Full Ab Initio Many-Electron Simulation of Attosecond Molecular Pump

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Dynamics

Full ab initio many-electron simulation of attosecond molecular pump-probe spectroscopy Marco Ruberti, Piero Decleva, and Vitali Averbukh J. Chem. Theory Comput., Just Accepted Manuscript • DOI: 10.1021/acs.jctc.8b00479 • Publication Date (Web): 04 Sep 2018 Downloaded from http://pubs.acs.org on September 8, 2018

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Journal of Chemical Theory and Computation

Full

ab initio

many-electron simulation of

attosecond molecular pump-probe spectroscopy M. Ruberti,

†1 Department

∗,†

P. Decleva,

‡

†

and V. Averbukh

of Physics, Prince Consort Road, London SW7 2AZ, United Kingdom

‡2 Dipartimento

di Scienze Chimiche, Via Giorgieri 1, I-34127 Trieste, Italy

E-mail: [email protected]

Abstract Here we present an ab initio approach to full simulation of an attosecond molecular pump-probe experiment. Sequential molecular double ionization by the pump and probe laser pulses with controlled delay is described from rst principles with full account of the continuum dynamics of the photoelectrons. Many-electron boundcontinuum dynamics is simulated using the time-dependent (TD) molecular B-spline algebraic diagrammatic construction (ADC) method. Our calculations give a quantitative prediction about the creation of a coherent superposition of molecular ionic states in the photo-ionization process and simulate the probe of the ensuing attosecond dynamics by a second ionising pulse within a single rst-principles many-electron framework. We thus demonstrate the capability to simulate and interpret the results of a prototypical molecular pump-probe experiment of interest in attoscience. As a particular example, we simulate and elucidate the interpretation of a pump-probe experiment in CO2 aimed at measuring strong eld induced hole dynamics via photoionization yields.

1

ACS Paragon Plus Environment

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Introduction The groundbreaking development of attosecond laser pulses 1 in the extreme ultraviolet (XUV) spectrum has enabled the experimental study of attosecond physics. 2 During the last few years considerable eort has been made in order to produce intense enough ultrashort XUV and VUV pulses that would open the way to attosecond pump - attosecond probe experiments. 35 The development of this capability will make it possible to study coherent many-electron dynamics both in the valence and core energy regions. This calls for the development of the matching rst principles theoretical tools capable of modeling and analysing the observed many-electron dynamics. Such tools would be indispensible for designing the upcoming experiments and exploring new physical phenomena. Spectacular experimental progress has also been made on the hybrid schemes involving both attosecond and few-femtosecond IR pulses, such as the ones based on transient absorption, 6,7 HHG spectroscopy 8,9 and XUV/VUV-IR pump-probe. 10 Another promising scheme for real-time tracking of impulsively created valence electronic wavepackets and electronic coherences has been suggested, based on nonlinear X-ray spectroscopy. 11 Such schemes have a proven potential for capturing both purely bound (e.g. hole migration 8,10 ), and boundcontinuum (e.g. Auger decay 12,13 ) many-electron dynamics in molecules on their intrinsic time scales. However, the outcomes of such molecular pump-probe experiments are often notoriously dicult to interpret unambiguously because of the richness of the dynamics encoded in the data. 8,10 Therefore, the interpretation of these studies has to rely on a substantial amount of theoretical input. Hence there is a need for a consistent general theory that would satisfy the following essential requirements. First it should be able to describe the correlation between the photoelectron and the parent ion in order to quantitatively predict the degree of coherence between the photo-ionized states that is essential for triggering an ultrafast electronic response. Furthermore, an accurate knowledge of the initial coherent superposition parameters, namely the relative phases between the ionic states and their relative populations, is also essential in 2


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order to interpret the measurement of the post-ionization dynamics. Coherence and eigenstate content of the initial ionic many-electron wave-function are long-standing issues for the theoretical analysis of the hole-migration dynamics, where simplied models, such as sudden ionization, often had to be used. 9,1417 Finally, any eect of the (possibly intense) probe pulse or harmonic generating eld on the measured attosecond dynamics needs to be accounted for theoretically in order to extract the eld-free evolution. Therefore, the required theory should describe quantitatively both pump and probe stages of a general attosecond experiment producing accurate predictions for the above quantities at the molecular level for a wide range of both pump and probe pulse parameters. In order to pursuit new physical eects at the attosecond time scale, this theory should also not rely on empirical elements but rather be completely rst-principles. So far

ab initio

studies of coherence formation have been

performed in atomic systems, 6,18 for dierent ionizing eld intensity regimes, while a recent molecular study based on real-time Nonequilibrium Green's Functions has been performed predicting the dynamics initiated by XUV-pump photoexcited Phenylalanine molecule. 19 Although physical eects such as hole migration are inherently molecular, no studies of coherence formation in the non-perturbative ionization regime nor full ab-initio many-electron simulation of pump-probe experiments have been performed on molecules. In this work we present a general

ab initio

theoretical framework for full simulation of a

prototypical attosecond molecular pump-probe experiment. Our

ab initio

method enables us to describe the interaction of the molecule with both

attosecond and femtosecond ionizing pump and probe pulses using exactly the same general framework ADC(n) for the description of electron-electron interaction in the neutral and ionized systems, while avoiding the complication of describing two electrons in the continuum by extracting the coherent ionic wave function produced after the rst ionization event. Consequently this approach accounts for any type of sequential two-electron ionization by two non-overlapping pump and probe laser pulses. The method presented here can be used to describe both attosecond/attosecond and hybrid attosecond/femtosecond pump-probe 3



schemes in both perturbative and non-perturbative (over the barrier and tunnel ionization dominated) eld regimes. Of course the level of theory employed in actual calculations will depend on the nature of the physical process (for example presence or absence of shake-up eects in photoionization at the given eld parameters). As a specic example, we simulate a pump-probe experiment where the CO2 molecule is rst ionized by an ultrashort infrared (IR) laser pulse (pump) and subsequently probed by an ionizing attosecond VUV pulse, see Fig. 1. Simulating the interaction between the generated ionic system and the ultrashort VUV laser pulses allows us to show how the strong eld ionization (SFI) induced ionic coherence and the resulting electron hole dynamics can be probed. In this case separate oneand two-photon absorption are needed to reach the same nal continuum from ionic states of dierent parity.

