Controlling the 2P Hole Alignment in Neon Via the 2S–3P Fano Resonance

Home , Autoionization

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Section II discusses briefly our TDCIS approach. In the Hartree-Fock energy EHF such that the Hartree- Sec. III, we present our results beginning in Sec. IIIA Fock ground state is at zero energy (for details see with a review of basic aspects of the 2s–3p Fano res- Ref. [21, 32]). The nuclear charge is given by Z and onance in the energy (Sec. IIIA1) and time domains the index n runs over all N electrons in the system. (Sec. IIIA2), and explaining in Sec. IIIB the mechanism Light-matter interaction for linearly polarized pulses in of the 2p hole alignment when targeting this Fano reso- the electric-dipole approximation is given in the length nance. The influence of spin-orbit interaction on the hole gauge by −E(t)ˆz withz ˆ = n zˆn [26]. All of the alignment is studied in Sec. IIIC. Section IV concludes electron-electron interactions thatP cannot be described the discussion. by the mean-field potential VˆMF are captured by Hˆ1 = 1 1 Atomic units are employed throughout unless other- ′ − VˆMF(ˆrn). Introducing a local com- 2 n,n |ˆrn−ˆrn′ | n wise indicated. plexP potential hasP the consequence that the symmetric inner product (·| , |·) must be used instead of the hermi- tian one h·| , |·i [33]. II. THEORY The second term in Eq. (2b), which describes the electron-electron interaction, is the only term within the Our implementation of the TDCIS approach to solve TDCIS theory that leads to many-body effects. Elec- the multichannel Schrödinger equation has been de- tronic correlation effects, which within TDCIS can only scribed in previous publications [21, 27]. We have ap- occur between the ionic state (index i) and the photoelec- plied our TDCIS approach to a wide spectrum of pro- tron (index a), are captured in the interchannel coupling cesses [26], ranging from attosecond photoionization [28] terms (i 6= j) where both indices (a and i) are changed to nonlinear x-ray ionization [29] and strong-field tunnel simultaneously. It means that the ionic state changes due ionization [19, 20, 30]. to the interaction with the excited electron. Intrachan- The N-body TDCIS wave function reads nel interactions do not change the ionic state (i = j) and describe the long-range −1/r Coulomb potential for the a a |Ψ(t)i = α0(t) |Φ0i + αi (t) |Φi i , (1) excited electron. Intrachannel interaction can be viewed Xa,i in terms of a one-particle potential and cannot lead to a electron-electron correlations. The importance of many- where |Φ0i is the Hartree-Fock ground state and |Φi i = † body correlation effects [28, 34] can be easily tested by câcî |Φ0i are singly excited configurations with an electron removed from the initially occupied orbital i and either allowing (full TDCIS model) or prohibiting (intra- placed in the initially unoccupied orbital a. Since Eq. (1) channel TDCIS model) interchannel interactions which ˆ describes all N electrons in the atom, an electron can be are captured in the H1. removed from any orbital. This multichannel approach is very helpful in describing ionization processes with XUV and x-ray light where more than one orbital is accessible. III. RESULTS By limiting the sum over i, specific occupied orbitals can be picked to be involved in the dynamics, thereby testing We begin in Sec. IIIA with a discussion of the spectral the multichannel character of the overall dynamics. In- and temporal properties of the 2s–3p Fano resonance in serting Eq. (1) into the full time-dependent Schrödinger neon, which we exploit in Sec. IIIB to control the hole equation, one finds the following equations of motion for alignment by tuning the XUV pulse across the Fano res- the CIS coefficients: onance.

a a b b The photoabsorption cross section, σ(ω), of neon − E(t) (Φi | zˆ|Φ0) α0(t)+ (Φi | zˆ Φj αj (t) , around the 2s–3p resonance obtained within TDCIS is Xb,j shown in Fig. 1, both with and without interchannel cou- (2b) pling between the 2s and 2p shells. They are both calcu- 2 lated via an autocorrelation function (see Refs. [3, 26]). pˆn Z where Hˆ0 = − + VMF(ˆrn) − iηW (|ˆrn|) − n 2 |ˆrn| Strictly speaking, the 2s–3p resonance has in principle no h i −1 EHF includes allP one-particle operators (kinetic energy, line width in the intrachanel model since the state 2s 3p attractive nuclear potential, the mean-field potential, cannot autoionize and, therefore, lives forever. In Fig. 1, VˆMF, contributing to the standard Fock operator [31], this resonance has a finite width that is artificial and has and the complex absorbing potential, −iηW (ˆr), prevent- been introduced by hand for better visualization [42]. ing artificial reflection from the boundaries of the nu- With the addition of interchannel coupling of the elec- merical grid. The entire energy spectrum is shifted by trons in the full model, the excited 2s−1 3p state autoion- 3

