Attosecond Dynamics of Molecular Electronic Ring Currents - The

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Attosecond Dynamics of Molecular Electronic Ring Currents Kaijun Yuan, Chuancun Shu, Daoyi Dong, and Andre Dieter Bandrauk J. Phys. Chem. Lett., Just Accepted Manuscript • Publication Date (Web): 03 May 2017 Downloaded from http://pubs.acs.org on May 5, 2017

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The Journal of Physical Chemistry Letters

Attosecond Dynamics of Molecular Electronic Ring Currents Kai-Jun Yuan,† Chuan-Cun Shu,∗,‡ Daoyi Dong,‡ and Andre´ D. Bandrauk∗,† †Département de Chimie, Faculté des Sciences, Université de Sherbrooke, Sherbrooke (Québec) Canada J1K 2R1 ‡School of Engineering and Information Technology, University of New South Wales, Canberra, Australian Capital Territory 2600, Australia E-mail: [email protected]; [email protected]

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Abstract Ultrafast charge migration is of fundamental importance to photoinduced chemical reactions. However, exploring such a quantum dynamical process requires demanding spatial and temporal resolutions. We show how electronic coherence dynamics induced in molecules by a circularly polarized UV pulse can be tracked by using a time-delayed circularly polarized attosecond X-ray pulse. The X-ray probe spectra retrieve an image at different time-delays, encoding instantaneous pump-induced circular charge migration information on attosecond time scale. A time-dependent ultrafast electronic coherence associated with the periodical circular ring currents shows a strong dependence on the helicity of the UV pulse, which may provide a direct approach to access and control the electronic quantum coherence dynamics in photophysical and photochemical reactions in real time.

Graphical TOC Entry UV

e X-ray

Eletronic Ring Cur rent


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Exploring quantum dynamics on their intrinsic time scales with ultrafast laser pulses is an area of current interest in many disciplines of science. Remarkable progress in femtochemistry has made it possible to access the photoinduced chemical reactions on the time scale of the nuclear motions of molecules. 1–5 The rapid development of the attosecond pulse techniques 6,7 provides a new tool necessary for investigating electron dynamics on its natural attosecond (1as=10−18 s) time scale and sub-nanometer dimension, leading to the emerging field of attosecond chemistry. 8–12 To this end, one can now have an insight into pure electronic quantum effects through attosecond electron movies and visualize the quantum dynamics of electron with spatio-temporal diffraction and microscopy in materials and biology. 13–16 Ultrafast negative charge migration illustrates electronic dynamics on reactivity on the attosecond time scale is a fundamental quantum process in many biological and chemical reactions. 17–24,26,42 Recent experiment has for the first time reconstructed attosecond charge migration with high-order harmonic generation (HHG) spectroscopy in ionized iodoacetylene. 22 Due to multiple excited states by circular femtosecond pulses, coherent superposition of electronic states with polarized ultrashort laser pulses leads to an electronic ring currents (ERC) and circular photochemical charge migration reaction can be created inside molecules, 27–29 leading to a number of potential applications in molecular magnetism, 30 the generation of attosecond magnetic field pulses, 31 and quantum devices. 32 However, it remains a challenge to access such circular ultrafast quantum coherent dynamics in real time. In this Letter, we present a theoretical scheme to show how the intramolecular ERCs in charge migration produced by a circularly polarized UV pulse can be tracked by using time-delayed circularly polarized attosecond X-ray pulses. To explicitly illustrate our scheme with the simplest description, we consider a benchmark system of the hydrogen molecular ion that is excited by a circularly polarized UV pulse with its polarization vector in the (x, y) plane, leading to charge migration from the electronic ground state |ψg (r)⟩ with the eigenenergy Eg to an electronic excited state |ψ±e (r)⟩ with the eigenenergy Ee . A coherent superposition of the two electronic states after one-photon absorption which leads to



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electron transfer on attosecond time scale 8,9 can be written as |ψ±0 (r, t)⟩ = cg |ψg (r)⟩e−iEg t/~ + ce |ψ±e (r)⟩e−iEe t/~ .