Description of the method We have developed a fully

ab initio

framework for sequentially solving the 3D molecular

many-electron time-dependent Schrödinger equation for both the neutral (parent) and ionic (daughter) systems interacting with the laser eld, molecular B-spline time-dependent algebraic diagrammatic construction (B-spline TD-ADC) that is based on the many-body Green's function ADC methods, 2022 B-spline basis set 25 and Arnoldi-Lanczos time propagation algorithm. 26 The B-spline implementation of the excitation ADC schemes for neutral atomic and molecular systems have been rst introduced for calculation of photo-ionization cross sections and high-harmonic generation (HHG) spectra of closed shell atoms, 25 transient-absorption dynamics in atoms 7 and molecular HHG electron dynamics. 27 Here, for the rst time, we present the B-spline implementation of the single ionization ADC schemes for singly-ionised closed shell molecular systems and we show how the two schemes can be combined together to simulate a prototypical molecular pump-probe experiment within a single rst-principles

4


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Figure 1: Top panel: schematic representation of the interaction of CO2 with the IR-pump ultrashort ionizing laser pulse. During the interaction with the strong ionizing IR pump + pulse, several states of CO+ 2 can be produced. The ionic states are denoted as CO2 i−1,p where the index i denotes the molecular orbital from which the electron has been ionized and the second index p denotes the overall parity of the ionic state. Bottom panel: schematic representation of the interaction of the CO+ 2 ion with the VUV-probe ultrashort laser pulses. Ionic states CO+ of dierent parity p can be further ionized by the VUV pulse into the 2 i−1,p 2+ − same nal state (CO2 + e ) by absorption of two photons from the ω eld and one photon from the 2ω eld respectively.

5



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many-electron framework. Within our time-dependent (TD) B-spline ADC approach to molecular pump-probe simulations, the 3D molecular many-electron time-dependent Schrödinger equation (TDSE) is solved sequentially for both the neutral (parent)

∂ | ΨN (t)i ˆ N (t) | Ψ (t)i , = H ∂t

(1)

∂ | ΨN −1 (t)i ˆ N −1 (t) | ΨN −1 (t)i , = H ∂t

(2)

ih ¯

and ionic (daughter) systems

ih ¯

interacting with the laser eld by making the following ansatz for the time-dependent electronic wave-function of the neutral

| ΨN (t)i = C0 (t) | Ψ0 i +

X

˜ I iN CI (t) | Ψ

(3)

I

and ionic system respectively

| ΨN −1 (t)i =

X

˜ I iN −1 CI (t) | Ψ

(4)

I

˜ I iN and | Ψ ˜ I iN −1 represent the correlated ground-state of the parent system Here | Ψ0 i, | Ψ and the ADC conguration basis states 2022 of the neutral and ionic system respectively. 23 Within the hierarchy of the ADC

ab initio

schemes the many-electron states are con-

structed starting from the so-called correlated excited states (CES). 24 These are dened as N/N −1 †

| ΨI iN/N −1 = CÎ

| Ψ0 i ,

(5)

N/N −1 † where the operators CÎ denote the physical excitation operators corresponding respec-

6


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tively to the dierent excitation classes of the neutral (1p1h, 2p2h etc...),

n o CÎN † = a ˆ†a a î ; a ˆ†a a ˆ†b a ˆj a ˆk (a < b, j < k) . . . . . . ,

(6)

and ionic (1h, 1p2h etc...) systems

ˆj a ˆk (j < k) . . . . . . , CÎN −1 † = a î ; a ˆ†a a

(7)

respectively. Here a and i indeces refer to virtual and occupied Hartree-Fock orbitals respectively. Recursive orthonormalization of the excitation classes by a two-step procedure leads to the nal expression for the ADC many-electron states, which in a compact notation can be written as

˜m ˆ m−1 |Ψ x i = Q

X

m | Ψm y i Syx

− 12

,

(8)

y m where Syx is dened as m ˆ m−1 | Ψm Syx = hΨm y | Q x i

and

ˆ m = ˆ1 − Q

m X

Pˆ l

(9)

(10)

l=0

is the projector operator onto the space orthogonal to the rst m excitation classes. The ADC secular and dipole matrices are the representation of the shifted electronic

ˆ − E0 and dipole operator D ˆ in the space of ADC intermediate states: 2023 Hamiltonian H ˜I | H ˆ − E0 | Ψ ˜ J i , DIJ = hΨ ˜I | D ˆ |Ψ ˜ Ji . HIJ = hΨ

(11)

Within ADC, Møller-Plesset (MP) perturbation theory is introduced to describe the ground state correlation, i.e. | Ψ0 i and E0 , and the hierarchy of ADC(n) approximations is obtained for each order n by truncating the excited state manifold at some limiting excitation class 7



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and, also, by consistently truncating the resulting perturbation expansions for the included classes. 2023 Specically, in the following study, the rst order ADC(1) scheme (including all the 1h1p excitations of the molecule) of the ADC(n) hierarchy has been used for the description of the neutral molecular system. Within the ADC(1) framework, the perturbation expansion of the single excitation classes is truncated at the rst order and the ADC(1) Hamiltonian reads ADC(1)

Hai,bj

= −Vaj[bi] = −haj || bii

(12)

where a and b represent the excited/ionized electron degree of freedom. The terms of Eq. (12) with i 6= j describe inter-channel couplings, i.e. the mixing of excitations originating from dierent cationic states, which are uncoupled at the independent particle level. Within ADC(1) the excited states energies are equivalent to the CIS ones, while the calculation of transition dipole moments between the ground state and the complete manifold of singly excited/ionized states includes the contribution, evaluated at the MP2 level, of double excitations in the ground state. The MP2-corrected transition dipole moments are expressed as

