Ultrafast X-ray photoelectron diffraction in triatomic molecules by circularly polarized attosecond light pulses

Abstract

We theoretically study ultrafast photoelectron diffraction in triatomic molecules with cyclic geometry by ultrafast circular soft X-ray attosecond pulses. Molecular frame photoelectron distributions show complex diffraction patterns, arising from molecular multiple center interference and circular charge migration. It is found that photoelectron diffraction patterns depend on the initial electronic state, encoding the information of molecular orbital symmetries. In a resonant coherent electron excitation process, time-resolved photoelectron diffraction patterns enables us to reconstruct the charge migration with highly spatiotemporal resolutions. We simulate and analyze the results from ab initio calculations of the single electron triangular hydrogen molecular ion H₃²⁺ which is used as a benchmark molecular system in combination with an ultrafast multi-center and multi-state photoionization model. This approach presents a general scheme which can be applied to explore circular charge migration from electron orbits and attosecond coherent electron dynamics in polyatomic systems by circular ultrafast laser pulses.

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1 Introduction

Imaging and manipulating nuclear and electronic dynamics on their natural time and spatial scales by ultrafast laser pulses is a fundamental issue in photophysics and photochemsitry.^1–6 The developments of ultrashort laser technology^7,8 have led to photoimaging techniques such as Coulomb explosion^9,10 for nuclear motion on femtosecond (1 fs = 10⁻¹⁵ s) time scale to laser-induced electron diffraction (LIED)^11–14 for coupled electron-nuclear motion on the attosecond (1 as = 10⁻¹⁸ s) time scale. Now the shortest 43 as pulses are available for new ultrafast optical imaging.¹⁵ Illuminating molecules from within with ultrafast lasers will allow one to rely on electron interference and diffraction where structural information is encoded in interference patterns that result from the intrinsic wave nature of the electron and its laser-assisted scattering.¹⁶ Photoelectron diffraction therefore has been shown to be a promising candidate for studies of molecular structural dynamics.^17–23 Such phenomena in diatomic molecules were predicted earlier by Cohen and Fano²⁴ in linear polarization perturbative single photon ionization. Based on multiple-scattering X-ray photoelectron diffraction, one can measure molecular geometry structure.²¹ Effects of nuclear motion in molecular photoionization have also been investigated.²⁵ Most recently, ultrafast X-ray photoelectron diffraction from diatomic molecules undergoing photodissociation has also been studied for tracking dissociation and elimination processes in chemical reactions.²⁶ It has been shown that the LIED technique can be used to monitor geometric and orbital structures of a triatomic molecule with a resolution of a fraction of an Ångström.^27,28 However, most of the work has been focused on linear molecular systems and single electronic state processes.

In a resonant excitation process, the pulse induces nonstationarity of electronic states in the molecule.²⁹ The corresponding coherent population in multiple electronic states leads to charge migration in molecules on the attosecond time scale.^30–44 The process can be used for selective laser excitation in molecules, leading to the concept of charge-directed chemistry.^45,46 Therefore, steering charge migration by ultrafast laser pulses offers a new approach to controlling chemical reactions. The charge migration is related to a strong coupling between electronic states in the molecule which can enhance ionization⁴⁷ and induce transient localization of electron wave packets,^48,49 thus allowing to control photoelectron emission spectra⁵⁰ and high-order harmonic spectra.⁵¹ It has also been shown that quantum control of electron flux during intramolecular charge migration is possible by designing ultrafast laser pulses that prepare the system in selective electronic states,^42,52 giving rise to ultrafast molecular magnetism^53–55 for new quantum devices.⁵⁶ Experiments have for the first time measured attosecond charge migration with linearly polarized high harmonic spectroscopy in ionized iodoacetylene via attosecond electron recollision processes.⁵⁷ The developments of attosecond laser pulses enable us therefore to obtain insights into electron dynamics by time-resolved photoelectron spectroscopy on the attosecond time scale and sub-nanometer dimension.^58–61 We have therefore proposed that one can monitor coherent excitation and attosecond charge migration in time-dependent photoelectron diffraction in molecular photoelectron distributions which encode information on the symmetry of electron orbitals and molecular bonding^62–64 by a model which describes the electron coherent excitation between two electronic states in the diatomic H₂⁺. In polyatomic molecules, effects of multiple center interference and electronic degeneracy make time-resolved photoelectron diffraction more complex. One thus requires modelling exact photoelectron diffraction patterns to explore coherent electron excitation in complex molecular systems.

In this paper, we demonstrate attosecond photoelectron diffraction patterns in triatomic molecular systems with cyclic geometry by a circularly polarized attosecond soft X-ray pulse. We first present photoelectron diffraction patterns from different initial electronic states, which are shown to be dependent on the molecular geometry and orbital symmetry. The ionization process from degenerate electronic states is also presented where electronic state interference effects have been shown to be important in molecular photoionization.^65,66 For a coherent superposition state in a resonant excitation process created by a pump pulse, we use a time-delayed circularly polarized soft X-ray probe pulse to produce photoelectron distributions. Results show that photoelectron diffraction patterns are dependent on the time delay between the pump–probe pulses, inferring the coherent intramolecular electron dynamics. A cyclic triatomic single electron molecular ion H₃²⁺ is adopted as a benchmark molecular system model. The trihydrogen cations, as well as their isotopes, due to the unique equilateral triangular geometry, provide a fundamental original system for the theoretical development of the nonperturbative response of nonlinear polyatomic molecules to intense laser pulses.^67–72 Simulation results are obtained by solving the time-dependent Schrödinger equation (TDSE). The molecular internuclear distance is always fixed during excitation and ionization processes. We analyze the photoelectron distributions by multi-center and multi-state diffraction models derived from ultrafast photoionization models.¹¹ Moreover, the molecular rotational and vibrational motions on picosecond (1 ps = 10⁻¹² s) and femtosecond time scales are ignored because these effects are much slower than the laser-molecule interaction that occurs within the attosecond time scale. We use a circularly polarized probe laser pulse with equal field amplitude in each orthogonal direction, which does not influence the symmetry of photoionization. Such ultrafast attosecond circularly polarized X-ray pulses have been generated by current laser techniques from circularly polarized high harmonics.^73–75

The paper is arranged as follows: in Section 2, we briefly describe the numerical and computational methods. Section 3 presents numerical results for photoelectron diffraction obtained by time-dependent quantum electron wave-packet calculations from the corresponding TDSE for oriented H₃²⁺. A theoretical analysis based on ultrafast multi-center and multi-state photoionization models is also derived. Finally, we summarize our findings in Section 4 (throughout this paper, atomic units, a.u., are used unless otherwise noted).

