Probing Attosecond Electron Coherence in Molecular Charge Migration by Ultrafast X-Ray Photoelectron Imaging

Electron coherence is a fundamental quantum phenomenon in today’s ultrafast physics and chemistry research. Based on attosecond pump–probe schemes, ultrafast X-ray photoelectron imaging of molecules was used to monitor the coherent electron dynamics which is created by an XUV pulse. We performed simulations on the molecular ion H 2 by numerically solving time-dependent Schrödinger equations. It was found that the X-ray photoelectron angular and momentum distributions depend on the time delay between the XUV pump and soft X-ray probe pulses. Varying the polarization and helicity of the soft X-ray probe pulse gave rise to a modulation of the time-resolved photoelectron distributions. The present results provide a new approach for exploring ultrafast coherent electron dynamics and charge migration in reactions of molecules on the attosecond time scale.


Introduction
Time-resolved photoelectron emission spectroscopy has been used widely as an efficient technique for investigating nuclear and electronic dynamics in molecular reactions [1][2][3][4][5][6][7][8].An ultrashort pump laser pulse initiates a coherent transition between ground and excited states in molecules and the time evolution is subsequently monitored after a variable time delay by a second probe pulse.Advances in synthesizing ultrashort intense pulses [9][10][11] allow one to visualize and control molecular reaction processes from femtosecond (1 fs = 10 −15 s) time scales for nuclear vibrations to attosecond (1 as = 10 −18 s) scales for electron motion.Currently, the shortest attosecond pulses with a duration of 43 as are available for such new photoelectron imaging techniques [12].Electrons have a great potential for probing the time-resolved transient structure of matter via ultrashort photoelectron spectroscopy.One now can envisage the emergence of laser-induced electron interference and diffraction [13][14][15][16][17][18][19][20] as efficient methods of measuring molecular electronic structure and electron motion for different fixed nuclear configurations.
Recently, ultrafast charge migration arising from a coherent population excitation in multiple electronic states has attracted considerable attention in the field of photophysics and photochemistry [21][22][23][24][25][26][27][28][29][30][31][32][33][34][35].It has been shown that quantum control of electron flux during intramolecular charge migration can be produced by designing ultrafast laser pulse that prepare the system in selective electronic state [36][37][38].The electron currents are sensitive to the polarization and helicity of the driving pulse, i.e., the symmetry of the excited electronic state [39].Recent work has shown how to control molecular electric symmetry via charge migration with ultrashort laser [40,41].By measuring high-order harmonic generation in ionized iodoacetylene via attosecond electron recollision, one can probe attosecond charge migration [42].Angular electronic fluxes can also be used to reconstruct electron charge migration in excited benzene by preparing a coherent electronic state [37,39], which can be monitored in photoelectron momentum spectra by a time-delayed high frequency attosecond pulses [43].It is also found that molecular photoelectron angular distributions depend on the symmetry of molecular orbitals and molecular bondings, thus allowing to monitor electron coherent and charge migration [44][45][46][47].Most recently, nonresonant ultrafast X-ray scattering from a molecular wave packet has been used to observe an adiabatic electron transfer in molecules [48].
Time-resolved photoelectron imaging as a variant of time-resolved photoelectron spectroscopy has been used to study molecular dynamics [49,50].In the present work, we theoretically studied time resolved photoelectron emission with attosecond resolutions in molecular coherent excitation and charge migration processes.A coherent superposition of quantum states was created by a linearly polarized XUV pump pulse.Subsequently, a time delayed soft X-ray pulse was used to ionize the excited molecules, as illustrated in Figure 1.Simulations were performed on the benchmark molecular ion H + 2 by numerically solving the corresponding time-dependent Schrödinger equation (TDSE).Results show that photoelectron momentum distributions exhibite asymmetric structure and evolve periodically with time, reflecting the electron coherence in molecular charge migration.It was found that the evolution of the photoelectron distributions is dependent on the polarization and helicity of the probe pulse.The paper is arranged as follows: In Section 2, we briefly describe the computation methods.The numerical results obtained by time-dependent quantum electron wave-packet calculations from the corresponding TDSE for a prealigned H + 2 are presented and discussed in Section 3. We describe the coherent excitation in molecules and analyze probing photoelectron angular and momentum distributions for imaging attosecond charge migration.Finally, we summarize our findings in Section 4 (throughout this paper, atomic units (a.u.) are used unless otherwise noted).

