Attosecond photoionization dynamics in neon Juan J. Omiste Chemical Physics Theory Group, Department of Chemistry, and Center for Quantum Information and Quantum Control, University of Toronto, Toronto, ON M5S 3H6, Canada Lars Bojer Madsen Department of Physics and Astronomy, Aarhus University, 8000 Aarhus C, Denmark (Dated: September 11, 2018) We study the role of electron-electron correlation in the ground-state of Ne, as well as in photoion- ization dynamics induced by an attosecond XUV pulse. For a selection of central photon energies around 100 eV, we find that while the mean-field time-dependent Hartree-Fock method provides qualitatively correct results for the total ionization yield, the photoionization cross section, the pho- toelectron momentum distribution as well as for the time-delay in photoionization, electron-electron correlation is important for a quantitative description of these quantities. I. INTRODUCTION mally updated in each time-step. The latter approach benefits not only from a reduction in the number of or- bitals, but also from an implicit description of multiple Ever since the early days of atomic physics, the role of ionization events, and from its flexibility to identify the electron-electron correlation has been a key topic. Even most important configurations for an accurate descrip- though ground and singly-excited states can be described tion of the system [8, 10, 11, 16, 17]. We will study by mean-field theory, it was soon recognized that a cor- the ground-state energy and the photoionization dynam- related basis is beneficial for convergence to an accu- ics and their sensitivity to the active orbital space, i.e., rate ground-state energy [1, 2]. For multiply excited to the RAS scheme. Finally, we will study the impor- states, correlation is crucial and theory based on single- tance of electron correlation on the time-delay in pho- configurations breaks down [3]. In strong-field and at- toionization. Time-delay studies have attracted much tosecond physics, the theoretical description is typically attention because of available experimental data [18, 19] explicitly time-dependent and processes often involve and the development of powerful theoretical and com- continua wavepackets carrying the temporal information putational methods [20]. Time-delays have been evalu- encoded by ultrafast pump and probe pulses [4]. The ated in different scenarios, including in photoionization presence of multiple continua challenges theory tremen- from first principles [21], as a function of angle of the dously. Needless to say carrying out a mean-field time- ejected electron [22{24] or in strong-field ionization [25]. dependent Hartree-Fock calculation is possible for phys- In particular, the experimental measurement of the time- ical systems so large that fully correlated configuration- delay in Ne between the photoemission of electrons from interaction calculations are not. One of the tasks for the 2s and 2p shells [18] has provoked a lot of interest theory is therefore to establish which level of approxi- due to a disagreement in the magnitude of the mea- mation is sufficient for a qualitative correct description sured and calculated time-delays, see, e.g., Refs. [26{35]. of a given observable. This is one of the questions that Note that very recently, analysis of interferometric mea- we address in this work. To this end we need a the- surements [36] suggests that a shake-up process, not re- ory where we can control the level of approximation, solved and accounted for in the streaking experiment of and we need to investigate a system which is possi- Ref. [18], could affect the experimental result and possi- ble to describe at such different levels. Thus, we de- bly bring the experimental time-delay in agreement with scribe the photoionization of the Ne ground-state [5{ the many-body calculations [34]. In this work, we prop- 7] mediated by an ultrashort XUV pulse using the agate the photoelectron wavepacket to directly measure time-dependent restricted-active-space self-consistent- the time of emission from each electronic shell for a set field (TD-RASSCF) approach [8{11]. This method per- of active spaces to probe the role of the electronic corre- arXiv:1712.00625v1 [physics.atom-ph] 2 Dec 2017 mits the introduction of restrictions on the number of lation [16]. excitations in the active orbital space and is a gen- The paper is organized as follows. In Sec. II, we eralization of the multiconfigurational time-dependent summarize the computational approach. In Sec. III, we Hartree-Fock method [12]. In contrast