Open Access Article

This Open Access Article is licensed under a Creative Commons Attribution-Non Commercial 3.0 Unported Licence

M.
Ruberti
*

Department of Physics, Imperial College London, Prince Consort Road, London SW7 2AZ, UK. E-mail: m.ruberti11@imperial.ac.uk; Tel: +44 (0)20 7589 47728

Received
11th September 2020
, Accepted 30th November 2020

First published on 8th December 2020

Here I present a fully ab initio time-resolved study of X-ray attosecond transient absorption spectroscopy (ATAS) in a prototypical polyatomic molecule, pyrazine, and demonstrate the possibility of retrieving the many-electron quantum ionic coherences arising in attosecond molecular photoionisation and pre-determining the subsequent charge-directed photochemical reactivity. Advanced first-principles many-electron simulations are performed, within a hybrid XUV pump/X-ray probe setup, to describe the interaction of pyrazine with both XUV pump and X-ray probe pulses, and study the triggered correlated many-electron dynamics. The calculations are carried out by means of the recently-developed ab initio method for many-electron dynamics in polyatomic molecules, the time-dependent (TD) B-spline Restricted Correlation Space-Algebraic Diagrammatic Construction (RCS-ADC). RCS-ADC simulates molecular ionisation from first principles, combining the accurate description of electron correlation of quantum chemistry with the full account of the continuum dynamics of the photoelectron. Complete theoretical characterisation of the atto-ionised many-electron state and photo-induced attosecond charge dynamics is achieved by calculating the reduced ionic density matrix (R-IDM) of the bipartite ion–photoelectron system, with full inclusion of the correlated shakeup states. Deviations from the sudden approximation picture of photoionisation, in the low-photon-energy limit, are presented. The effect of the multi-channel interaction between the parent-ion and the emitted photoelectron on the onset of the quantum electronic coherences is analysed. Moreover, I show how the Schmidt decomposition of the R-IDM unravels the many-electron dynamics triggered by the pump, allowing for the identification of the key channels involved. Finally, I calculate the X-ray attosecond transient absorption spectra of XUV-ionised pyrazine. The results unveil the mapping of the ATAS measurement onto the quantum electronic coherences, and related non-zero R-IDM matrix elements, produced upon ionisation by the XUV pump laser pulse.

The study of the coherent many-electron dynamics upon photoionisation is a central topic in attosecond science. Since molecular bonds are more likely to break where the hole is localised, direct access to the ultrafast charge redistribution underpinned by electronic coherence in the ion could be ultimately exploited to control photoionisation-induced charge-directed reactivity on unprecedented ultrafast (attosecond) time-scales, steering the reaction dynamics at a very early stage of its quantum evolution by acting only on the electronic degrees of freedom, i.e. before the nuclei start to move and the de-phasing of electronic coherences occurs. This represents a unique opportunity, which was previously unavailable on the much longer (femto/pico-second) time-scales intrinsic to nuclear dynamics, to find new ways of controlling photochemical reactions by acting on the quantum electronic coherences, rather than just on the populations. Moreover, since the ultrafast charge redistribution underpinned by electronic coherences in the ion initiates all photochemical change,^{12,13} and as a result also pre-determines radiation damage, the study of quantum electronic coherence and entanglement in photoionisation deepens our understanding of the basic processes leading to photosynthesis, radiation damage, mutation and, more generally, the alteration of molecular biological functions.

To this end, the goal should be to develop quantum protocols, based on attosecond measurement techniques, for the tomographic reconstruction of the attosecond ionised quantum states. Indeed, the existence of long-enough lived quantum electronic coherences would not only make their experimental observation possible, but could also enable quantum state tomography of the ionic density matrix emerging from a molecular attosecond ionisation event. In this direction, the recent spectacular advances in ultrafast light-source technology have opened up the possibility of unprecedented direct experimental access to electronic quantum coherences through cutting-edge attosecond pump–probe spectroscopy techniques.^{14,15}

However, in order to reach these goals, a crucial role is played by the development of a comprehensive theoretical understanding, based on an accurate theoretical modelling of the ultrafast molecular ionisation process and the role of the measurement, which can strongly support and guide attosecond experiments and their interpretation. The present theoretical picture lacks a full understanding of the level of entanglement produced in the photoionisation process and the consequent role of measurement in the observation of the ensuing dynamics. Moreover, in spite of the fact that predicting the quantum ionic coherences arising in photoionisation is essential for modelling attosecond hole migration,^{14} until very recently the density matrix characterising the ionic state emerging from attosecond ionisation could not be predicted theoretically or characterised experimentally, except for 2-level atomic cases.^{16} This is because predicting the ionic coherence in attosecond molecular ionisation represents a major theoretical challenge, where the multi-channel and multi-centre many-electron problem, in the presence of a laser field, must be solved with an accurate description of the photoelectron and parent-ion.

