Ultrafast Magnetic Field Generation in Molecular $\pi$-Orbital Resonance by Circularly Polarized Laser Pulses

Abstract

Optically induced magnetic fields,from the femtosecond nuclear to attosecond electron time scales are shown to be produced by intense ultrashort laser pulses due to highly nonlinear nonperturbative optical response. The light-matter interaction results in coherent electron currents, giving rise to magnetic field generation. Schemes with bichromatic high-frequency co-rotating and counter-rotating circularly polarized UV light pulses are used to produce the spatial and temporal evolution of the generated magnetic field. The one-electron molecular ion H$_2^+$ as a benchmark model is used to describe the ultrafast photophysics process. Under the condition of molecular resonance excitation, results obtained from ab-inito simulations show a strong dependence on the molecular alignment. In bicircular polarization processes, the interference effects between multiple resonant excitations modulate the evolution of the generated magnetic field, thus leading to pulse relative phase dependence. It is found that the modulation of generated magnetic fields is dependent on the pulse frequency and helicity combination. Molecular resonant excitation induces coherent ring electron currents, resulting in suppression of the phase dependence. Pulse helicity effects illustrate laser induced electron dynamics in bichromatic circular polarization processes. These highly nonlinear phenomena are described by attosecond ionisation and coherent electron current models. The results offer a guiding principle for generating ultrafast magnetic fields and for studying coherent electron dynamics in complex molecular systems.

Kai-Jun Yuan is a deceased author.

Appendices

Appendix A: Interference in Multi-pathway Photoionization

We here derive the multi-pathway electron interference modes in bichromatic fields based on the perturbation ionization theory [60,61,62]. The pulse frequencies $\omega _2=2\omega _1$, where $\omega _1<I_p$ and $\omega _2>I_p$, and $I_p$ the molecular ionization potential, are used. The bichromatic circularly polarized laser pulses can produce photoelectron wave packets with the same kinetic energies $E_e=2\omega _1-I_p=\omega _2-I_p$, which then interfere each other in the continuum.

For direct one $\omega _2$ photon ionization processes by laser pulses, the relevant transition matrix element $\mathscr {A} ^{(2)}$ can be expressed simply in the dipole form

$$\begin{aligned} \mathscr {A}^{(2)} = \langle \psi _I | \mathbf{D }\cdot \mathbf{F }_2(\omega _2 )|\psi _0 \rangle =\mathscr {W}^{(2)}\mathscr {F}_2(\omega _2), \end{aligned}$$

(6.10)

with the first order ionization amplitude

$$\begin{aligned} \mathscr {W}^{(2)}=\langle \psi _I |\mathbf { {D}} \cdot \mathbf{e }|\psi _0 \rangle ,\end{aligned}$$

(6.11)

where $\psi _0$ and $\psi _I$ are respectively the initial ground state and the continuum state. $\mathbf{D }$ is the electric dipole operator. $\mathbf{F} _2(\omega _2)=\mathbf{e}\mathscr {F}_2(\omega _2)=E(\omega _2)e^{i\phi _2}$ is the ionizing pulse amplitude with unit vector $\mathbf{e }$ and field $E(\omega _2)$.

For the $\omega _1$ laser pulse, at least a two $\omega _1+\omega _1$ photon absorption is required to ionize H$_2^+$. The transition matrix element reads as

$$\begin{aligned} \mathscr {A}^{(1)}=\mathscr {W}^{(1)} \mathscr {F}^2_1(\omega _1) \end{aligned}$$

(6.12)

with the amplitude

$$\begin{aligned} \mathscr {W}^{(1)}\sim & {} \int dE \frac{\langle \psi _I |\mathbf {D} \cdot \mathbf {e} |\psi _n\rangle \langle \psi _n|\mathbf {D}\cdot \mathbf {e}|\psi _0\rangle }{E_{1s\sigma _g}-E_{ni}+\omega _1+i\Delta \epsilon }, \end{aligned}$$

(6.13)

where $\psi _n$ and $E_{ni}$ are the wavefunction and energy of the intermediate (virtual) electronic state and $\Delta \epsilon$ is the level width, and laser fields $\mathscr {F}_1(\omega _1)=E(\omega _1)e^{i\phi _1}$. In equation (6.13) we see that the ionization amplitude $\mathscr {W}^{(1)}$ is determined by the field-free structure of the molecule and the laser pulse [63].

