Imaging and manipulating molecular electron dynamics is one of the main goals in photophysical processes and photochemical reactions. Advances in synthesizing ultrashort intense laser pulses [1, 2] allow one to visualize and control electrons on their natural attosecond (1 as = 10⁻¹⁸ s) timescale and sub-nanometer dimension [3–6]. One important application of ultrashort circularly polarized attosecond pulses is to produce strong magnetic field pulses from electronic ring currents in atomic and molecular systems [7–13]. By creating unidirectional constant valence-type electronic currents in molecules with circularly polarized UV laser pulses, static magnetic fields [7–9] can be efficiently generated by the excitation of resonant degenerate orbitals. These laser-induced magnetic fields are much larger than those obtained by traditional static field methods [14]. In [8], it has been found that for the hydrogen-like atom, the existence of ring currents is related to the presence of the states having nonzero magnetic orbital momentum magnetic quantum numbers. Surprisingly, the strongest magnetic field originates from the 2 p ± orbital in the hydrogen-like atom, which can be prepared via resonant 1 s → 2 p ± transition. One also finds that, in general, ring electronic currents are dependent on the symmetry of the molecular orbitals. The helicity of driving circularly polarized pulses [15] can be used to reconstruct attosecond charge migration [16–19]. Linearly polarized laser pulses can also induce excited ring currents by controlling the rotation direction of π electrons in planar/nonplanar aromatic molecules [20–22]. We have proposed methods previously to create “spinning” continuum electrons which can be generated and remain localized on sub-nanometer molecular dimensional scales [23, 24], offering a way to produce high-order harmonic generation (HHG). Time-dependent circular coherent electron wave packets (CEWPs) and currents are created as superposition of bound-continuum states. They thus become the source of intense time-dependent internal magnetic fields generated on attosecond timescale. The induced attosecond magnetic fields have been shown to be a function of various laser pulse parameters, such as the pulse intensity, wavelength, and duration [25, 26], thus providing new tools for control of ultrafast optical magnetism generation [27–31].

Investigating ultrafast electron dynamics by bichromatic circularly polarized attosecond laser pulses with corotating or counter-rotating components has been attracting considerable attention in the field of light–matter interactions. It has already been shown that counter-rotating intense ultrafast circularly polarized pulses can induce re-collision, thus ensuring efficient HHG [32–35], the new source of circularly polarized X-ray attosecond pulses. These counter-rotating laser fields are now being adopted to produce circularly polarized HHG with nonzero initial angular momenta [36–38]. With counter-rotating circularly polarized laser pulses, the technique of double optical gating can be efficiently employed for producing isolated elliptically polarized attosecond pulses [39]. Bichromatic laser fields have also been adopted to probe atomic and molecular structure by photoelectron momentum distributions [40]. By combination of two circularly polarized attosecond ultraviolet (UV) pulses, spiral electron vortices in photoionization momentum distributions have been predicted theoretically in both atomic [41–43] and molecular systems [44–46], which are shown to be sensitive to the helicity of the bichromatic fields. Recent experiments have demonstrated this fact by focusing on multiphoton femtosecond ionization of potassium atoms [47, 48]. Most recently, above-threshold ionization obtained previously by a bicircular field has been reported [49–52].

In this work, we present attosecond magnetic field generation and electron currents under molecular resonant excitation in bichromatic attosecond circular polarization processes. Such ultrafast attosecond pulses have been generated by current laser techniques from circularly polarized HHG [53–55]. Numerical simulations are performed on the aligned molecular ion H 2 + at equilibrium by numerically solving the corresponding three-dimensional (3D) time-dependent Schrödinger equation (TDSE). Ultrafast magnetic field generation has been studied previously by bichromatic circularly polarized laser pulses [56, 57]. It has shown that the interference effect between multiple ionization pathway influences the magnetic field generation. However, the effect of the coherent electron currents in bicircular magnetic field processes under molecular resonant excitation has not been presented. We focus here on the pulse phase and helicity-dependent magnetic field generation. We demonstrate molecular resonant excitation effects by comparing the dependence of the generated ultrafast magnetic field on the relative carrier-envelope phase (CEP) and the helicity of pulses at different molecular alignments. It is also found that attosecond charge migration arising from coherent resonant excitation induces coherent electron ring currents in molecules, leading to an absence of the CEP dependence. Induced electron ring currents resulting from molecular coherent resonant excitation are shown to be an important factor in bichromatic magnetic field generation. These results allow to control ultrafast magnetic fields, leading to molecular attosecond charge migration dynamics. Since molecular vibrational and rotational effects occur on the femtosecond (1 fs = 10⁻¹⁵ s) and picosecond (1 ps = 10⁻¹² s) timescales, fixed nuclei simulations are valid and used to describe ultrafast magnetic field generation processes on the attosecond timescale.