˜ ai | D ˆ | Ψ0 i = dai + hΨ

XX v

o

hva || oii dov o + i − v − a

(13)

where hva || oii is the antisymmetrised two-particle Coulomb integral in physicists' notation and the two indices v and o run over the virtual and the occupied canonical (HartreeFock) orbitals respectively. Despite this is the only MP2 correction of ADC(1), it aects exactly those matrix elements which embody the physical process we describe in the pump stage: ground state photoionization. On the formal side, ADC(1) should be regarded as a Hermitian approximation to Random Phase Approximation (RPA), please see 28 for the detailed discussion and comparison. Notice that, although within the ADC(1) scheme for the neutral molecule the ionic system is described at the Koopmans approximation level, electron-electron interaction (e-e) is described also at this level of theory by the couplings

8


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between the excited/ionized electron and the parent ion (which, among other eects, accounts for rescattering-induced excitations of the parent ion by the returning photo-electron in the strong-eld regime) and also by the modied transition dipole moments from the ground state to the complete manifold of excited/ionized states. In other words, in a picture of separate channel excitations, customary in simplied descriptions of strong eld eects, the couplings due to dipole transitions induced by the eld, and the transitions induced by the residual e-e interaction are both included. In this work, the description of the cation is given within the second order ADC(2)x scheme of the ionic ADC(n) hierarchy. Therefore, in the simulation of the interaction of the ion with the probe ionizing eld, all the ionic two hole one particle (2h-1p) congurations are also included and electron correlation within the ionic system is taken into account. To summarize we have used the lowest level of ADC compatible with a correct description of the excitation of the system by the laser pulses: the conguration manifold included in the description of the neutral system excitation by the pump pulse via TDSE is the singly excited congurations (1h-1p), from which the density matrix of the nal cationic states is obtained tracing over the excited (positive energy) electron. The successive sequential ionization by the probe pulse, which generates single excitations from the cationic states, is described within the manifold of the (1h, 2h-1p) congurations, further dressed by perturbation theory. Single-electron orbitals are represented in the monocentric B-spline basis set. 29 The reason for employing B-splines as radial basis functions is that, with respect to our purpose, the description of the continuum states requires basis sets which could reproduce the oscillating behaviour of the electronic wave-function up to big distances from the molecular region and for a wide range of kinetic energy. Although traditional L2 basis functions, such as Slater Type Orbitals (STOs) or Gaussian Type Orbitals (GTOs), can in principle provide a good description of the low kinetic energy ionised-electron wave-function in the proximity of the parent molecular ion, they are not adequate to accurately describe the strongly oscillating continuum orbitals of a fast ionised electron far away from the molecular region, since numer9



Page 10 of 33

ical linear dependencies rapidly occur as the basis set increases. 25,30,31 B-splines, conversely, are very exible functions and overcome this diculties by providing a very accurate representation of both bound and continuum states without running into numerical dependencies even for large bases. The use of such a pre-determined set of one-particle orbitals eliminates the need of any a priori procedure in selecting the parameters for the Gaussian-type or Slater-type functions required in the standard L2 quantum chemistry calculations which often depend upon non-trivial treatment to minimise the linear dependence embedded in the choice of basis functions. 29 Therefore the basis functions used in this work are composed of a spherical harmonics for the angular part and B-splines for the radial coordinate,

1 ψilm = Bi (r)Ylm (θ, φ). r

(14)

The calculations have been performed using a parabolic-linear B-spline knot sequence 25,29 with a radial box radius Rmax = 130 a.u.. The number of B-spline radial functions used in this calculation is 180. The rst step in the calculation is the solution of the discretised closed-shell Hartree-Fock (HF) equations which take into account all the N electrons of the molecule and that, in the molecular case is solved for every irreducible representation (IRREP) of the molecular point symmetry group. In the case of the carbon dioxide molecule, whose point symmetry group is D∞h , the number of spherical harmonics belonging to each IRREP is equal to Lmax + 1 for Σg , Lmax for Σu and Πu , Lmax − 1 for Πg and ∆g , etc. Here Lmax represents the maximum value of the orbital angular momentum used in the calculation, which in this work has been set to be Lmax = 60. Solving the molecular HF equations self-consistently, we obtain a quasi-complete set of canonical occupied and virtual HF molecular orbitals, expressed in terms of B-spline basis functions. In this work, the multi-electron ADC congurations built with the HF orbitals are both spin-adapted and point-symmetry group adapted ones. Therefore we take into account the

10


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full symmetry of the molecule, i.e. D∞h , not only in the HF one-particle calculation but also in the many-electron ADC one. The calculation has been performed for CO2 molecules perfectly aligned along the IR laser eld polarization direction. Using a laser eld linearly polarised along the molecular axis leads to conservation of the quantum number M representing the total electronic axial angular momentum and, consequently, in the simulation of the pump ionization step it is enough to consider only the two multi-electron spaces of the + neutral system corresponding to the irreducible representations Σ+ g and Σu .

A complex absorbing potential (CAP) has been used in order to eliminate wave-packet reection eects from the radial grid boundaries. The form of the CAP used was the following:

ˆ = η(r − rCAP )2 , r ≥ rCAP W

(15)

The CAP used in this calculation has a strength η = 0.0005 and starts at rCAP = 80 a.u. With the addition of the CAP term the total time-dependent Hamiltonians of Eqs. (1),(2) for the time-evolution of both the parent and ionic systems interacting with the pump and probe pulses respectively become complex symmetric and read

ˆ H Nˆ(t) = Hˆ0N + Dˆz Epump (t) − iW

(16)

