Ultrafast X-ray Photoelectron Imaging of Attosecond Electron

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A: Kinetics, Dynamics, Photochemistry, and Excited States

Ultrafast X-ray Photoelectron Imaging of Attosecond Electron Dynamics in Molecular Coherent Excitation Kaijun Yuan, and Andre Dieter Bandrauk J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.8b12313 • Publication Date (Web): 23 Jan 2019 Downloaded from http://pubs.acs.org on January 29, 2019

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The Journal of Physical Chemistry

Ultrafast X-ray Photoelectron Imaging of Attosecond Electron Dynamics in Molecular Coherent Excitation Kai-Jun Yuan∗,† and André D. Bandrauk∗,‡ †Institute of Atomic and Molecular Physics, Jilin University, Changchun, 130012, China ‡Laboratoire de Chimie Théorique, Faculté des Sciences, Université de Sherbrooke, Sherbrooke, Québec, Canada, J1K 2R1 E-mail: [email protected]; [email protected]

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Abstract Ultrafast photoelectron imaging allows to measure information about coherent electron dynamic processes in materials or chemical compounds on femtosecond to attosecond time scales. We show that molecular time-resolved photoelectron diffraction produced by a time-delayed soft X-ray attosecond pulse can be used to monitor the ultrafast coherent excitation induced by a resonant UV pump pulse with variable carrierenvelope phases. Asymmetric diffraction angular patterns illustrate coherent electron dynamics of charge migration with spatiotemporal resolution on the attosecond and ˚ Angström scale. This allows the temporal reconstruction of phases and amplitudes of electronic states and geometry of molecules as a function of time delay of the probe pulse and carrier-envelope phases of the pump pulse. Results are obtained from solutions of time-dependent Schrödinger equations of the hydrogen molecular ion, and analyzed by ultrafast photoelectron diffraction models for coherent superposition of electronic states. The present demonstration provides a guiding principle for monitoring ultrafast spatiotemporal coherent electron dynamics and imaging molecular electronic structure in complex systems by ultrafast pump-probe experiments and their dependence on carrier-envelope phases and time delays.

1 INTRODUCTION Imaging and controlling electron and nuclear dynamics in molecular reactions is a main goal in photon induced ultrafast chemical and biological processes. 1–5 Time-resolved photoelectron spectroscopy has been used widely as an efficient technique for investigating ultrafast molecular dynamics. 6–12 An ultrashort pump laser pulse initiates a coherent superposition of ground and excited electronic states in molecules and the time evolution of the initial coherence is subsequently monitored after a variable time delay by a second probe pulse. Advances in synthesizing ultrashort intense laser pulses 13,14 allow one to visualize and control molecular reaction processes from femtosecond (1 fs=10−15 s) time scales for nuclear

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vibrations to attosecond (1 as=10−18 s) scales for electron motion. To date, the shortest attosecond pulse with a duration 43 as has been produced. 15 Moreover, it has been shown that by compensating for attosecond chirp in gaseous media, the generation of near transformlimited 63 as pulses can be achieved. 16 These ultrafast laser pulses are available for such new photoelectron imaging techniques. Electrons have a great potential for probing the timeresolved transient structure of matter via such ultrashort photoelectron spectroscopy. One can now envisage the emergence of time resolved laser-induced electron diffraction (LIED) from electron wavepackets, 17–21 which was predicted earlier by Cohen and Fano, 22 as efficient methods of measuring molecular electronic structure and electron motion for different fixed nuclear configurations at different times. Most recently, effects of nuclear motion in molecular photoionization have also been investigated. 23 Photoelectron imaging of coherent excitation and charge migration which is a fundamental and important quantum process in photochemistry of molecules 24–40 by photoionization of a coherent population of multiple electronic states offers therefore a new approach to probe molecular orbitals and controlling chemical reactions. Attosecond charge migration has been, for the first time, measured in experiments by linearly polarized high-order harmonic generation in ionized iodoacetylene via laser control of attosecond electron recollisions. 36 Angular electronic fluxes can also be used to reconstruct electron charge migration in excited cyclic molecules such as benzene by preparing coherent electronic states, 41–43 which can then be monitored by photoelectron momentum distributions with time-delayed high frequency attosecond pulses. 44 Molecular photoelectron angular distributions can also be used to monitor electron coherent excitation. 45,46 Measuring phases and amplitudes of coherent electron wave packets in molecular charge migration is still a challenge and remains a difficulty in ultrafast time-resolved photoimaging spectroscopy. Nuclear vibrations and width of the vibrational wave packets also give rise to the dephasing of electron coherence. Such dephasing leads to suppression of frequency resolution, but can improve time resolution 47 in photoelectron imaging. In this work we