30 2. Temporal Features Intrachannel Full

25 ω intra=45.511 eV In order to investigate the temporal character of the ω max=45.525 eV autoionization process, we resonantly excite neon with a 20 relatively short 2.4 fs pulse of frequency ωres and a peak intensity of 5.6×1013 W/cm2. The duration of this pulse + 6 Mb is purposely chosen to be much shorter than the lifetime

) (Mb) 15 ω ω =45.538 eV −1 ( res of the 2s hole given by T2s 3p = 1/Γ = 20.7 fs, in our σ calculations. 10

ω min=45.559 eV 5 (a) total (b) 0.155 total 0 0.13 0.135 45.35 45.4 45.45 45.5 45.55 45.6 45.65 45.7 Photon Energy ω (eV) (c) 0.065

2p0 Figure 1: (color online) Photoabsorption cross section of neon 0.062 in the vicinity of the 2s → 3p resonance for the intrachannel (d) 0.039 0.07 2p0

TDCIS model (blue-dashed line) and the full TDCIS model Depopulation 2p±1 (red-solid line). Fano profile fits [24] give the resonance fre- 0.037 2p±1 quency for the intrachannel TDICS model ωintra, and the resonance frequency for the full TDCIS model ωres. The curve 0.012 for the intrachannel model is shifted up by +6 Mb for better 2s 0.01 2s (e) visualization. 0.006 0 10 20 7 15 23 Time (fs) izes to a singly charged ionic state 2p−1 εl. This indirect ionization of a 2p electron (2s22p6 + γ → 2s−13p → Figure 2: (color online) (a) Hole population of the 2s (red- 2p−1 εl) and the direct one-photon ionzation of a 2p elec- solid line), 2p0 (green-dashed line), and 2p±1 (blue-dashed line) orbitals as well as the ground state depopulation (pink- tron (2s22p6 + γ → 2p−1 εl) can now interfere, resulting dashed line) for the full CIS model. The pulse has a Gaussian in an asymmetric Fano line shape [1] (see Fig. 1). We fit shape and is 2.4 fs long (FWHM of the intensity) centered both curves (with and without interchannel interactions) around t = 0, and has the carrier frequency ωres. Also shown to the characteristic Fano profile [1, 24] given by are scaled close-ups of the ground state depopulation (b), and of the hole populations of the different orbitals (c-e). 2 (q + ǫ) ω − ωres σ(ω)=σ + σ , with ǫ = , (3) a 1+ ǫ2 b Γ/2 The hole population for the 2s,2p0, and 2p±1 orbitals as well as the depopulation of the neon ground state are where q is the Fano parameter describing the asymmetry presented in Fig. 2. Note that for linearly polarized light of the line shape, ωres is the resonance frequency of the the sign of the magnetic quantum number m is unim- transition, and Γ is the width of the resonance structure. portant and the +m and −m electrons behave exactly in These fits give the transition frequencies for both models the same way when the initial state is an M = 0 state as well as the transition width and Fano parameter for as it is the case for closed-shell atoms. At the end of the the full model: ωintra = 45.511 eV, ωres = 45.538 eV, pulse, all 2pm depopulations increase while that of the Γ = 31.8 meV, and q = −1.32. The experimentally ob- 2s decreases. The total depopulation, which is the sum tained value for the resonance position is 45.546 eV, for of the 2s and all 2pm orbitals, remains constant, indi- the line width is 13 meV, and the Fano parameter is cating that the 2p and 2s hole populations vary equally q = −1.58 [22, 24]. Since the experimental line width but oppositely. Note that in Fig. 2(b-e) the time scale is more than twice as small as our theoretical one, the is changed to visually emphasize these temporal trends. spectral features presented in Figs. 3-4 will be in reality This is also consistent within TDCIS, where the depop- not as broad. Qualitatively, however, this line width dis- ulation of the ground state can no longer change when crepancy has no effect on the results and the conclusions. the pulse is over [see Eq. (2a)]. Only the hole rearranges At frequencies below ωres, the two ionization pathways with time from the 2s orbital to the 2p orbitals. constructively interfere and the overall 2p ionization is This hole rearrangement is the resonant Auger decay increased. Above ωres, the two pathways destructively (or the autoionization process). The energy released by interfere and the overall 2p ionization is suppressed. The the hole movement, 26.9 eV, is sufficient to knock the ex- photon energies at the minimum ωmin = 45.559 eV and cited electron residing in the 3p shell, which has a binding the maximum ωmax = 45.525 eV are determined visually. energy of 2.9 eV, into the continuum [43]. 4

