Molecular rotation movie filmed with high-harmonic generation

TL;DR

This paper demonstrates that machine learning combined with angle-resolved high-harmonic spectroscopy can fully reconstruct molecular rotational dynamics in space and time, advancing molecular imaging techniques.

Contribution

It introduces a novel approach integrating machine learning with HHS to decode complex molecular rotational dynamics from harmonic signals.

Findings

Successful reconstruction of molecular rotational wave packets.

Good agreement between experimental data and simulations.

Enhanced understanding of molecular rotation in space and time.

Abstract

Direct imaging of molecular dynamics is a long-standing goal in physics and chemistry. As an emerging tool, high-harmonic spectroscopy (HHS) enables accessing molecular dynamics on femtosecond to attosecond time scales. However, decoding information from the harmonic signals is usually painstaking due to the coherent nature of high-harmonic generation (HHG). Here we show that this obstacle can be effectively overcome by exploiting machine learning in HHS. Combining the machine learning with an angle-resolved HHS method, we demonstrate that the rich dynamics of molecular rotational wave packet is fully reconstructed from the angular distributions of HHG measured at various time delays of the probe pulse. The experimental retrievals are in good agreement with the numerical simulations. These findings provide a comprehensive picture of molecular rotation in space and time which will facilitate the development of related researches on molecular dynamics imaging.

Figures

Equations

$${\Psi }_{JM}(\theta ,\varphi ,t)=\sum _{J^{\prime }{M}^{\prime }}{C}_{JM,J^{\prime }{M}^{\prime }}(t)Y_{J^{\prime }}^{M^{\prime }}(\theta ,\varphi ),$$

$$\rho (\theta ,\varphi ,t)=\sum _{JM}{\Gamma }_{JM}|{\Psi }_{JM}(\theta ,\varphi ,t){|}^2,$$

$$I_q(\alpha ,t)={\left|{\int }_{\varphi =0}^{2\pi }{\int }_{\theta =0}^{\pi }{S}_q(\beta )\rho (\theta ,\varphi ,t)\sin \theta d\theta d\varphi \right|}^2.$$

$$i\frac{\partial {\Psi }_{JM}(\theta ,\varphi ,t)}{\partial t}$$

Full text

I Introduction

Resolving molecular dynamics in time and space, videlicet, making a “molecular movie”, has been long desired for its potential in revealing the intermediate processes of chemical reactions and biological transformations film1 ; film2 ; film3 . This enticing prospect, however, has encountered grand challenges over the last century due to the formidable spatiotemporal scales of molecular dynamics. Coherent diffraction imaging techniques based on X-ray and electron beam have enabled identifying the atomic positions in molecules with sub-ångstrom spatial resolution xray1 ; xray2 ; xray3 ; xray4 ; xray5 . However, the achievable temporal resolutions of these methods are currently limited to few tens to hundreds of femtoseconds.

Recent advances in strong-field physics suggest that electrons liberated from a molecule can be driven back by the oscillating electric field to revisit its parent ion three1 ; three2 ; QRS1 . Such an encounter gives rise to a number of electron-ion collisional phenomena and offers new opportunities for the probe of molecular dynamics phe1 ; phe2 ; phe3 ; phe4 ; phe5 ; phe6 ; phe7 ; phe8 . In particular, high-harmonic generation (HHG) from molecules provides a snapshot of molecular structure and dynamics at the recombination instant of the returning electrons. The information is encoded in the amplitudes, phases, and polarization of the harmonic fields. Extracting information from HHG signals systematically is known as high-harmonic spectroscopy (HHS), which recently has been an emerging tool for ultrafast molecular detection. In HHS, the temporal resolution comes from the interval between ionization and recombination events, and the spatial resolution is related to the de Broglie wavelength of the recombining electron, which promises resolving molecular dynamics on attosecond time scale d1 ; d2 ; d3 ; d4 ; d5 and imaging molecular struture with ångstrom spatial resolution o1 ; o2 ; o3 . However, despite these superiorities, HHS also encounters rigorous challenges in the information decoding from the HHG signals. Especially for the complex-valued molecular information, of which both the amplitude and phase are coherently embroiled in the harmonic spectrum, the decoding is rather arduous or even unattainable due to the poor ability of the conventional iterative algorithm commonly-used in HHS in dealing with the phase problem (complex-valued data) o3 ; wx . Thus in many previous works on HHS, the phase information of the dynamics is either left out ky ; vm or addressed under certain approximations ren , which makes it very challenging to make a full molecular movie.

In recent years, with the emergence of artificial intelligence, the modern machine learning community has developed techniques with remarkable abilities to recognize, classify, and characterize complex sets of data with a high degree of accuracy, which have been widely applied in diverse domains, e.g., image processing ml1 ; ml2 , face recognition ml3 , optical tomography ml4 , and playing the game Go ml5 , etc. More recently, machine learning has been explored as a tool in genetics eml1 , condensed-matter physics eml2 , and material science eml3 . Owing to the excellent ability in complex data-processing, the machine learning is also expected to be a powerful candidate for the decoding in HHS studies.