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present a method of phase sensitive photoelectron diffraction to monitor coherent excitation and charge migration processes. Angular diffraction patterns depend on phases and ampli48 tudes of the coherent electron wavepackets. The molecular ion H+ 2 as a benchmark model

is used to illustrate the molecular laser induced electron diffraction and dynamics. A linearly polarized UV pump pulse with different phases is used to excite the molecule, by which an electron coherence is created between the ground and excited states. Subsequently a timedelayed soft X-ray attosecond probe pulse ionizes the excited molecule. Results obtained from numerical solutions of time-dependent Schrödinger equations (TDSE) show signatures of asymmetric molecular photoelectron distributions, which are found to be dependent on phases and intensities of the pump pulse, encoding the information of the time-dependent electron coherence. Since the photoelectron distributions are sensitive to the pulse phase, carrier envelope phase (CEP) stable lasers are necessary in the pump-probe processes.

2 THEORETICAL MODELS The pump pulse induces nonstationarity of states by laser induced coherence in the molecule. 49 For the molecular hydrogen ion H+ 2 at equilibrium, coherent superposition of the two electronic states, the ground 1sσg ψσg (r) state with the eigenenergy Eσg and the excited 2pσu state, ψσu (r) with the eigenenergy Eσu , is therefore created by a parallel transition ψc (r, t) = cσg (t)ψσg (r)e−iEσg t/¯h + cσu (t)ψσu (r)e−iEσu t/¯h .

(1)

|cσg (t)|2 and |cσu (t)|2 are the orbital occupation probabilities. At the initial time t = 0, the molecular system is in a stationary state ψc (r, t = 0) = ψσg (r), i.e., cσg (t = 0) = 1 and cσu (t) = 0. According to first order perturbation theory, the occupation coefficient cσu in the excited state is then given by Z

t

cσu (t) = −i

0

dt0 V (t)ei∆Et /¯h ,

−∞

4


(2)

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where ∆E = Eσu − Eσg = 0.45 a.u. (atomic units) corresponds to 12 eV. The interaction term V (t) = hψσg (r)|r · E(t)|ψσu (r)i is a time-dependent matrix element, where the external pump field E(t) = eE0 ei(ωt+φ) , includes the time factor t and the CEP φ of the pulse. The R Fourier transform of the pulse, E(ε) = E(t)eiεt dt, in frequency domain, is composed of two components, E(ε) = eφ E + ε + e∗φ E − ε. 50 The factor E ± (ε) are CEP and polarizationR independent scale parameters of the pulse E ± (ε) = 12 E0 ei(ε∓ω)t dt and the unit vector eφ = ee−iφ . As a result, the time-dependent excited state occupation coefficient cσu (t) in the superposition state in Eq. (1) is a function of the CEP of the pump pulse, cσu (t) = c0σu (t)eiφ . Then the corresponding time dependent coherent electron density distributions are described by,

D(r, t) = |ψc (r, t)|2 = D(g) (r) + D(u) (r) + D(g,u) (r, t) = |cσg (t)ψσg (r)|2 + |cσu (t)ψσu (r)|2 +2|cσg (t)c0σu (t)| 10τpu , the occupation coefficients are constant 10


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Figure 2: Time t and CEP φ dependent electron density probabilities D(x, y) of the molecular ion H+ 2 aligned along the x axis after excitation by a linearly polarized λ = 100 nm pump pulse with its field vector along the x axis at intensity I0 = 2.75 × 1013 W/cm2 (E0 = 2.84 × 10−2 a.u.) and duration T = 10τpu = 3.31 fs. The CEPs of the pump pulse are (from top to bottom arrows) φ = 0, π/2, π, and 3π/2, and the times are (from left to right columns) t = 11.0τpu , 11.25τpu , 11.5τpu , and 11.75τpu . at |cσg (t)|2 = |cσu (t)|2 = 0.5. The same electron probabilities result in the maximum of the interference effect between the two electronic states. From Fig. 2 one sees that the electron density distributions D(x, y) show an oscillation in