With the availability of the new B-spline restricted correlation space (RCS)-ADC many-body theoretical approach,^{17,18} complete theoretical characterisation of the attosecond ionised states in molecular systems has become possible.^{18} This has opened up the possibility of calculating, in a fully ab initio fashion and including the correlated ionic shakeup states, the reduced ionic density matrix (R-IDM) of the bipartite ion–photoelectron system produced by molecular attosecond ionisation. It also allows us to fully capture the onset of the quantum electronic coherences in the ionic system, generated during the laser-induced dynamics, as well as the ion–photoelectron entanglement. The B-spline restricted correlation space (RCS)-ADC method for many-electron dynamics allows one to model ionisation of much larger molecular systems than previously possible within the standard B-spline ADC method,^{25–28} at the same or higher level of accuracy. It successfully combines the full description of the photoelectron continuum states (described in terms of a multi-centre B-spline basis set) with a realistic quantum chemistry approach to the modelling of electron correlation. Moreover, the new method goes beyond the previous theoretical studies where the ionisation step (and the photoelectron degree of freedom) was either completely neglected,^{3–5,8–11,19,20} or modelled without taking into account important electron correlation effects such as interchannel couplings and ionic shakeup.^{14,21–24} RCS-ADC is based on the separation of the orbital space into correlation and ionisation spaces and naturally bridges the gap between standard ADC multi-configurational ab initio techniques and closed-coupling schemes based on the limited number of essential, physically relevant, ionic states, combining the key advantages of both.

In this article, I apply the B-spline RCS-ADC method to describe, in a fully ab initio fashion, a complete attosecond pump–attosecond probe experiment in a prototypical polyatomic molecule, i.e. pyrazine. In particular, I present a time-resolved study of X-ray attosecond transient absorption spectroscopy (ATAS) in pyrazine and demonstrate the possibility of retrieving the many-electron quantum ionic coherences arising in attosecond photoionisation and pre-determining the subsequent charge-directed reactivity. Advanced first-principles many-electron simulations are performed, within a hybrid XUV/X-ray pump–probe setup, to describe the interaction of pyrazine with both the XUV pump and X-ray probe pulses. First, during the pump step, I calculate the density matrix of the bipartite ion–photoelectron system, from which I extract a R-IDM fully including the correlated shakeup ionic states. I perform a Schmidt decomposition of the R-IDM in order to analyse the coherent ionic states emerging from the ionisation event. As a result, the pump-triggered charge dynamics and predominant timescales for many-electron motion are identified and mapped onto the quantum coherences between the populated ionic eigenstates. In addition, I analyse the effects of the residual interaction between the emitted photoelectron and the parent-ion on the quantum coherent many-electron dynamics, as well as its deviations from the predictions of the sudden approximation, in the low-photon-energy regime, i.e. close to the ionisation thresholds of the many-body system. In this way, this work provides a complete theoretical characterisation of the atto-ionised many-electron state and photo-induced charge dynamics.

Second, I study how this dynamics is reflected in the actual pump–probe signal by calculating the X-ray transient absorption spectra of XUV-ionised pyrazine. The ab initio simulation of the ATAS measurement is performed by solving the von Neumann equations for the characterised R-IDM prepared by the pump interacting with the X-ray probe field. The results of the simulation show how the ATAS measurement observables can be mapped onto the quantum electronic coherences corresponding to the non-zero R-IDM matrix elements, and related charge dynamics, produced upon ionisation by the pump laser pulse. The chemically site-specific information is obtained by using X-ray photon energies able to resonantly excite the ionic system to the nitrogen and carbon K-edges. The results presented here extend the work reported in a recent theoretical study on the simulation of pump–probe experiments in CO_{2}^{28} to a larger molecular system and, at the same time, at a higher level of accuracy in the description of electron correlation. Furthermore, it represents an important step further with respect to previous reports where the calculations were restricted to describing the electron dynamics following the ionisation induced by the pump pulse, but ignored the effect of the probe pulse and the role of the attosecond measurement.^{18,29} The calculations are performed within the frozen-nuclei approximation and focus only on the electronic degree of freedom, which is a valid approach for the first few femtoseconds after ionisation.^{7,11,13} On longer time-scales, these calculations provide a frozen-nuclei benchmark to which the experimental results can be compared, in order to identify the effects of nuclear motion.