B. Hole Alignment As we can see from Fig. 3(a) the 2pm populations do vary across the resonance. Especially the ionization for As we have seen in Sec. IIIA1, the indirect ioniza- 2p±1 is much more suppressed at ωmin than for 2p0. In tion pathways via the autoionizing 2s−13p state interferes the following, we investigate in more details why the ion- constructively or destructively with the direct ionization ization of the 2pm orbitals behave so differently by having pathway depending on the detuning of the photon en- a closer look at the partial-wave contributions leading to ergy. The spectral information (in Fig. 1) does, however, s-wave and d-wave photoelectrons. not contain channel-resolved cross sections. Particularly, it cannot answer the question as to which extent the in- 1. d-wave photoelectron terference affects all 2pm ionization channels equivalently or whether there is a preferred m ionization channel. A non-uniform behavior would result in different effective The 2p±1 ionization is much more suppressed than ionization rates for 2p0 and 2p±1 and, consequently, in a 2p0 around ωmin [see Fig. 3(a)]. For 2p±1, the destruc- modified ratio between 2p0 and 2p±1 hole populations tive interference is so strong that it leads to a sup- compared to the ratio expected for non-resonant one- pression of almost 2 orders of magnitude compared to −1 photon ionization at similar photon energies. non-resonant photon energies. All 2pm εdm partial-wave By studying theoretically and experimentally the an- channels show the same degree of suppression. To be −1 −1 gular distribution of the photoelectron [24], a large vari- more precise, the ratio between 2p0 εd0 and 2p±1εd±1 ation of the asymmetry parameter β has been found. is exactly 4/3. A detailed analysis shows that this ra- Therefore, we also expect an variation in the ionic hole tio between the m = 0 and |m| = 1 appears in both, states. However, it is not possible to connect directly the the direct and the indirect, ionization pathways and can angular photoelectron distribution with the ionic hole be explained by the Wigner-Eckart theorem [35]. Con- state. Theoretical studies [24] showed that at ωmin an sequently, the behavior of constructive and destructive asymmetry parameter of β = 0 is expected for the 2s– interference is exactly the same for all d-wave channels, −1 2p Fano resonance, meaning the photoelectron is in a 2pm εdm. pure s-wave state. For this special case, the photoelectron angular distribution can be related to the ionic hole alignment, since an s-wave photoelectron can only origi- 2. s-wave photoelectron nate from the 2p0 orbital. Such a connection to the ionic state has, however, not been made in earlier studies. To generate 2p0 holes there exists another ionization −1 channel leading to an s-wave photoelectron, i.e., 2p0 εs0.

−1 The behavior of this partial-wave channel is diﬀerent 10 (a) 2p−1 2p−1εd (b) full TDCIS ±1 0 0 intrachannel TDCIS −1 −1 −1ε than the behavior of the 2pm εdm partial-wave channels 2p0 2p0 s0 1 1 − −2 [see Fig. 3(a)]. For 2p εs0, the destructive interference 10 10 0 / 2p 0 happens at ωmax and constructive interference occurs at

