechanisms dominate double ionization and at which energies the different mechanisms are important [8]. One may argue that a satisfying definition of shake-off would be one based on physical principles in addition to mathematical rigor. Hence, a “good” definition should separate shake-off as much as possible from other ionization mechanisms. Clearly, such a separation is not strictly possible in the presence of other available routes to ionization and can only be approximate, which makes the discussion of a shake-off mechanism a somewhat delicate issue. In comparing calculations with experiments, e.g., one should always keep in mind that no strict one-to-one correspondence between a shake-off mechanism as an approximate physical picture and a separate calculation of shakeoff can be expected due to the neglect of interference between possible decay routes. Nevertheless, since such simple physical pictures, to the extent of their applicability, can be very valuable for our intuitive understanding of physical processes, a definition separating shake-off from “non-shake-off” would seem most rewarding conceptually. One such definition has recently been given by Schneider et al. [7] (hereafter referred to as SCR), where the single-photon double ionization process has been described in terms of two separate contributions, namely “shake-off” and “knock-out”. The method used in SCR was shown to lead to excellent agreement with experiment and ab-initio calculations for double ionization from the ground state [7, 9], and very recently also from excited states [10], of helium. Thus, it suggests itself as a more or less natural “operational” definition of shake-off in the framework of the “half-collision” picture of single-photon multiple ionization [11, 4]. The calculation reported in SCR starts from a mixed quantum-classical ansatz that is based on the separation of the photoabsorption process (which is not treated explicitly in the calculation) from the subsequent evolution of the system. It treats this Letter to the Editor 3 evolution (i.e. the redistribution of the energy between the two electrons) in the spirit of an (e,2e)–like process with the additional possibility of shake-off. Such a “half-collision” picture has been originally suggested by Samson [11] and elaborated by Pattard and Burgdörfer [4], allowing for shake-off processes which are not taken into account in Samson’s original model. In the SCR ansatz, the (e,2e)–like (“knock-out”) part of the cross section is calculated using a classical trajectory Monte Carlo method, to which the shake-off as a purely quantum mechanical process is added on top. In this spirit, shake-off is introduced as a more or less ad-hoc quantum correction to an essentially classical treatment. Here, we start from a fully quantum mechanical expression and see which kind of approximations lead to an SCR-like ansatz. In this way, further insight into the validity of the ansatz, concerning both technical details of the calculation as well as the approximate separation of physical mechanisms (shake-off and knock-out), can be obtained. In ref. [4], a Born series for the transition amplitude from the ground state ψi to a final state ψ (0) f of a two-electron target following single-photon absorption has been derived. It was shown that, under the assumption of negligible electron–electron correlation in the ionized final state, the transition amplitude can be written as afi = −2πi δ (Ef − Ei − ω) 〈ψ f | 1− i ∞ ∫ 0 dt e0 Tee e −iH0t Vpe|ψi〉 . (1) In the above equation, Vpe is the photon-electron interaction, usually taken in dipole approximation, H0 is the final-state Hamiltonian H0 = Hat − Vee and Tee denotes the Coulomb T -matrix for electron-electron scattering. ψ (0) f is an eigenfunction of H0, i.e. a product of two one-electron states, due to the assumption of vanishing electron-electron correlation in the final state (where at least one electron is ionized), while ψi is the fully correlated initial (ground) state of the target. Introduction of a complete set of intermediate states then allows for a separation (on the amplitude level!) of the initial photon absorption from the subsequent propagation afi = −2πi δ (Ef −Ei − ω) ∑∫ a 〈ψ f |S+|ψa〉 〈ψa|Vpe|ψi〉 , (2)

@inproceedings{PattardOnTR,
title={On the role of shake-off in single-photon double ionization},
author={Thomas Pattard and J . - M . Rost}
}