The article is organized as follow: We briefly describe the computational method for solving TDSEs of the aligned molecular ion H 2 + to simulate electron currents and magnetic field generation in Section 2. The results of ultrafast magnetic fields by intense bichromatic circularly polarized attosecond XUV laser pulses are presented and discussed in Section 3. In Section 4, we finally summarize our findings. Throughout this article, atomic units (au) which are defined by setting ℏ = e 2 = m e = 1 are used unless otherwise stated.

For the aligned molecule ion H 2 + within Born–Oppenheimer approximation static nuclear frames, the corresponding 3D TDSE in the cylindrical coordinate r = ( ρ , θ , z ) reads as i ∂ ∂ t ψ ( r , t ) = [ − 1 2 ▽ r 2 + V e n ( r , R ) + V L ( r , t ) ] ψ ( r , t ) , (1) with the Laplacian − 1 2 ▽ r 2 = − 1 2 ρ ∂ ∂ ρ ( ρ ∂ ∂ ρ ) − 1 2 ρ 2 ∂ 2 ∂ θ 2 − 1 2 ∂ 2 ∂ z 2 , (2) where V e n ( r , R ) is the electron-nuclear potential in which we fixed the internuclear separation to the equilibrium separation and |R| = 2 au. The circularly polarized laser pulse propagates along the z axis, perpendicular to the (x,y) plane, with x = ρ cos θ and y = ρ sin θ , as shown in Figure 1. The radiative interaction between the laser field and the electron V L ( r ) = r ⋅ E ( t ) is described in the length gauge for circularly polarized pulses of frequencies ω 1 and ω 2 . E ( t ) = E 1 ( t ) + E 2 ( t ) = E f ( t ) { e ^ x [ cos ( ω 1 t + ϕ 1 ) + cos ( ω 2 t + ϕ 2 ) ] e ^ y [ sin ( ω 1 t + ϕ 1 ) + ϵ ⁡ sin ( ω 2 t + ϕ 2 ) ] , (3) where e ^ x / y is the laser polarization direction, and the symbol ϵ = ± 1 denotes the helicity of combined fields, that is, corotating ( ϵ = + 1 ) or counter-rotating ( ϵ = − 1 ) components. ϕ 1 and ϕ 2 are CEPs of the pulses E 1 ( t ) and E 2 ( t ) , respectively. A smooth sin 2 ( π t / T l p ) pulse envelope f ( t ) for maximum amplitude E, intensity I = I x = I y = c ε 0 E 2 / 2 , and duration T l p = n τ is adopted, where one optical cycle period τ 1,2 = 2 π / ω 1,2 .