(1)

where cg and ce are the corresponding expansion coefficients of the two states, and |ψ±e (r)⟩ denote the degenerated quantum states induced by the left-handed (-) and the right-handed (+) circularly polarized pulses, respectively. By describing the electronic motions in the cylindrical coordinate space with r = (ρ, θ, z) with (x = ρ cos θ, y = ρ sin θ) and aligning the molecule along the z axis, the wavefunction of the degenerate excited state can be written as 1 e±e (ρ, z)e±iθ , ψ±e (ρ, θ, z) = √ [ψex (ρ, z) ± iψye (ρ, z)] = ψ 2

(2)

e±e (ρ, z)|2 = [|ψex (ρ, z)|2 +|ψye (ρ, z)|2 ]/2 and ψex (ρ, z) and ψye (ρ, z) are the degenerate real orbital. where |ψ The corresponding time dependent electron density can be given by e±e (r)|2 + 2cg ce ψg (ρ, z)ψ e±e (ρ, z) cos(∆Et/~ ∓ θ), A± (ρ, θ, z, t) = |cg ψg (r)|2 + |ce ψ

(3)

with the energy difference ∆E = Ee − Eg , and the UV laser induced ERCs in molecules can be described by the electronic angular continuity equation 23 ∂ ± d ± Aθ (t) + j (t) = 0. dt ∂θ θ

(4)

We find that the time-dependent electronic density A± (t) and currents j±θ (t) in Eqs. (3) and (4) refocus with a period of τ = 2π/∆E. Since the period τ is extremely short, its observation requires the demanding spatial and temporal resolutions. To track the coherent electronic dynamics in Eqs. (3) and (4) which evolve as a function of the azimuth angle θ, we measure molecular angular and energy-resolved photoelectron spectra by employing a time-delayed circularly polarized attosecond X-ray pulse to ionize the coherent excit-


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ed state in the UV-excited molecule. The angular and energy-resolved photoelectron spectra as a function of a time delay allow us to both spatially and temporally reconstruct the circular electronic dynamics involved in molecular charge migration. The use of the X-ray pulse has the following advantage in probing the electronic coherence. First, the high-photon energy of the X-ray can release electrons directly into the continuum after one-photon absorbtion, so that additional excitations to other lower electronic states can be suppressed. Second, the extremely short duration of the X-ray pulse on attosecond time scale can be utilized to resolve the coherent electron rotation with the period of few hundred attoseconds. Third, the circularly polarized laser pulse with equal field amplitude in each orthogonal direction does not influence the symmetry of photoionization. Such ultrafast attosecond X-ray pulses have been generated by current laser techniques from circularly polarized HHG. 33–35 In our simulations, we employ a circularly polarized UV pulse as the pump pulse to resonantly couple the ground 1sσg electronic state and the excited 2pπu electronic state of aligned H+2 at equilibrium with corresponding energies Eg = −0.58 a.u. (atomic units) and Ee = 0.07 a. u. A circularly polarized X-ray pulse is used as the probe pulse. The total pump and delayed probe laser field pulse E(t) = E pu (t) + E pr (t − ∆t) with its polarization vector in the (x, y) plane, propagating along the z axis, has the form       eˆ x [cos ω pu t + cos ω pr (t − ∆t)] E(t) = E0 f (t)  ,     eˆ y [ξ pu sin ω pu t + ξ pr sin ω pr (t − ∆t)]

(5)

where ω pu and ω pr are the pulse frequencies, ∆t presents the time delay between the peak times of the pump E pu (t) and probe E pr (t) pulses with the maximum amplitude E0 (the peak intensity I0 = cεE02 /2), and ξ pu/pr denotes the pulse helicity (+ for right and − for left). A temporal slowly varying envelope f (t) = sin2 (πt/T pu/pr ) is used with duration T pu/pr = 10τ pu/pr with one optical ∫ cycle τ pu/pr = 2π/ω pu/pr , ensuring zero field area E(t)dt = 0. The evolution of the total electronic wavefunction ψ(ρ, θ, z, t) is described by the time-dependent Schrödinger equation (TDSE) with the Hamiltonian H(t) = T r + Vc (r) − r · E(t), where T r = − 21 ∇2r denotes the kinetic energy operator 5 ACS Paragon Plus Environment