For a two $\omega _1$ photon ionization with an intermediate resonant excited state, the transition matrix element $\mathscr {A}^{(1)}$ can be written as [64]

$$\begin{aligned} \mathscr {A}^{(1)}=\mathscr {A}^{iR}T_R\mathscr {A}^{Rc}, \end{aligned}$$

(6.14)

where $T_R$ is the life time of the intermediate resonant state. $\mathscr {A}^{iR}$ is the one photon transition between the initial ground state and the intermediate resonant electronic state, which is determined by the pulse area theorem. $\mathscr {A}^{Rc}$ is the photoionization of the intermediate resonant electronic state.

For photoionization by bichromatic laser pulses, that is, simultaneous two $\omega _1 + \omega _1$ photon and one $\omega _2=2\omega _1$ photon ionization, the total transition probability ${\mathscr {P}}$ is the square of the two amplitudes with an interference term of the cross products of the two one- and two-photon ionization amplitudes, that is,

$$\begin{aligned} \mathscr {P}=\mathscr {P}^{(1)} +\mathscr {P}^{(2)}+\mathscr {P}^{(1,2)}, \end{aligned}$$

(6.15)

where $\mathscr {P}^{(1)}=|\mathscr {|}A^{(1)}|^2$, $\mathscr {P}^{(2)}=\mathscr {A}^{(2)}are|^2$, and $\mathscr {P}^{(1,2)}$ is the interference term which can be simply written as

$$\begin{aligned} \mathscr {P}^{(1,2)}= & {} \mathscr {A} ^{(1) *}\mathscr {A}^{(2)}+\mathscr {A} ^{(1)}\mathscr {A}^{(2)*} \nonumber \\= & {} 2\mathscr {W}^{(1)}\mathscr {W}^{(2)}E ^2(\omega _1)E (\omega _2)\cos (\Delta \eta ), \end{aligned}$$

(6.16)

where $\mathscr {W}^{(1)}$ and $\mathscr {W} ^{(2)}$ are the two $\omega _1$ and one $\omega _2$ photon transition amplitudes corresponding to (6.11) and (6.13). The phase difference $\Delta \eta =\Delta \phi +\Delta \xi$. It should be noted that the phase difference depends on the helicity of bichromatic fields. For a co-rotating case, the relative pulse phase difference $\Delta \phi =\phi _2-2\phi _1$ and $\Delta \xi =\xi _{\omega _2}-\xi _{\omega _1}$, where $\xi _{\omega _1}$ and $\xi _{\omega _2}$ are respectively the phases of the continuum electron wave packets from two $\omega _1$ and one $\omega _2$ photoionization, whereas for a counter-rotating case, $\Delta \phi =\phi _2+2\phi _1$ and $\Delta \xi =\xi _{\omega _2}+\xi _{\omega _1}$. The phase difference $\Delta \xi$ is determined by the molecular intrinsic characteristic.

Appendix B: Numerical Methods

For the aligned molecule ion H$_2^+$ within Born-Oppenheimer approximation and static nuclear frames, the corresponding 3D TDSE in cylindrical coordinates $\mathbf{r} =(\rho ,\theta ,z$) reads as,

$$\begin{aligned} i \frac{\partial }{\partial t} \psi (\mathbf{r} ,t)=\left[ -\frac{1}{2}\bigtriangledown _\mathbf{r}^2 +V_{en}(\mathbf{r} )+V_L (\mathbf{r} ,t)\right] \psi (\mathbf{r} ,t). \end{aligned}$$

(6.17)

$\bigtriangledown _\mathbf{r}$ is the Laplacian operator, and $V_{en}$ is the electron-nuclear potential. The circularly polarized laser pulse propagates along the z axis, perpendicular to the (x, y) plane with $x=\rho \cos \theta$ and $y=\rho \sin \theta$, Fig. 6.1. The radiative interaction between the laser field and the electron $V_L(\mathbf{r} )=\mathbf{r} \cdot \mathbf{E} (t)$ is described in the length gauge for a single color circularly polarized pulse,

$$\begin{aligned} \mathbf{E} (t)= & {} E f(t) [\hat{e}_x \cos (\omega t) +\hat{e}_y \xi \sin (\omega t) ], \end{aligned}$$

