⇦Chinese Physics Letters, 2022, Vol. 39, No. 12, Article code 123402 Quantum Engineering of Helical Charge Migration in HCCI ChunMei Liu (刘春梅)1, Jörn Manz (源满)2,3,4*, Huihui Wang (王慧慧)3,4, and Yonggang Yang (杨勇刚)3,4* Affiliations 1Crystal Physics Research Center, College of Science, Nanjing University of Posts and Telecommunications, Nanjing 210023, China 2Institut für Chemie und Biochemie, Freie Universität Berlin, 14195 Berlin, Germany 3State Key Laboratory of Quantum Optics and Quantum Optics Devices, Institute of Laser Spectroscopy, Shanxi University, Taiyuan 030006, China 4Collaborative Innovation Center of Extreme Optics, Shanxi University, Taiyuan 030006, China Received 11 October 2022; accepted manuscript online 8 November 2022; published online 22 November 2022 *Corresponding authors. Email: ygyang@sxu.edu.cn; jmanz@chemie.fu-berlin.de Citation Text: Liu C M, Manz J, Wang H H et al. 2022 Chin. Phys. Lett. 39 123402    Abstract Electronic charge of molecules can move on time scales when the nuclei stand practically still, from few hundreds of attoseconds to few femtoseconds. This ultrafast process is called “charge migration”. A typical consequence is rapid change of electronic dipole, which points to the center of charge. Corresponding linear (one-dimensional, 1D) and planar (2D) dipolar motions have already been well documented. Here we construct the first case of charge migration which causes chiral 3D dipolar motion, specifically along a helix about oriented iodo-acetylene (HCCI). Quantum dynamics simulations show that this can be induced by well-designed laser pulses.
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 DOI:10.1088/0256-307X/39/12/123402 © 2022 Chinese Physics Society Article Text Charge migration has been discovered as ultrafast process of electrons, first in linear molecules[1-15] or in quasi-linear ones (e.g., in peptides[16-19]). For example, a laser pulse may photo-ionize the molecule, thus creating an electron hole at one end; the charge will then migrate to the opposite end, on time scale from few hundreds of attoseconds (as) to few femtoseconds (fs). This has, or it may have several consequences: In particular, it induces motions of the electronic dipole (which points to the center of the moving charge) along a line (1D), and we call this “linear charge migration” (LCM). It may also induce sequel processes such as selective bond breaking.[16-18] Subsequently, it was discovered that a laser pulse can also induce circular charge migration (CCM), e.g., about an oriented diatomic molecule[5,20,21] or in a planar molecule (e.g., in benzene[22,23] or in a metal-porphyrin[24-26]). This has consequences which are different from the 1D case of LCM. In particular, CCM induces dipolar motions in a plane (2D).[5,20-39] Moreover, it may generate giant magnetic fields, see, e.g., Refs. [25,26,35,39]. There are also exceptional cases of charge migration which do not change the dipole, cf. Refs. [4,40]. After the familiar 1D and 2D cases of LCM[1-19,28,41] and CCM,[5,20-39] the main purpose of this Letter is to demonstrate, as a proof-of-principle, that charge migration may also induce three-dimensional (3D) motions of the electronic dipole, specifically along a helix about an oriented linear molecule. This will be called “helical charge migration” (HCM). The present HCM should open new applications, e.g., ultrafast manipulation of chirality. The secondary purpose of this study is to present two different yet equivalent strategies for the design of laser pulses which yield HCM in an oriented linear molecule, and this will be called “quantum engineering of HCM”, in analogy to classical engineering of helical motion. The proof of principle will be demonstrated for iodo-acetylene HCCI. This molecule is chosen because it satisfies an important pre-requisite,[42] namely researchers have already demonstrated that it can be oriented along the laboratory $z$-axis, with subsequent ionization and induction of LCM in the cation HCCI$^+$.[8,11,12] Those results stimulated the discovery of decoherence and recoherence of charge migration in HCCI$^+$.