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of electron, and Vc (r) is the two-center electron Coulomb potential. The field-molecule interaction r · E(t) is treated in the length gauge and dipole approximation, since for exact solutions of the TDSE, the numerical results are gauge invariant. 36 We numerically solve the TDSE by using a five-point finite difference method and fast Fourier transform technique combined with high-order split-operator methods. 37,38 For intense high frequency laser pulses, a quite large grid range must be used to obtain via Fourier transforming the high energy of the ejected electron in the ionization spectra. In this work, we calculate molecular-frame photoelectron angular distributions at the kinetic energy E by a Fourier transformation of the time-dependent electronic flux as 39,40 [∫ J(θ, E) ∼ Im

] ∂ψ(ρ, θ, z) dzψ (ρ, θ, z) |ρ=ρi , ∂ρ −∞

∫

∞

dte

iEt

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∗

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where ρi is an asymptotic point before the electronic wave packet is absorbed at the grid boundary, and t p is the time after the probe pulse turns off. From the relation between the kinetic energy E and the momentum p with E = p2 /2 and transformation between the cylindrical (ρ, θ) and Cartesian (x,y) with p x = p cos θ and py = p sin θ, we obtain the two-dimensional (2D) momentum distribution spectrum of the photoelectron. Furthermore, the probability of observing fractional photoelectrons in a small angle region ∆θ can be calculated by ∫ P=

∫

E+~ω pr /3

dE

∆θ

dθJ(θ, E),

E−~ω pr /3

(7)

0

where E = ~ω pr − I p , corresponding one ~ω photon absorption. Figure 1 shows the fractional probability P as a function of the delay time ∆t within the angular region ∆θ (0 ≤ θ ≤ π/12) for left-handed circularly polarized pump-probe pulses at ξ pu = ξ pr = −1. The peak intensities of both pulses are fixed at I0 = 1 × 1014 W/cm2 (E0 = 5.34 × 10−2 a.u.). The wavelength of the pump pulse is λ pu =70 nm (~ω pu = 0.65 a.u.), resonant with the 1sσg − 2pπu transition, and the attosecond X-ray probe pulse wavelength is λ pr = 5 nm (ω pr = 9.11 a.u.). The pulse durations T are ten cycle corresponding to full width at half maximum (FWHM) of 1.16 fs and 83 as for the pump and probe pulses. The kinetic energy E of photoelectron in Eq. 6 ACS Paragon Plus Environment

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Probability (arb. units)

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Figure 1: The probability P, Eq. (7), of observing fractional photoelectrons in a small region ∆θ = π/12 as a function of the time delay ∆t by left-handed (ξ pu = ξ pr = −1) circularly polarized λ pu = 70 nm UV pump and λ pr = 5 nm attosecond X-ray probe pulses. Blue points indicate the four time delays ∆t illustrated in Fig. 2. (7) is chosen at ~ω pr − I p = 8.03 a.u. with the molecular ionization potential of I p = 1.08 a.u., which corresponds to one ~ω pr photon absorption from the 1sσg − 2pπu superposition state. The corresponding coefficients in Eq. (1) are respectively cg = 0.4 and ce = 0.6 by the UV pump pulses. In Fig. 1 we can clearly see that the observed fractional probability P is critically sensitive to the time-delay ∆t. By fixing the UV pump pulse E pu (t), varying the probe time, i.e., ∆t, leads to a smooth oscillation of the observation with a period of τ = τ pu =232 as. The detected photoelectron signal produced by the X-ray probe pulse illustrates the electronic coherence between the two 1sσg and 2pπu electronic states created by the pump pulses. Equations (3) and (4) show that the evolution of the coherent electronic dynamics satisfies ∼ cos(∆Et/~ + θ). The electron rotates with a period of 2π/∆E = 232 as in angle, in good agreement with the simulation results in Fig. 1. These time-dependent observing fractions provide the necessary attosecond temporal resolutions for electron charge migration. In order to intuitively view the probe signal, in Fig. 2(a) we plot the 2D X-ray photoelectron momentum distributions at four different time delays ∆t as marked in Fig. 1. It is found that the 2D distributions are asymmetric with respect to the molecular center and exhibit a semiarc structure. Varying the time-delay gives rise to a clockwise rotation of the semiarc distribution with a period of τ = 232 as. The dependence of the 2D X-ray photoelectron distributions on the time-delay ∆t enables determination of the coherent electron with spatial and temporal solutions. The extreme