to many-body present our results. First, we study the ground-state for methods based on time-independent single-particle or- different RAS schemes. Next, we analyze the ionization bitals such as time-dependent configuration-interaction dynamics induced by the laser. This analysis consists with singles (TD-CIS) [13], time-dependent restricted- of the calculation of the ionization cross section and the active-space configuration-interaction (TD-RASCI) [14] description of the main features of the single photoion- or time-dependent general-active-space configuration- ization, in particular, the contribution of each ioniza- interaction (TD-GASCI) [15], the TD-RASSCF is based tion channel obtained by considering the photoelectron on time-dependent single-particle orbitals that are opti- spectrum. Finally, we present computed time-delays be- 2 tween the electrons ejected from the 2s and 2p subshells the matrix elements are given by and compare with available experimental and theoretical p ∗ values. We conclude in Sec. IV. Atomic units are used hq (t) = d~rφp(~r; t)h(~r; t)φq(~r; t); (3) throughout unless indicated otherwise. Z φ∗(~r; t)φ∗(r~0; t)φ (~r; t)φ (r~0; t) pr ~0 p r q s vqs (t) = d~rdr :(4) ~r r~0 ZZ j − j The TD-RASSCF methodology is a generalization of II. SUMMARY OF THE TD-RASSCF METHOD MCTDHF [38, 39] in the sense that it includes the pos- sibility to impose restrictions on the excitations in the active space [8, 10], i. e., the many-body wavefunction In this section, we summarize the TD-RASSCF reads method used to propagate the many-electron wavefunc- tion. We refer to previous works for details [8{10, 16]. Ψ(t) = CI (t) ΦI (t) ; (5) j i j i I2V We propagate the dynamics of an Ne-electron atom in X the laser field in the length gauge within the dipole ap- where the sum runs over the set of configurations , and proximation. The dynamics of this system is described V by the Hamiltonian not necessarily the full configuration space, and CI (t) and ΦI (t) are the amplitudes and Slater determinants of thej configurationi specified by the index I, which con- tains direct products of spin-up and spin-down strings, Ne p2 Z Ne Ne 1 j ~ i.e., I = I" I#, each of them including the indices of the H = + E(t) ~rj + = ⊗ 2 − rj · ~rj ~rk spatial orbitals [40, 41]. Each Slater determinant is built j=1 ! j=1 k>j j − j X X X from M time-dependent spatial orbitals φ (~r; t) M , in N N N j j=1 e e e 1 the active orbital space . In the casef of MCTDHF,g = h(~rj; t) + ; (1) P ~rj ~rk FCI, that is, the full configuration space [39]. On j=1 j=1 k>j X X X j − j theV ≡ other V hand, in the TD-RASSCF, the configurations are taken from the restricted active space, RAS, which is defined as a subset of by imposingV ≡ V re- where the first sum is over one-body operators and the FCI strictions on the excitations in theV active space. In this second over two-body operators. The nuclear charge is method, the active orbital space is divided into 3 sub- denoted by Z and the external electric field of the laser P ~ spaces: 0; 1 and 2. 0 constitutes the core, and its pulse is E(t). To formulate and apply the TD-RASSCF orbitalsP areP fully occupied.P P All the different ways to theory, it is convenient to work in second quantization. form configurations by combination of orbitals in 1 are We work in the spin-restricted framework, which implies P allowed. The orbitals in 2 are filled with restrictions that a given Slater determinant, ΦI (t) , is formed by P j i by excitations from 1. The number of orbitals in 0, Ne=2 spatial orbitals for each spin specie. In second P P 1 and 2 are denoted by M0, M1 and M2, and the quantization, the Hamiltonian reads P P total number of orbitals equals M = M0 + M1 + M2. The single-particle Hilbert space is completed by the - Q 1 space such that the unit operator can be resolved as H(t) = hp(t)Eq + vpr(t)Eqs; (2) q p qs pr 1 = P (t) + Q(t), with P (t) = j φj(t) φj(t) and pq 2 pqrs j ih j Q(t) = φa(t) φa(t) , with φj(t) belonging to - X X a j ih j j P i P space and φa(t) to -space. In thisP work,j i we doQ not consider a core, i. e., we do where we use the spin-free excitation operators [37] q qs q y qs not have a 0 subspace. We apply the TD-RASSCF- Ep and Epr , defined as Ep = bpσbqσ; and Epr = P σ=";# D method, i. e., include double (D) excitations from the y y y P active space partition 1 to 2. The TD-RASSCF-D bpσbrγ bsσbqγ with bpσ and bpσ the creation and P P σ=";# γ=";# method was shown to be numerically efficient and stable annihilationP P operators of a single spin-orbital φp(t) σ in the case of photoionization of Be [16].