Intensity-dependent angular distribution of low-energy electrons generated by intense high-frequency laser pulse

: By solving the three-dimensional time-dependent Schrödinger equation, we investigate the angular distributions of the low-energy electrons when an intense high-frequency laser pulse is applied to the hydrogen atom. Our numerical results show that the angular distributions of the low-energy electrons which generated by the nonadiabatic transitions sensitively depend on the laser intensity. The angular distributions evolve from a two-lobe to a four-lobe structure as the laser intensity increases. By analyzing nonadiabatic process in the Kramers-Henneberger frame, we illustrate that this phenomenon is attributed to the intensity-dependent adiabatic evolution of the ground state wavefunction. When the laser intensity further increases, the pathway of nonadiabatic transition from the ground state to the excited state and then to the continuum states is non-negligible, which results in the ring-like structure in the photoelectron momentum distribution. The angular distributions of the low-energy electrons provide a way to monitor the evolution of the electron wavefunction in the intense high frequency laser fields.


Introduction
The development of the free-electron laser technology, which is capable of generating extreme ultraviolet (XUV) lasers pulses with unprecedented intensity [1][2][3][4][5][6], has attracted quite a lot of attentions on the laser-matter interaction in high-frequency regime. Many remarkable and counterintuitive phenomena in high-frequency regime have been actively studied in recent years. One of them is the atomic stabilization, which has been understood as the formation of metastable Kramers-Henneberger (KH) states of the dressed potential [7][8][9][10][11][12]. Experimentally, this phenomenon has been observed in Rydberg states [13][14][15].
Numerous investigations on the atomic stabilization in high-frequency regime focus on monochromatic laser pulse, and the high-frequency Floquet theory (HFFT) has been employed to explore the underlying dynamics. However, for real laser pulse, the pulse duration is finite and there are the rising and falling edges, which could induce many interesting effects not accounted for by the HFFT. An adiabatic version of the HFFT taking account of the pulse envelope has been proposed [16,17]. This approach has been successful in understanding the pulse envelope variation induced phenomena, such as the dynamic interference, which originates from the interference of the electron wave packets ejected on the rising and falling edges of the laser pulse [17][18][19][20][21][22][23][24][25].
Another interesting issue due to the finite pulse duration is the ionization by nonadiabatic transition. As one of the most basic mechanisms of the universe, nonadiabatic transition occurs when the adiabatic parameter changes quickly [26,27]. One of the most well-known examples of the nonadiabatic transition is the transition between the adiabatic electronic states defined by the Born-Oppenheimer approximation when the nuclear motion is involved [28,29]. In the intense high-frequency laser pulses with short duration, the rapid turn-on and turn-off of the laser pulse could induce nonadiabatic transition. It has been shown that the turn-on of the short pulse could induce nonadiabatic transition from the ground state to the excited states, which has been revealed by tracing the time-dependent electrons wave packet of the bound state in the intense high-frequency pulses [7,[30][31][32][33][34]. The nonadiabatic transition also occurs from ground state to continue states, resulting in low-energy electrons generation in the intense high-frequency laser pulse [35][36][37][38]. These low-energy photoelectrons have attracted considerable attention in the past decades. For example, the modulation in the energy spectrum of the low-energy electrons, which originates from interference of the electrons generated by nonadiabatic transition on the rising and falling edges of the laser pulse, has been detaildedly surveyed [39]. The similar nonadiabatic ionization of Rydberge states has been theoretically and experimentally observed [40,41]. The investigations on the nonadiabatic transition deepen the understanding of the electrons dynamics in the high-frequency regime. In most of previous investigations on the nonadiabatic ionization, attentions mainly focus on the energy spectrum of the electrons [37][38][39]. With the development of the experimental technique, it is feasible to detect the photoelectron momentum distributions (PEMDs) induced by the high-frequency laser pulse. The PEMDs can provide more information about the electrons dynamics in the nonadiabatic transitions.
In this work, we investigate the angular distributions of the low-energy electrons generated due to nonadiabatic ionization in the intense high-frequency laser pulse. The PEMDs obtained by numerically solving the three-dimensional (3D) time-dependent Schrödinger equation (TDSE) show that the low-energy photoelectron angular distributions (LEPADs) sensitively depend on the laser intensity. It changes from a two-lobe structure to a four-lobe structure as the laser intensity increases. By analyzing the nonadiabatic transitions in the KH frame, we show that the change of the LEPAD originates from intensity dependent evolution of the ground-state wavefunction. As the laser intensity further increases, a ring-like structure appears in momentum distribution of the low-energy electrons, which is attributed to the nonadiabatic transition from the ground state to an excited state followed by the nonadiabatical ionization. The momentum distributions of the low-energy electrons are the manifestation of the adiabatic evolution wavefunction, and thus it provides us a way to monitor the evolution of the electron wavefunction. This paper is organized as follows. In Sec. 2., we introduce our method for numerically solving the 3D TDSE and the method of treating the nonadiabatic transition. Then we show our numerical results and discussions in Sec. 3. Section. 4. provides a brief summary.