(1) |

(2) |

|^{N}_{μ,n}〉 = ĉ^{†}_{μ}|Ψ^{N−1,RCS}_{n}〉, | (3) |

Ĥ^{N−1}_{RCS-ADC(n)}|Ψ^{N−1,RCS}_{n}〉 = I^{n}_{p}|Ψ^{N−1,RCS}_{n}〉, | (4) |

(5) |

A complex absorbing potential (CAP) of the form Ŵ = η(r − r_{CAP})^{2}, r ≥ r_{CAP} is used to eliminate wave packet reflection effects from the radial grid boundaries. With the addition of the CAP term, the total time-dependent Hamiltonian of eqn (1) reads as

Ĥ^{N}(t) = Ĥ^{N}_{RCS-ADC(n)} + ^{N}_{RCS-ADC(n)}E_{Pump}(t) − iŴ, | (6) |

(7) |

The R-IDM fully describes the ionic state prepared by attosecond molecular ionisation; its diagonal elements correspond to the TD population of the different ionic eigenstates, P_{n}(t) = |ρ_{n,n}(t)|, while the off-diagonal ones, ρ_{m,n}, are related to the degrees of coherence, 0 ≤ G_{m,n} ≤ 1, between any pair of populated ionic channels m and n

(8) |

(9) |

The time evolution of the hole-density (r,t) created upon removal of one electron by the pump pulse is calculated in the RCS as , where the hole-density matrix Ñ_{pq}(t)^{17} depends on the R-IDM and reads as

(10) |

(11) |

(12) |

Ĥ^{Ion}(t) = Ĥ^{N−1}_{RCS-ADC(n)} + ^{N−1}_{RCS-ADC(n)}E_{X}(t), | (13) |

(14) |

The initial condition used to solve eqn (12) is , where is the R-IDM as calculated in eqn (7), τ_{d} > 0 is the pump–probe time-delay and T_{Probe} is the time-duration of the X-ray probe pulse. Note that, prior to the arrival of the probe pulse , the time-dependence of the R-IDM is given by , which is in general different from the free-evolution solution of eqn (12) in the same time interval because of the residual interaction between the parent-ion sub-system and the photoelectron “bath” after the pump ionisation event. On the contrary, this interaction is neglected at later times , where the time evolution of the R-IDM interacting with the X-ray probe pulse is given by the solution of eqn (12). The validity of this approximation requires the effect of the interaction with the X-ray probe pulse to dominate over the residual interaction with the emitted photoelectron and to mainly govern the many-electron dynamics of interest in the cation. This is confirmed, for values of the pump–probe time-delay τ_{d} approximately >1 fs, by the results presented in Section 3. The frequency-dependent absorption cross-section σ(ω;τ_{d})^{30} is calculated from the expectation value of the molecular electric dipole moment, , and the incident X-ray probe field E_{X}(t) as

(15) |

The ab initio simulation of the pump step has been performed using a linearly polarised low-photon-energy XUV pulse, characterised by a Gaussian temporal envelope, a central frequency ℏω_{Pump} = 25 eV, a peak intensity I_{Pump} = 6 × 10^{11} W cm^{−2} and a time duration T_{Pump} ∼ 500 as at full width at half maximum (fwhm). The latter value corresponds to a few-cycle pulse with a broad, e.g. ≈7.5 eV fwhm, spectral-energy bandwidth. Results for the pump-triggered dynamics are presented for one specific orientation of the laser polarisation direction with respect to the molecular frame, namely the one parallel to the C–C bonds (and the N–N axis, z-direction). Convergence of the results has been achieved by using a Δt = 5 time-step and a K = 40 Krylov-space dimension in the Arnoldi/Lanczos time propagation.