2 Computational and numerical methods

The excitation and ionization dynamics of the molecular ion H₃²⁺ can be exactly described by numerically solving the three-dimensional (3D) TDSE, thus allowing to describe ultrafast electron dynamics in molecular reaction processes.⁷⁶ For the molecular ion H₃²⁺ oriented in the (x,y) plane, Fig. 1, the corresponding TDSE with a static nuclei frame described in cylindrical coordinates (ρ, θ, z) with (x = ρ cos θ, y = ρ sin θ) as the molecular plane is written as,The kinetic energy operator isThe molecular electron Coulomb potential is given by,The nuclear coordinates N_ix and N_iy are fixed by the relations N_1x = 0, N_2x = −R sin(Θ/2), N_3x = +R sin(Θ/2), N_1y = (2/3)R cos(Θ/2), N_2y = −(1/3)R cos(Θ/2), N_3y = −(1/3)R cos(Θ/2), R being the distance between two protons and the bond angle Θ = 60°, see Fig. 1(a). A soft parameter β = 0.35 is used to remove the Coulomb singularity and produce accurately the electronic state potential energies of H₃²⁺.⁷⁷ The field-molecule interaction V_L(r,t) = r·E(t) is treated in the length gauge and dipole approximation, since by exact solutions of the TDSE, numerical results are gauge invariant.⁷⁸ The laser fields E(t) = E_pu(t) + E_pr(t) always propagate along the z axis, perpendicular to R and the molecular (x,y) plane. The forms, for the circularly polarized pump pulse E_pu(t) are defined asand for the circularly polarized probe pulse E_pr(t − Δτ),ê_x/y is the laser field polarization vector. ω_pu and ω_pr are frequencies of the pump and probe pulses. A temporal slowly varying envelope f(t) = sin²(πt/T_pu/pr) is used with duration T_pu/pr = nτ_pu/pr, where τ_pu/pr = 2π/ω_pu/pr, one optical cycle of pump/probe pulses. Multiple optical cycle duration T_pu/pr ensures the field area to eliminate effects of static fields and carrier envelope phases of pulses. The time delay Δτ = t_pr0 − t_pu0 defines the time interval between the peak times t_pu0 and t_pr0 of the pump E_pu(t) and probe E_pr(t) pulses.

Fig. 1 (a) Molecular geometry of the cyclic triangular molecular ion H₃²⁺ oriented in the (x,y) plane. s_i denotes the s orbital of the atomic H on proton i, where i = 1, 2, and 3. R_ij = R_j − R_i presents the molecular internuclear distance and Θ = 60° is the bond angle between two molecular bonds. (b) Illustration of the molecular photoionization process in H₃²⁺ by an attosecond soft X-ray probe pulse initiated from the three electronic states, ψ_ie, ie = A′, E⁺, or E⁻, in eqn (10)–(12), where the E⁺ and E⁻ states are degenerate, and ψ_co, a superposition state in eqn (16) created by a circularly polarized resonant pump pulse. (c) The density distributions of the three electronic states, and arbitrary units are used.

We numerically solve the 3D TDSE of H₃²⁺ in eqn (1) by using a five-point finite difference method and fast Fourier transform technique combined with high-order split-operator methods.^79,80 The time step is fixed at Δt = 0.01 a.u. = 0.24 as and the spatial discretization is Δρ = Δz = 0.25 a.u. (1 a.u. = 1 a₀, Bohr radius) for radial grid sizes 0 ≤ ρ ≤ 128 a.u. and |z| ≤ 64 a.u., and angle grid size Δθ = 0.01 radian. The grid sizes exceed largely the maximum field induced electron displacements α_d = E₀/ω_pu/pr². To prevent unphysical effects due to the reflection of the ionized wave packet from the boundary, a mask function with the form cos^1/8[π(ρ − ρ_a)/2ρ_abs] is used at ρ_abs = ρ_max − ρ_a = 24 a.u. with ρ_max = 128 a.u.

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:⁸¹

where t_p is the time after the pulse turn off and ρ_f = 100 a.u. is an asymptotic point before the wavepacket is absorbed. E is the kinetic energy of an ionized electron with wave vector k = p = 2π/λ_e, p = (p_x² + p_y²)^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 and the x polarization axis. With the transformation p_x = p cos θ and p_y = p sin θ, we then obtain the two-dimensional (2D) momentum distributions of photoelectrons from eqn (6). Photoelectron angular distributions at photoelectron kinetic energy E_e are obtained by integrating over the one photon energy, where the spectra width of the probe pulse Δω ≈ ω_pr/3,where E_e = p²/2 = ω_pr − I_p, corresponding to the main one photon ω_pr frequency of absorption with the orbital ionization potential I_p. In this work, the wavelength of photoelectron λ_e < R leads to photoelectron diffraction effects¹¹ in photoelectron distributions obtained from eqn (7).

3 Results and discussions

The purpose of the present work is to study ultrafast photoelectron diffraction patterns in molecular photoionization by circularly polarized attosecond X-ray pulses. The result provides a way to monitor molecular structure and orbital symmetry and coherent electron dynamics. We adopt the triangular molecular ion as a benchmark system to illustrate the photoelectron imaging process. The single electron molecular ion H₃²⁺ system can be considered as the fundamental element of triatomic molecular systems. The molecule is excited and ionized by high-frequency pump and probe pulses. Such high-frequency pulse can not lead to tunnelling ionization and electron recollision. The ponderomotive energy U_p = E₀²/4ω_pu/pr² is weak and the Keldysh parameter .⁷ Therefore, the modification of the ionization potential by laser-induced Stark shifts can be ignored. The dipole approximation, in which the spatial dependence and magnetic component of the external field are neglected, remains valid. Moreover, the non-adiabatic coupling effects due to electron-nuclear combined dynamics in triatomic molecules⁶⁸ are also ignored in the ultrafast photoionization processes.

3.1 Electronic state symmetry dependent photoelectron diffraction

Photoelectron diffraction has been shown to be determined by the initial molecular symmetry, thus providing a tool for imaging of molecular orbitals.^11,27,28 We first compare the photoelectron diffraction patterns with different molecular orbital symmetries in the equilateral triangular molecular ion H₃²⁺, as illustrated in Fig. 1(b). The three lowest electronic states of H₃²⁺, A′, E⁺, and E⁻, are used respectively as the initial state. The excited electronic states E⁺ and E⁻, are degenerate with E-symmetry. The complex wavefunctions can be expressed in terms of two real ones,andcorresponding to angular momentum m = ±1 around the z axis.⁵³ The two orthogonal real components ψ⁽¹⁾_E and ψ⁽²⁾_E are obtained numerically from imaginary time propagation of the field-free Hamiltonian eqn (1). In Fig. 1(c) we also display the corresponding electron density distributions |ψ_ie(x,y)|², ie = A′, E⁺, or E⁻, of the three electronic states. The corresponding ionization potentials are respectively I_p(A′) = 1.08 a.u. (29.38 eV) and I_p(E⁺) = I_p(E⁻) = 0.92 a.u. (25.02 eV) for the molecular internuclear distance R = 4 a.u. and the bond angle Θ = 60°.