Numerically solving TDSE
We obtain the PEMDs by numerically solving the 3D TDSE. The TDSE in velocity gauge is written as (atomic units are used unless otherwise stated): where V(r) = −1/r is Coulomb potential of H. The laser field is described as where A 0 and ω are the amplitude of the vector potential and center frequency of the laser pulse, respectively. f (t) is the envelope of the laser pulses. In our calculations, we consider two types of pulses, the Gaussian shape laser pulse, and the sin 2 shape laser pulse, Here, ∆ and τ are the full width at half maximum (FWHM) for the Gaussian shape laser pulse and the pulse duration for the sin 2 shape laser pulse, respectively. In our simulation, the TDSE in Eq. (1) is solved in the spherical coordinates, in which the wave function Ψ(r, t) is expanded by spherical harmonics |l, m⟩, where R l,m (r, t) is the radial part of the wave function. This radial wave function is discretized by the finite-element discrete variable representation (FE-DVR) method [42]. The angular momentum number l is chosen up to 30, and the convergence has been ensured by varying the number of included partial waves. The time propagation of the TDSE is calculated by the split-Lanczos method [43] with the time step fixed at ∆t = 0.01 a.u.. The maximal box size for the radial coordinates is chosen to be 400 a.u.. An absorbing mask function has been applied in each step of time propagation of the wavefunction, which is written as F(r) = 1 − 1/(1 + e (r−R c )/L ) with R c = 300 a.u. and L = 2 a.u.. The wavefunction Ψ(r, t) is split into the inner part Ψ in (r, t) = Ψ(r, t)F(r) and the outer part Ψ out = Ψ(r, t) − Ψ in (r, t) by the absorbing mask function. The inner wave function is still evolved strictly as TDSE, while the outer part Ψ out is propagated by Coulomb-Volkov propagator [44]. The initial wavefunction is prepared by imaginary-time propagation which is chosen as the ground state of H atom. The ionization amplitude is extracted from the final wavefunction by projecting it to the scattering state [43], Here, Ψ p (r) is the normalized scattering state for H atom, which is normalized by ∫ drΨ * p (r)Ψ p ′ (r) = 2πδ(p − p ′ ). Note that the nondipole effect is not considered here, because the nondipole effect does not affect the angular distributions of the photoelectrons in the polarization plane (the slices of 3D PEMDs at p y = 0) for the laser parameters considered in this work, as discussed in Appendix A..

Floquet envelope Hamiltonian approach
We adopt the Floquet envelope Hamiltonian approach [16,[45][46][47] to investigate the intensitydependent LEPAD. By the unitary transformation Ψ KH (t) = exp[iα(t) · p]Ψ(r, t), the Hamiltonian in Eq. (1) can be transformed to the Hamiltonian in KH frame, which is written as where α(t) = − ∫ t A(t ′ )dt ′ is the excursion amplitude of the laser pulse. For the linearly monochromatic laser pulse considered in HFFT, the excursion amplitude is written as where α 0 = A 0 /ωe z . The electrons dynamics in the high-frequency regime are described by expanding the laser-distorted Coulomb potential as [48] V(r + α(t)) = ∑︂ n V n (r; α 0 )e −inωt .