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molecules which are dependent on the time t and the phase φ of the pump pulse. The coherent electron wave packets moves periodically between the two nuclear centers at (±1, 0, 0) a.u.. At time t = 11τpu and φ = 0, a strong asymmetry is obtained and the coherent electronic wave packet mainly lie on the left center of the molecule, x = −1 a.u.. Increasing t and/or φ leads to a variant of the electron distribution. The electron moves to the right center at time t = 11τpu and φ = π, or t = 11.5τpu and φ = 0. As the time t or the CEP φ increases further, electrons come back. The period of the oscillation is 2π for the phase φ and τ = τpu = 331 as for the time t, in good agreement with the prediction in in Eq. (3). At ∆Et/¯ h + φ = nπ, where n is an integer, the superposition term cos(∆Et/¯ h + φ) = ±1 gives rise the maximum asymmetry of electron density distributions, whereas at ∆Et/¯ h + φ = (n + 1/2)π, i.e., cos(∆Et/¯ h + φ) = 0, symmetric electron distribution are produced, as illustrated in Fig. 2. The evolution of the electron density distribution therefore describes the dependence of the attosecond electron coherence on the time t and the CEP φ of the pump pulse in charge migration, i.e., the superposition term D(g,u) (r) in Eq. (3). Figure 3 displays molecular photoelectron momentum distributions produced by a circularly polarized λpr = 5 nm attosecond probe pulse. We use the same linearly x polarized λpu = 100 nm pump pulse to excite the x aligned H+ 2 molecule resonantly to the 2pσu state, as shown in Fig. 2. The CEPs of the pump pulse Epu vary from φ = 0 to 2π and the Table 1: The angles of photoelectron diffraction patterns as functions of the CEP φ and the time delay ∆τ for H+ 2 at equilibrium R = 2 a.u., obtained by theoretical analysis in Eq. (5) and Fig. 4, and numerical simulation in Fig. 3 ∆τ = 6.0τpu

Eq. (5) Fig. 3

∆τ = 6.5τpu

Eq. (5) Fig. 3

φ=0 ±11.5◦ , ±78.5◦ , ±126◦ ±8.5◦ ,±80.0◦ , ±128.0◦ ±54◦ , ±101.5◦ , 180◦ ± 11.5◦ ±53.5◦ ,±101.0◦ , 180◦ ± 6◦

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φ=π ±54◦ , ±101.5◦ , 180◦ ± 11.5◦ ±53.5◦ ,±98.5◦ , 180◦ ± 5.5◦ ±11.5◦ , ±78.5◦ , ±126◦ ±3◦ ,±77.5◦ , ±124.0◦

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Figure 3: Photoelectron momentum distributions as functions of the time delay ∆τ and the CEP φ of the pump pulse. From top to bottom rows: the phases of the pump pulse are chosen as φ = 0, π/2, π, 3π/2, and 2π. The time delays are (from left to right columns) ∆τ = 6.0τpu , 6.25τpu , 6.5τpu , and 6.75τpu . The molecule H+ 2 is aligned along the x axis. The 13 2 −2 pulse intensities I0 = 2.75×10 W/cm (E0 = 2.84×10 a.u.), duration Tpu = 10τpu = 3.31 fs (1.655 fs FWHM) and Tpr = 10τpr = 165.4 as (83 as FWHM), are used. time delay between the pump and probe pulses ranges from ∆τ = 6.0τpu to ∆τ = 6.75τpu , corresponding to the ionization times t = ∆τ + Tpu /2 as illustrated in Fig. 1. The other laser parameters are intensities I0 = 2.75 × 1013 W/cm2 (E0 = 2.84 × 10−2 a.u.) and durations 13



