2.1. Numerical methods

We numerically solve the two-dimensional time-dependent Schrödinger equation (TDSE) for neon atom and its model atom in rectangular plane coordinates r ≡ (x,y). The TDSE reads as (atomic units are used throughout this paper unless otherwise stated)

which is described by the Hamiltonian H(r, t) = H₀ (r,t) + H_int(r,t) with the field-free term H₀(r) = T(r) + V_C(r), where the atomic kinetic operator is expressed as

and the soft-core Coulomb potential

We obtain the bound states of the field-free Hamiltonian H₀ by numerically solved the time-independent Schrödinger equation with the sine discrete-variable representation (sine-DVR).^[28,29] For neon atom, the soft parameters Z(x,y) = 1+9 exp (–x² – y²) and a = 2.88172 are used to remove the Coulomb singularity and to obtain the ionization potential I_p = 0.793 a.u. (atomic unit) for the 2p orbitals. From the time-independent Schrödinger equation, we obtain two real 2p orbitals in the 2D Cartesian coordinate, that is, 2p_x and 2p_y. The current-carrying orbital 2p₊ (2p_–) with magnetic quantum number m = +1 (m = –1) is taken as the initial electron wave function ψ (r,t = 0), which is defined as . The quantum numbers m = 1 and m = –1 are projections of the angular momentum in the quantization axis (z axis, which is in the light propagation direction), which means that the electron ring currents in the (x, y) plane are counter-rotating and co-rotating with respect to the helicity of the laser pulse.^[5] For reference calculations we use 1s as the initial state but keep the same ionization potential by taking Z (x, y) = 1 and a = 0.1195.^[5] The term H_int (r,t) is the laser–electron interaction term, in the length gauge, which can be expressed as

where E_x (t) and E_y (t) are the x and y components of the laser electric field. The laser electric field is

with

where t_d represents the time delay between the two elliptically polarized attosecond laser pulses and ε is the ellipticity of the laser pulses. The sign “±” indicates that the two pulses are co-rotating (“+”) or counter-rotating (“–”) polarization. The laser pulse envelope f(t) = sin²(πt/τ) is used with a duration of τ = n_p T, where T = 2π/ω is the period of the laser field. E₀ is the maximum amplitude of the field, corresponding to a pulse intensity which is fixed at 6 × 10¹³ W/cm².

The TDSE is propagated on a 2D Cartesian grid, obtained by using the parallel quantum wave-packet computer code LZH-DICP.^[30,31] In our simulation, we set the length of the integration grid to be 480 a.u. A cos^1/8 mask function is placed at x, y = ±200 a.u. after each time evolution step in order to avoid the reflection of the wave function from the boundary. After the end of the laser pulse, the wave function further propagates for an additional ten optical cycles to ensure that all the ionized components move away from the core. Then we obtain the ionized wave function, ψ_ion (x,y) = M(r_b)ψ_final(x,y), in the range of x² + y² > 50 a.u., where ψ_final(x,y) is the wave packet at final time, M(r_b) is an absorption function in the form of^[32]

In the present simulation, we choose γ = 0.125 a.u. and r_b = 50 a.u. Finally, the final photoelectron momentum distribution (PMD) is obtained by applying Fourier transform to the ionized wave function as obtained below:

2.2. Perturbation theory

In the present work, the simulations are conducted on the ionization process of neon atom and its model atom with elliptically polarized attosencond XUV pulses. The frequencies of the pulses are set to be ω = 1.0 a.u. The Keldysh parameter , where I_p is the ionization potential, implying that this is a multiphoton ionization process. Since I_p = 0.793 a.u., the single-photon ionization process is dominated. The first-order TDPT can be adopted to describe the ionization.^[18,33] The first-order amplitude A₁ for the single-photon ionization of an atom by the electric field E (t) is defined as

where | ψ₀〉 is the initial state with energy –E₀, | ψ_c 〉 is the final continuum state of the electron with momentum p ≡ (p_x, p_y) and energy E = p²/2, ω_c0 = E + E₀, and D = nD is the dipole moment of the atom.

In this paper, the electric field E(t) in Eq. (5) can be written as

where (j = 1,2) is the complex polarization vector normalized as e_j* ⋅ e_j = 1. Thus, the first-order TDPT amplitude in Eq. (10) for single-photon ionization of an atom under the pair of pulses takes the factorized form

where the parameter A_γ is in the form of

and the relative phase, Φ, is given by

The relative phase Φ in Eq. (14) consists of two factors: ̩ t_d, the difference in the phase accumulation during the temporal evolution of the electronic wave packets for a time delay t_d, and the difference between the CEPs of the pair of pulses (ϕ₁ – ϕ₂). The α (p) = | α|e^{i χ} depends on the ionizing laser pulse and the initial electronic state.