Numerical Methods
We briefly describe the numerical and computational methods for simulating the photoelectron emission process in aligned H + 2 by pump-probe pulses, as illustrated in Figure 1.The corresponding three-dimensional (3D) TDSE within static nuclear frames reads as, We describe the electron dynamics in cylindrical coordinates r = (ρ, θ, z) with x = ρ cos θ and y = ρ sin θ.Then, the kinetic (Laplacian) operator gives V en is the electron-nuclear potential.The molecule is aligned along the z-axis.The pump and probe laser pulses propagate along the z-axis, parallel to the molecular axis.The radiative interaction between the laser field and the electron V L (r) = r • E(t) is described in the length gauge.The total fields E(t) = E pu (t) + E pr (t) have the forms, for a linearly polarized pump pulse with its field vector along the x-axis, and for a circularly polarized probe pulse with its field vector in the (x, y) plane.The symbol ξ = ±1 denotes the helicity of the probe field, i.e., right (ξ = +1) or left (ξ = −1) handed circular polarization and ξ = 0 presents the linear polarization.A smooth sin 2 (πt/T pu/pr ) pulse envelope f (t) for maximum amplitude E 0 , intensity I 0 = I x = I y = cε 0 E 2 0 /2 and duration T pu/pr = 10τ pu/pr are used, where one optical cycle period τ pu/pr = 2π/ω pu/pr .This pulse satisfies the total zero area E pu/pr (t)dt = 0 in order to exclude static field effects [9].The 3D TDSE in Equation ( 1) for aligned H + 2 is numerically solved by a second-order split operator method that conserves unitarity in the time step δt combined with a fifth order finite difference method and Fourier transform technique in the spatial steps δρ, δz, and δθ [51,52].The initial electron wavefunction ψ(r, t = 0) is prepared in the ground 1sσ g state calculated by propagating an initial appropriate wavefunction in imaginary time using the zero-field TDSE in Equation (1).The time step is taken to be δt = 0.01 a.u.= 0.24 as.The spatial discretization is δρ = δz = 0.25 a.u.for a radial grid range 0 ≤ ρ ≤ 128 a.u.(6.77 nm) and |z| ≤ 32 a.u.(1.69 nm), and the angle grid size δθ = 0.025 radian.One notes that reflections of the electron wave packet from the edges of the numerical grid can cause artificial effects [53,54].To prevent these unphysical effects, we multiply ψ(ρ, θ, z, t) by a mask function or absorption potential in the radial coordinates ρ with the form cos 1/8 [π(ρ − ρ a )/2ρ abs ].The important ultrafast electron dynamics in intense fields occur on a spatial volume close to the molecule.These studies can therefore be performed on a relatively small grid neglecting the exact form of the outgoing ionizing parts of the wave function.For all results reported here, we set the absorber domain at ρ a = ρ max − ρ abs = 104 a.u. with ρ abs = 24 a.u., exceeding well the field induced electron oscillation α d = E 0 /ω 2 pu/pr of the electron.Molecular frame photoelectron distributions are calculated by a Fourier transform of the 3D time dependent electronic wavefunction ψ(ρ, θ, z, t), which exactly describe the electron dynamics in the continuum [55,56]: where t p is the time after the pulse turns off and ρ f = 100 a.u. is an asymptotic point before the wavepacket is absorbed.E = p 2 e /2 is the kinetic energy of an ionized electron with wave vector k = p e = 2π/λ e , p e = (p 2 x + p 2 y ) 1/2 is the momentum of a photoelectron of wavelength λ e .Since the ionization occurs in the laser polarization (x, y) molecular plane, we define θ as the angle between the electron momentum p e and the x polarization axis.With the transformation p x = p e cos θ and p y = p e sin θ, we then obtain the two-dimensional (2D) momentum distributions of photoelectrons from Equation (5).Molecular photoelectron angular distributions at photoelectron kinetic energy E e are obtained by integrating over the one photon energy, where the spectral width of the probe pulse ∆ω ≈ ω pr /3, corresponding to the main one photon ω pr frequency of absorption.