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Figure 2: Electronic dynamic in attosecond charge migration. (a) Molecular photoelectron momentum distributions at four different time delays ∆t marked in Fig. 1 by using left-handed (ξ pu = ξ pr = −1) circularly polarized pump-probe pules. (b) The corresponding electronic probability densities A− (t) and (c) angular currents j−θ (t) at four different probe moments t = ∆t by a single λ pu = 70 nm pump pulse with left helicity ξ pu = −1. Arbitrary units of distributions are used. values of the photoelectron distributions indicate the amplitude and phase of the coherent superpositions. Figure 2(b) and 2(c) display the corresponding time-dependent electronic probability densities A− (t) and angular currents j−θ (t). We present the charge migration processes by the λ = 70 nm pump pulse at moments t = ∆t, where the X-ray probe pulse is absent. These time-dependent ∫ electronic dynamical results are numerically calculated by A− (t) = dzψ∗ (ρ, θ, z, t)ψ(ρ, θ, z, t) and ∫ i~ j−θ (t) = − 2m dz[ψ∗ (t)∇θ ψ(t) − ψ(t)∇θ ψ∗ (t)]. The similar distributions are produced Fig. 2. As e predicted in Eqs. (3) and (4), A− and j−θ depend on the function of cos(∆Et + θ) showing the maxima of the densities occur at θ ∼ −∆Et (i.e., cos(∆Et + θ) = 1). As a result, the evolution of the electronic densities A(t) and electronic currents jθ (t) will exhibit a periodical and clockwise rotation with a period of τ = 232 as, as shown in Figs. 2 (b) and (c). The negative and positive do8 ACS Paragon Plus Environment

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Figure 3: Electronic dynamic in attosecond charge migration. (a) Molecular photoelectron momentum distributions at four different time delays ∆t by using right-handed (ξ pu = ξ pr = +1) circularly polarized pump-probe pules. (b) The corresponding electronic probability densities A+ (t) and (c) angular currents j+θ (t) at four different probe moments t = ∆t by a single λ pu = 70 nm pump pulse with right helicity ξ pu = +1. Arbitrary units of distributions are used. minions of the electronic currents are mirror images of each other at times t → t + τ pu /2, indicating strictly the period of the electronic coherence. In Fig. 2(a), the observed photoelectron distributions rotate clockwise as a function of the time delay, which therefore reconstruct the evolutions of the electronic density and currents in Figs. 2 (b) and (c). Different helicities of pump pulses produce an electron with different angular momentum Λ = ±1 for circular polarization perpendicular to the molecular axis. We next show in Fig. 3 results by using the right-handed (ξ pu = ξ pr = +1) circularly polarized pump-probe pulses. The other laser parameters are the same as in Fig. 2. The photoelectron distributions and electronic densities and currents exhibit similar asymmetric behaviours as already observed for the left-hand pulses in Fig. 2. It is however found that these distribution show an anticlockwise rotation. By a right-handed



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Figure 4: Comparison of molecular photoelectron angular distributions J(θ) at four different time delays ∆t by using (a) left-left (ξ pu = ξ pr = −1), (b) left-right (ξ pu = −1, ξ pr = +1), (c) right-left (ξ pu = +1, ξ pr = −1) and (d) right-right (ξ pu = ξ pr = +1), handed circularly polarized pump-probe pules. The other laser parameters are the same as in Figs. 2 and 3. Arbitrary units of distributions are used. pump pulse, a coherent superposition |ψ+0 (r, t)⟩ of the electronic states with angular momentum m = +1 is created. In Eqs. (3) and (4), the maxima of the densities and currents occur at θ ∼ ∆Et. As a result, the sign change of the helicity of the pump pulse leads to an opposite rotation effect of electronic coherence on the ERCs. To further examine the dependence of the electronic coherence on the pump pulse, we compare a series of simulations with four different combinations of helicities ξ pu = ±1 and ξ pr = ±1 for the UV pump and X-ray probe pulses. Figure 4 shows the simulation results of the photoelectron angular distributions J(θ) at different time delays by intergrading over the kinetic energy E, where ∫ E+~ω pr /3 J(θ) = E−~ω /3 J(θ, E)dE. For the left-hand ξ pu = −1 circularly polarized UV pump pulse, pr