(6.18)

where $\hat{e}_{x/y}$ is the laser polarization direction and the symbol $\xi =\pm 1$ denotes the helicity of the field, $\xi =1$ for right hand and $\xi =-1$ for left hand, or a bichromatic circularly polarized pulse

$$\begin{aligned} \mathbf{E} (t)= & {} \mathbf{E} _1(t)+\mathbf{E} _2(t) \nonumber \\= & {} E f(t)\left\{ \begin{array}{c} \hat{e}_x [ \cos (\omega _1 t+\phi _1)+\cos (\omega _2 t+\phi _2)] \\ \hat{e}_y [ \sin (\omega _1 t+\phi _1)+ \xi \sin (\omega _2 t+\phi _2) ] \end{array}\right. , \end{aligned}$$

(6.19)

where $\xi$ presents the helicity of the combined field, i.e., co-rotating ($\xi =1$) and counter-rotating ($\xi =-1$) components. $\phi _{1}$ and $\phi _2$ are CEPs of the pulses $\mathbf{E} _1(t)$ and $\mathbf{E} _2(t)$. A smooth $\sin ^2(\pi t /T_{lp})$ pulse envelope f(t) for maximum amplitude E, intensity $I =I_x=I_y={ c\varepsilon _0 E ^2}/2$ and duration $T_{lp}=n\tau _{1}$ is adopted, where one optical cycle period $\tau _{1,2}=2 \pi /\omega _{1,2}$, n=5 cycles. This pulse satisfies the total zero area $\int E(t) dt=0$ in order to exclude static field effects [1].

The 3D TDSE in (6.17) is propagated by a second order split operator method which conserves unitarity in the time step $\Delta t$ combined with a fifth order finite difference method and Fourier transform technique in the spatial steps $\Delta \rho$, $\Delta z$, and $\Delta \theta$ [65, 66]. The initial electron wavefunction $\psi (\mathbf{r} , t = 0)$ is prepared in the ground $1s\sigma _g$ state calculated by propagating an initial appropriate wavefunction in imaginary time using the zero-field TDSE in (6.17). The time step is taken to be $\Delta t=0.01$ a.u.=0.24 as. The spatial discretization is $\Delta \rho =\Delta z=0.25$ a.u. for a radial grid range $0 \le \rho \le 128$ a.u. (6.77 nm) and $|z|\le$ 32 a.u. (1.69 nm), and the angle grid size $\Delta \theta =0.025$ radian. To prevent unphysical effects due to the reflection of the wave packet from the boundary, we multiply $\psi (\rho ,\theta ,z,t)$ by a “mask function" or absorber in the radial coordinates $\rho$ with the form $\cos ^{1/8}[\pi (\rho -\rho _\text {a})/2\rho _{\text {abs}}]$. For all results reported here we set the absorber domain at $\rho _{\text {a}}=\rho _{\text {max}}-\rho _{\text {abs}}$=104 a.u. with $\rho _{\text {abs}}=24$ a.u., exceeding well the field induced electron oscillation $\alpha _d={E }/{\omega _{1/2}^2}$ of the electron [35].

The time dependent electronic current density is defined by the quantum expression in the length gauge,

$$\begin{aligned} \mathbf{j} (\mathbf{r} ,t)=\frac{i}{2}[\psi (\mathbf{r} ,t) \nabla _\mathbf{r } \psi ^*(\mathbf{r} ,t) -\psi ^*(\mathbf{r} ,t)\nabla _\mathbf{r }\psi (\mathbf{r} ,t)], \end{aligned}$$

(6.20)

$\psi (\mathbf{r} ,t)$ is the exact Born-Oppenheimer (static nuclei) electron wave function obtained from the TDSE and $\nabla _\mathbf{r }=\mathbf{e} _\rho \nabla _\rho +\mathbf{e} _\theta \frac{1}{\rho }\nabla _\theta +\mathbf{e} _z\nabla _z$ in cylindrical coordinates. Then the corresponding time dependent magnetic field is calculated using the following classical Jefimenko’s equation [67],

(6.21)

where $t_r=t-r/c$ is the retarded time and $\mu _0=4\pi \times 10^{-7}$ NA$^{-2}$ (6.692$\times 10^{-4}$ a.u.) is called the permeability of free space. Units of $B(\mathbf{r} ,t)$ are Teslas (1T=$10^{4}$ Gauss). For the static zero-field time-independent conditions occurring after the pulse, then (6.21) reduces to the classical Biot-Savart law [67].