Here, V n (r; Then the Hamiltonian can be divided into time-independent H 0 and time-dependent H I terms In the lowest-order of the HFFT, the electron is in a stationary state of H 0 . The interaction determined by H I represents multiphoton decay channels, which is taken into account in the higher order of the HFFT. For the short intense laser pulse considered in this work, effect of the pulse envelope is non-negligible and correction is implemented by the substitution [17] where f (t) is envelope of the laser pulse. It indicates that the laser distorted potential V 0 (r; α 0 ) is time-depend and follows the envelope of the laser pulse. Then the instantaneous eigenvalue E n (α 0 ) and eigenstate |ϕ n (α 0 )⟩ of H 0 are determined by The time-dependent electron wave packet |Ψ KH (t)⟩ can be expanded by the instantaneous eigenstates of H 0 , Inserting the wavefunction |Ψ(t)⟩ into the Schrödinger equation with the Hamiltonian H KH in Eq. (10), the evolution of time-dependent amplitudes C n (t) is determined bẏ The term ⟨ϕ n |H I |ϕ m ⟩ represents the photon absorption or emission, and the term ⟨ϕ n | ∂ ∂α 0 |ϕ m ⟩ provides nonadiabatic coupling. In this work, we focus on the nonadiabatic transition originating from the derivatives of the laser pulse envelope and dynamics of the low-energy electrons generated by the nonadiabatic transition, which is independent on the term ⟨ϕ n |H I |ϕ m ⟩ [37,45]. Thus the term ⟨ϕ n |H I |ϕ m ⟩ is not considered in the rest of discussions. Then the time-dependent amplitudes of the bound and the continuum states in the nonadiabatic transitions can be written as and Here, ϕ p is approximately chosen as the normalized scattering state for H atom and ∂ ∂α 0 |ϕ m ⟩ is derivative of the wavefunction. Eq. (16) indicates that if the eigenenergies and eigenstates of H 0 are acquired by solving Eq. (12), the PEMDs of the low-energy electrons generated by the nonadiabatic transition can be obtained. In addition, this approach also helps us identify the contributions of the different nonadiabatic transition channels. Figure 1 shows the PEMDs for the ionization of H obtained by solving the 3D TDSE. The parameters for the Gaussian shape laser pulses are ω = 3 a.u. and ∆ = 5T. The laser intensities for Figs. 1(a)-1(d) are 1 × 10 14 W/cm 2 , 4 × 10 18 W/cm 2 , 8 × 10 18 W/cm 2 and 2 × 10 19 W/cm 2 , respectively. For the lowest intensity [ Fig. 1(a)], the PEMD reveals a ring-like structure and the radius of the ring is p = √︁ 2(ω − I p ). This is characteristic of the single-photon ionization from ground state of H atom. As the intensity increases to 4 × 10 18 W/cm 2 [ Fig. 1(b)], two ring-like structures appear in the high energy region, which originate from the single-and two-photon ionization. The modulations on the ring-like structures originate from the dynamic interference [17][18][19][20]. Surprisingly, low-energy photoelectrons appear in the PEMD. Because the photon energy is much higher than the ionization potential of the atom, these electrons can not be attributed to the photon absorption process. In previous works, the low-energy electrons have been observed in the investigations on the ionization of H − [37,39] and clusters [38]. It is demonstrated that the low-energy electrons are generated via the nonadiabatic transitions due to the finite length of the laser pulse. This can be verified by calculating the ionization of the atom for the longer laser pulse, as shown in Fig. 2. With the increasing of the duration of the laser pulse, the yield of the low-energy photoelectrons decreases [ Fig. 2(a)]. When the pulse duration further increases to ∆=20T, as shown in Fig. 2(b), the low-energy photoelectrons disappear. To further verify that the low-energy electrons are generated by the nonadiabatic transitions, we change the envelope of the laser pulse. For the sin 2 shape laser pulse, as shown in Figs. 2(c) and 2(d), the low-energy photoelectrons appear in the PEMD for τ=30T and disappear for a longer laser pulse. More interestingly, our results show that the angular distributions of the low-energy electrons sensitively depend on the laser intensity as displayed in Fig. 1. For example, the LEPAD exhibits a two-lobe structure for I 0 = 4 × 10 18 W/cm 2 [ Fig. 1(b)], and a four-lobe structure for more intense laser pulse [ Fig. 1(c)]. When the laser intensity further increases, as shown in Fig. 1(d), a ring-like structure appears in the PEMD. As far as we know, the intensity-dependent LEPAD has not been observed and investigated previously. In the following, we will focus on the intensity-dependent LEPADs and choose the Gaussian shape laser pulse used in Fig. 1 as an example. We adopt the Floquet envelope Hamiltonian approach as shown in Sec. 2.2 to understand the intensity-dependent LEPADs. In this approach, the evolution of the wavefunction is described on  12), is solved by diagonalization of the Hamiltonian matrix in Lagrange interpolating polynomials basis [42] and the eigenenergies for three lowest-lying bound states are shown in Fig. 3(a). As indicated by Eq. (16), the nonadiabatic transition rates of the continuum states depend on the term ∂α 0 /∂τ. It implies that the nonadiabatic transition depends on the derivatives of the laser pulse envelope rather than the envelope itself. For the Gaussian shape laser pulse, the derivative of excursion amplitude reveals a double-peak structure and the peaks locate at -t s and t s , as shown in Fig. 3(b). The contributions of transitions at these two instants are the same and dominate in the entire pulse duration. Thus, we only consider the nonadiabatic transition at one of these two peaks, i.e., the transition at t s . Since the laser pulse is short and the laser intensity is low enough, the interference effect caused by the phase accumulated as the time propagation,

Results and discussions
, is ignored. The term C m (τ) describes the depletion of the states. We choose the ground state of H as the initial state, and the depletion of 1sσ g state is neglected in the zero-order approximation. Therefore, the momentum distributions of electrons ionized by the nonadiabatic transitions from the ground state can be approximately written as It implies that the angular distributions of the low-energy electrons are mainly determined by the derivatives of wavefunction at transition moment t s . For the Gaussian shape laser pulse, the transition moment is given by t s = ∆/2 √ ln 2. Then α 0 (t s ) = e −1/2 √ I 0 /ω 2 , which monotonously increases with the laser intensity. It indicates that the derivative of wavefunction at transition moment t s depends on the laser intensity. Therefore, the LEPAD sensitively depends on the laser intensity.