ˆ (t) = H Nˆ −1 + Dˆz Eprobe (t) − iW ˆ H N −1 0

(17)

and

ˆ −1 N/N

for the neutral and ionic system respectively. Here H0

is the eld-free many-electron

ADC Hamiltonian describing the neutral and ionic system respectively. The laser-molecule interaction is described in the length form and using the dipole approximation Dˆz E(t), where P the z component of the dipole operator is Dˆz = N j=1 zˆj and the summation over the j index runs over all the N electrons of the molecule. The time propagation of the unknown coecients C0 (t), CI (t) of the B-spline ADC manyelectron neutral and ionic wave-functions (see Eqs. (3),(4)) is performed by means of the 11



general complex Lanczos, or Arnoldi-Lanczos, algorithm. 26 In the computational implementation of this scheme used in TD B-spline ADC, it is possible to x the maximum number of iterations requested, i.e. the maximum allowed dimension of the Krylov space. Moreover, the algorithm is stopped when the norm of the (j + 1)-th Krylov vector candidate becomes smaller than a certain xed threshold value, which, in this work, has been taken as the numerical precision of the calculator. The error, i.e. the dierence between the propagated and exact wave-functions, when using the Lanczos integrator is sensitive to both the time-step size ∆t or the maximum order K allowed. In the pump (IR) and probe (VUV) calculations we used 500 and 40 time steps per laser eld cycle respectively. Since in this work we used a single-cycle IR pump and a 3 cycles VUV probe pulses, the total number of time steps used in this calculation is 620. The maximum Krylov space dimension K used in the pump and probe calculations is equal to K = 40, at which value the convergence of our nal results has been obtained. Using this algorithm, the time propagation reduces to repetitive diagonalization, at each time step, of Hessenberg matrices whose dimension is insignicant with respect to the full ADC Hamiltonian. This allows the treatment of dynamical problems in which the size of the resulting system Hamiltonian is very large and time-propagation by full diagonalization of the secular matrix at each time step becomes practically inaccessible. Another very important feature of the Arnoldi-Lanczos algorithm is that, when computing the tridiagonal matrix

HK , the full Hamiltonian matrix H is needed only for a simple matrix vector product and remains unchanged during the whole procedure. Each of these reduced single calculations has been massively parallelized, both inter-nodes, with standard MPI (message passing interface) technology in parallel computing environment, and intra-node with OpenMP (shared memory) technology. In order to guarantee the convergence of the Arnoldi-Lanczos algorithm, which depends crucially on the spectral radius of the Hamiltonian, we removed from the numerical simulation the ADC congurations with a zeroth-order energy higher than a certain threshold 12


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value. The threshold value for the cuto energy which has been used in the calculations performed in this work is 40 a.u.; with this choice the Arnoldi-Lanczos algortihm converges well at each time step and, moreover, the removal of these high energy electron orbitals has no eect on the ionic coherences and wave-packet parameters we are looking at. The coherence and nature of the cation system produced after ionization by the pump pulse is obtained by calculating the reduced ionic density matrix (R-IDM) of the CO2 cation system produced in the non-perturbative ionization of CO2 molecules by high-intensity ultrashort IR laser pulses. The time-dependent R-IDM ρR−IDM (t) is obtained from the total time-dependent density matrix of the N-electron system

ρˆ(t) =| Ψ(t)ihΨ(t) |

(18)

by tracing out the unobserved photoelectron degree of freedom

ρR−IDM (t) = T ra [ˆ ρ(t)],

(19)

where T ra stands for the trace over all virtual orbitals. Within the ADC(1) framework

ρR−IDM (t) can be espressed as [ρR−IDM (t)]i,j =

X

˜ a | Ψ(t)ihΨ(t) | Ψ ˜ ai hΨ i j

(20)

a

=

X

Cia (t)[Cja (t)]∗ .

a

From now on we shall omit the R-IDM superscript and denote the reduced ionic density matrix as ρi,j . A quantitative prediction on the coherence degree between couples of nal ionic channels i and j formed during the pump-stage ionization of the molecular system is then given by 33

| ρ(t)i,j | . ρ(t)i,i ρ(t)j,j

Gi,j (t) = p

13


(21)


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Totally incoherent statistical mixtures result in Gi,j (t) = 0 while the maximum achievable (perfect) coherence corresponds to Gi,j (t) = 1. As we are interested in looking for the probability of ionising the system and to form ionic species, the bound excited part of the neutral wave-function must be removed from the total wave-function of the system after the interaction with the pump ionizing pulse, before the trace is computed. This means that formally the formulas remain the same, but the coecients on the singly excited ADC states, Cia , must now refer only to the ionised part of the wave-function:

| Ψ(t)ionised i =| Ψ(t)i −

X

hΨbound | Ψ(t)i | Ψbound i. n n

(22)

n

It should be noted that the bound states are the eigenstates of the ADC(1), i.e. CIS Hamiltonian, that have excitation energy below the rst ionization potential. In fact all virtual orbitals have positive energies so that one cannot simply take the trace over positive orbital energies. In the strong eld regime the bound state excitation of the system is usually much smaller and negligible with respect to its ionization; therefore this modication has a very small (negligible) eect onto the resulting R-IDM diagonal matrix elements. However the odiagonal matrix elements can be very sensitive to the presence of a bound state component in the total wave-function; this is due to the fact that, in the bound excited states, the excited electron remains close to the ion causing the coherence factor to continue oscillating even a long time after the interaction of the system with the laser pulse is over. With the introduction of a CAP the norm of the density matrix experiences a decay and therefore ρ(t) must be corrected for the loss of norm. The corrected ionic reduced density matrix that does not experience any damping 33 reads as

ρi,j (t) =

X

Cia (t)[Cja (t)]∗ +

a

14


(23)

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j i )t i(IP −IP

t

Z

X

+2e

j

0

0

i

0

0

wb,a Cia (t )[Cjb (t )]∗ e−i(IP −IP )t dt ,

−∞ a,b

where IPi is the ionisation potential of the ionic state i and wb,a is the CAP matrix element between photo-electron orbitals a and b. Notice that, although the term of Eq. (23) accounts for the loss of norm of the R-IDM due to the absorption of the photo-electron wavepacket by the CAP, in order to allow the ionized electrons to be driven back by the strong IR eld and re-collide with the parent ion, the value of the absorbing radius rCAP has to be set bigger than the largest classical quiver amplitude of the recolliding electrons rquiver =

Emax . ω2

Within the range of ionizing

eld parameters used in this work, we satised the rCAP > rquiver requirement. The resulting ionic wave-packet coecients and phases are also calculated using the diagonal and o-diagonal elements of the reduced-ionic density matrix ρi,j respectively, as follows. Here, we used laser elds linearly polarised along the molecular axis of the aligned CO2 molecule and therefore, when the nal produced ionic states show coherence, two distinct ionic wave functions arise: one within the Π (| M |= 1) symmetry space of the ion