Tpu = 10τpu = 3.31 fs (1.655 fs FWHM, full width at half maximum) and Tpr = 10τpr = 165.4 as (83 as FWHM). The pump photon h ¯ ωpu = ∆E leads to a σg − σu parallel transition, creating a superposition state ψc in Eq. (1). Since we focus on the probing photoelectron distributions produced by the circular soft X-ray pulse, only the photoelectron distributions p hωpr − Ip ) corresponds to a single around the momentum pe = 4.01 a.u., where pe = ± 2me (¯ ωpr photon absorption, are presented in Fig. 3. In Fig. 3 we see that asymmetric photoelectron momentum distributions with multiple angular patterns are produced. Varying the pulse CEP φ and time delay ∆τ gives rise to modulation of photoelectron distributions. Time-delay dependent photoelectron signals have been used to measure CEP of pump pulses. 61 Since the ionized electron wavelength λe =1.56 a.u. (λe = 2π¯ h/pe ) of the photoelectron is smaller than the molecular internuclear distance R = 2 a.u., diffraction occurs, 17 thus leading to multiple angular patterns. In this work, the pulse field strength E in Eq. (5) can be ignored to describe the effect on the electron movement in high frequency photoionization since the chosen intensities (2.75×1013 W/cm2 ) 2 in the simulations produce negligible ponderomotive energies Up = e2 E02 /4me ωpr Ip , where

Ip is the ionization potential, and E p2e /2me . After the pump pulse duration, the numerical occupation coefficients are constants, |cσg (t)|2 = 0.5 and |cσu (t)|2 = 0.5 but the total density oscillates as a charge migration even though there are no Rabi oscillations. In Fig. 4, we show the dependence of the photoelectron angular distributions on the time t and the phase φ of the pump pulse predicted from Eq. (5). The photoelectron momentum is chosen as pe = 4.01, corresponding to one ωpr =9.11 a.u. photon absorption. The theoretical analysis shows strictly asymmetry with respect to the angle θ = π and CEP φ = π, corresponding to evolution of coherent electron wave packets in Eq. (1) with the time t and the CEP φ of the pump pulse. Therefore photoelectron diffraction well describes the electron coherent in charge migration. The diffraction angles obtained from theoretical predictions in Fig. 4 and numerical simulations in Fig. 3 are in good agreement. For example, table 1 shows results of pho-

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t=11.0tpu

t=11.25tpu

t=11.5tpu

t=11.75tpu

Figure 4: Photoelectron momentum distributions F(θ, t, φ) in Eq. (5) as functions of the photoelectron ejection angle and the CEP φ of the pump pulse at different probe times (a) t = 11.0τpu , (b) 11.25τpu , (c) 11.5τpu , and (d) 11.75τpu . The photoelectron momentum pe = 4.01 a.u., molecular internuclear distance R = 2 a.u., and the energy difference ∆E = 0.455 a.u. are always fixed. toelectron diffraction angles obtained from theoretical analysis in Eq. (5) and Fig. 4 and numerical simulations in Fig. 3. We choose the CEPs φ = 0 and π and the time delays ∆τ = 6.0τpu and 6.5τpu , where the photoelectron momentum distributions show an opposite symmetry. Due to the coherent superposition of the two electronic states, their interference term cos(∆φ) leads to asymmetry in the photoelectron momentum distributions. For the case of ∆τ = 6.0τpu , the distributions in the right momentum plane px > 0 are dominant at φ = 0 while at φ = π, the diffraction patterns in the left momentum plane px < 0 dominate. A reverse process occurs at ∆τ = 6.5τpu . For these cases, the photoelectron diffraction patterns in the left and right plane are nearly mirror images of each other, as predicted in Eq. (5). The maximum asymmetry of photoelectron diffraction reflects the phase difference ∆φ = φ − ∆E∆τ /¯ h = nπ of the electronic states. Of note is that at time 15