To verify the validity of Eq. (17) in describing the LEPADs, we compare the TDSE data with the results obtained by Eq. (17) in Fig. 4. The dot dashed lines represent the LEPADs, which are obtained by the integration of the PEMDs from p=0 a.u. to p=0.5 a.u. The results show that the amplitude of the angular distribution along the polarization direction increases as the increasing of laser intensity, and then the momentum distributions change from a two-lobe structure to a four-lobe structure. Apparently, the results calculated by Eq. (17) agree well with the TDSE data. It indicates that Eq. (17) is valid to describe the LEPADs, and the low-energy electrons are generated mainly by the nonadiabatic transition from the ground state in these laser intensities.  Actually, the nonadiabatic transition occurs not only from the ground state to the continuum state but also to the excited states. Figure 5 shows time-dependent populations of the excited states for the nonadiabatic transition from the ground state 1sσ g obtained by Eq. (15). Since the operator ∂/∂α 0 does not change parity of the states, the nonadiabatic transition only exists between the states with the same parity. Therefore, the population of 2pσ u state is zero for both laser intensities, as shown by the blue lines. For 2sσ g state, as shown by the red dashed line, the population of this state is less than three percent for I 0 = 4 × 10 18 W/cm 2 . When the laser intensity increases to 2 × 10 19 W/cm 2 , the population of 2sσ g state is nearly twenty percent. Therefore, the nonadiabatic ionization from 2sσ g state should be considered. In Fig. 6(a), we show the PEMD obtained by solving the TDSE with the laser intensity of I 0 = 2 × 10 19 W/cm 2 . As indicated by the dot dashed lines, the angular distributions in the lower (black line) and higher (red line) energy region reveal different structures, i.e., a four-lobe structure for the lower energy region and a ring-like structure for the higher energy region. The difference of the angular distributions implies that there are two different nonadiabatic transition channels for the low-energy electrons. We calculate the PEMDs for the nonadiabatic ionization from 1sσ g state and 2sσ g state by Eq. (16), as shown in Figs. 6(b) and 6(c). The dot dashed lines represent the angular distributions for corresponding PEMDs. It indicates that the nonadiabatic transition from 1sσ g state exhibits four-lobe structure and the transition from 2sσ g state reveals ring-like structure. The difference of the angular distributions reflects the different wavefunction derivatives at the nonadiabatic transition moment t s of these two states. As shown by the insets of Figs. 6(b) and 6(c), the derivative of wavefunction with respect to α 0 exhibits a three-lobe structure for 1sσ g state and a ring-like structure for 2sσ g state.
As discussed above, the LEPADs are mainly determined by the wavefunction derivative at the transition instant t s . To further understand the dependence of the angular distributions on the derivative of wavefunction with respect to α 0 , we calculate the weights of the partial waves of the wavefunction and the wavefunction derivative of the 1sσ g state by and Here, |l, m⟩ is the spherical harmonics. Since the laser pulse is linearly polarized along the z axis, the magnetic quantum number m is conserved. As a consequence, we only need to consider the component of m=0. As shown in Fig. 7(a), for the wavefunction, the weights of the partial waves are roughly unchanged as α 0 increases. While for the wavefunction derivatives with respect to α 0 , the weight of l=0 decreases and the weight of l=2 increases as the increasing of α 0 . The l=0 partial wave produces a isotropy structure and the l=2 partial wave exhibits a four-lobe structure in the momentum space. The PEMDs for the low-energy electrons can be thought as the superposition of these two components. Therefore, as shown in Fig. 7(b), the momentum distributions of the low-energy electrons change from a isotropy structure to a four-lobe structure as α 0 increases. To quantitatively identify the contributions of the s (l=0) and d (l=2) wave to the LEPADs, we use the following formula to describe the LEPADs Here, C is the normalization constant, β 2,4 are the second-and fourth-order anisotropy parameters, and P 2,4 are the second-and fourth-order Legendre polynomials [50]. β 2,4 are obtained by projecting the angular distributions of low-energy electrons (Integration of the PEMD from 0 a.u. to 0.5 a.u.) to the corresponding Legendre polynomials. The ratio β 2 /β 4 as a function of laser intensity is shown in Fig. 7(c), in which the circle solid line and the triangle dashed line are for the TDSE data and the results obtained by Eq. (17), respectively. Apparently, the agreement between these results is remarkable, which further verifies the validity of Eq. (17) to describe the angular distribution of the low-energy electrons. Moreover, the increasing of the ratio as laser intensity represents that the contributions of d wave increases as the laser intensity and then the LEPAD evolves from the two-lobe structure to the four-lobe structure.