ΨIonic (t) = Π

X

CnΠg e−iEnΠg t+iφnΠg | nΠg i

(24)

n

+

X

CnΠu e−iEnΠu t+iφnΠu | nΠu i

n

and one within the Σ (M = 0) symmetry space of the ion

ΨIonic (t) = Σ

X

−iEnΣ+ t+iφnΣ+

CnΣ+u e

u

u

| nΣ+ ui

n

+

X

−iEnΣ+ t+iφnΣg

CnΣg e

g

| nΣ+ g i.

n

15


(25)


Within the range of ionizing eld parameters used in this work, a total of four ionization channels of the CO2 molecule, 32 corresponding to creating the CO2 ion in the ground 12 Πg 2 + (channel X), rst 12 Πu (channel A), second 12 Σ+ u (channel B) and third excited state 1 Σg

(channel C) respectively, play the main role in the triggered electron dynamics and contribute to the harmonic emission of the CO2 molecule, while the contribution of deeper ionization channels was found to be negligible. Within ADC(1) these channels consist of, in the molecular orbital picture, the removal of a bound electron from the HOM O(1πg ), HOM O − 1(1πu ), HOM O − 2(3σu ) and HOM O −

3(4σg ) HF orbitals. Indeed the orbitals relevant for ionization and recombination are the Dyson orbitals, the overlaps between the multi-electron ground state wave-function of the neutral system and the ground/excited state wave-function of the ion and, within the ADC(1) method for the N-electron system, they coincide with the occupied HF orbitals of the neutral system. While the corresponding experimental ionization potentials are respectively 13.8,

17.3, 18.1 and 19.36 eV and therefore span a 5.56 eV energy range, the ADC(1) ionization potentials obtained in this calculation are 15.03, 18.81, 19.6 and 20.7 eV spanning a 5.7 eV energy range. In Table 1 a comparison is shown between the experimental orbital/ionization energies of the CO2 molecule and the theoretical results obtained by means of HF calculations performed using our B-spline basis set and a cc-pcvTZ Gaussian basis set respectively. The GTO HF calculation was performed using the MOLPRO quantum chemistry package. Table 1: Orbital/ionization energies of the CO2 molecule. Comparison between the experimental values and the theoretical results obtained by means of HF calculations performed using our B-spline basis set and a cc-pcvTZ Gaussian basis set respectively.

Ionic state 12 Πg 12 Πu 12 Σ+ u 2 + 1 Σg

Experimental I.P.s (eV) [24] 13.8 17.3 18.1 19.36

HF B-spline basis I.P.s (eV) 15.03 18.81 19.62 20.84

16


HF cc-pcvTZ GTO basis I.P.s (eV) 14.74 19.31 20.13 21.76

Page 16 of 33

Page 17 of 33 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


The ionic wave-packet coecients and phases calculated in this work are given in terms of the reduced-ionic density matrix ρi,j elements by the following expressions:

C12 Πg

√ ρ12 Πg ,12 Πg =p ρ12 Πg ,12 Πg + ρ12 Πu ,12 Πu

(26)

√

ρ12 Πu ,12 Πu ρ12 Πg ,12 Πg + ρ12 Πu ,12 Πu

C12 Πu = p e

−iφ12 Πg −12 Πu

=√

ρ12 Πg ,12 Πu ρ12 Πg ,12 Πg ρ12 Πu ,12 Πu

for the populated Π symmetry states of the ion, and by

√ C12 Σ+u = p

ρ12 Σ+u ,12 Σ+u

ρ12 Σ+g ,12 Σ+g + ρ12 Σ+u ,12 Σ+u

(27)

√ρ

2 + 12 Σ+ g ,1 Σg

C12 Σ+g = p

ρ12 Σ+g ,12 Σ+g + ρ12 Σ+u ,12 Σ+u

e

−iφ12 Σ+ −12 Σ+ u

g

ρ12 Σ+u ,12 Σ+g =√ ρ12 Σ+g ,12 Σ+g ρ12 Σ+u ,12 Σ+u

for the populated Σ symmetry states of the ion. Full characterization of the initial ionic state produced after the rst pump ionization event allows us to simulate the interaction between the ionic system and the probe pulse by providing us with the initial condition for numerically solving the ionic TDSE Eq. (2) with the Hamiltonian of Eq. (17). The propagation of the ionic wave-packet (see Eqs. (4)) on the congurational space including all the ionic one hole (1h) and two hole one particle (2h-1p) congurations within the ADC(2)x theoretical framework, both before and during the interaction of the ionic system with the probe pulse, is performed again by means of the general complex Lanczos, or Arnoldi-Lanczos, algorithm.

17


Page 18 of 33

b 0.8

1 a

0.7 0.6 0.5 0.4 0.3 0.2

0.8

0.6 0.4

0.2

G1 Σ+ - 1Σ+ u g G1Π - 1Π

0.8

0.6

g

u

u

g

0 1

φ +- + 1 Σ u 1 Σg φ 1Π - 1Π d

c

1 0.8

C1Π g C1Π

0.6

C1Σ+ u C1Σ +

u

0.4

0.4

g

0.2

0.2

Σ Wave-function coeffs Relative phases (π)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Π Wave-function coeffs Coherence degrees


0.3 0.6 0.9 1.2 1.5 1.8 0.3 0.6 0.9 1.2 1.5 1.8 14

2

IR-pump peak-intensity (10 W/cm ) Figure 2: Top panel: Intensity dependence of G12 Πg −12 Πu and G12 Σ+u −12 Σ+g (panel a). For comparison, in the same graph, we also show the coherence degrees calculated without taking into account the inter-channel couplings of Eqs. (28),(12) (dashed-dotted lines); Intensity dependence of the relative phases φ12 Πg −12 Πu and φ12 Σ+u −12 Σ+g of Eqs. (26),(27) (panel b). Bottom panel: Intensity dependence of the Π symmetry C12 Πg , C12 Πu (panel c) and Σ symmetry C12 Σ+u , C12 Σ+g (panel d) wave-function normalised coecients of Eqs. (24),(26),(25),(27). The central wavelength (in the pulse energy envelope distribution) of the single-cycle IR pulse used is 800 nm.