[13] The present investigation adapts the scenario of Refs. [8,11,12], i.e., HCCI is oriented along $z$, with the center of mass at the origin and with the nuclei frozen in the global minimum geometry of the electronic ground state. Effects of moving nuclei will be discussed finally. Note, however, that we consider the neutral HCCI (not HCCI$^+$). Helical motions are coaxial superpositions of linear and circular motions. By analogy, HCM in a linear molecule oriented along $z$ is a coaxial superposition of LCM along $z$ and CCM about the $z$-axis, in brief: HCM = LCM + CCM. This motivates our first strategy for quantum engineering of HCM in the oriented linear molecule. Specifically, we employ a superposition of a linearly $z$-polarized laser pulse and a right circularly polarized laser pulse which propagates in $z$-direction. It is well known that the individual laser pulses induce LCM and CCM. cf., Refs. [1-19,28,41] and Refs. [5,20-39], respectively. As a consequence, the superposition of the two laser pulses should yield the target superposition of charge migration LCM + CCM = HCM. To demonstrate the quantum engineering of HCM in the oriented HCCI by strategy I (and later on also by strategy II), we shall now present some important properties of the molecule and of the laser pulses. The geometry and the electronic levels of the ground state $1\varSigma^+$ and the degenerate excited states $1\varPi_x,1\varPi_y$ and $2\varPi_x,2\varPi_y$ of HCCI oriented along the laboratory $z$-axis are illustrated in Figs. 1(a) and 1(b). The values are listed in the Supporting Information (SI). Linearly $x$-, $y$- and $z$-polarized laser pulses can induce selective symmetry-allowed transitions between these states, as indicated in Fig. 1(b). The corresponding transition matrix elements (TMEs) are also listed in the SI. The levels are the same as those for the set of states $1\varSigma^+$, $1\varPi_+ $ and $2\varPi_+ $, cf., Fig. 1(c). These states play a key role for our demonstration of the quantum engineering by strategies I and II. For convenience, they are labeled by $j = 0,\,1,\,2$, respectively. The corresponding levels $E_0 < E_1 < E_2$ and the electronic eigenfunctions $\varPsi_0$, $\varPsi_1$, and $\varPsi_2$ are calculated as solutions of the time-independent electronic Schrödinger equation (TISE) $H_{\rm e} \varPsi_j = E_j \varPsi_j$. The corresponding one-electron densities are denoted $\rho_{jj}$. The Hamiltonian $H_{\rm e}$ accounts for the electronic kinetic energy and for the Coulomb interactions of all electrons and nuclei. The quantum chemical methods for solving the TISE and for calculating the $\rho_{jj}$ and the TME are explained in the SI.
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Fig. 1. Properties of HCCI oriented along the laboratory $z$-axis. (a) Geometry. (b) Energy levels of the ground state $1\varSigma^+ (0.00$ eV) and the degenerate excited states $1\varPi_x, 1\varPi_y (5.60$ eV) and $2\varPi_x, 2\varPi_y (7.88$ eV). Symmetry-allowed resonant transitions between these states can be induced by means of linearly $x$-, $y$- or $z$-polarized laser pulses, as indicated by the vertical double-headed arrows. (c) Same energy levels of states $1\varSigma^+$, $1\varPi_+ =(1\varPi_x + i \cdot 1\varPi_y)/\sqrt{2}$ and $2\varPi_+ = (2\varPi_x + i \cdot 2\varPi_y)/\sqrt{2}$. Strategy I (left) employs a superposition of right (+) circularly and linearly $z-$polarized laser pulses. Strategy II (right) employs right circularly polarized laser pulses, only. The laser pulses induce symmetry-allowed resonant transitions between the states, as indicated by the vertical double-headed arrows.
The laser induced resonant transitions for strategies I and II are indicated in Fig. 1(c). The electric fields of the employed transform limited right (+) circularly and linearly $z$-polarized laser pulses are designed as \begin{equation} \begin{split} \boldsymbol{\epsilon}_+(t; lp_+) =\,& \epsilon_x(t; lp_+) \boldsymbol{e_x} + \epsilon_y(t; lp_+) \boldsymbol{e_y}\\ \equiv \,&\epsilon_+ \exp(-t^2/\tau_+^2) \cdot [ \cos(\omega_+ t + \eta_+) \boldsymbol{e_x}\\ &+ \sin(\omega_+ t + \eta_+) \boldsymbol{e_y}] \\ \boldsymbol{\epsilon_z}(t; lp_z) =\,& \epsilon_z(t; lp_z) \boldsymbol{e_z}\equiv \epsilon_z \exp(-t^2/\tau_z^2) \cos(\omega_z t + \eta_z) \boldsymbol{e_z}. \end{split}\tag {1} \end{equation} The laser parameters $lp =\{\epsilon, \tau, \omega, \eta\}$ include the field strength $\epsilon$, the duration $\tau$, the resonant carrier frequency $\omega$, and the carrier