Results and Discussions
We use the molecule H + 2 as a benchmark system, which can be fully investigated [57] to describe the photoelectron imaging process of coherent electron dynamics.The molecule at equilibrium aligned along the z-axis is excited by a linearly polarized λ pu = 70 nm XUV pump laser pulse with its field vector along the x-axis.Subsequently, a λ pr = 5 nm soft X-ray attosecond probe pulse with its field vector polarized in the (x, y) plane ionizes the excited molecule.For the processes by high-frequency pump-probe pulses, multi-photon ionization dominates with the Keldysh parameter γ 1 [9].Since the ponderomotive energy U p = E 2 0 /4ω 2 pr is very weak, the modification of the ionization potential by laser induced Stark shifts can be ignored as well.Moreover, the dipole approximation, in which the spatial dependence and magnetic component of the external field are neglected, remains valid.
We first present the coherent excitation and charge migration by a pump pulse.With a linearly polarized pump pulse with its field vector polarized along x-axis, perpendicular to the molecular axis, a σ g − π u parallel resonant pump excitation occurs from the electronic ground 1sσ g state, ψ σ g (r) with the eigenenergy E σ g to the electronic excited 2pπ u state, ψ π u (r) with the eigenenergy E π u .In our numerical models, the energies for the 1sσ g and 2pπ u states are, respectively, E σ g = −1.08 a.u. and E π u = −0.43a.u.To induce molecular resonant excitation, the wavelength λ pu = 70 nm (ω pu = 0.65 a.u.) of the pump pulse is required, where ω pu = E π u − E σ g = 0.65 a.u.A coherent superposition of the two electronic states is then created due to a strong charge-resonance excitation [58], c σ g (t) and c π u (t) are the occupation coefficients.ψ π u (r) is the degenerate quantum electronic state.
For the case excited by the linearly polarized 70 nm XUV laser pulse with its vector along the x-axis, the wavefunction of the degenerate excited state is given by [59], 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 wave function of the degenerate excited state can be written as where The coherent electron dynamics is composed of two electronic state densities, P (g) (r) = |c σ g (t)| 2 |ψ σ g (r)| 2 and P (u) (r) = |c π u (t)| 2 |ψ π u (r)| 2 , and their interfering superposition P (g,u) (r, t) = 2|c σ g (t)||c π u (t)|ψ σ g (r)ψ π u (r) cos(∆Et) cos(θ).After the pump pulse, the occupation coefficients c σ g (t) and c π u (t) are constant and P (g) (r) and P (e) (r) are symmetric and are insensitive to the time, whereas the coherent superposition term P (g,u) (r) is determined by the time t and the angle θ.Note that, for the linear polarization resonant excitation, the distribution of the coherent superposition term is mainly along the laser polarization direction with a function cos(θ).
Figure 2 shows numerical results of electron density distributions P (x, y) of the superposition state created by the λ pu = 70 nm (ω pu = 0.65 a.u.) pump pulse as a function of the time t, where P (x, y) = dz|ψ 2 (x, y, z)| 2 in Equation ( 1).We use the pulse intensity I 0 = 1.0 × 10 14 W/cm 2 (E 0 = 5.34 × 10 −2 a.u.), and durations T pu = 10τ pu = 2.32 fs (1.16 fs full width at half maximum, FWHM).Of note is that the occupation coefficients, c σ g (t) and c π u (t), are dependent on the intensity of the pump pulse.Varying I 0 leads to a variance of the electron coherence in Equation (7).At such intensity I 0 = 1.0 × 10 14 W/cm 2 , one obtains the maximum interference effects.After the pump pulse, the occupation coefficients are constant.Figure 2 shows that the electron density distributions P (x, y) are asymmetric with respect to the molecular center and oscillate with the time t.The coherent electron wave packets move periodically in the left (x < 0) and right (x > 0) half planes along the laser x polarization axis.At time t = 9.75τ pu , the coherent electron wave packets are mainly localized in the left plane of the molecule.As the time t increases, the electron moves to the right.At time t = 10.25τpu , the density distributions of the coherent electron wave packets lie in the right plane.As the time t increases further, the electron wave packets come back to the left plane.