Fig. 2 shows photoelectron momentum distributions of the three electronic states, (a) A′, (b) E⁺, and (c) E⁻, of H₃²⁺ by the single circularly polarized attosecond soft X-ray probe pulse E_pr(t), eqn (5). The molecular ion H₃²⁺ with internuclear distance R = 4 a.u. is oriented in the (x,y) plane, Fig. 1. The circularly polarized attosecond probe pulse with its field vector in the (x,y) plane propagates along the z axis. The pulse wavelength λ_pr = 5 nm (ω_pr = 9.11 a.u.), duration T_pr = 10τ_pr = 165.4 as, corresponding to 83 as full width at half maximum (FWHM), and intensities I₀ = 2.75 × 10¹³ W cm⁻² (E₀ = 2.84 × 10⁻² a.u.) are used. Inserts (white lines) are the corresponding photoelectron angular distributions calculated from eqn (7). From Fig. 2 we see that photoelctron momentum distributions exhibit ring structure at momentum a.u. With such high frequency X-ray pulse, the wavelength of the photoelectron is λ_e = 2π/p = 1.57 a.u., less than the molecular internuclear distance R. As a result, diffraction of the photoelectron occurs¹¹ where multiple angular nodes are produced in photoelectron distributions. Of note is that the geometries of the photoelectron distributions are different for various electronic states, i.e., the photoelectron diffraction patterns depend on the orbital symmetry of the initial electronic state. One further finds that for the two degenerate excited electronic states, E⁺, and E⁻, photoelectron momentum and angular distributions are mirror image with respect to the molecule center, as shown in Fig. 2(b) and (c), since the angular momentum of each is m = +1 and −1, i.e., reflection images of each current.

Fig. 2 Photoelectron momentum (p_x,p_y) distributions of different H₃²⁺ initial electronic states, (a) A′, (b) E⁺, and (c) E⁻, (Fig. 1) by the circularly polarized attosecond soft X-ray probe pulse E_pr(t). The corresponding electron momentum is about p = 4.0 a.u. The molecular ion H₃²⁺ with internuclear distance R = 4 a.u. is oriented in the (x,y) plane. The probe pulses are polarized in the (x,y) plane, propagating along the z axis. The probe pulse wavelength λ_pr = 5 nm (ω_pr = 9.11 a.u.), duration T_pr = 10τ_pr = 165.4 as (83 as FWHM), and intensities I₀ = 2.75 × 10¹³ W cm⁻² (E₀ = 2.84 × 10⁻² a.u.) are used. Inserts (white lines) are the corresponding photoelectron angular distributions. Arbitrary units are used.

We describe the ultrafast photoelectron distributions of H₃²⁺ by deriving the delta function photoionization model which has been used successfully in ultrafast H₂⁺ ionization processes.¹¹ For the molecule H₃²⁺, each H atom has a 1s orbital. According to molecular orbital theory, the electronic states are linear combinations of the three 1s orbitals. The molecular wavefunctions of the bound atomic configurations are given by⁸²

for the A′ state,for the E⁺ state, andfor the E⁻ state, where ε = e^i2π/3. ψ_s1, ψ_s2, and ψ_s3 denote the wavefunctions of the 1s atomic orbital on protons 1, 2, and 3. The E⁺ and E⁻ states are doubly-degenerate with the same eigenenergies and are reflection of each other ψ_E⁺ = ψ_E⁻*. According to the ultrafast photoionization model^11,83 derived in Appendix B, photoelectron distributions are dependent on the orbital symmetry of the initial state. In this work, the pulse field strength E₀ can be ignored to describe the effect on the photoelectron dynamics in high-frequency photoionization since the chosen intensities (I₀ = 2.75 × 10¹³ W cm⁻² and E₀ = 2.84 × 10⁻² a.u.) in the simulations produce negligible ponderomotive energies U_p ≪ I_p and E₀ ≪ p. The transition amplitude for the ground A′ electronic state in eqn (31) can be simply expressed aswhere ϑ_p defines the angle between the photoelectron momentum p and the x axis. Similarly, for the two degenerate excited E⁺ and E⁻ electronic state, the corresponding photoelectron distributions are,and

From eqn (13)–(15) one notes that in the nonlinear ultrafast three center photoionization model, the interference is simply the sum of the contributions from all centers. The ionized electron wave packets from different centers will interfere with each other. For the triatomic molecular system, the initial orbitals are separated by R. The terms of sine and cosine triangle functions present the interference effects of the three outgoing electron wave packets emanating from the three nuclear centers.

In Fig. 3 (left column) we show results of molecular photoelectron momentum (p_x,p_y) distributions obtained from eqn (13)–(15) for the three lowest electronic states with the photoelectron momentum ranging from 0 to 7.5 a.u. The molecular internuclear distance is set as R = 4 a.u. It is found that for the three electronic states, A′, E⁺, and E⁻, a set of alternating bright and dark interference fingers with points are obtained in Fig. 3. We also plot the molecular photoelectron angular distributions at a certain momentum p = 4 a.u., corresponding to the ionization by a λ = 5 nm X-ray pulse, right column in Fig. 3. From Fig. 3 we see that the predictions are in good agreement with the simulation in Fig. 2. The mirror images of photoelectron distributions for the degenerate excited E⁺ and E⁻ electronic states satisfy [scr F, script letter F]^(E+)(p,ϑ_p) = [scr F, script letter F]^(E−)(p,ϑ_p ± π). Multiple angular nodes with different amplitudes are produced, arising from the multiple center interference of photoelectron. The different diffraction patterns illustrate the various symmetries of the initial electronic state, thus allowing for imaging of molecular orbitals.

Fig. 3 (left column) Photoelectron momentum (p_x,p_y) distributions of the H₃²⁺ three electron states, (a) A′, (b) E⁺, and (c) E⁻, predicted in eqn (13)–(15). The photoelectron momentum ranges from 0 to 7.5 a.u. The molecular ion with intercnulear distance R = 4 a.u. is oriented in the (x,y) plane. (right column) Photoelectron angular distribution at a certain momentum p = 4 a.u., corresponding the photoelectron wavelength λ_e = 1.57 a.u. obtained by a 5 nm photon absorption. Arbitrary units are used.

3.2 Time-resolved photoelectron diffraction in molecular coherent resonant excitation

We next consider the photoelectron diffraction in a coherent resonant excitation process. In the molecule H₃²⁺, a resonant excitation from the ground A′ electronic state, ψ_A′ with the eigenenergy E_A′, to the two degenerate electronic excited E⁺ and E⁻ states, ψ_E⁺ and ψ_E⁻, with the eigenenergies E_E⁺ = E_E⁻ can be triggered by a frequency ω_pu = E_E⁺/E⁻ − E_A′ pump laser pulse. A coherent superposition of the three electronic states created by the ω_pu pump pulse, ψ_co(r,t) in eqn (36) can be obtained due to a strong charge-resonance excitation,²⁹ as shown in Fig. 1(b). The corresponding ionization potentials of the three electronic states are respectively, I_p(A′) = 1.69 a.u., and I_p(E⁺) = I_p(E⁻) = 1.08 a.u. at R = 2 a.u. We choose a smaller internuclear distance R, in order to produce larger energy difference between the ground electronic and the two degenerate excited states. As a result, a shorter wavelength resonant pump pulse can be used, thus giving rise to better energy and angular momentum resolutions.