10−3 ωmin. ratio 2p

hole population −1 The overall trend is dominated by 2pm εdm, since the 10−4 probability of ejecting an electron from a p-orbital into 1 45.5 45.55 45.6 45.5 45.55 45.6 an s-continuum is generally much smaller than ejecting energy [eV] energy [eV] the electron into a d-continuum [36]. Only around ωmin, where the ionization into a d-continuum is strongly sup- Figure 3: (color online) (a) Hole populations are shown as pressed, the situation changes and ionization into the a function of the photon energy for the 2p0 orbital (dashed- s-continuum becomes the dominant ionization channel dark-blue line), the 2p±1 orbital (solid-red line) as well as −1 (corresponding to an asymmetry parameter of β = 0). for the two 2p0 ionization channels 2p0 εd0 (dotted-light-blue −1 −1 The relative enhancement of the 2p εs0 partial-wave line) and 2p εs0 (dashed-dotted-violet line). (b) The ratio 0 0 channel results in a ten times smaller overall suppression 2p0/2p±1 is shown for the full TDCIS model (solid-orange line) and the intrachannel TDCIS model (dashed-green line). for 2p0 ionization than for 2p±1 ionization [see Fig. 3]. A Gaussian pulse with a peak intensity of 3.5 × 1013 W/cm2 and a FWHM-duration of 174 fs has been used. 3. The ratio of 2pm hole populations In Fig. 3(a), the m-resolved hole populations of the 2p-shell are shown (thick lines). Next to the hole pop- In Fig. 3(b), the hole population ratio 2p0/2p±1 is ulations for 2p0 (dashed dark-blue line) and 2p±1 (solid shown as a function of the photon energy for the full −1 red line), the two partial-wave channels 2p0 εs0 (dashed- TDCIS model (orange-solid line) and the intrachannel −1 dotted violet line) and 2p0 εd0 (dotted light-blue line) TDCIS model (green-dashed line). This ratio is a di- are shown as well. For the 2p±1 hole ionization, there rect measure of hole alignment, where 1 stands for an exists only one ionization channel where the continuum isotropic hole distribution, ∞ for perfect hole alignment −1 electron is a d-wave (i.e., 2p±1εd±1). along the polarization direction, and 0 for perfect hole 5 antialignment in the plane perpendicular to the polariza- (a) [ 2p±0.5 ]−1 (b) full TDCIS 0.003 0.5 intrachannel TDCIS tion direction. ±0.5 −1 [ 2p1.5 ] ±1.5 −1 ±1.5 1.5 [ 2p1.5 ] 10

Strong variations of the hole alignment across the Fano 0.002 / 2p resonance are found resulting in ratios that vary by more ±0.5 1.5

than one order of magnitude (between 1.6 and 18). A hole population 0.001 ratio of 18 means the 2p hole is primarily located in the ratio 2p 2p0 orbital, and only a 10% chance exists to find the 0 1 45.5 45.55 45.6 45.5 45.55 45.6 hole in either the 2p+1 or 2p−1 orbital. Such strong hole energy [eV] energy [eV] alignment is normally only encountered in the strong field regime where tunnel ionization almost exclusively ionizes Figure 4: (color online) (a) Hole population is shown as a ±0.5 the outermost p0 orbital (when using linearly polarized function of the photon energy for the 2p0.5 orbital (solid-red ±0.5 ±1.5 light) [25, 27]. line), the 2p1.5 orbital (dashed-blue line), and the 2p1.5 ±1.5 ±0.5 orbital (dotted-light-blue line). (b) The ratio 2p1.5 /2p1.5 In the off-resonance limit, the intrachannel TDCIS is shown for the full TDCIS model (solid-brown line) and the model and the full TDCIS model approach the same intrachannel TDCIS model (dashed-green line). The same value for the 2p0/2p±1 ratio (1.6). Such values are very pulse parameters as in Fig. 3 has been used. common in the XUV and x-ray regimes where an almost isotropic distribution of the hole is found with a slight preference for the polarization direction (i.e., m = 0). The maximum hole alignment is reached when the pho- ±0.5 2 1 ton energy is ω , located at the minimum of the Fano 2p = ± 2p 1 ∓ 2p 1 (4a) min 0.5 r3 ±1,∓ 2 r3 0,± 2 resonance, which is exactly the energy where the sup- E E pression of the dominant ionization channels (leading to ±0.5 1 2 2p =+ 2p 1 + 2p 1 (4b) 2p−1 εd ) is most pronounced, and only s-wave photo- 1.5 r3 ±1,∓ 2 r3 0,± 2 m m E E electrons are formed which leave a 2p0 hole behind. ±1.5 1 2p1.5 = 2p±1,± 2 (4c) E