The evolution of the electron density distributions in Figure 2 describes the dependence of the attosecond electron coherence on the time t in charge migration, i.e., the superposition term P (g,u) (r) in Equation (10).The period of the oscillation is τ = τ pu = 232 as for the time t, in good agreement with the prediction in Equation (10).Since P (g,u) (r) ∼ cos(∆Et) cos(θ), the density distributions of the coherent electron wave packets are asymmetric along the x-axis and the extreme value occurs at θ = 0 or π (180 • ).Varying the time t leads to a modulation of the coherent electron density distribution following a form cos(∆Et).The time dependent electron density distributions reflects the electron coherence.
To visualize the coherent electron wave packets in Figure 2, we present the process of time resolved photoelectron imaging by pump-probe pulses.Because the oscillation period of the coherent electron is very short, τ = 232 as, a soft X-ray attosecond probe pulse is necessary.We use a λ pr = 5 nm (ω pr = 9.11 a.u.) and T pr = 10τ pr (83 as FWHM) probe pulse to ionize the excited molecule prepared by the linearly x polarized λ pu = 70 nm pump pulse in Figure 2. The use of the soft X-ray pulse has the following advantage in probing the electronic coherence.First the high frequency single photon process can avoid additional excitation effects resulting from other lower electronic states.Second, the extremely short duration of the X-ray pulse on the attosecond time scale can be utilized to resolve the coherent electron rotation with the period of a few hundred attoseconds.Figure 3 displays molecular photoelectron momentum J (p x , p y ) and angular J (θ) distributions produced by a linearly polarized λ pr = 5 nm attosecond probe pulse with its field vector along the x-axis, i.e., ξ = 0.The time delay ∆τ between the pump and probe pulses varies from ∆τ = 4.75τ pu to 6.0τ pu with a time interval of 0.25τ pu .The other laser parameters are always fixed at pulse intensities I 0 = 1.0 × 10 14 W/cm 2 (E 0 = 5.34 × 10 −2 a.u.) and durations T pu = 10τ pu = 2.32 fs (1.16 fs FWHM) and T pr = 10τ pr = 165.4as (83 as FWHM).The proper intensity of the probe pulse should be chosen.With lower intensities of the X-ray pulse, the probing photoelectron signal will be weak due to small transition dipole moments, whereas, with very strong intensities, the broad spectral width of the X-ray pulse will induce additional unexpected strong field phenomena.Since we focus on the probing photoelectron emission produced by the soft X-ray pulse, only the photoelectron distributions around the momentum p e = 4.01 a.u., where p e = ± 2(ω pr − I p ) corresponds to one ω pr photon absorption, are presented in Figure 3.The angular distributions calculated in Equation ( 6) are also displayed.The polarization of the probe pulse can influence the probing photoelectron imaging of the coherent electron dynamics.We next present the photoelectron imaging process by a circularly polarized probe laser pulse.Figure 4 shows results of photoelectron momentum J (p x , p y ) distributions by a left-handed ξ = −1 circularly polarized λ pr = 5 nm probe pulse.The corresponding photoelectron angular J (θ) distributions obtained from Equation ( 6) are also plotted (yellow curves).We use again a linearly polarized λ pu = 70 nm pump pulse with its field vector polarized along the x-axis to excited the molecule H + 2 aligned along the z-axis, the same as in Figures 2 and 3. We show the helicity effects of the probe pulse on the photoelectron angular and momentum distributions of the coherent excitation.