The PEMD is a manifestation of the wavefunction derivative, as indicated by Eq. (17). Thus, it provides us a way to trace the evolution of the wavefunction derivative. For the Gaussian pulses, the low-energy photoelectrons are mainly released at two instants, -t s and t s . For the short pulses in our calculations, the interference between electrons released at these two instants is negligible. Then we reconstruct the wavefunction derivative according to Eq. (17). Note that there is no phase information in the PEMDs and thus in our reconstruction we do not consider the phase of the factor ⟨ϕ p | ∂ ∂α 0 (t s ) |1sσ g ⟩. The results are shown in Fig. 7(e). The exact wavefunction derivatives with respect to α 0 are also shown for comparison in Fig. 7(d). The reconstructed results agree reasonably well with the exact results. The additional structures appearing in the large coordinate range might be due to the limited bandwidth of the electron spectrum. The similar problem also exists in the tomographic imaging of molecular orbitals [51].

Conclusion
In conclusion, we have investigated the angular distributions of the low-energy electrons generated by nonadiabatic transition. By numerically solving the 3D TDSE, we find that the LEPAD sensitively depends on the laser intensity. It changes from a two-lobe structure to a four-lobe structure as the laser intensity increases. By analyzing the nonadiabatic transition in the KH frame, we find that this phenomenon is attributed to the intensity-dependent wavefunction derivatives of the ground state at the transition instant. For the more intense laser pulse, the non-negligible nonadiabatic transition from the excited state results in a ring-like structure in the angular distribution. It indicates that the angular distributions of the low-energy electrons provide a way to directly identify the contributions of different nonadiabatic transition channels. Moreover, utilizing the relationship between the PEMDs of the low-energy electrons with the wavefunction derivative at the transition instant, we successfully monitor the evolution of the ground state wavefunction by the momentum distributions of the low-energy electrons. Our work deepens the understanding of the nonadiabatic transitions in the intense high-frequency laser pulse.

A. Nondipole effect of the low-energy electrons
For the intense laser field, the nondipole effect is important. In previous works [36,52,53], the nondipole effect for the low-energy electrons are investigated. To test the validity of the dipole approximation in analyzing the generation of the low-energy electrons, the lowest-order nondipole correction is considered. The lowest-order nondipole corrected Hamiltonian in the velocity gauge is written as Here, A(t) and F(t) are the usually adopted vector potential and electric field of the laser pulse in the dipole approximation, respectively. In our simulation, the laser pulse propagates along the positive y-axis and is polarized along the z-axis.
We numerically solved the 3D TDSE within nondipole correction in spherical coordinates and the details have been shown in our previous work [20]. Figure 8 shows the PEMDs for the ionization of H atom with the highest laser intensity considered in our work; i.e., I 0 = 2 × 10 19 W/cm 2 . The upper and bottom panels are the results with dipole approximation and the lowest nondipole correction, respectively. Obviously, the nondipole effect results in the asymmetry distribution in the plane with the laser propagation direction as shown in Figs. 8(b) and 8(d).
The similar asymmetry is also displayed in previous works [36,52,53]. However, the nondipole effect does not affect the angular distribution of low-energy electrons in the polarization plane (p x -p z ) for the laser parameters considered here, as shown in Figs. 8(a) and 8(c). As indicated by previous work [52], when the amplitude of electric field is larger than 30 a.u., the dipole approximation begins to break down due to strong influence of the generation of the low-energy electrons by nondipole correction and then the PEMD in polarization plane is different from the results with dipole approximation. However, the amplitude of electric field for the highest laser intensity considered in this work (I 0 = 2 × 10 19 W/cm 2 , E 0 =23.8 a.u.) is less than 30 a.u.. Therefore, it is valid to analysis the PEMD in polarization plane with dipole approximation for the laser parameter considered in our work.