Fig. 4 displays numerical results of electron density distributions [scr D, script letter D](x,y) of the superposition state created by the λ_pu = 75 nm (ω_pu = 0.608 a.u.) pump pulse in eqn (4) as a function of time t, where in eqn (1). We use a left handed circularly polarized XUV pulse to excite the molecular ion H₃²⁺. The pulse intensity and duration are chosen as I₀ = 2.75 × 10¹³ W cm⁻² (E₀ = 2.84 × 10⁻² a.u.), T_pu = 10τ_pu = 2.48 fs (1.24 fs FWHM). Fig. 5 displays the evolution of the probabilities of the two ground A′ and excited E⁺ electronic states, |c_i(t)|², where c_ie(t) = 〈ψ_ie(r)|ψ(r,t)〉 and ie denotes the A′, E⁺, or E⁻ electronic state, with time t. By this pump pulse, a resonant excitation between the ground A′ and excited E⁺ states occurs. After the pump pulse t > 10τ_pu, the occupation coefficients are nearly constant. As displayed in Fig. 5, the occupation in the excited E⁺ state is |c_E⁺(t)|² = 0.5. The population remaining in the ground A′ state is |c_A′(t)|² = 0.42. Of note is that during the excitation processes, the molecule is also excited E⁺ to higher excited state and ionized to the continuum after absorptions of one and two ω_pu from the excited state. As a result, the total population in the three electronic state is less than 1, i.e., |c_A′(t)|² + |c_E⁺(t)|² < 1.

Fig. 4 Evolution of the coherent electronic wave packets with time in the molecular ion H₃²⁺ oriented in the (x,y) plane excited by a λ_pu = 75 nm (ω_pu = 0.608 a.u.) circularly polarized laser pulse with its field vector polarized in the (x,y) plane, eqn (4). The molecular internuclear distance is R = 2 a.u. The pulse intensity I₀ = 2.75 × 10¹³ W cm⁻² (E₀ = 2.84 × 10⁻² a.u.) and duration T_pu = 10τ_pu = 2.48 fs (1.24 fs FWHM). Arbitrary units are used.

Fig. 5 Time-dependent occupations in the ground A′ and the excited E⁺ electronic states in H₃²⁺ excited by a λ_pu = 75 nm (ω_pu = 0.608 a.u.) circularly polarized pump laser pulse with its field vector polarized in the (x,y) plane. The molecule is oriented in the (x,y) plane and its internuclear distance is R = 2 a.u. The pulse intensity I₀ = 2.75 × 10¹³ W cm⁻² (E₀ = 2.84 × 10⁻² a.u.) and duration T_pu = 10τ_pu = 2.48 fs (1.24 fs FWHM). Insert (top): the pump–probe scheme for time-resolved photoelectron imaging. Δτ denotes the time delay between the pump and probe pulses for monitoring coherent excitation processes. The used time delay is about 1.2–1.5 fs.

From Fig. 4 one sees that the electron density distribution [scr D, script letter D](x,y,t) shows an asymmetric structure in molecules which is dependent on the time t. The coherent electron wave packet moves periodically among the three nuclear centers. At time t = 10.2τ_pu, a strong electron localization is obtained and the coherent electronic wave packet mainly lies on proton 3. Increasing t leads to a variation of the electron distribution. The electron moves to proton 2 through proton 1. At time t = 10.8τ_pu, the electron is mainly localized on proton 2. As the time t increases further, the electron moves back to proton 1, and then proton 3. The period of the oscillation is approximately τ⁽⁰⁾ = τ_pu = 248 as, one cycle of the circular pump pulse at 75 nm.

The evolution of the superposition state with time t in Fig. 4 arises from the coherent electron wave packet interference between the two A′ and E⁺ electronic states. As shown in Appendix C, the distributions of the coherent electronic states are time-dependent after the pulse excitation. The coherent superposition between the ground A′ state and the excited E⁺ state, is given by

As shown in eqn (37), the time-dependent coherent electron density distributions come from the A′–E⁺ state interference, i.e.,

[scr D, script letter D]^int_co(r,t) ∼ |c_A′(t)c_E⁺(t)‖ψ_A′(r)ψ_E⁺(r)|cos(ΔEt), (17)

where ΔE is the energy difference between the excited E⁺ states and the ground A′ electronic states, ΔE = E_E⁺ + E_A′. Eqn (17) shows that the electron density distributions oscillate periodically with τ⁽⁰⁾ = 2π/ΔE = 10.37 a.u. = 248 as. From eqn (17) one also sees that the electron density probability [scr D, script letter D](r,t) will evolve among protons 1 ← 2 ← 3, i.e., rotating around the molecular center with an anti-clockwise direction, corresponding to a left handed current, the helicity of the pump pulse. Comparing with Fig. 4 and eqn (17) shows that the numerical simulations are in good agreement with the theoretical predictions. At ΔEt = nπ, where n is an integer, the superposition term cos(ΔEt) = ±1 gives rise the maximum asymmetry of electron density distributions, whereas at ΔEt = (n + 1/2)π, i.e., cos(ΔEt) = 0 and t = (n + 1/2)τ_pu, symmetric electron distribution are produced, as illustrated in Fig. 4. The evolution of the electron density distribution therefore corresponds to the dependence of the attosecond electron coherence and resulting currents on the time t in charge migration.

The corresponding time resolved photoelectron distribution can be derived based on a δ-function model¹¹ for the ultrafast multi-center photoionization process in Appendix C which is given by, [eqn (39)],

[scr F, script letter F]^int_co(p,t) ∼ [scr F, script letter F]^int₁₂ + [scr F, script letter F]^int₂₃ + [scr F, script letter F]^int₁₃, (18)

where

[scr F, script letter F]^int₁₂ = {cos[pR cos(ϑ_p − Θ) + π/3] + 1/2} × cos(ΔEt − π/3)ψ_s²(p), (19)

[scr F, script letter F]^int₂₃ = {cos[pR cos(ϑ_p) − π/3] + 1/2}cos(ΔEt)ψ_s²(p), (20)

and

[scr F, script letter F]^int₁₃ = {cos[pR cos(ϑ_p + Θ) − π/3] + 1/2} × cos(ΔEt + π/3)ψ_s²(p), (21)

where 1, 2, and 3 are the three protons in Fig. 1(a). In eqn (18) the superposition term [scr F, script letter F]_co(p,θ,t) ∼ cos(ΔEt) leads to the evolution of the photoelectron distributions with the time t. The photoelectron distributions encode the information on the time dependent electron motion in molecules, which is much faster than nuclear motion. Fig. 6 illustrates photoelectron angular distributions obtained from the photoionization model, theoretically predicted in eqn (18), at two different moments (a) t = 0 and (b) π/ΔE, i.e., ΔEt = 0 and π. The proton internuclear distance is R = 2 a.u. and the photoelectron momentum p = 4.0 a.u., corresponding to a 5 nm photon soft X-ray photon absorption. The time resolved photoelectron angular distributions show asymmetric structure and mirror image, indicating the coherent coupling between the ground A′ and the excited E⁺ electronic states.

Fig. 6 Time-resolved photoelectron diffraction patterns (angular distributions) obtained from ultrafast photoionization models for a coherent electronic state predicted in eqn (18) at moments (a) t = 0 and (b) π/ΔE = 124 as (ΔE = 0.608 a.u.). The photoelectron momentum p = 4.0 a.u. and the molecular internuclear distance R = 2 a.u., cf., Fig. 7.