envelope phase (CEP) $\eta$. The unit vectors $\boldsymbol{e_x}, \boldsymbol{e_y}, \boldsymbol{e_z}$ point to the laboratory $x,y,z$ directions. The $lp$'s and the corresponding periods $T = 2\pi/\omega$ are labelled either by subscripts $+$ and $z$ for the polarizations, or by the corresponding target transitions between levels $j$ and $k$, e.g., $\omega_+ = \omega_{01} = (E_1 - E_0)/\hbar$ for the resonant frequency of the transition between states $j=0$ and 1. Strategies I and II employ superpositions of the laser pulses with total field strengths \begin{equation} \begin{split} &\boldsymbol{\epsilon}_{\rm I}(t)= \boldsymbol{\epsilon}_+(t; lp_{01}) + \boldsymbol{\epsilon_z}(t; lp_{12}), \\ &\boldsymbol{\epsilon}_{\rm II}(t)= \boldsymbol{\epsilon}_+(t; lp_{01}) + \boldsymbol{\epsilon}_+(t; lp_{02}). \end{split}\tag {2} \end{equation} The values of the $lp$'s are listed in the SI. The laser pulses are negligible and hence they are set equal to zero before the initial time $t_{\rm i} = - 5$ fs and after the final time $t_{\rm f} = 5$ fs. The laser pulses drive the electronic wavefunction from the initial ground state $\varPsi(t = t_{\rm i}) = \varPsi_0$ to the superposition state \begin{equation} \begin{split} \varPsi(t) = c_0(t) \varPsi_0 + c_1(t) \varPsi_1 + c_2(t) \varPsi_2 , \end{split}\tag {3} \end{equation} with populations $P_j(t) = |c_j(t)|^2$ of the electronic eigenstates $\varPsi_j$. The wavefunction (3) is obtained as solution of the time-dependent Schrödinger equation (TDSE) $i \hbar \dfrac{\partial}{\partial t}\varPsi(t)= H(t) \varPsi(t)$, with $H(t) = H_{\rm e} - \hat{\boldsymbol{d}} \cdot \boldsymbol{\epsilon}(t)$, in semiclassical dipole ($\hat{\boldsymbol{d}}$) approximation. The methods for solving the TDSE and for calculating the coefficients $c_j(t)$ are explained in the SI. The SI also has the method for designing the laser pulses such that they yield arbitrarily chosen target values of the coefficients at the end of the laser pulse, $C_j = c_j(t_{\rm f})$. Thereafter, i.e., for times $t' = t - t_{\rm f} > 0$, the populations are robust, $P_j(t') = P_j(t_{\rm f}) = |C_j|^2$, and the coefficients evolve with time-dependent phases, $c_j(t') = C_j \exp(- i E_j t'/\hbar)$. The wavefunction (3) then represents the laser-induced HCM. Following the examples of Refs. [5,28,43,44], this shall be documented by the time-dependent dipole $\boldsymbol{d}(t')$. The calculations of $\rho(t')$ and $\boldsymbol{d}(t')$ are explained in the SI. Supplementary snapshots of the one-electron density $\rho(t')$ or the difference $\Delta \rho(t') = \rho(t') - \bar{\rho}$ between $\rho(t')$ and its mean $\bar{\rho} = \varSigma_j P_j(t_{\rm f}) \rho_{jj}$ are also shown in the SI.
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Fig. 2. Laser engineering of helical charge migration (HCM) of HCCI oriented along the laboratory $z$-direction by a superposition of right circularly and linearly $z$-polarized laser pulses (Strategy I). (a) Electric field components, cf., Eqs. (1) and (2) with parameters in the SI. (b) The resulting populations $P_j(t)$ of states $1\varSigma^+ (j=0)$, $1\varPi_+(j=1)$, and $2\varPi_+(j = 2)$. The subsequent HCM is illustrated in Fig. 3 and Fig. SI-1.
The $x$-, $y$-, $z$-components of the electric fields of the superimposed right (+) circularly and linearly $z$-polarized laser pulses for strategy I are illustrated in Fig. 2(a). The $lp$'s are listed in the SI. The resulting populations $P_j(t)$ of eigenstates $\varPsi_j$ are shown in Fig. 2(b). The laser pulses are designed such that at the end, they yield $P_0(t_{\rm f}) = 0.97$, $P_1(t_{\rm f}) = 0.02$, $P_2(t_{\rm f}) = 0.01$, with real-valued coefficients $C_{j} = c_j(t_{\rm f}) = \sqrt{P_j}(t_{\rm f})$. As is expected, the superposition of the right circularly and linearly $z$-polarized laser pulses for strategy I yields a superposition of circular plus linear charge migrations, i.e., helical charge migration, in brief, CCM + LCM = HCM. This is also obvious from the time evolution of the dipole $\boldsymbol{d}(t')$.[5,28,43,44] The present strategy I yields the helical path of $\boldsymbol{d}(t')$ shown in Fig. 3. The individual components $d_x (t'),d_y (t')$ and $d_z (t')$ of the chiral $\boldsymbol{d}(t')$ are shown in Fig. SI-2. They confirm the superposition HCM = CCM + LCM.