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the rotation of the photoelectron momentum angular distributions shows a clockwise rotation for both left ξ pr = −1 and right ξ pr = +1 handed circularly polarized X-ray prob pulses, whereas for the right-handed ξ pu = +1 pump pulse an anticlockwise rotation is observed for the both probe cases. The dependence of the rotation of the X-ray spectra on the helicity of the pump/probe pulses indicates the permanent rotational property of the coherent superposition state in the charge migration induced by the pump pulse. The observed photoelectron distributions are mapped from the UV pulse-induced ERCs, and therefore the electronic coherence effect encoded in the ERCs can be tracked in real time. It should be noted that the excited 2pπu state is dissociative, leading to an attenuation of the electronic coherence, i.e., the periodicity of the ERCs, due to the loss of the overlap between the initial and the dissociating wave packets. As discussed previously in H+2 , the effects of the nuclear movement are to lower the amplitude of oscillations as function of the time delay. 41 This therefore allows to measure the decay of the ERCs by probing X-ray spectra. In summary, we have theoretically demonstrated an optical scheme to explore ultrafast negative charge migrations by monitoring the intramolecular electronic coherence dynamics with attosecond time and sub-nanometer resolutions. To this end, a coherent superposition of electronic states was constructed by using a circularly polarized UV laser pule, and therefore an attosecond timedependent ERC is induced inside molecules. Our theoretical analysis combined with ab initio simulations in hydrogen molecular ions has shown that such ultrafast intramolecular electronic motions can be tracked and controlled in real time by using an attosoecond X-ray probe pulse, where the calculated angular and energy-resolved distributions of the photoelectrons can be measured experimentally by using velocity map photoelectron imaging techniques. The present demonstration of tracking attosecond electronic coherence and ring currents by probing circular polarization X-ray spectra paves the way to control and image charge migration by intense ultrafast laser pulses. As shown here the electronic rings depend on the helicity of the pump pulse. This suggests the symmetry of the molecular orbitals and the molecular alignment and orientation as essential parameters for complete measurement and control of electron dynamics. Laser pulses can therefore be used to achieve molecular imaging and control by monitoring



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photoelectron spectra during the charge migration on attosecond time scale. The control of charge migration by laser pulses can be further refined with polarization. Finally, one can also measure the decay or the life time of the repulsive excited electronic state. Related to this are new measurements of time delay of attosecond ionization in molecules. 42 The present scheme in principle can be extended to other complex molecular systems, and therefore has a great of potential for probing intramolecular attosecond electronic dynamics involved in circular polarized photoinduced chemical reactions. 43

ACKNOWLEDGEMENTS K.J.Y. acknowledges enlightening discussions with Dr. S. Chelkowski. C.-C.S acknowledges the financial support by the Vice-Chancellor’s Postdoctoral Research Fellowship of University of New South Wales. D.D. acknowledges partial supports by the Australian Research Council under Grant Nos. DP130101658. The authors also thank RQCHP and Compute Canada for access to massively parallel computer clusters.

Author information Corresponding Author (C.-C.S) Email:[email protected] (A.D.B) Email:[email protected] The authors declare no competing financial interest.

References (1) Bandrauk, A. D. Molecules in Laser Fields, Marcel Dekker: New York, USA, 1994. (2) Manz, J.; Wöste, L. Femtosecond Chemistry, Wiley-VCH:Weinheim, Germany, 1995. 12 ACS Paragon Plus Environment