According to the first-order TDPT, for the two time-delayed elliptically identically polarized pulses in the polarization plane- (x, y) plane, i.e.,

We define also the degree of linear and circular polarization of the pulse, l and ξ as shown in Ref. [18]:

Then

where φ is the angle between n and the x axis and is also the ejection angle of the photoelectron defined in this work. The final PMD for ionization is

where |α (p)|²[ 1 + lcos (2φ)]cos (Φ) is the interference term which is determined by the relative CEP, the time delay and the ellipticity of the laser pulses. For the two time-delayed elliptically oppositely polarized pulses, i.e.,

The final PMD for ionization is

where tan (α) = ε sin (φ)/cos (φ), and

is the interference term. According to previous research in Ref. [18], for the atom in its 1s ground state, the parameter | α (p)|² depends only on the electron energy E = p²/2 and is independent of the momentum direction. Therefore, the angular distribution for the 1s state is determined only by the parameter | A_γ |² which depends on pulse parameters e₁, e₂, t_d, and (ϕ₁ – ϕ₂).

In this work, we present numerical results of neon atom and its model atom in different initial states for two laser cases: (i) the elliptically identically polarized pulses e₂ = e₁, i.e., the co-rotating polarization case; (ii) the elliptically oppositely polarized pulses, e₂ = e₁*, i.e., the counter-rotating polarization case. We find that besides the laser pulse, the initial electronic state has also great influence on PMDs.

3. Numerical results and discussion

We first consider the single-photon ionization process of the model atom with the 1s state as the initial state. Figure 1 displays results of the single-photon ionization by a pair of elliptically polarized pulses at different values of ellipticity ε. According to the first-order perturbation theory, we show the relationship between the parameter |A_γ|² and the ejection angle φ in Figs. 1(i) and 1(j). For the case of co-rotating polarization, i.e., e₁ = e₂, as l increases, the circular symmetry of the PMD will be broken, the maxima appear at φ = 0 and φ = π, and the minima appear at φ = π / 2 and φ = 3π / 2. The magnitudes of the maxima decrease and those of minima increase as ε increases, which is presented in Fig. 1(i). For the case of counter-rotating polarization, i.e., , from Fig. 1(j) we see that the positions of the maxima and the minima are same as those for the case e₁ = e₂, the magnitudes of maxima still decrease as ε increases, but those of minima do not change as ε increases and always stay 0. From Figs. 1(a)–1(h), one sees that the PMDs are sensitive to the ellipticities of the laser fields, as ε changes, the changes of magnitudes of the maxima and the minima are consistent with the TDPT results presented in Figs. 1(i) and 1(j).

We also study the effect of the pulse relative CEP ϕ₁ – ϕ₂ and the time delay t_d on the PMDs under a pair of identically polarized pulses at fixed ellipticity ε = 0.25, and the results are shown in Fig. 2. It can be seen from the figure that as predicted by Eq. (19), only the magnitudes of PMDs change with CEPs and time delay, while the distribution patterns remain unchanged.

The numerical results of the model atom under a pair of time-delayed elliptically oppositely polarized pulses with different values of ellipticity ε are shown in Fig. 3. We also take the 1s state as an initial state. The other laser parameters are the same as those used for the identical case in Fig. 2. The PMDs at two pulse ellipticities, ε = 0.75 [Figs. 3(a)–3(d)] and ε = 0.25 [Figs. 3(e)–3(h)], four time delays, t_d = 0.25T [Figs. 3(a) and 3(e)], t_d = 0.5T [Figs. 3(b) and 3(f)], t_d = 0.75T [Figs. 3(c) and 3(g)] and t_d = 1.0T [Figs. 3(d) and 3(h)] are presented, respectively. As one can see from Fig. 3, for the elliptically oppositely polarized case, PMDs, especially the PADs, are sensitive to the pulse ellipticity ε and the time delay t_d. It is found that the time delay varying from t_d = 0.25T to t_d = 1.0T causes the PMDs to rotate counterclockwise different angles (different values of Δφ) under the laser pulses with different ellipticities. We next also use the first-order TDPT to describe the ejection angles of PMDs in Figs. 3(a)–3(h). From Eq. (23) one sees that W_ε(p) is a function of ejection angle φ, relative pulse CEP (ϕ₁ – ϕ₂), pulse ellipticity ε and time delay t_d. At fixed ellipticities ε = 0.75 and ε = 0.25, when the time delay changes, the corresponding | A_γ|² values are present in Figs. 3(i) and 3(j). As shown in Figs. 3(i) and 3(j), one sees that when the ellipticity is fixed, not only the magnitude, but also the ejection angle φ for each of the maxima and minima change with time delay. The corresponding angle shift Δφ is a function of ε and t_d. The result is different from that for the case of two oppositely circularly polarized pulses at the same frequencies in Ref. [18]. The main reason for the difference is the dependence of the interference term on the ellipticity as indicated in Eq. (23).