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Fig. 3. Perspective view of the helical path of the time-dependent dipole $\boldsymbol{d}(t')$, mapping helical charge migration (HCM) in HCCI oriented along $z$ in the laboratory frame, for times $t' = t - t_{\rm f}>0$ after the laser pulses for strategies I or II. The sequence of rainbow colors is for the periods $[0, T_{12}/2]$ (red), $[T_{12}/2, T_{12}]$ (yellow), $[T_{12}, 3T_{12}/2]$ (green), and $[3T_{12}/2, 2T_{12}]$ (violet), $T_{12}/2=900$ as.
Let us now proceed to the alternative strategy II. It is motivated by classical engineering which substitutes the linear motion of a piston by the rotation of a wheel. By analogy, the present quantum engineering substitutes the linearly polarized laser pulse for the transition from $1\varPi_+ (j=1)$ to $2\varPi_+ (j=2)$ of strategy I by another right circularly polarized laser pulse, for the resonant transition from $1\varSigma^+ (j=0)$ directly to $2\varPi_+ (j=2)$, cf., Fig. 1(c). We keep the right circularly polarized laser pulse of strategy I for resonant transition from the ground state $1\varSigma^+ (j=0)$ to the first excited state $1\varPi_+ (j=1)$, albeit with re-adjusted $lp$'s. The two circularly polarized laser pulses for strategy II are documented in Fig. 4(a). The resulting populations $P_j(t)$'s are in Fig. 4(b). The $lp$'s are in the SI. The laser pulses are designed such that at the end, the populations $P_j(t_{\rm f})$ and also the coefficients $c_j(t_{\rm f})$ of strategy II are the same as those for strategy I. As a consequence, the subsequent HCM (Fig. 3 and Figs. SI-1 and SI-2) is exactly the same for both strategies I and II. In conclusion, the results shown in Figs. 2–4 and Fig. SI-1 document equivalent quantum engineering of helical charge migration (HCM) in oriented HCCI, by means of two entirely different strategies. Strategy I is based on considering HCM as superposition of circular plus linear charge migrations, HCM = CCM + LCM. Accordingly, it prepares HCM by means of a superposition of circularly and linearly polarized laser pulses. Strategy II substitutes the linearly polarized laser pulse of strategy I by another circularly polarized pulse. This provides a quantum engineering analogue of the classical substitution of the linear motion of a piston by the rotation of a wheel.
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Fig. 4. Laser engineering of helical charge migration (HCM) of HCCI oriented along the laboratory $z$-direction by means of two right circularly polarized laser pulses (strategy II). The notations are the same as those for Fig. 2. The subsequent HCM induced by the laser pulses for strategy II is the same as for strategy I, cf., Fig. 3 and Figs. SI-1 and SI-2.
The present proof of the principle should be considered as first step into mostly unexplored territory of research on helical or alternative 3D charge migrations in oriented linear molecules, or in other suitable molecular systems. The field has many challenges: (i) Search for molecules with larger charges which can be driven along helices. (ii) Investigations of the role of nuclear motions and distributions. These tend to induce decoherence of charge migration.[6,43,45] This calls for the search for helical charge migration in systems with long decoherence times. It is encouraging that the literature has prominent examples of charge migrations which last for longer than 10 fs (i.e., for longer than the total duration of the present laser pulses),[22,46] or even with recoherences after tens of femtoseconds.[13] (iii) Monitoring helical charge migrations. Again, it is encouraging that the literature has several methods which could be applied, or developed further for this purpose.[4,47-51] (iv) Exploring the consequences of the transient chirality imposed by helical charge migration. For example, we consider the recent demonstration of time-resolved neighbor roles in the dynamics of molecules in a dimer as stimulating.[52] As analogous working hypothesis, helical charge migration induced in an achiral molecule may effect different dynamics in neighboring molecules with different chiralities. Work along these lines is in progress. Acknowledgments. This work was supported by the National Key Research and Development Program of China (Grant No. 2017YFA0304203), the Program for Changjiang Scholars and Innovative Research Team (Grant No. IRT17R70), the National Natural Science Foundation of China (Grant Nos. 12004193 and 11904215), the 111 Project (Grant No. D18001), the Fund for Shanxi 1331 Project Key Subjects Construction, the Hundred Talent Program of Shanxi Province, and NJUPT-SF (Grant No. NY220089). 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