The pulse intensities I 0 = 1.0 × 10 14 W/cm 2 (E 0 = 5.34 × 10 −2 a.u.) and duration T pu = 10τ pu = 2.32 fs (1.16 fs FWHM) and T pr = 10τ pr = 165.4as (83 as FWHM) are always fixed while the time delay ∆τ between the pump-probe pulse varies from 4.75τ pu to 6.0τ pu .Figure 4 shows that asymmetric photoelectron angular and momentum distributions are produced again which evolve periodically with the time delay ∆τ.Comparing to the process by a linearly polarized probe pulse in Figure 3, the time resolved photoelectron distributions rotate periodically with an anti-clockwise direction around the molecular R or z-axis.As shown in Figure 4a at ∆τ = 4.75τ pu , the photoelectron distributions are mainly localized in the right half plane.As the time delay ∆τ increases, the distribution rotate to the upper half plane (p y > 0) and then to the left.At ∆τ = 5.5τ pu , the photoelectron distributions in the lower half plane (p y < 0) dominate.At ∆τ = 5.75τ pu the distributions move back to the right.The evolution period of the photoelectron distributions is τ pu = 2π/∆E = 232 as, as predicted in Equation (10) for the coherent electronic wave packets.
The rotation of the probing photoelectron distributions with the time delay ∆τ corresponds to the helicity of the probe pulses.For the circularly polarized probe pulse, its field vector is isotropic in the (x, y) polarization plane.Moreover, for the molecule H + 2 aligned along the z-axis, the influence of the molecular potential on the high frequency photoelectron emission is negligible.As a result, the asymmetric photoelectron distributions are determined by the electric field vector of the probe pulse, leading to a rotation around the molecular axis.That differs from the linear probe process in Figure 3 where the photoelectron distributions are mainly localized along the laser polarization direction since the field-molecule interaction term follows the form cos(θ), i.e., ∼ r • E.
We finally show the effect of the helicity of the circularly polarized probe pulse on the photoelectron emission.Figure 5 displays results of photoelectron momentum J (p x , p y ) and angular J (θ) distributions with right handed helicity ξ = 1.The other laser parameters are the same as those used in Figure 4. Figure 5 shows that similar phenomena are produced, except with a clock-wise rotation.Asymmetric photoelectron angular and momentum distributions resulting from the coherent excitation by the λ pu = 70 nm pump pulse are produced as well.Varying the time delay ∆τ gives rise to a periodical rotation of photoelectron distributions with period 232 as, as predicted in Equation (10).It is found that the rotation with a clock-wise direction follows the right handed helicity of the circularly polarized probe pulse.The combination of Figures 4 and 5 shows that the evolution of the probing photoelectron angular and momentum distributions with time arises from the coherent excitation between the ground and excited electronic state by the pump pulse while the rotation corresponds to the helicity of the circularly polarized attosecond probe pulse.Comparison of results in Figures 3-5 shows that the time-resolved photoelectron angular and momentum distributions depend on the polarization and helicity of the probe pulse.By a linearly polarized pump pulse, the coherent electronic state is degenerate with m = ±1.The subsequent ionization from the coherent superposition state are dependent on the properties of the probe pulse.The evolution of the probing photoelectron angular and momentum distributions with the time delay depends on the polarization and helicity of the soft X-ray probe pulse, as illustrated in Figures 3-5.This differs from the coherent excitation process by a circularly polarized pump pulse, in which the excited 2pπ u state is non-degenerate, with m = 1 or −1.As we reported previously [43], the rotation of time-delay dependent photoelectron distributions is determined by the pump pulse.Altering the helicity of the probe pulse cannot vary the rotation of the time-resolved photoelectron distribution.The dependence of the time-resolved photoelectron distributions on the polarization and helicity of the soft X-ray probe pulse indicates the permanent property of the coherent superposition state in the coherent excitation and charge migration.Of interest is that, in the present work, we consider the process of molecular resonant excitations with σ g − π u symmetries in the molecular ion H + 2 .By a linearly polarized XUV pulse with its field polarized along the x-axis, the σ g − π u transition occurs only in the case of the molecular axis perpendicular to the laser polarization direction.Therefore, only the excitations that are mainly localized along the laser polarization direction dominate the spectrum and evolve periodically with time, as shown in Figure 2. Therefore, the time-dependent signatures of the coherent electronic dynamics can also be observed even when averaging over all molecular alignments.