18


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Results and discussion Coherence between dierent ionic states requires them to ionize in the same photoelectron continuum as this leads to the complete separability of the N-electron wave function into the antisymmetrized product of a photoelectron and a (N-1)-electron ionic wave functions. The question about the coherence of the ionic wave packet formed after ionization, which is a highly non-trivial consequence of several photo-ionization mechanisms, is especially relevant in molecules due to the much smaller energy separation between the eigenstates of the ionic system with respect to the atomic case. Various physical mechanisms may lead dierent ionic states to ionize in the same photoelectron continuum and therefore interplay in establishing the coherence within the produced ionic system. The rst one consists of direct SFI, with dierent probabilities, in both ionic channels followed by the strong laser eld driven acceleration of the respective photoelectrons in the same nal continuum state. The second one consists of electron rearrangement within the ionic system caused by laser-driven dipole couplings between the dierent ionic states i and j

Dai,bj E(t) = −δa,b hi | dˆ | jiE(t) i 6= j .

(28)

Here a and b represent the photo-electron degree of freedom. Moverover, also inter-channel couplings of Eq. (12) driven by e-e interactions can aect the nal coherence of the ionic system. All these mechanisms are described by the ADC(1) level of theory. The calculations reported in this work have been performed using the following time prole for the laser pulses electric eld:

E(t) = Epeak × (cos(t/T ) × cos(t/2T n)2 − sin(t/T ) × cos(t/2T n) × sin(t/2T n))

(29)

where Epeak and n stand for the maximum value of the electric eld and the number of cycles respectively. The form of Eq. (29) does not contain a source term as it has been derived

19



a

14

I = 0.8 x 10 W/cm

2

0.8

G1Π - 1Π u g G1Σ+- 1Σ+

0.6 0.4 0.2

g

u

1

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3

1.2

1.4

b

1.6

1.8

2

2.2

Central photon energy (eV) c d

C1Π C1Πug

2.4

φ1Πg -1Π φ1Σu+ -1Σg+u

1.6

2

0.8 0.6

C1Σ u+ C1Σ +

0.4

g

1.2

1

2.4 1.2

1.6

2

Central photon energy (eV)

2.4 1.2

1.6

2

Relative phases (π)

Coherence degrees

1

Wave-functions coeffs

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 20 of 33

0.2 2.4

Central photon energy (eV)

Figure 3: Top panel (panel a) central frequency dependence of G12 Πg −12 Πu and G12 Σ+u −12 Σ+g ; for comparison, in the same graph, we also show the coherence degrees calculated without taking into account the inter-channel couplings of Eqs. (28),(12) (dotted lines). Bottom panel central frequency dependence of the Π symmetry C12 Πg , C12 Πu (panel b) and Σ symmetry C12 Σ+u , C12 Σ+g (panel c) ionic wave-function normalised coecients of Eqs. (24),(26),(25),(27); central frequency dependence of the relative phases φ12 Πg −12 Πu and φ12 Σ+u −12 Σ+g of Eqs. (26),(27) (panel d). The IR peak intensity and pulse duration used in the calculation are respectively 0.8 × 1014 W/cm2 and single-cycle for each central frequency.

20


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directly from the following vector potential

A(t) = Apeak × sin(t/T ) × cos(t/2T n)2 ) .

(30)

Figs. 2, 3 show the dependence of the nal degrees of coherence G12 Πg −12 Πu and G12 Σ+u −12 Σ+g (panel a) on the IR eld peak intensity and central frequency (in the pulse energy envelope distribution) respectively, together with the corresponding dependencies of both the Π and

Σ ionic wave-packet parameters of Eqs. (24),(26),(25),(27) (panels b,c,d). In the calculation presented in Figs. 2 and 3 the IR frequency and peak intensity have been xed to 800 nm and 0.8 × 1014 W/cm2 respectively. Ultrashort single-cycle laser pulses have been used in both calculations for the pump stage. Notice that, within the range of ionizing pump eld parameters used in this work, the Keldysh parameter of the pump laser eld, given in terms q I of the ponderomotive potential Up and the CO2 ionization potential Ip as γ = 2Upp , varies roughly between the values 0.7 and 1.6. As can be observed in Figs. 2a, 3a, strong eld ionization of a molecule such as CO2 by an ultrashort laser pulse in the IR frequency domain can produce a high coherence degree between multiple pairs of ionic electronic states. The dependency of the coherence degree on the parameters of the ionizing laser eld varies considerably for dierent pairs of ionic states. This is due to the fact that such a dependency is a non-trivial consequence of the specic ionic states properties such as their energy gap, relative ionization probabilities and symmetries, dipole transition and interchannel e-e interaction coupling strengths. Therefore, in order to accurately predict the nal degrees of coherence, it is necessary to fully include multi-channel interaction eects (Eqs. 28. 12) in the theoretical description of the SFI dynamics. Indeed, the degree of coherence calculated using the full ADC(1) method can deviate strongly from the predictions of the independent electron model, see G12 Σ+u −12 Σ+g in Figs. 2a, 3a. Moreover, the dependency of the ionic wave-packet parameters on the ionizing laser eld gives the possibility of controlling the resulting ionic states superposition by appropriately tuning the