Fig. 7 displays the time-resolved molecular photoelectron distributions from 3D TDSE simulations produced by a left handed circularly polarized λ_pr = 5 nm (τ_pr = 16.5 as) attosecond probe pulse. The molecule H₃²⁺ is first excited by a circularly polarized λ_pu = 75 nm pump pulse. The time delays between the pump and probe pulses vary from Δτ = 5.0τ_pu to 6.0τ_pu with a time interval 0.25τ_pu (τ_pu = 248 as), i.e., the ionization time of the excited molecule by the probe pulse ranges from t = 10τ_pu to 11τ_pu. The other laser parameters are always fixed at pulse intensities I₀ = 2.75 × 10¹³ W cm⁻² (E₀ = 2.84 × 10⁻² a.u.) and durations T_pu = 10τ_pu = 2.48 fs (1.24 fs FWHM) and T_pr = 10τ_pr = 165.4 as (83 as FWHM). With such 5 nm X-ray pulse, the extremely short duration on attosecond time scale can be utilized to resolve the coherent electron motion with the period since T_pr(FWHM) ≈ τ⁽⁰⁾/3. It should be noted that the pump pulse can also ionize the molecular ion H₃²⁺. The corresponding photoelectron energy is nω_pu − I_p, much less than that of the X-ray ω_pr photon absorption due to ω_pr/ω_pu = 15. Therefore we only focus on probing photoelectron distributions produced by the soft X-ray puls, i.e., the photoionization yields around the momentum p = 4.0 a.u., where corresponding to one ω_pr photon absorption to the continuum.

Fig. 7 Photoelectron momentum (p_x,p_y) distributions at different time delays Δτ between the pump E_pu(t) and probe E_pr(t − Δτ) pulses. The molecular ion H₃²⁺ with internuclear distance R = 2 a.u. is oriented in the (x,y) plane. The pump and probe pulses are polarized in the (x,y) plane, propagating along the z axis. The pulse wavelengths λ_pu = 75 nm (ω_pu = 0.608 a.u.) and λ_pr = 5 nm (ω_pr = 9.11 a.u.), duration T_pu = 10τ_pu = 2.48 fs (1.24 fs FWHM) and T_pr = 10τ_pr = 165.4 as (83 as FWHM), and intensities I₀ = 2.75 × 10¹³ W cm⁻² (E₀ = 2.8 × 10⁻² a.u.) are used. Inserts (white lines) are the corresponding photoelectron angular distributions. Arbitrary units are used.

From Fig. 7 one sees that similar results of X-ray photoelectron distributions with multiple angular nodes are obtained. With such λ_pr = 5 nm (ω_pr = 9.11 a.u.) soft X-ray probe pulse, the photoelectron momentum and wavelength are p = 4.0 a.u. and λ_e = 2π/p = 1.57 a.u. less than the molecular internuclear distance R = 2 a.u. As a result, diffraction effects with multiple angular nodes appear in photoelectron distributions. It is, however, found that for these time-delayed cases, the photoelectron distributions show significant asymmetry with respect to the molecular center. Varying the time delay Δτ leads to a rotation of the photoelectron momentum and angular distributions in an anti-clockwise direction. The period of the rotation is τ_pu = 2π/ΔE = τ⁽⁰⁾. The periodical rotation feature of the asymmetric photoelectron distributions indicates the electronic coherent excitation by the pump pulses in Fig. 4. One sees that due to the transformer of the electron between the protons, asymmetric distributions of the coherent superposition state evolve with the time t. The density distributions evolve with a period of τ⁽⁰⁾ = 2π/ΔE = τ_pu. Consequently, altering the time-delay gives rise to a rotation of photoelectron diffraction patterns. The numerical results in Fig. 7 are in consonance with the predictions of photoelectron diffraction in coherent electron excitation processes in eqn (18) and Fig. 6, illustrating the evolution of the electron wave packets with time in molecules. Therefore the time-resolved photoelectron diffraction in Fig. 7 and eqn (18) allows us to explore the coherent electron dynamics of molecular attosecond circular charge migration.

Similar results of time-resolved coherent electron dynamics and photoelectron diffraction can also be obtained by a right-handed circularly polarized pump pulse. A resonant excitation between the ground A′ and E⁻ electronic states is triggered, thus leading to a superposition state ψ_co(r,t) which reads as

The corresponding electron density distributions and photoelectron momentum distributions evolve with time as well, as predicted in eqn (37) and (40). Comparing to the previous left handed case in Fig. 4 and 7, a phase difference of ±π occurs in right-handed cases, due to the photoelectron mirror images of the degenerate E⁺ and E⁻ electronic states in Fig. 2 and eqn (14) and (15).

Of note is that in the present work, we only consider a single electron molecular model. It has also been shown that in the diatomic H₂ at large internuclear distances where entanglement of atomic configurations, corresponding to the Heitler–London wave functions, photoelectron diffraction patterns can also be influenced due to the effect of the electron–electron correlation, as reported previously.⁸⁴ The photoelectron angular distributions exhibit signature of electron correlation. Therefore, for multiple electron molecular systems, electron entanglement via exchange due to spin strongly may influence photoionization emission in circular polarization. Moreover, to clearly observe photoelectron diffraction patterns in triatomic molecular systems, perfect alignments are required. As derived analytically in Appendixes, photoelectron angular distributions are functions of the molecular internuclear distance and the angle between the molecular axis and the photoelectron momentum direction. Current laser technology allows for orienting such molecules using coherently prepared rotational wave packets.⁸⁵

4 Conclusions

In the present work, we study molecular photoelectron diffraction by circularly polarized X-ray laser pulses. Simulations are performed on the oriented molecular ion H₃²⁺ by numerically solving TDSEs. We first present the photoionization by a circular X-ray pulse for stationary electronic states. Photoelectron distributions exhibit diffraction patterns with multiple angular nodes, which are determined by the initial electronic state symmetry. We also present the photoionization in a charge migration process. The molecule is excited by a UV pulse, creating a coherent superposition of the ground and degenerate excited electronic states of different helicities, E⁺ or E⁻. Subsequently, a time-delayed circularly polarized soft X-ray attosecond probe pulse is used to ionize the excited molecule. X-ray photoelectron diffraction distributions display asymmetric structure, which is shown to be sensitive to the time delay between the pump and probe pulses.

We present ultrafast multiple center and multiple electronic state molecular photoionization models to describe the photoelectron diffraction patterns. Theoretical analysis confirms these numerical simulations from TDSEs. Molecular orbital and coherent state symmetries determine the diffraction fringes in photoelectron distributions. In a coherent excitation process, the time-resolved asymmetric photoelectron diffraction patterns result from the evolution of the coherent electronic wave packets in the molecule. The time-dependent probing photoelectron diffraction illustrates the electron coherence, thus allowing to explore time-dependent charge migration in molecular reaction processes. Relating to new experiments in attosecond photoionization of molecules,⁸⁶ our results pave a way to image molecular orbital and monitor intramolecular coherent electron dynamics by photoelectron diffraction which can be extended to more complex systems and laser control of chemical reactions.^87–90

Conflicts of interest

There are no conflicts to declare.

Appendix

A Circularly polarized δ-function field pulses

The Dirac δ-function is expressed asFor a circularly polarized laser field with its polarization vector in the (x,y) plane, the form can be defined as E(t) = E[x̂ cos(ωt) ± ŷ sin(ωt)], where the sign ± presents left/right handed helicity. From eqn (23), one obtains the following relations by integrating over all ω,andConsequently, the δ-function circularly polarized laser pulse is

E(t) = x̂Eδ(t)/2π, (26)

for both left and right handed helicities, similar to a linear process.