Table 2: The angles of photoelectron diffraction patterns as functions of the amplitudes |cσg | and |cσu | at the CEP φ = 0 and the time delay ∆τ = 6τpu for H+ 2 at equilibrium R = 2 a.u., obtained from theoretical analysis in Eq. (5), and numerical simulation in Fig. 5. |cσg |2 0.82

|cσu |2 0.18

0.22

0.62

Eq. (5) Fig. 5 ◦ ◦ ±26.6 , ±83.5 , ±27.0◦ ,±85.5◦ , ◦ ±132.5 ±136.0◦ 0, ±75.0◦ , ±121.5◦ , 0, ±71.0◦ , ±117.0◦ , ±180◦ ±180◦

delays ∆τ = 6.25τpu , interference effects disappear since cos(∆φ) = 0 as predicted in Eqs. (3) and (5), and Fig. 4. However for the σg and σu state, the electron density distributions are not equivalent, |ψσg |2 6= |ψσu |2 , Fig. 2. Therefore diffraction patterns are also produced in Fig. 3. The photoelectron diffraction patterns in Fig. 3 illustrate the coherent electron excitation dynamics in Fig. 2. The dependence of the photoelectron diffraction patterns produced by the probe pulse on the pulse CEP φ and the time delay ∆τ arises from the coherent electron motion, i.e., the charge migration in molecule. The asymmetry of the probing distributions reflects to the phase difference ∆φ, i.e., φ − ∆Et/¯ h. Therefore, the pulse CEP dependence corresponds to the time delay ∆τ dependence. The distributions from the two individual electronic states, σg and σu , are time independent and symmetric, as shown in Eq. (5). Their interference term cos(∆φ) leads to the oscillation of the photoelectron distributions with the time t and the CEP φ. For example at ∆τ = 6.0τpu , the extreme values of the interference term occur at ∆φ = nπ, i.e., cos(∆φ) = ±1. One has the evolution time t = h ¯ (nπ − φ)/∆E. However the time delay ∆τ 0 = 6.5τpu leads to a reverse evolution process. At φ = 0 the distribution mainly lies in the left plane whereas at φ = π those in the right dominate. This results from the difference of ∆E(∆τ 0 − ∆τ )/¯ h = π, thus leading to a sign change of the superposition term cos(∆φ) in Eq. (5), the phase of the coherent electron wavepackets ∆φ0 = ∆φ + π. The CEP-dependent photoelectron diffraction also allows us to illustrates the evolution of the coherent electron in molecules with time.

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(b) 1.5E0

(a) 0.5E 0 0.0

0.0

4

2

0

0.2

-2

0.3

py (a.u.)

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-4 -4

-2

0

2

4

-4

-2

0

2

4

px (a.u.)

px (a.u.)

Figure 5: Photoelectron momentum distributions at different intensities of the pump pulse: (a) 0.5E0 and (b) 1.5E0 . The CEP of the pump pulse is φ = 0 and the time delay is ∆τ = 6.0τpu . The other laser parameters are the same as in Fig. 3 We next present the dependence of the photoelectron diffraction on the amplitude of the coherent electron wave packets. Figure 5 displays photoelectron distributions at different pump pulse intensities (a) 0.5E0 (0.69 × 1013 W/cm2 ) and (b) 1.5E0 (6.19 × 1013 W/cm2 ). We fix the probe pulse at E0 (2.75 × 1013 W/cm2 ). The CEP and the time delay are fixed at φ = 0 and ∆τ = 6.0τpu . It is found that the diffraction angular patterns are strongly modulated. This modulation is mainly due to the variance of amplitudes of the coherent excitation. As shown in Eq. (5), the amplitudes of each electron wavepacket, i.e., cσg and cσu which are determined by the pump pulse Epu (t), also influence the interference of quantum electronic states. After the pump pulses, the corresponding numerical populations |cσg (t)|2 and |cσu (t)|2 are 0.82 and 0.18 at 0.5E0 and 0.22 and 0.62 at 1.5E0 . The σg and σu dominant diffraction thus leads to different angular patterns. We show in table 2 the angles of photoelectron diffraction patterns at various amplitudes cσg and cσu of the electron wavepackets in the two quantum ground and excited states. From table 2 one sees that for the molecule H+ 2 at R = 2 a.u., the photoelectron momentum pe = 4.01 a.u., the corresponding diffraction angle predicted by Eq. (5) are in good agreement with simulated results in Fig. 5. This confirms that the photoelectron diffraction angular sectors enable us to measure the amplitude of the coherent excitation. The photoelectron diffraction angular patterns also provide the information of the bond

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(a) R=5 a.u.