As a result of this rich electronic structure and the high density of states in the electronic spectrum of the molecular cation, the broad bandwidth of the laser pulse can, in principle, create quantum coherence between multiple pairs of ionic many-electron states. Fig. 2 shows the TD RCS-ADC(2)x calculated final (at about 40 fs after ionisation) degree of coherence G_{m,n} between each pair of ionic eigenstates populated by the XUV pump pulse. The final stationary populations P_{n}(∞) of the RCS-ADC(2)x states of the pyrazine cation are also shown in the vertical and horizontal side panels of Fig. 2. In the calculations presented here, all the ionic time-dependent populations P_{n}(t) have converged to their final stationary value at ∼2 fs after ionisation. The relative phases Φ_{m,n} between the partially coherently populated eigenstates of energies E_{m} and E_{n} are shown in Fig. 3. The Φ_{m,n} relative phases are extracted from the phases of the off-diagonal matrix elements of the R-IDM and read as Φ_{m,n} = arg|ρ_{m,n}(t)e^{+i(Em−En)t}|. Note that the hermiticity of the density matrix implies that Φ_{m,n} = −Φ_{n,m}. The P_{n}, G_{m,n} and Φ_{m,n} quantities represent a complete set of information for the description of the ionic state, which is fully characterised by its density matrix, prepared by ionisation in the pump step. In Fig. 2, it is possible to identify islands of strong coherence close to the diagonal (the highest degree of generated coherence being 0.9), mainly for ionic states with ionisation potential up to 23 eV. In general, a robust degree of coherence is produced between pairs of states with an energy gap up to approximately the value of the pump–pulse bandwidth, while for larger energy separations, the coherence rapidly decays to very low (<0.2) values. Furthermore, the potentially very high number of energetically-allowed coherences is reduced by the high symmetry of the molecule. Indeed, when ionisation occurs by one-photon absorption from the ground-state of the molecule, as is mainly the case with the pump pulse parameters used in this work, a degree of quantum coherence different from zero is possible only between ionic eigenstates of the same molecular point group symmetry. This is because the ionic density matrix is calculated by tracing the full N-electron neutral wavefunction over all the spatial-symmetry quantum numbers of the photoelectron, implying a measurement set-up where the direction of the emitted photoelectron is either not measured or integrated out in the analysis of the experimental data.

The charge dynamics triggered by the partially-coherent population of ionic eigenstate pairs upon XUV pump ionisation is analysed in Fig. 4, which shows the B-spline RCS-ADC(2)x calculated time evolution, as well as its Fourier-transform, of the hole-occupation numbers ñ_{p}(t) (positive eigenvalues of the hole-density matrix, see eqn (11)) for the most contributing natural charge orbitals. Fig. 4 also provides a snapshot of the difference between the pyrazine hole-densities at times t_{2} and t_{1} as indicated in the upper panel (a). Note that the two times correspond to the minimum and maximum positions in the hole-occupation number evolution for the natural charge orbitals, showing a ∼5.5 fs period oscillation. The dominant frequencies of the charge dynamics are highlighted in panel b. As it is possible to see in Fig. 4, the emerging charge dynamics in the pyrazine cation is characterised by two predominant timescales; a main, relatively slow one with period ∼5.5 fs, and a faster one, with period ∼500 as. The former dynamics is driven exclusively by electron correlation, i.e. by the coherence between the main ionic state characterised by 1h 3b_{2u} content at 16.2 eV and its satellite at 15.45 eV. On the contrary, the faster timescale is a result of the partially-coherent superposition of the ground (8.7 eV) and first excited (16.25 eV) ionic states of A_{g} symmetry, mainly representing ionisation from the two different 6a_{g} and 5a_{g} molecular orbitals, respectively. Therefore, in spite of the multitude of channels involved in pump ionisation, the theoretical analysis presented here makes it possible to disentangle the triggered dynamics and attribute it to a few specific, coherently-populated, ionic channels. In terms of charge redistribution across the molecular backbone, the periodic ∼5.5 fs oscillation mode originating from electronic correlation consists of hole-density fluctuations that occur both off-plane and in-plane along the pyrazine skeleton, between the molecular (C–C and C–N) bonds and the carbon and nitrogen atomic centres. The 500 as charge dynamics consists of anti-phase oscillations of the hole-density around the nitrogen and carbon atoms, as can be seen, in Section 3.2, from the spatially-resolved information that can be obtained from the energy-integrated ATAS spectra. In the present work, molecular ionisation is modelled beyond the sudden approximation (SA)^{3} by fully describing the details of the interaction with the laser pulse as well as the residual interaction between the ionised electron and the remaining ionic core. Therefore, it is interesting to compare the full TD B-spline RCS-ADC(2)x results with those obtained using the SA ansatz.^{3} This is done in Fig. 4, where the hole-occupation numbers ñ_{p}(t) calculated within the SA framework are also shown. The results demonstrate that, in the low-photon-energy limit, the deviations from the sudden approximation results are strong and more pronounced with respect to those present at higher photon energies.^{18} The pronounced deviations visible in Fig. 4 can be explained by both the different values of the degrees of coherence, which are assumed maximal in the sudden approximation ansatz, and, predominantly, the differences in the amplitudes, i.e. populations and phases, of the created ionic wavepacket. This is due to the fact that, in the low-photon-energy limit, the ionisation partial cross-sections of the various open channels (and respective transition dipole matrix elements in the continuum) usually present a strong non-monotonous energy dependence, which is more pronounced than at higher photon energies. This feature, as well as the specific laser pulse parameters and polarisation, is completely neglected within the sudden approximation and causes the relative ionic-state populations to differ considerably from the spectral intensity values in Fig. 1. The correct description of the ionisation amplitudes, which also define the relative phases imprinted in the ionic wavepacket (see Fig. 3) and thus the characteristics of the prepared many-electron system, is necessary for the accurate simulation and interpretation of ionisation experiments. Therefore, for pulses characterised by photon energies close to the ionisation thresholds of the system, a quantitative description of the induced many-electron dynamics requires modelling of the ionisation step beyond the sudden approximation, including an explicit time-dependent description of the process and, in addition to the correlated motion of the bound electrons, the molecular electronic continua.