B Ultrafast photoelectron diffraction models of triatomic molecular systems

For an impulsive field model, the wavefunction in momentum space right after the pulse E(t) = Eδ(t = 0) is given by⁹¹

ψ(p,0⁺) = ψ₀(p + E) (27)

where ψ₀(p) is the wavefunction of the initial electronic state in momentum p space. Transforming the wavefunction ψ(p,0⁺) to coordinate space, we then obtain

ψ(r,0⁺) = exp(−iE·r)ψ₀(r). (28)

The transition amplitude to the continuum then can be obtained by a Fourier transform of the function ψ(r,0⁺),

For the ground A′ electronic state of H₃²⁺ at internuclear distance R, the corresponding wavefunction is

In eqn (30) the electron wavefunction ψ_A′(r) is a linear combinations of hydrogenic 1s orbital located at positions R_i, i = 1, 2, 3. Setting the molecular orbital ψ₀(r) = ψ_A′(r), one then obtains the transition amplitude [scr F, script letter F](p), i.e., the momentum distribution of photoelectron in eqn (29) which can be written as

[scr F, script letter F]^(A′)(p) = [cos²(P·R₁₂/2) + cos²(P·R₂₃/2) + cos²(P·R₁₃/2)]ψ_s²(P), (31)

where the vector P = p + E and R_ij = R_j − R_i, i, j = 1, 2, 3 and R_i,j = R, the molecular internuclear distance, as illustrated in Fig. 1(a). The wavefunctions of the two degenerate excited states are given byfor the E⁺ electronic state, andfor the E⁻ electronic state. Of note is that ψ_E⁺(r) = ψ_E⁻*(r). Then the corresponding photoelectron distributions in eqn (29) are given by

[scr F, script letter F]^(E+)(p) = [cos²(P·R₁₂/2 + π/3) + cos²(P·R₂₃/2 + 2π/3) + cos²(P·R₁₃/2 + π/3)]ψ_s²(P), (34)

and

[scr F, script letter F]^(E−)(p) = [cos²(P·R₁₂/2 − π/3) + cos²(P·R₂₃/2 − 2π/3) + cos²(P·R₁₃/2 − π/3)]ψ_s²(P), (35)

From eqn (31)–(35) one sees that, in the ultrafast photoionization model, the effect of the delta-function pulse is to shift the total photoelectron distributions by the amount of the field vector E. The phase of [scr F, script letter F]^(E+)(p) = [scr F, script letter F]^(E−)(−p), i.e., opposite helicity. This corresponds to opposite current helicity since the current is j(r,t) = ψ(r,t)∇ψ*(r,t) − ψ*(r,t)∇ψ(r,t) = −[ψ*(r,t)∇ψ(r,t) − ψ(r,t)∇ψ*(r,t)] with ψ_E⁺(r) = ψ_E⁻*(r).

C Time-resolved photoelectron diffraction patterns

We next consider the case for a photoionization process from a time-dependent coherent superposition state ψ₀ = ψ_co of the three electronic states. A resonant excitation between the A′ and E⁺ (or E⁻) can be obtained by a circularly polarized pump pulse. The corresponding electronic wavefunction reads aswhere c_ie(t), ie = A′, E⁺ or E⁻, is the occupation coefficients of the ie electronic state, and E_ie is the eigenenergy of the ie electronic state. Then the corresponding time-dependent coherent electron density distributions are described bywith ℜ being the real part. ΔE = E_E⁺ − E_A′ = E_E⁻ − E_A′ is the energy difference between the excited E⁺/E⁻ state and the ground A′ electronic states. From eqn (37) one sees that the total coherent electron dynamics are composed of three components, two electronic state densities, [scr D, script letter D]^A′(r,t) and [scr D, script letter D]^(E+/E−)(r,t), and their interfering superposition terms [scr D, script letter D]^{(A′,E+/E−)}(r,t). In general, the population probabilities |c_ie(t)|² are constant after pump pulses. Therefore, the state densities [scr D, script letter D]^(A′)(r,t) and [scr D, script letter D]^(E+/E−)(r,t) do not depend on the time. The interference term between electronic states, [scr D, script letter D]^{(A′,E+/E−)}(r,t) is shown to be dependent on the time t with oscillation period τ⁽⁰⁾ = 2π/ΔE. The evolution of the superposition terms describes the attosecond electron coherence.

According to the ultrafast photoionization model, the instantaneous transition amplitude in eqn (29) can be written as

[scr F, script letter F]_co(p,t) = |c_A′(t)|²[scr F, script letter F]^(A′)(p) + |c_E⁺/E⁻(t)|²[scr F, script letter F]^(E+/E−)(p) + |c_A′(t)c_E⁺/E⁻(t)|[scr F, script letter F]^{(A′,E+/E−)}(p,t) (38)

[scr F, script letter F]^(ie)(p) is the distribution for the ie single electronic state, in eqn (31), (34), and (35). Their interference terms are given bybetween the A′ and E⁺ states, andbetween the A′ and E⁻ states. It is also found that the interference terms [scr F, script letter F]^(A′,E+)(p,t) and [scr F, script letter F]^(A′,E−)(p,t) are also time dependent with an oscillation period τ⁽⁰⁾ = 2π/ΔE_ij, the same as in eqn (37). The time-dependent photoelectron distribution [scr F, script letter F]_co(p,t) in eqn (38) describe the coherent excitation between the electronic states.

Acknowledgements

This work is in part supported by the National Natural Science Foundation of China (Grants No. 11974007) and the Natural Sciences and Engineering Research Council of Canada (NSERC), Fonds de Recherche du Québec-Nature et Technologies (FRQNT). The authors also thank Compute Canada for access to massively parallel computer clusters.