(b) R=7 a.u. 0.0

0.0

4

2

0

-2

0.8

0.4

py (a.u.)

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-4 -4

-2

0

2

4

px (a.u.)

-4

-2

0

2

4

px (a.u.)

Figure 6: Photoelectron diffraction patterns of H+ 2 at larger molecular internuclear distances R = 5 a.u. and 7 a.u. from coherent electronic states in Eq. (1). The CEP of the linearly x polarized pump and the circularly polarized soft X-ray attosecond probe pulse are φ = 0. The time delay is fixed at ∆τ = 6.0τpu . The molecule H+ 2 is aligned along the x axis. The pulse 13 2 −2 intensities I0 = 2.75 × 10 W/cm (E0 = 2.84 × 10 a.u.) and duration Tpu = 10τpu = 3.31 fs (1.655 fs FWHM) and Tpr = 10τpr = 165.4 as (83 as FWHM) are used. length R of the molecule. As shown in Eq. (5), the angular patterns are determined by the term pe R cos θ. In Fig. 6 we display photoelectron diffraction at larger bond lengths R = 5 a.u. and 7 a.u. where charge-resonance-enhanced ionization occurs. 62,63 Photoionization initiates again from coherent electronic states of the ground 1sσg and excited 2pσu states. It is found that more diffraction angular patterns are produced in Fig. 6. The diffraction angles satisfy the condition sin(pe R cos θ) = ±1, as predicted in Eq. (5). Therefore by probing the photoelectron diffraction of the coherent excitation at different time we can obtain the information of not only the phase and amplitude of the coherent electronic state, but also the temporal change of bond lengths of molecules. We present X-ray photoelectron imaging of molecular excitation and ionization dynamics within the fixed nuclear frame. The vibrational motion of nuclei may reduce the electronic coherence. 23 The time delay ∆τ ∼ 6τpu (∼ 2 fs), is far less than the molecular vibrational 64 period of ∼ 15 fs of the molecule H+ Consequently, the simulation of X-ray distributions 2.

remains valid to tracking coherent electron wave packets, and electronic oscillations should be be experimentally observable at these short time delays. As reported previously, 65 the effects of the nuclear movement are to lower the amplitude of oscillations as a function of

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the time delay, thus allowing us to measure the decay of the electronic coherence by phase and time dependent X-ray photoelectron distributions.

5 CONCLUSIONS In this work, we have shown that by exploring time and CEP dependent ultrafast photoelectron diffraction, attosecond electron dynamics of coherent electron excitation in molecules can be reconstructed. A resonant linearly polarized UV pump pulse creates a time-dependent coherent superposition of electronic states and a subsequent time-dependent circularly polarized attosecond soft X-ray probe pulse is used to ionize the excited molecule. Using the CEP φ dependence, without varying time delay, enables us in principle to minimize the dephasing and relaxation of the electron coherence. The electron coherence leads to photoionization asymmetry which varies with the CEP φ and time t of the X-ray photoelectron diffraction that enable us to extract the phase of the excited electronic state φσu = nπ − φ + ∆Et/¯ h at extreme values. By varying the CEP φ of pump pulses one can monitor electron coherence from phtoelectron diffraction patterns in X-ray probing distributions, thus following the evolution of the coherent electron wavepackets with attosecond time resolutions. Diffraction angle patterns illustrate time-dependent phases and amplitudes of the coherent excitation and also the varying bond lengths of a molecule. Our theoretical analysis combined with ab initio simulations shows that such attosecond intramolecular coherent electron motions can be monitored by ultrafast X-ray photoelecton diffraction. The present proposal in which the pulse CEP φ determines the evolution time and state population of the coherent electron wave function allows for the phase dependence of photoelectron imaging of charge migration. Using time delay further allows for measure dephasing. The experimental advances in molecular attosecond photoionization 66,67 make the present temporal imaging possible to monitoring electron charge migration, which in principle can be applied to more complicated molecular systems.

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Notes The authors declare no competing financial interest.

Acknowledgement The authors thank Compute Canada for access to massively parallel computer clusters and NSERC, FQRNT for support of this research.

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Ultrafast X-ray Photoelectron Imaging of Attosecond Electron

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