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(14) Shao, H. C.; Starace, A. F. Detecting Electron Motion in Atoms and Molecules Phys. Rev. Lett. 2010, 105, 263201 (15) Baum, P.; Zewail, A. H. Attosecond Electron Pulses for 4D Diffraction and Microscopy. Proc. Natl. Acad. Sci. USA 2007, 104, 18409-18414. (16) Shorokhov, D.; Zewail, A. H. 4D Ultrafast Electron Microscopy - Evolutions and Revolutions. J. Chem. Phys. 2016, 144, 080901. (17) Weinkauf, R.; Schanen, P.; Yang, D.; Soukara, S. Schlag, E. W. Elementary Processes in Peptides: Electron Mobility and Dissociation in Peptide Cations in the Gas Phase. J. Phys. Chem., 1995, 99, 11255-1265 (18) Cederbaum, L. S.; Zobeley, J. Ultrafast Charge Migration by Electron Correlation. Chem. phys. Lett. 1999, 307, 205-210. (19) Mignolet, B.; Levine, R. D.; Remacle, F. Charge Migration in the Bifunctional PENNA Cation Induced and Probed by Ultrafast Ionization: A Dynamical Study. J. Phys. B 2014, 47, 124011 (20) Despré, V.; Marciniak, A.; Loriot, V.; Galbraith, M. C. E.; Rouzée, A.; Vrakking, M. J. J.; Lépine, F; Kuleff, A. I. Attosecond Hole Migration in Benzene Molecules Surviving Nuclear Motion. J. Phys. Chem. Lett. 2015, 6, 426?31. (21) Calegari, F.; Trabattoni, A.; Palacios, A.; Ayuso, D.; Castrovilli, M. C.; Greenwood, J. B.; Decleva, P.; Mart´ın, F.; Nisoli, M. Charge Migration Induced by Attosecond Pulses in Biorelevant Molecules. J. Phys. B 2016, 49, 142001. (22) Kraus, P. M.; Mignolet, B.; Baykusheva, D.; Rupenyan, A.; Horný, L.; Penka, E. F. et al. Measurement and Laser Control of Attosecond Charge Migration in Ionized Iodoacetylene. Science 2015, 350, 790-795


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Probability (arb. units)

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 1.0 58 59 60 0.5

ACS Paragon Plus Environment

0.0 4.0

4.5

5.0

Delay time ∆t (τpu)

5.5

6.0

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∆t=4.5τ

∆t=4.25τ

∆t=4.0τ

(a)

∆t=4.75τ

4

py (a.u.)

0

2

0

1

-2

-4 -4

-2

0

2

-4

4

px (a.u.)

(b) 6 t=4.0τ

-2

0

2

4

px (a.u.)

t=4.25τ

-4

-2

0

2

t=4.5τ

-4

4

px (a.u.)

-2

0

2

4

px (a.u.)

t=4.75τ

0

4

0 -2

0.08

y (a.u.)

2

-4 -6

(c)

6

-0.2

4 2 0 -2

0.26

y (a.u.)

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


-4 -6

-6

-4

-2

0

x (a.u.)

2

4

6 -6

-4

-2

0

2

4

6 -6

-4

-2

x (a.u.)


0

x (a.u.)

2

4

6 -6

-4

-2

0

x (a.u.)

2

4

6


∆t=4.5τ

∆t=4.25τ

∆t=4.0τ

(a)

∆t=4.75τ

4

py (a.u.)

0

2

0

1

-2

-4 -4

-2

0

2

-4

4

px (a.u.)

(b) 6 t=4.0τ

-2

0

2

t=4.25τ

-4

4

px (a.u.)

-2

0

2

px (a.u.)

t=4.5τ

-4

4

-2

0

2

4

px (a.u.)

t=4.75τ

0

4

0 -2

0.08

y (a.u.)

2

-4 -6

(c)

6

-0.2

4 2 0 -2

0.26

y (a.u.)

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

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-4 -6

-6

-4

-2

0

x (a.u.)

2

4

6 -6

-4

-2

0

x (a.u.)

2

4

6 -6

-4

-2

0

x (a.u.)


2

4

6 -6

-4

-2

0

x (a.u.)

2

4

6

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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


(a) ξ pu=−1, ξ pr =−1 1.0

0.5

(b) ξ pu=−1, ξ pr =+1 90

90 120

120

60

150

210

330 240

1.0

270

0.5

120

4.0τ pu 4.25τ pu 4.5τ pu 4.75τ pu

150

1.0

0

330 240

270

300

300

0.0

0.5

1.0

(d) ξ pu=+1, ξ pr =+1 90

1.0

60

150

30

210

270

120

60

0.0 180

0.5

330 240

90

0.5

0

210

300

(c) ξ pu=+1, ξ pr =−1 1.0

30

0 180

0.0 180

0.5

150

30

1.0

60

30

180

0.5

0

210

330 240

270


300

0.0

0.5

1.0

Attosecond Dynamics of Molecular Electronic Ring Currents - The

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