Notes and references

1. A. H. Zewail, J. Phys. Chem. A, 2000, 104, 5660–5694.
2. W. Becker, X. Liu, P. J. Ho and J. H. Eberly, Rev. Mod. Phys., 2012, 84, 1011–1043.
3. T. Bredtmann, D. J. Diestler, S. D. Li, J. Manz, J. F. Pérez-Torres, W. J. Tian, Y. B. Wu, Y. Yang and H. J. Zhai, Phys. Chem. Chem. Phys., 2015, 17, 29421–29464.
4. P. Baum and F. Krausz, Chem. Phys. Lett., 2017, 683, 57–61.
5. S. Beaulieu, A. Comby, D. Descamps, B. Fabre, G. A. Garcia, R. Géneaux, A. G. Harvey, F. Légaré, Z. Masin, L. Nahon, A. F. Ordonez, S. Petit, B. Pons, Y. Mairesse, O. Smirnova and V. Blanchet, Nat. Phys., 2018, 14, 484.
6. C. C. Shu, K. J. Yuan, D. Dong, I. Petersen and A. D. Bandrauk, J. Phys. Chem. Lett., 2017, 8, 1.
7. F. Krausz and M. Ivanov, Rev. Mod. Phys., 2009, 81, 163.
8. Z. Chang, P. B. Corkum and S. R. Leone, J. Opt. Soc. Am. B, 2016, 33, 1081–1097.
9. S. Chelkowski, P. B. Corkum and A. D. Bandrauk, Phys. Rev. Lett., 1999, 82, 3416–3419.
10. C. Beylerian and C. Cornaggia, J. Phys. B: At., Mol. Opt. Phys., 2004, 37, L259–L265.
11. T. Zuo, A. D. Bandrauk and P. B. Corkum, Chem. Phys. Lett., 1996, 259, 313–320.
12. M. Meckel, D. Comtois, D. Zeidler, A. Staudte, D. Pavičić, H. C. Bandulet, H. Pépin, J. C. Kieffer, R. Dörner, D. M. Villeneuve and P. B. Corkum, Science, 2008, 320, 1478–1482.
13. C. I. Blaga, J. Xu, A. D. DiChiara, E. Sistrunk, K. Zhang, P. Agostini, T. A. Miller, L. F. DiMauro and C. D. Lin, Nature, 2012, 483, 194–197.
14. K. Amini, M. Sclafani, T. Steinle, A.-T. Le, A. Sanchez, C. Müller, J. Steinmetzer, L. Yue, J. R. Martnez Saavedra, M. Hemmer, M. Lewenstein, R. Moshammer, T. Pfeifer, M. G. Pullen, J. Ullrich, B. Wolter, R. Moszynski, F. J. Garca de Abajo, C. D. Lin, S. Gräfe and J. Biegert, Proc. Natl. Acad. Sci. U. S. A., 2019, 116, 8173–8177.
15. T. Gaumnitz, A. Jain, Y. Pertot, M. Huppert, I. Jordan, F. Ardana-Lamas and H. J. Wörner, Opt. Express, 2017, 25, 27506–27518.
16. M. J. J. Vrakking, Phys. Chem. Chem. Phys., 2014, 16, 2775–2789.
17. M. G. Pullen, B. Wolter, A.-T. Le, M. Baudisch, M. Hemmer, A. Senftleben, C. D. Schröter, J. Ullrich, R. Moshammer, C. D. Lin and J. Biegert, Nat. Commun., 2015, 6, 7262.
18. Y. Ito, C. Wang, A. T. Le, M. Okunishi, D. Ding, C. D. Lin and K. Ueda, Struct. Dyn., 2016, 3, 034303.
19. T. T. Nguyen-Dang, M. Peters, J. Viau-Trudel, E. Couture-Bienvenue, R. Puthumpally-Joseph, E. Charron and O. Atabek, Mol. Phys., 2017, 115, 1934–1943.
20. M. Peters, T. T. Nguyen-Dang, C. Cornaggia, S. Saugout, E. Charron, A. Keller and O. Atabek, Phys. Rev. A: At., Mol., Opt. Phys., 2011, 83, 051403(R).
21. M. Kazama, T. Fujikawa, N. Kishimoto, T. Mizuno, J. Adachi and A. Yagishita, Phys. Rev. A: At., Mol., Opt. Phys., 2013, 87, 063417.
22. H. C. Shao and A. F. Starace, Phys. Rev. A: At., Mol., Opt. Phys., 2016, 94, 030702.
23. D.-D. T. Vu, N.-L. T. Phan, V.-H. Hoang and V.-H. Le, J. Phys. B: At., Mol. Opt. Phys., 2017, 50, 245101.
24. H. D. Cohen and U. Fano, Phys. Rev., 1966, 150, 30–33.
25. A. Palacios, A. González-Castrillo and F. Martín, Proc. Natl. Acad. Sci. U. S. A., 2014, 111, 3973–3978.
26. S. Tsuru, T. Fujikawa, M. Stener, P. Decleva and A. Yagishita, J. Chem. Phys., 2018, 148, 124101.
27. M. Peters, T. T. Nguyen-Dang, E. Charron, A. Keller and O. Atabek, Phys. Rev. A: At., Mol., Opt. Phys., 2012, 85, 053417.
28. R. Puthumpally-Joseph, J. Viau-Trudel, M. Peters, T. T. Nguyen-Dang, O. Atabek and E. Charron, Mol. Phys., 2017, 115, 1889.
29. R. S. Mulliken, Phys. Rev., 1937, 51, 310–332.
30. R. Weinkauf, P. Schanen, S. Yang, D. Soukara and E. W. Schlag, J. Chem. Phys., 1995, 99, 11255–11265.
31. L. S. Cederbaum and J. Zobeley, Chem. Phys. Lett., 1999, 307, 205–210.
32. I. Barth and J. Manz, Angew. Chem., Int. Ed., 2006, 45, 2962–2965.
33. F. Remacle and R. D. Levine, Proc. Natl. Acad. Sci. U. S. A., 2006, 103, 6793–6798.
34. W. Li, A. A. Jaron-Becker, C. W. Hogle, V. Sharma, X. Zhou, A. Becker, H. C. Kapteyn and M. M. Murnane, Proc. Natl. Acad. Sci. U. S. A., 2010, 107, 20219–20222.
35. H. Mineo, M. Yamaki, Y. Teranishi, M. Hayashi, S. H. Lin and Y. Fujimura, J. Am. Chem. Soc., 2012, 134, 14279–14282.
36. B. Mignolet, R. D. Levine and F. Remacle, J. Phys. B: At., Mol. Opt. Phys., 2014, 47, 124011.
37. V. Despré, A. Marciniak, V. Loriot, M. C. E. Galbraith, A. Rouzée, M. J. J. Vrakking, F. Lépine and A. I. Kuleff, J. Phys. Chem. Lett., 2015, 6, 426–431.
38. G. Hermann, C. Liu, J. Manz, B. Paulus, J. F. Pérez-Torres, V. Pohl and J. C. Tremblay, J. Phys. Chem. A, 2016, 120, 5360–5369.
39. K.-J. Yuan and A. D. Bandrauk, J. Chem. Phys., 2016, 145, 194304.
40. H. C. Shao and A. F. Starace, Phys. Rev. Lett., 2010, 105, 263201.
41. H. C. Shao and A. F. Starace, Phys. Rev. A, 2018, 97, 022702.
42. D. Jia, J. Manz, B. Paulus, V. Pohl, J. C. Tremblay and Y. Yang, Chem. Phys., 2017, 482, 146–159.
43. G. Hermann, C. Liu, J. Manz, B. Paulus, V. Pohl and J. C. Tremblay, Chem. Phys. Lett., 2017, 683, 553–558.
44. D. J. Diestler, G. Hermann and J. Manz, J. Phys. Chem. A, 2017, 121, 5332–5340.