Fig. 4 (a) Time-dependent hole-occupation numbers ñ_{p}(t) (eqn (11)) of the hole-type natural charge orbitals most contributing to the ultrafast charge dynamics emerging from the pump step ionisation of the C_{4}H_{4}N_{2} molecule, calculated using the full TD B-spline RCS-ADC(2)x theoretical method (full lines) and the sudden approximation ansatz (dashed lines). The natural charge orbitals with positive occupation number correspond to the removal of one electron from one of the occupied orbitals in the HF ground state, with different coloured lines describing the fraction of the removed bound electron from the HOMO (1b_{1g}) (green curve), HOMO−1 (6a_{g}) (red curve), HOMO−2 (1b_{2g}) (dark green curve), HOMO−3 (5b_{1u}) (black curve), HOMO−5 (1b_{3u}) (orange curve), HOMO−6 (4b_{2u}) (magenta curve), HOMO−8 (3b_{2u}) (blue curve) and HOMO−10 (3b_{3g}) (purple curve) occupied HF orbitals. (b) Fourier-transform of the TD B-spline RCS-ADC(2)x results in (a); the dominant frequencies are highlighted with circles. (c) Charge density representing the difference between the pyrazine hole-densities at times t_{2} and t_{1} as indicated in (a); the red and green isosurfaces enclose negative and positive charge densities, respectively. |

Moreover, when ionisation happens at photon energies close to the various ionisation thresholds of the molecular system, it is possible to expect a fraction of slow photoelectrons to be produced, which will then be interacting in close proximity with the parent-ion for relatively long times compared to the natural bound-electron timescales. It is therefore interesting to investigate, in the present case of XUV pump ionisation, the effect of the residual inter-channel interaction between the parent-ion and the photoelectron “bath” on the quantum ionic coherence, the ion–photoelectron entanglement and, ultimately, on the ionic charge dynamics while the emitted photoelectron is moving away from the molecular region. In Fig. 5, the full TD RCS-ADC(2)x result for the pump-induced charge dynamics in pyrazine is compared to that obtained by neglecting the inter-channel ion–photoelectron interaction at times t > 170 as after the ionisation event. The results show a quantitative deviation between the two results, demonstrating that, for low-photon-energy pulses, the effect of the interaction with the slower photoelectron on the ionic coherences is stronger than that calculated in ref. 18, where higher photon energies (60–80 eV) were considered. The differences in charge dynamics, when neglecting the residual ion–photoelectron interaction at t > 170 as, are due to the fact that, as can be seen in panel (d) of Fig. 5, the driving quantum electronic coherences keep changing in an appreciable way for longer times. The main coherences responsible for the charge dynamics, G_{1Ag,2Ag}(t) and G_{3B2u,4B2u}(t), become effectively stationary at t ∼ 1 fs after ionisation, whereas the values of other coherences either drop or oscillate for times up to ∼2 fs. At times t > 2 fs after the XUV pump ionisation event, the values of the relevant coherences show an effectively negligible variation, which explains why the charge dynamics computed with the full TD B-spline RCS-ADC(2)x method does not experience any effective damping. The very small residual variation at t > 2 fs reflects the fact that, in agreement with the results presented in ref. 21, the fraction of photoelectrons with “infinitely”-low kinetic energy, that can remain close to the parent-ion for longer times, is effectively negligible. Moreover, this result confirms the validity of using eqn (12), which neglects the residual ion–photoelectron interaction, to describe the interaction of the pump-ionised system with the probe pulse at time-delays τ_{d} > 2 fs. The time evolution of all the individual coherences is reflected in that of the purity Tr[ρ^{2}](t) (shown in panel (b)), which shows a fast decay over the first fs after ionisation (with consistent increase in the ion–photoelectron entanglement as measured by the von Neumann entropy s(t)) and a further, slower and relatively smaller decay which persists for the entire time-range of the simulation.