The 3D TDSE in Eq. 1 is propagated by a second-order split operator method which conserves unitarity in each time step Δ t combined with a fifth-order finite difference method and Fourier transform technique in the spatial steps Δ ρ , Δ z , and Δ θ [59, 60]. The initial electron wave function ψ ( r , t = 0 ) is prepared in the ground 1 s σ g state calculated by propagating an initial appropriate wave function in imaginary time using the zero-field TDSE in Eq. 1. The time step is taken to be Δ t = 0.01  au = 0.24 as. The spatial discretization is Δ ρ = Δ z = 0.25  au for a radial grid range 0 ≤ ρ ≤ 128  au (6.77 nm) and |z| ≤ 32 au (1.69 nm), and the angle grid size Δ θ = 0.025 radian. To prevent unphysical effects due to the reflection of the wave packet from the boundary, we multiply ψ ( ρ , θ , z , t ) by a “mask function” or absorber in the radial coordinate ρ with the form cos 1 / 8 [ π ( ρ − ρ a ) / 2 ρ abs ] . For all results reported here, we set the absorber domain at ρ _a = ρ _max −ρ _abs = 104 au with ρ _abs = 24 au, exceeding well the field-induced electron oscillation α d = E / ω 1 / 2 2 of the electron [25]. The time-dependent probability current density is defined by the quantum expression in the length gauge, j ( r , t ) = i ℏ 2 m e [ ψ ( r , t ) ∇ r ψ * ( r , t ) − ψ * ( r , t ) ∇ r ψ ( r , t ) ] , (4) where ψ ( r , t ) is the exact Born–Oppenheimer (static nuclei) electron wave function obtained from the TDSE in Eq. 1, and ∇ r = e ρ ∇ ρ + e θ ∇ θ / ρ + e z ∇ z is the momentum operator in cylindrical coordinates. Then the corresponding time-dependent magnetic field is calculated using the following classical Jefimenko equation [61]: B ( r , t ) = − e μ 0 4 π ∫ ​ [ j ( r ′ , t r ) | r − r ′ | 3 + 1 | r − r ′ | 2 c ∂ j ( r ′ , t r ) ∂ t ] × ( r − r ′ ) d 3 r ′ , (5) where t r = t − r / c is the retarded time and μ 0 = 4 π × 10 − 7  NA⁻² (6.692 × 10⁻⁴ au) is the permeability of free space. Units of B ( r , t ) are teslas (1 T = 10⁴ Gauss) if the elementary charge e is in Coulombs. For the static zero-field time-independent conditions occurring after the pulse duration, Eq. 5 reduces to the classical Biot–Savart law, that is, B ( r , t ) = − e ( μ 0 / 4 π ) ( v × r ) / r 3 [61]. Note that the retardation effects due to r/c = 0.35 attoseconds (where for an estimate r = R = 2 au. is used) are negligible.

Of note is that equation Eq. 4 defines the quantum probability current (not the electric current) as defined in any quantum mechanics textbook. The electron electric current used in the Biot–Savart law in Eq. 5 is therefore j electric = − e j . This explains the sign (−) in Eq. 5, e = 1 in atomic units.

The ground and excited states of H 2 + ion and its electron potentials, 1 s σ g , 1 s σ u , 2 s σ g , 2 s σ g , 1 π u , etc., are well documented in [65]. Numerical solutions of the TDSE for H 2 + aligned with the laser polarization (x–y) plane, that is, R ∥ to the x-axis with the electric field vector E ( t ) rotating in the (x,y) plane (Figure 1A), are used for obtaining time-dependent probabilities P(t) of the 1 s σ g and 2 p π u states at equilibrium R = 2 au. excited by a 70-nm pulse at two different intensities, I = 2× 10¹⁴ W/cm² and I = 1 × 10 15 I = 1×10¹⁵ W/cm², by a five-cycle pulse (one cycle = 0.234 fs = 234 as). Thus, at I = 2 × 10 14 , Figure 2A, one sees no significant ionization, whereas at I = 1 × 10¹⁵ W/cm², Figure 2B, the 2 p π u state is 85% occupied, and still one sees little ionization. In conclusion, short few-cycle intense and resonant pulses contribute little ionization with major excitation of the resonant state, such as the 2 p π u state at 70 nm and some Rydberg states above the 2 p π u state shown in 1.

We investigate laser-induced highly nonlinear optical effects using pairs of bichromatic circularly polarized laser pulses. We use λ ₁ = 70 nm (ω ₁ = 0.65 au) circularly polarized pulse in combination with λ ₂ = 35 nm (ω ₂ = 2ω ₁ = 1.3 au) circularly polarized pulse. Pairs of circularly polarized harmonics of different frequency and helicity can easily be prepared by a combination of pairs of counter-rotating circularly polarized laser pulses at different frequencies [62]. The molecular ion H 2 + is aligned along the x/z axis, the two X-ray ultra violet (XUV) pulses with their field polarization vectors in the (x,y) plane propagate along the z axis, as illustrated in Figure 1A, for in-plane excitation, and Figure 1B for around axis excitation. With the pulse frequency ω 1 = Δ E and the energy difference Δ E = E π u − E σ g = 0.65  au, a resonant excitation with the ground 1 s σ g state and the excited π u electronic states occurs. Moreover, with ω 2 = 2 ω 1 , such bichromatic laser pulses can also produce CEWPs with the same kinetic energies, E e = 2 ω 1 − I p = ω 2 − I p , by combination of multiple multiphoton transitions to Rydberg and the continuum state, thus leading to electron interference between the two ionization pathways. Since the induced electron currents are localized in the laser ( x , y ) polarization plane, the generated magnetic field is concentrated along the z axis. We therefore only present the results of the magnetic field B along the z axis.