Conclusions
In summary, we present time resolved photoelectron emission in electron coherent excitation and charge migration processes.Simulations were performed on the aligned molecular ion H + 2 by numerically solving the molecular TDSE.The molecule was excited by a linearly polarized XUV pulse, creating a coherent superposition of the ground and excited electronic states.Subsequently, a linearly/circularly polarized soft X-ray attosecond probe pulse was used to ionize the excited molecule.Probing molecular photoelectron momentum distribution displayed an asymmetric structure, which was shown to be dependent on the time delay between the pump and probe pulses, and the polarization and helicity of the probe pulse.
The asymmetry of the time resolved photoelectron angular and momentum distributions arose from the coherent excitation of the molecule induced by the pump pulse.The dependence of the probing photoelectron emission yield on the time-delay illustrated the evolution of the coherent electron wave packets with time.Our numerical simulations combined with theoretical analysis therefore showed that the time resolved X-ray photoelectron imaging allows monitoring the electron coherence in molecular reaction processes.We showed how the polarization and helicity of the soft X-ray probe pulse influence the probing photoelectron emission, illustrating the property of the coherent superposition state in molecular coherent excitation and charge migration.Relating to new experiments in attosecond time-delayed photoionization of molecules [60], our results pave the way to visualize and manipulate electron motion inside molecules by time-dependent photoelectron spectroscopy imaging, which can be extended to more complex systems and laser control of chemical reactions [61,62].

Figure 1 .
Figure 1.Illustration of time-resolved photoelectron imaging of coherent electron excitation in molecules by an XUV pump pulse and a time delayed soft X-ray probe pulse.The molecule H + 2 is aligned along the z-axis and the pump-probe pulses with their field vectors in the (x, y) plane propagate along the z-axis.The time delay ∆τ is the time interval between the peak times of the pump and probe pulses.

Figure 2 .
Figure 2. Evolution of the electron density probabilities P (x, y) with the time t in the molecular ion H + 2 aligned along the z-axis excited by a linearly polarized λ = 70 nm pump pulse with its field vector along the x-axis at intensity I 0 = 1.0 × 10 14 W/cm 2 (E 0 = 5.34 × 10 −2 a.u.) and duration T = 10τ pu = 2.32 fs.

Figure 3
Figure3shows that asymmetric photoelectron angular and momentum distributions are produced, which are shown to be dependent on the time delay ∆τ.Altering the time delay ∆τ results in a change of the asymmetry.At ∆τ = 4.75τ pu , the distributions in the right half angular and momentum plane dominate.Increasing ∆τ leads to a decrease of the right momentum ring and an increase of the left in amplitude.At ∆τ = 5.0τ pu the amplitudes of the momentum distributions in the left and half planes are comparable, and at ∆τ = 5.25τ pu the left one reaches the maximum value.Increasing the time delay ∆τ further, a reverse process occurs where the left distribution decreases while the right one increases.At ∆τ = 5.5τ pu , the comparable distributions in the left and right planes are obtained, and at ∆τ = 5.75τ pu , the right one dominates again.The periodical evolution of the photoelectron momentum distributions with the time delay ∆τ illustrates the electronic coherent in the resonant excitation by the pump pulse.The evolution period is τ = τ pu = 2π/∆E, the same as the coherent electron wave packets in Equation (10) and Figure2.The electronic coherence gives rise to asymmetric electron density distributions of the coherent superposition state ψ 0 (r, t).
and |ψ x π u (ρ, z)| 2 and |ψ y π u (ρ, z)| 2 are the degenerate real orbitals.The electron density distribution of the coherent superposition electronic state in Equation (7) is described by,