Fig. 5 (a) Time-dependent hole-occupation numbers ñ_{p}(t) (eqn (11)) of the “occupied” natural charge orbitals triggered by attosecond ionisation of the C_{4}H_{4}N_{2} molecule in the pump step; full lines: full TD RCS-ADC(2)x results; dashed lines: results obtained by switching off the residual interaction between the parent-ion and the photoelectron at t_{free} = 170 as after ionisation. Time evolution of (b) the purity Tr[ρ^{2}](t) and (c) the von Neumann entropy of entanglement s(t) of the bipartite ion–photoelectron system after pump ionisation of the C_{4}H_{4}N_{2} molecule. (d) Time evolution, upon ionisation by the pump laser pulse, of the degrees of coherence G_{m,n}(t) of the pairs of ionic states mostly contributing to the density-matrix purification (see Fig. 6); the time evolution of the G_{1Ag,2Ag}(t) (black curve) and G_{3B2u,4B2u}(t) (red curve) coherences, mostly contributing to the pump-triggered charge dynamics, are highlighted. |

As already mentioned, the results presented in Fig. 4 and 5 have revealed that, despite the high number of open ionisation channels (see Fig. 1) and triggered ionic coherences (see Fig. 2 and 3), the charge dynamics in the pyrazine cation can be effectively traced back to a few quantum electronic coherences, in particular G_{1Ag,2Ag}(t) and G_{3B2u,4B2u}(t), by which it is mainly driven. The question is then: can the description of the full state of the ionic system, i.e. the density matrix, also be simplified and reduced to a few key, physically-relevant, ionic eigenstates? The key theoretical tool that allows us to answer this question is the Schmidt decomposition, i.e. the purification of the ionic density matrix (see eqn (9)). Diagonalization of the ionic density matrix emerging from XUV pump ionisation of pyrazine yields a few pure-state channels with weight greater than 0.1, i.e. whose contributions, more than 10%, dominate the Schmidt decomposition, while the majority of the resulting pure-state channels have negligible contributions (weight r_{n} ≤ 0.01). Therefore, the answer to the above question is affirmative. This is shown in Fig. 6, which shows the weights r_{n} and expansion coefficients |c^{n}_{i}|^{2}, in the basis of ionic eigenstates i, of the 10 pure states that contribute most to the purification. Each pure-state channel is a coherent superposition of the ionic eigenstates with a non-zero coefficient |c^{n}_{i}|^{2}. In particular, the two coherences G_{1Ag,2Ag}(t) and G_{3B2u,4B2u}(t) arise in the two principal channels with weights r_{1} and r_{2}, respectively. The r_{4} channel consists of a coherent superposition of the 1B_{1u} and 2B_{1u} ionic eigenstates, while the r_{3} and r_{5} channels in Fig. 6 mainly coincide with the 1B_{3g} and 1B_{2g} ionic eigenstates, respectively. Finally, the coherently-populated ionic eigenstates contributing to the pure-state channels of weight r_{1}, r_{2} and r_{3} are the same as those characterising the lower-weighted r_{6}, r_{7} and r_{8} pure-state channels shown in Fig. 6. The theoretical simulation presented here therefore shows that the state of the ionic system prepared by XUV pump ionisation of pyrazine can be approximated by a reduced, limited number of physically-relevant pure-state channels, to which the triggered many-electron dynamics can be ascribed, and further allows one to identify and characterise them.