21



pump pulse parameters. 34 Full characterization of the initial ionic state allows us to simulate the interaction between the ionic system and the probe pulse. In order to see the interference eects induced by the coherent part of these superpositions, the two eigenstates need to be ionized by the probe pulse into the same nal state of a certain energy and symmetry. Due to the high symmetry of the CO2 molecule and to the parameters of the ionizing pump pulse considered in this work, only pairs of ionic states with opposite parities are coherently populated. Since singlephoton absorption preserves their parity dierence, we use a two colour probe in which two-photon absorption from one of the two eigenstates leads to the same nal state as single-photon absorption from the other eigenstate in the superposition. The probe consists of the superposition of two ultrashort pulses in the VUV range, one moderately intense (I1 = 0.8 × 1013 W/cm2 ) pulse with central frequency ω = 20 eV and another weaker pulse (I2 = 0.5 × 1011 W/cm2 ) centered at its second harmonic 2ω . The duration of the two pulses used in the calculation is 3 and 5 cycles respectively. The ω and 2ω VUV pulses have broad enough frequency bandwidth for transitions from two populated cationic states of distinct −1 energies En and Em to produce the same continuum state ΨN . Interference between two E

such transitions creates a modulation in the double-ionization yield signal as a function of the pump-probe delay τ , with a period inversely proportional to the energy separation of the two ionic eigenstates En − Em . In a perturbative regime, this can be shown by writing down the perturbative formula for the total dication yield PVTUOT V (τ ) as a function of pump-probe delay τ . Within time-dependent perturbation theory, by writing the time-dependent electric eld (1)

as E(t) = ξω e−iωt fω (t) t > τ , the rst- AV U V,ω (τ ; Ef ) and second- order ω -eld induced (2)

AV U V,ω (τ ; Ef ) amplitudes for the transition from a general superposition of cationic eigen−1 −1 states ΨN to a nal continuum state ΨN of energy Ef above the CO2 double ionization n Ef

22


Page 22 of 33

Page 23 of 33 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


potential (DIP) can be written, as a function of pump-probe delay τ , as (1)

AV U V,ω (τ ; Ef ) = ξω

X

Cn e+iφn e+i(ωEf −n −ω)τ

(31)

n

D −i lim h ¯ t→∞

Z

−1 ΨN Ef

t−τ

E ˆ N −1 × Dz Ψn 0

0

0

dt e+i(ωEf −n −ω)t fω (t + τ )

0

and (2)

AV U V,ω (τ ; Ef ) = ξω2

X

Cn exp(+iφn )e+i(ωEf −n −2ω)τ ×

(32)

n

Z D X

−1 ΨN Ef

ED E ˆ N −1 N −1 ˆ N −1 ΨJ Dz Ψn × D z ΨJ

J

−1 lim lim h ¯ 2 t→∞ t0 →∞

Z

t−τ

0

0

0

0

dt e+i(ωEf −J −ω)t fω (t + τ )

Z

t −τ

00

00

00

dt e+i(ωJ−n −ω)t fω (t + τ )

0

0

respectively, where the eld strength and time-prole function of the ω laser probe pulse have been denoted as ξω and fω respectively and the identity operator Iˆ, expressed in terms of a complete set of discrete and continuum CO+ 2 intermediate states, has been introduced

R P −1 −1 Iˆ = J |ΨN i ΨN | . In Eqs.( 31, 32), Cn and φn are the coecients and phases of the J J ionic states superposition respectively and ωEf −n = Ef − En is the energy dierence between −1 −1 the nal doubly-ionized continuum state ΨN and the energy of the cationic state ΨN in n Ef

the initial superposition. T OT The total CO2+ 2 yield PV U V (τ ) as a function of pump-probe delay τ is given by

PVTUOTV (τ )

Z

∞

dE ×

= DIP

2 (1) (1) (2) (2) AV U V,ω (τ ; E) + AV U V,2ω (τ ; E) + AV U V,ω (τ ; E) + AV U V,2ω (τ ; E) .

23


(33)


Page 24 of 33

Eqs.( 31, 32, 33) generalize the rst-order perturbation theory expression for the ionization probability of a general superposition of cationic eigenstates of 35 to include second-order two-photon transitions. The inteference term PVINU VT (τ ) responsible for the oscillatory behaviour of PVTUOT V (τ ) as (1)

a function of the pump-probe delay τ comes from the cross terms involving AV U V,2ω (τ ; E) (2)

and AV U V,ω (τ ; E), as these are the two amplitudes that can ionize two dierent states of the −1 initial ionic superposition to the same nal state ΨN . E

In the present work, by denoting the two

gerade

and

ungerade

states in the initial cation

−1 −1 IN T superposition by ΨN and ΨN 1,g 2,u respectively, PV U V (t) can be expressed as

(34)

PVINU VT (t) ∝ 2C1 C2 cos[(E1 − E2 )t − φ1−2 ]× Z

∞

dE Re

hD

DIP

Re

hD

ED Ei N −1 N −1 −1 ˆ (2) N −1 ˆ (1) Ψ Ψ ΨN O Ψ O + 1,g 2,u ω,1−>E 2ω,E−>2 E,g E,g

Ei ED N −1 N −1 −1 ˆ (1) N −1 ˆ (2) Ψ . Ψ Ψ O ΨN O 1,g 2,u ω,E−>2 E,u 2ω,1−>E E,u

ˆ (1) where we have introduced the following notation for the single-photon O 2ω,n−>E and the ˆ (2) parity-conserving two-photon O ω,n−>E transition operators: ˆ (1) O 2ω,n−>E

ˆ (2) ˆ O ω,n−>E = Dz

= Dˆz lim

t→∞

Z X

Z

t−τ

0

0

0

dt e+i(ωE−n −2ω)t f2ω (t + τ ), Z

−1 |ΨN i lim J t→∞

t−τ

0

0

0

dt e+i(ωE−J −ω)t fω (t + τ )

0

J 0

Z lim

t0 →∞

t −τ

00

dt e

(35)

0

00

+i(ωJ−n −ω)t

00

fω (t + τ )

D

0

−1 ΨN J

ˆ Dz .