45. R. Weinkauf, P. Schanen, A. Metsala, E. Schlag, M. Bürgle and H. Kessler, J. Phys. Chem. A, 1996, 100, 18567–18585.
46. R. Weinkauf, E. W. Schlag, T. J. Martinez and R. D. Levine, J. Phys. Chem. A, 1997, 101, 7702–7710.
47. M. R. Miller, Y. Xia, A. Becker and A. Jaron-Becker, Optica, 2016, 3, 259–269.
48. B. Mignolet, R. D. Levine and F. Remacle, Phys. Rev. A: At., Mol., Opt. Phys., 2014, 89, 021403.
49. B. Mignolet, R. D. Levine and F. Remacle, J. Phys. Chem. A, 2014, 118, 6721–6729.
50. A. D. Bandrauk, S. Chelkowski, P. B. Corkum, J. Manz and G. L. Yudin, J. Phys. B: At., Mol. Opt. Phys., 2009, 42, 134001.
51. T. Bredtmann, S. Chelkowski and A. D. Bandrauk, J. Phys. Chem. A, 2012, 116, 11398–11405.
52. H. Mineo, S. H. Lin and Y. Fujimura, J. Chem. Phys., 2013, 138, 074304.
53. I. Barth, J. Manz, Y. Shigeta and K. Yagi, J. Am. Chem. Soc., 2006, 128, 7043–7047.
54. K. J. Yuan and A. D. Bandrauk, Phys. Rev. A: At., Mol., Opt. Phys., 2013, 88, 013417.
55. K. J. Yuan, J. Guo and A. D. Bandrauk, Phys. Rev. A, 2018, 98, 043410.
56. E. Räsänen, A. Castro, J. Werrschnik, A. Rubio and E. K. U. Gross, Phys. Rev. Lett., 2007, 98, 157404.
57. P. M. Kraus, B. Mignolet, D. Baykusheva, A. Rupenyan, L. Horný, E. F. Penka, G. Grassi, O. I. Tolstikhin, J. Schneider, F. Jensen, L. B. Madsen, A. D. Bandrauk, F. Remacle and H. J. Wörner, Science, 2015, 350, 790–795.
58. P. Hockett, C. Z. Bisgaard, O. J. Clarkin and A. Stolow, Nat. Phys., 2011, 7, 612–615.
59. T. Okino, Y. Furukawa, Y. Nabekawa, S. Miyabe, A. Amani Eilanlou, E. Takahashi, K. Yamanouchi and K. Midorikawa, Sci. Adv., 2015, 1, e1500356.
60. F. Lépine, M. Y. Ivanov and M. J. J. Vrakking, Nat. Photonics, 2014, 8, 195–204.
61. K. Ramasesha, S. R. Leone and D. M. Neumark, Annu. Rev. Phys. Chem., 2016, 67, 41–63.
62. K. J. Yuan and A. D. Bandrauk, Phys. Chem. Chem. Phys., 2017, 19, 25846–25852.
63. K. J. Yuan and A. D. Bandrauk, J. Phys. Chem. A, 2018, 122, 2241–2249.
64. K. J. Yuan and A. D. Bandrauk, J. Phys. Chem. A, 2019, 123, 1328–1336.
65. W. Hu, Y. Liu, S. Luo, X. Li, J. Yu, X. Li, Z. Sun, K.-J. Yuan, A. D. Bandrauk and D. Ding, Phys. Rev. A, 2019, 99, 011402.
66. Y. Liu, W. Hu, S. Luo, K.-J. Yuan, Z. Sun, A. D. Bandrauk and D. Ding, Phys. Rev. A, 2019, 100, 023404.
67. J. McKenna, A. M. Sayler, B. Gaire, N. G. Johnson, K. D. Carnes, B. D. Esry and I. Ben-Itzhak, Phys. Rev. Lett., 2009, 103, 103004.
68. B. Mukherjee, D. Mukhopadhyay, S. Adhikari and M. Baer, Mol. Phys., 2018, 116, 2435.
69. S. Ghosh, S. Mukherjee, B. Mukherjee, S. Mandal, R. Sharma, P. Chaudhury and S. Adhikari, J. Chem. Phys., 2017, 147, 074105.
70. C. Lefebvre, H. Z. Lu, S. Chelkowski and A. D. Bandrauk, Phys. Rev. A: At., Mol., Opt. Phys., 2014, 89, 023403.
71. K. J. Yuan, H. Lu and A. D. Bandrauk, ChemPhysChem, 2013, 14, 1496–1501.
72. M. Paul, L. Yue and S. Gräfe, Phys. Rev. Lett., 2018, 120, 233202.
73. X. Xie, A. Scrinzi, M. Wickenhauser, A. Baltuška, I. Barth and M. Kitzler, Phys. Rev. Lett., 2008, 101, 033901.
74. K.-J. Yuan and A. D. Bandrauk, Phys. Rev. Lett., 2013, 110, 023003.
75. P.-C. Huang, C. Hernandez-Garcia, J.-T. Huang, P.-Y. Huang, C.-H. Lu, L. Rego, D. D. Hickstein, J. L. Ellis, A. Jaron-Becker, A. Becker, S.-D. Yang, C. G. Durfee, L. Plaja, H. C. Kapteyn, M. M. Murnane, A. H. Kung and M.-C. Chen, Nat. Photonics, 2018, 12, 349–354.
76. K. J. Yuan, H. Lu and A. D. Bandrauk, Struct. Chem., 2017, 28, 1297–1309.
77. D. S. Tchitchekova, H. Lu, S. Chelkowski and A. D. Bandrauk, J. Phys. B: At., Mol. Opt. Phys., 2011, 44, 065601.
78. A. D. Bandrauk, F. Fillion-Gourdeau and E. Lorin, J. Phys. B: At., Mol. Opt. Phys., 2013, 46, 153001.
79. A. D. Bandrauk and H. Shen, J. Chem. Phys., 1993, 99, 1185–1193.
80. A. D. Bandrauk and H. Lu, J. Theor. Comput. Chem., 2013, 12, 1340001.
81. K. J. Yuan, H. Z. Lu and A. D. Bandrauk, Phys. Rev. A: At., Mol., Opt. Phys., 2011, 83, 043418.
82. G. C. Schatz and M. A. Ratner, Quantum Mechanics in Chemistry, Dover Publications Inc., New York, 2002.
83. K. J. Yuan and A. D. Bandrauk, Struct. Dyn., 2015, 2, 014101.
84. K.-J. Yuan, H. Lu and A. D. Bandrauk, Phys. Rev. A: At., Mol., Opt. Phys., 2009, 80, 061403.
85. K. F. Lee, D. M. Villeneuve, P. B. Corkum, A. Stolow and J. G. Underwood, Phys. Rev. Lett., 2006, 97, 173001.
86. M. Huppert, I. Jordan, D. Baykusheva, A. von Conta and H. J. Wörner, Phys. Rev. Lett., 2016, 117, 093001.
87. M. F. Kling and M. J. Vrakking, Annu. Rev. Phys. Chem., 2008, 59, 463–492.
88. S. Beaulieu, A. Comby, A. Clergerie, J. Caillat, D. Descamps, N. Dudovich, B. Fabre, R. Géneaux, F. Légaré, S. Petit, B. Pons, G. Porat, T. Ruchon, R. Taeb, V. Blanchet and Y. Mairesse, Science, 2017, 358, 1288–1294.
89. K. Wilma, C.-C. Shu, U. Scherf and R. Hildner, New J. Phys., 2019, 21, 045001.
90. L. Gallmann, I. Jordan, H. J. Wörner, L. Castiglioni, M. Hengsberger, J. Osterwalder, C. A. Arrell, M. Chergui, E. Liberatore, U. Rothlisberger and U. Keller, Struct. Dyn., 2017, 4, 061502.
91. M. D. Girardeau, K. G. Kim and C. C. Widmayer, Phys. Rev. A: At., Mol., Opt. Phys., 1992, 46, 5932–5937.

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