To answer these questions, the density matrix of XUV-ionised pyrazine interacting with the X-ray probe pulse is time-propagated here by solving the TD von Neumann equation in eqn (12). This is done for the three independent linear-polarisation directions of the probe field relative to the molecular frame, namely perpendicular to the molecular plane (x-direction), and along (z-direction) and perpendicular to (y-direction) the C–C bond of the fixed-in-space C_{4}H_{4}N_{2} molecule. In what follows I focus on orientation-averaged cross-sections for unaligned molecules. The size of the ionic ADC matrices is ∼4000, making the TD Hamiltonian matrix amenable to full diagonalization. Therefore, in this work, the equations of motion (eqn (12)) discussed in the previous section are solved by fully diagonalizing the time-dependent ionic Hamiltonian at each time-step. The calculation is performed for 160 values of the pump–probe delay, ranging from 1 to approximately 22 fs. The time-delay sampling used is 140 as, which is in principle small enough to allow for every coherent oscillation triggered by the pump pulse bandwidth to be resolved. The simulations are carried out for two different Gaussian probe pulses with photon energy centred at 272 eV and 393 eV, i.e. at the average transition-energy between the valence-ionised states of pyrazine and the C(1s) and N(1s) core-ionised states, respectively. Since the C(1s) to valence transitions and the N(1s) to valence transitions are energetically well separated, ATAS allows one to study valence coherent many-electron dynamics in pyrazine with atomic spatial resolution. Moreover, extremely high time-resolution is achieved by using ultrashort pulses of fwhm duration T^{C}_{Probe} = 150 as (C(1s) pulse) and T^{N}_{Probe} = 120 as (N(1s) pulse), corresponding to bandwidths ≥30 eV. The intensity of the probe fields is 5 × 10^{14} W cm^{−2}. The latter is in fact a relatively low intensity compared to the much higher intensities achievable in X-ray FEL experiments, and presents the advantage of operating in the linear interaction regime. I also assume a uniform intensity distribution over the ionised sample and thus do not perform spatial averaging over the pulse. Fig. 7 shows the time-delay dependent absorption cross-section integrated over selected spectral regions, i.e. the C K-edge and N K-edge energy windows centred at 272 eV and 393 eV, respectively. Results are shown both for the orientation-averaged cross-section and for the three independent linear-polarisation directions of the probe pulse relative to the molecular frame. The orientation-averaged absorption signals, energy-integrated over the carbon and nitrogen windows, reflect the time-evolution of the partial hole-density^{45} around the C and N atomic centres, respectively, whereas the polarisation-dependent signals are sensitive to the local oscillations of the atomic partial charge along the specific directions. The results show that the charge dynamics takes place at both the carbon and nitrogen atomic centres of the C_{4}H_{4}N_{2} ring. In particular, it is noticeable that the fast ∼500 as period oscillation characterises both the carbon- and the nitrogen-window curves, but with opposite phases, reflecting a charge dynamics which involves an oscillation of the hole-density between the C and N centres. On the contrary, the two C- and N-window curves show an in-phase ∼5.5 fs oscillation. This indicates that the related charge dynamics (shown in Fig. 4) corresponds to an in-phase oscillatory evolution of the hole-density around the C and N atomic centres.