where |2pm,σi refers to the spatial 2pm orbital with the spin projection σ. Note that in Sec. IIIB we focused only the spatial part of the orbitals because the spin-up C. Spin-orbit coupling and spin-down components behave exactly the same [45]. The spin-orbit interaction is treated here in degenerate perturbation theory (see Ref. [20, 37]) where only the Up to now, we have ignored that the 2p shell is actu- impact on the angular momentum is considered. The ra- ally split due to spin-orbit coupling into two subshells 2pj dial part is unaffected by the spin-orbit interaction which with j = 1/2 and j = 3/2. As a result, the hole align- leads to errors of few per cent [38]. ment has to be defined with respect to mj and not ml. By using Eqs. (4a–4c), all populations shown in mj In particular, the 2p3/2 hole populations for mj = ±1/2 Fig. 4(a) can be written in terms of the non-spin-orbit- and mj = ±3/2 have to be compared. Here, mj refers split populations shown in Fig. 3(a), and, consequently, to the projection of the total angular momentum j along also the alignment ratio in the case of spin-orbit splitting the XUV polarization axis. In our TDCIS approach, we can be expressed in terms of the ratio without spin-orbit consider only the spin-orbit interaction within the ion, splitting as done earlier. where it is the strongest, and we neglect it for the photoelectron (see Ref. [20] for details). IV. CONCLUSION ±0.5 Figure 4 (a) shows the hole populations of 2p0.5 , ±0.5 ±1.5 2p1.5 , and 2p1.5 , and (b) shows the ratio between 0 5 1 5 We have shown that resonant excitation of the au- 2p± . and 2p± . defining the hole alignment. Figure 4 1.5 1.5 toionizing 2s−13p state leads to a second ionization path- shows the same trends as Fig. 3. The mixing of 2p and 0 way that can interfere with the direct 2p photoionization 2p orbitals in the spin-orbit case reduces the maximum ±1 pathway and strongly influences the state of the parent hole alignment within the 2p -shell by ∼ 2/3 in com- 3/2 ion. This interference is well known as the origin of the parison to the non-spin-orbit case [44], which results in characteristic Fano profile. Also the asymmetry param- a maximum alignment ratio of ∼13 instead of 18. eter β measuring the angular distribution of the photo- The reduction factor of 2/3 can be easily explained electron varies strongly across the Fano resonance but a when expressing the spin-orbit-split orbitals in terms of direct relation to the hole alignment cannot be made. the non-spin-orbit-split orbitals. Specifically, the trans- We showed that this interference has destructive char- formation between the spin-orbit-split (coupled basis) acter at ωmin and creates a dark-state in the photoelec- and non-spin-orbit-split (uncoupled basis) orbitals reads: tron continuum. As a result, the 2p−1εd ionization chan- 6 nel is strongly suppressed, and the photoelectron is emit- Controlling the hole alignment via the 2s–3p Fano res- ted as a pure s-wave. Consequently, the only orbital that onance serves as an example of how correlation effects is ionized is the 2p0 orbital. The imbalance of ionizing can be explicitly targeted and exploited to create new 2p0 and 2p±1 orbitals leads to a large hole alignment and exotic electronic states in atoms and molecules. Si- along the XUV polarization direction. miliarly other Fano resonances can be used where the The ratio between the populations of 2p0 and 2p±1 goes strength of the resonance determines how strongly the as high as 19—localizing the hole in the 2p0 orbital—and hole alignment can be tuned. Furthermore, with a sec- is significantly different than the off-resonant value (1.6), ond pulse the Fano resonance could be modified within which possesses only a slight hole alignment. Strong hole attoseconds [11] to gain an even larger control of the alignments are usually only encountered after tunnel ion- electronic motion. Also the extension to high-intensity ization with strong-field IR pulses, where the Keldysh pulses is interesting, which can be realized with currently parameter is well below 1 [26]. Here, we used XUV available seeded free-electron lasers like FERMI [39] or pulses and we are in the perturbative one-photon regime, sFLASH [40]. First preliminary results we have obtained where the Keldysh parameter is well above 1 and large suggest that completely different ionization behavior oc- anisotropies in the hole states are not expected. curs when a Fano resonance is driven by a high-intensity When disabling interchannel coupling effects, i..e, dis- pulse. abling the correlation-driven autoionization mechanism of the excited 2s−13p state, no interference of the ionization pathways occurs and no hole alignment modulation Acknowledgments appears when tuning across the 2s–3p resonance. Includ- ing spin-orbit interaction within the ion does not change E. H.-J. would like to thank the DAAD RISE program the picture. Only the strong hole alignment within the and her mother Christa Heinrich-Josties for financial sup- 2p3/2-shell is reduced by a factor 2/3, which still results port. This work has been supported by the Deutsche in a strong hole alignment with ratios up to 13:1 between Forschungsgemeinschaft (DFG) under grant No. SFB ±1/2 ±3/2 2p3/2 and 2p3/2 hole populations. 925/A5.

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