In Eq.( 34) φ1−2 is the relative phase of the ionic states superposition. 24


(36)

Page 25 of 33 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


In this work the total dication yield PVTUOT V (t) is computed by solving again the TDSE for the ionic system (Eq. (2)) interacting with the probe pulse (Eq. (17)). In Fig. 4a the pump-probe time-delay dependence of the nal double ionization yield is shown for the cases 2 + of a 12 Σ+ u , 1 Σg CO2 ionic wave packet created by two 800 nm pump single-cycle pulses

with peak intensity 1.1 × 1014 W/cm2 and 2 × 1014 W/cm2 respectively. The dependence of the oscillation amplitude, for both Σ and Π symmetry wave packets, on the parameters of the IR pump eld used in this work is given in Fig. 4b,c. As qualitatively predicted by Eq. (34), the TD B-spline ADC probe simulation shows how the total double ionization yield oscillates as a function of the pump-probe time-delay with a period given by the inverse of the energy separation between the two eigenstates, as a result of the interference between one- and two-photon ionizations from the two dierent eigenstates into the same nal state. Moreover, our pump stage simulation shows that the pump pulse dependence of the relative phase can be extracted from the phase shift of the total double ionization yield curves φΣ+u −Σ+g . Indeed, as shown in the top panel of Fig. 4a, dierent double ionization yield curves are shifted according to the dierent relative phases

φIΣ1+ −Σ+ and φIΣ2+ −Σ+ . Therefore using a single reference measurement in a case where the u

g

u

g

relative phase is known through

ab initio

simulation, such a measurement gives information

about the relative phase of the produced ionic wave packets for all pump ionizing eld parameters. The amplitude of the total double ionization yield oscillation, predicted by our ab initio

probe simulation, is proportional to the product of the population coecients of the

two ionic states in the superposition C1 C2 and to the coherence degree of such superposition

G1−2 . The results of our pump simulation show that, in the case studied in this work, the rst quantity has a very weak dependence on the laser pulse parameters. Therefore, as can be seen in Fig. 4b,c, measurement of the CO2+ 2 oscillation amplitude gives access to the dependence of the coherence of the produced ionic wave packets on the laser eld parameters, making it a directly measurable quantity.

25


I

1

14

2 14 2 = 1.1 x 10 W/cm IR pump

2

I

= 2 x 10 W/cm

IR pump

4

a

∆φ=

2

Page 26 of 33

I

IR

IR 1

- φI - + 1Σ+-1Σ+

φ2+

1Σu 1Σg

u

g

0 -2 -4

0

1

ω = 1.55 eV

1

pump

2

3

4

5

VUV-IR delay τ (fs)

+

CO2 coherence

Π Σ

6 14

2

Ipump = 0.8 x 10 W/cm

b

1

c

0.8

0.8

0.6 0.4

1Πg- 1Πu

0.2

1Σg-1Σu+

0

0.4

2+

+

0

0.6

1Πg -1Π u 1Σg+- 1Σu+

VUV CO2 Yield Osc. Ampl. (arb. units)

2+ 2+


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

VUV CO2 Yield Osc.(%)


0.3 0.6 0.9 1.2 1.5 1.8

1.2

1.6

2

2.4

0.2

14 2 IR pump intensity (10 W/cm ) Pump central photon energy (eV)

Figure 4: Top panel: (gure a) ADC(2)x results for the pump-probe time-delay dependence 2 + of the nal dication yield for the particular case of the 12 Σ+ u , 1 Σg CO2 ionic wave packet created by a 800 nm pump single-cycle pulse with peak intensity 1.1 × 1014 W/cm2 (black curve) and 2 × 1014 W/cm2 (red curve); the percentage value of the oscillation amplitude with respect to the average total absorption is plotted. Bottom panel: dependence of the oscillation amplitude of the VUV probe absorption signal on the IR pump peak intensity (gure b) and carrier frequency (gure c) respectively; for comparison, in the same graphs, we also show the respective coherence degrees from Figs. 2, 3 (dashed lines)

26


Page 27 of 33 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Conclusions In conclusion, we presented a full pump-probe experiment. Our

ab initio

ab initio

approach to simulating an attosecond molecular

method enabled us to treat the interaction with the

ionizing pump and ionizing probe within a single rst-principles many-electron framework for the description of both the neutral and ionized molecular systems. While fully taking into account the continuum dynamics of the photoelectrons, we avoided the complication of describing two electrons in the continuum by accurately extracting the coherent ionic wave function produced after the rst ionization event, which also enabled us to solve the longstanding problem of the initial state of post-ionization molecular electron dynamics (hole migration, etc...). We have thus demonstrated the capability to simulate and interpret the results of a full prototypical molecular pump-probe experiment of interest in attoscience. Our methodology opens the way for a qualitatively new level of planning and interpretation of the future attosecond pump-probe experiments with both table top 3 and XFEL sources. 4 The theory presented in this work is formulated within the xed-nuclei approximation. On timescales longer than the ones considered here, the nature of the initial coherent superposition of ionized states predicted by our theory drives nuclear dynamics that in its turn aects the electronic coherence in the molecule. 3638 Thus, at these longer time scales, our calculations provide a frozen-nuclei benchmark to which the experimental results can be compared in order to identify the eects of nuclear motion. Moreover, the nature of the initial photo-ionized molecular state strongly aects not only the early stage oscillations of the electronic charge taking place on femto- or subfemtosecond time scales, but also the following ionization-induced nuclear motion within the ionized molecule. 37 Therefore, the possibility of fully characterizing the latter as a function of the ionizing pump pulse parameters, opens up the possibility for tailoring the nal nuclear relaxation pathways of the molecular structure by controlling the initial electronic state superposition of the molecular cation.

27



Page 28 of 33

Acknowledgement This work was funded by EPSRC/DSTL MURI grant EP/N018680/1.

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I

1

14

2 14 2 = 1.1 x 10 W/cm IR pump

2

I

= 2 x 10 W/cm

IR pump

4

a

∆φ=

2

I

IR

IR 1

I φ + + + 1Σ -1Σ

φ2+

1Σu 1Σg

u

g

0 -2 -4

0

1

ω = 1.55 eV

1

pump

2

3

4

5

VUV-IR delay τ (fs)

+

CO2 coherence

Π Σ

6 14

2

Ipump = 0.8 x 10 W/cm

b

1

c

0.8

0.8

0.6 0.4

1Πg- 1Πu

0.2

1Σg-1Σu+

0

0.4

2+

+

0

0.6

1Πg -1Π u 1Σg+- 1Σu+


2+ 2+


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


VUV CO2 Yield Osc.(%)

Page 33 of 33

0.3 0.6 0.9 1.2 1.5 1.8

1.2

1.6

2

2.4

0.2

14 2 IR pump intensity (10 W/cm ) Pump central photon energy (eV)

Figure 5: For Table of Contents Only

33


Full Ab Initio Many-Electron Simulation of Attosecond Molecular Pump

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