The spectrally-resolved transient absorption spectra at the carbon and nitrogen K-edges are shown in Fig. 8 together with their Fourier-transforms along the time-delay axis, which reveal the quantum electronic beatings resolved by the ATAS measurements. The absorption’s interference pattern, characterised by a rapidly oscillating signal as a function of the pump–probe time-delay, is a result of the electronic beatings within the pump-ionised wavepacket. In particular, the peaks associated with the main core-ionised final states of 1h character and which reflect the predominant quantum beatings governing the charge dynamics of the ionic system are marked in Fig. 8. The time-resolved spectrum in the nitrogen window presents a richer structure compared with that of the carbon-window, with the corresponding time-dependent absorption extending over broader spectral-energy ranges with respect to the very well-defined spectral lines visible in the carbon-window energy range. This is due to the presence of final core-ionised states of mainly 2h1p character in the energy region around the N(1s) K-edge. These states are absent in the vicinity of the C(1s) K-edge, i.e. the carbon-window energy range is predominantly characterised by the main core-ionised states with C(1s) 1h content greater than 0.9. As a consequence, the ATAS spectrum in the carbon-window effectively probes only the 1h content of the valence-ionised initial states. This is demonstrated in Fig. 9, which shows the spectrally-resolved transient absorption spectrum at the nitrogen K-edge, together with its Fourier-transform along the time-delay axis, calculated by neglecting the final core-ionised states of 2h1p character. The 2h1p states describe shakeup resonances in the core-ionised spectrum, as well as providing a very approximate model (based on the discrete set of GTO basis functions forming the RCS) of the double ionisation continuum for dicationic states with a hole in the core orbitals. Thus, their inclusion not only allows one to probe the 2h1p content of the initial pump-ionised channels in the valence energy region, but also gives rise to a background signal that models further ionisation directly from the core orbitals ([|_{i}〉|^{a}_{i;j}〉] → |^{a}_{i;core}〉). The latter transitions include possible single-photon laser-enabled Auger decay (sp-LEAD) transitions^{46,47} from the initial superposition of states directly to the final |_{i;core}〉 dicationic states. These results demonstrate that, when spectrally resolved, the time-resolved absorbance is sensitive to the different excitation classes participating in the coherent many-electron dynamics and thus provides highly-detailed insight into the correlated many-electron dynamics of a polyatomic molecule such as pyrazine. In particular, as shown in Fig. 10 for the carbon-window, it becomes possible to unveil the mapping between the spectrally-resolved frequency beatings observable in the Fourier-transformed absorption spectrum, and the predominant pump-triggered quantum electronic coherences as given by the ionic-density-matrix purification procedure (see Fig. 6). As can be seen in Fig. 10, by using a short-enough laser pulse and sampling the time-delay with sufficient temporal resolution, down to the attosecond regime, it is possible to resolve all the interference patterns that arise from the main quantum beatings in the system under investigation. The ionic coherences unveiled by the ATAS measurement are in fact the ones which characterise the principal, dominating pure-state channels of the ionic system shown in Fig. 6. This also shows that the X-ray probe pulse used in this work allows the retrieval of the quantum electronic coherences without substantially perturbing the dynamics induced by the pump pulse, e.g. without introducing new coherences. Achieving the necessary temporal resolution in the X-ray regime, by using pulses of attosecond duration with controlled time-delay, has recently become possible both in lab-scale experiments, where HHG sources are used to produce attosecond pulses in the water energy window,^{48–51} as well as at X-ray FEL facilities such as the Linac Coherent Light Source (LCLS) at Stanford where, following the successful development of the X-ray laser-enhanced attosecond pulse generation (XLEAP) set-up at the LCLS-II machine, recent experiments have demonstrated the production of high-power, isolated sub-femtosecond pulses in the soft X-ray regime.^{52} Thus, by using appropriate pulses capable of retrieving the calculated ionic coherences G_{m,n}, the use of the ATAS technique, combined with advanced theoretical simulations of the kind presented above, allows one to unveil the mapping of the measurement results onto the non-zero off-diagonal matrix elements of the ionic density matrix ρ_{m,n}, related to the coherences by eqn (8).

Fig. 10 Map between the measurement observable, i.e. the spectrally-resolved frequency beatings detectable in the carbon K-edge transient absorption spectrum, and the predominant pump-triggered quantum electronic coherences as obtained by the ionic-density-matrix purification procedure; the relevant pairs of coherently-populated ionic eigenstates for the pure-state channels of weight r_{1}, and r_{2} and r_{4} are indicated by the horizontal arrows on the upper and lower panels, respectively. These pairs coincide with the ones characterising the lower-weighted r_{6}, r_{7} and r_{8} pure-state channels shown in Fig. 6. Since the r_{3} and r_{5} channels in Fig. 6 mainly coincide with the 1B_{3g} and 1B_{2g} ionic eigenstates, respectively, no correspondent, appreciable, quantum beatings are detected at the presented resolution. |

Moreover, the analysis in Section 3.1 demonstrated that the physical observables can be described well within a limited Hilbert space of reduced dimensionality, consisting of the physically-relevant pure-state channels. Under these conditions, the results in Fig. 10 offer the exciting prospect of performing quantum state reconstruction of the ionic density matrix in a molecular system by generalising the same techniques already successfully used to treat an atomic 2-level case^{16} to systems described by a few additional states. The extent to which this result is affected by also explicitly taking into account the nuclear degrees of freedom remains to be seen, and will mainly depend on the timescales of the resulting decoherence, i.e. whether the electronic coherences live long enough before they are damped as a result of the chemical change that they pre-determine,^{53,54} and the spatial delocalisation of the ground-state nuclear wavefunction.^{55} Nevertheless, the importance of these results is in the fact that they demonstrate, by means of advanced ab initio calculations, that the ATAS technique, combined with the use of sub-fs X-ray pulses of the kind that have become available at X-ray FELs,^{52} is perfectly suitable for imaging the complex many-electron dynamics which emerges from attosecond ionisation of polyatomic molecules, including correlation-driven charge dynamics, and retrieving the quantum electronic coherences that are not completely washed out by the dephasing effect of nuclear motion.

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