As illustrated in Figure 1C, the coherent resonant excitation between the ground 1 s σ g state and the excited π u state by the λ ₁ = 70 nm circularly polarized laser pulse is dependent on the molecular alignment. In the case of R ∥ E , the in-plane degenerates perpendicular excitation, π u ± ← σ g for different angular momenta Δ m = ± 1 dominates, and the electron density distribution evolves nearly perpendicular to the molecular axis [63], whereas in the case of R ⊥ E , the produced electron wave packets move around the molecular axis due to individual π u + / − ← σ g transitions Δ m = 1 or − 1 [64]. Meanwhile, two ω 1 and single ω 2 transitions to the continuum ( 2 ω 1 = ω 2 > I p ) occur, leading to interference effects between these two excitation ionization channels. We show how coherent resonant excitation influences the interference effect on the magnetic field generation. We present the effects of the pulse phase on the attosecond magnetic field generation under the resonant excitation with various molecular alignments. Processes with different helicities ϵ, that is, corotating ( ϵ = + 1 ) and counter-rotating ( ϵ = − 1 ) pulse combinations, are compared as well.

We present the ultrafast magnetic field generation in molecules from one electron molecular TDSE simulation under effects of coherent resonant excitation in bichromatic ( ω 2 = 2 ω 1 ) co- and counter-rotating circularly polarized laser fields. Numerical results are obtained for the aligned molecular ion H 2 + , which can be fully and exactly studied. We evaluate the generated magnetic field B ( r = 0 ) at the molecular center at different relative CEP ϕ of the two circularly polarized pulses with both counter-rotating and corotating combinations. It is found that altering the CEP ϕ varies the maximum values of the generated magnetic field, B ( r = 0 ) , which is shown to be dependent on the molecule alignment, and is maximum for in-plane bichromatic excitation.

In a bichromatic (frequency ω 1 ≠ ω 2 ) circularly polarized field, a σ g − π u ± resonant excitation of individual magnetic components m = ± 1 is triggered by the resonant ω 1 (70 nm) pulse. As a result, coherent electron currents between the ground σ g state and the degenerate excited π u ± electronic state are induced in molecules. We compare two nonlinear responses for different molecular alignments:

• For the around-axis case, R ⊥ E , the molecule aligned along the z axis, perpendicular to the laser ( x , y ) polarization plane, Figure 1B, and the magnetic field mainly results from the coherent electron currents which are induced by one resonant ω 1 photon around R. The contribution from two ω 1 photons or one ω 2 photon is negligible. Varying the relative CEP ϕ does not influence the generated magnetic field. Similar results are obtained for both counter-rotating ε = − 1 , Figure 3B, and corotating ε = + 1 , Figure 6B, excitations, confirming the main role of the around-axis circular coherent electron current.

The present results in principle provide the importance of coherent electron dynamics and of control magnetic fields by bichromatic circularly polarized laser pulses. The dependence of the generated magnetic field on the relative phase and helicity of driving laser pulses also allows to characterize the property of laser pulses and probe coherent electron currents and to charge migration in molecules. Although a simple single electron molecular ion H 2 + is used, similar electron dynamics phenomena should be predicted in more complex molecular systems [7, 27, 30, 73], thus offering an approach for controlling ultrafast magnetic field generation.

The above laser-induced molecular magnetic field generation on the electron’s quantum timescale, the asec, was studied in the Born–Oppenheimer Approximation, that is, with static nuclei. Nuclear motion effects, that is, non–Born–Oppenheimer, are now being pursued on the near femtosecond timescale in order to include nuclear motion effects with bound and dissociation molecular states [83], de-and re-coherence in charge migration [84] and isotope effects in HD+ ultrafast ionization [85]. In the case of laser pulses propagating perpendicular to the molecular R-axis with the pulse electric fields in the molecular plane, Figure 1A, re-collision of electron currents with nuclei is an important nonlinear optical effect shown in Figure 6 to be examined in detail for moving nuclei. Finally, the strong magnetic fields generated by intense ultrafast laser pulses are expected to interact with the electron currents themselves. Proton beams have been shown recently to be useful tools to measure intense magnetic field directions generated by current solenoids [86], thus confirming that laser-generated magnetic fields can interact also with nuclei in matter.