Next, we study the effect of magnetic quantum number of the initial state on the patterns in PMDs by using a pair of time-delayed elliptically oppositely polarized pulses. We numerically simulate the photoionization process of neon atom of which the highest-occupied orbital is 2p orbital. The PMDs and the corresponding PADs of neon atom with different initial states under a pair of counter-rotating polarization laser pulses at different time delays with an ellipticity of 0.75 are displayed in Fig. 4. Comparing with the ejection angles of the maxima and minima of the model atom in Figs. 3(a)–3(d), there are angular shifts in the 2p cases. We also find that the different 2p states differ in angular shift from each other. The angular shift difference reflects the effect of the magnetic quantum number of the initial state on PMD. We also use the first-order TDPT to understand the angular shifts in Fig. 4. As shown in Eq. (10), the PMD is determined by the initial state. We notice that the electron probability density of the initial state | ψ_2p+ 〉 (| ψ_2p– 〉 ) exhibits toroidal distribution and the electron rotates anticlockwise (clockwise). Based on the relationship

where ± η is the argument of the complex wavefunction | ψ_2p±〉. One can obtain the first-order amplitude A_± from Eq. (10) as follows:

From Eq. (25), we can see that the transition matrix element 〈 ψ_c|D⋅ E| | ψ_2p|exp (± iη) 〉 is also a function of electronic state, which can encode the information about the sign of the magnetic quantum number. However, the argument of the complex wavefunction, ±η, cannot be added directly to the cos term such as cos (2α – Φ ± η) because η is a function of (x, y). We surmise that the angular shift is related to the sign of the 2p state and the simulation results confirm our suspicions. We rewrite Eq. (23) as follows:

where θ is the deflection angle compared with the result from Eq. (23) in the 1s case. From Figs. 4(b) and 4(d), we note that without the time delay, the maxima of the photoelectron angular distribution appear at around 60° and –120° for the 2p₊ state, while the maxima appear at around –60° and 120° for the 2p_– state. This conclusion is consistent with our conjecture. We also find that the direction of angular shift is the same as the rotation direction of ring current in the case of 2p orbital.

In order to explore the relation between the angular shifts and the time delay for neon atom, we present the PMDs and corresponding photoelectron angular distributions (PADs) of the 2p₊ and 2p_– at two different time delays t_d = 0.25T [Figs. 4(e)–4(h)] and t_d = 0.75T [Figs. 4(i)–4(l)]. For the time delay of t_d = 0.25T, we obtain the PADs shown in Figs. 4(f) and 4(h) for the 2p₊ and 2p_– orbitals, respectively. The result in Fig. 4(f) shows that for the 2p₊ case, the maxima appear at around 100° and 280°. Compared with the case without time delay, the corresponding angle shift Δφ ≈ 40°. For the 2p case, as shown in Fig. 4(h), the maxima appear at around −20° and 160°, of which the corresponding angle shift is also Δφ ≈ 40° compared to the case without time delay. When time delay t_d = 0.75T, for both of the 2p orbitals, the corresponding angle shifts Δφ ≈ 140°, which is compared with the case without time delay [as shown in Figs. 4(j) and 4(l)]. From the above results it can be found that the varying of the time delay causes the PMDs to rotate counterclockwise an angular shift Δφ, of which the value is the same as that of 1s orbital under the same conditions.

To verify our conclusion, we display the PMDs and the corresponding PADs of photoionization for neon atom with the 2p₊ initial state under a pair of time-delayed elliptically oppositely polarized pulses with an ellipticity of 0.25 in Fig. 5. One can see that the angular shifts [shown in Figs. 5(e)–5(h)] are also in good agreement with the predictions in Eq. (23) [as shown in Fig. 3(j)]. We also compare the magnitudes of PMDs in Fig. 4 with those in Fig. 5. It is found that the magnitudes of PMDs change with time delay and ellipticity for the 2p₊ orbital in the same way as those for the 1s orbital in Fig. 3.

In addition, we carry out a lot of other calculations and find that for the same 2p orbital in the same laser frequency conditions, the deflection angle θ is the same, that is, θ is a constant. For example, the PMDs of 2p₊ orbital under a pair of co-rotating polarization pulses are also calculated, the results show that the changing of the time delay will only result in the change of PMD magnitude and the ejection angles of the maxima are always stay at 60° and –120°. Therefore, under a pair of time-delayed laser pulses, the rotation direction of photoionization yield only depends on time delay and but not on the sign of the initial 2p state, which is consistent with the result in Eq. (26).