In the present work, we demonstrate the application of machine learning in HHS to probe molecular rotational dynamics with high resolution and fidelity. Molecular rotational control by coherently manipulating the rotational wave packet (RWP) has been proved an efficient way to connect the molecular and laboratory frames ro1 ; ro2 ; ro3 ; ro4 ; ro5 , which is significant for molecular structure imaging o1 ; o2 ; o3 and chemical reaction steering d3 ; chem . In experiment, the molecular frame dynamics is usually convolved with the rotational distribution of molecules in laboratory. A throughout knowledge of molecular rotation in space and time therefore is prerequisite for resolving the molecular frame dynamics from the measurement. By using an angle-resolved HHS method, we show that the rotational dynamics of the molecular RWP excited by a pump pulse is recorded in the angular distributions (ADs) of HHG measured at various polarization directions and time delays of the probe pulse. The time-dependent angular probability density of the RWP, is then retrieved from the measured harmonic ADs by using a machine learning algorithm, which provides a full movie of the spatiotemporal evolution of the molecular RWP. The underlying mechanism of the molecular rotational dynamics is further revealed by analyzing the populations and phases of the excited rotational states. All the experimental retrievals agree well with the numerical simulations of the time-dependent Schrödinger equation (TDSE).

II Results

Experimental measurement of the angle-resolved HHG. HHG through laser-molecule interaction occurs under a much faster timescale than molecular rotation. The instantaneous rotational state of molecules can be encoded in the generated harmonic spectra. In our work, the HHG experiment is performed with a pump-probe scheme as sketched in Fig. 1(a). A pump pulse with moderate intensity, linearly polarized along the z-axis [as illustrated in Fig. 1(d)], is first applied to the molecular ensemble to create molecular RWP through the stimulated Raman transitions ro1 . Subsequently, an intense linearly polarized probe pulse with adjustable time delay is applied to interact with the excited molecules to generate high-order harmonics (for more experimental details, see Methods). Impulsively excited by the pump pulse, the molecules are temporarily well confined in a narrow cone around the polarization direction of the pump pulse (i.e., the z-axis). Afterwards the created RWP disperses and evolves under the field-free condition [as illustrated in Fig. 1(b)]. To characterize the spatial distribution of the molecular RWP, we measure the angular distributions (ADs) of the generated harmonics by scanning the polarizing angle $\alpha$ of the probe pulse with respect to the pump pulse in the xz plane spanned by the field vectors of these two pulses [see the geometry in Fig. 1(d)]. The temporal evolution of molecular RWP is traced by repeating this process at various time delays of the probe pulse. The time-dependent harmonics ADs then provide a series of snapshots of the molecular RWP during the evolution [see Fig. 1(c)]. The measurement at each time delay will make a “frame” in the “movie” of the evolution of molecular RWP.

To demonstrate our scheme, we choose the commonly-used nitrogen (N2) molecule as a prototype to do the HHG experiment. Figure 2(a) shows the measured time-dependent ADs of harmonic 17 (H17) from N2. The harmonic ADs show strong delay dependence at the rotational revivals, e.g., around t=8.4 ps (1Trev, T${}_{rev}=\frac{1}{2B_{0}c}$ is the revival period of N2 with $B_{0}$ the rotational constant and $c$ the velocity of light) and 12.6 ps (1.5Trev). In Fig. 2(b) and (c), we plot the results (dots) measured at t= 8.2 ps and 8.45 ps [indicated by the blue and red arrows in Fig. 2(a)], respectively. One can see that the angle-dependent HHG yields measured at t=8.2 ps exhibit a maximum at $\alpha=90^{\circ}$. Driven by a linearly polarized probe pulse, HHG from N2 is most pronounced when the laser is parallel to the molecular axis due to the $\sigma_{g}$ symmetry of the highest-occupied molecular orbital (HOMO) of N2 homo1 ; homo2 . The maximal HHG yield at $\alpha=90^{\circ}$ in Fig. 2(b) implies that most molecules are aligned perpendicularly to the pump pulse, which just corresponds to an antialignment revival at t=8.2 ps. While at t=8.45 ps, the maximal HHG yield appears at $\alpha=0^{\circ}$ (or $180^{\circ}$) where the probe pulse is parallel to the pump pulse, indicating an alignment revival at this moment. From the time-dependent harmonic ADs, the rapid quantum transformation of the molecular RWP from antialignment to alignment is directly identified. However, it’s important to note that the measured harmonic AD is not an honest presentation of the distribution of molecular RWP, since the orientation-dependent electronic response is also convoluted in the HHG signals. For instance, at the same revivals (alignment/antialignment), HHG from CO2 molecule shows different ADs from N2 (see Supplementary Note 2). To gain a molecular movie of the evolution of molecular RWP, a further decoding from the measured harmonic signals is required.

Recuperation of molecular rotation movie from the angle-resolved HHG signals.

After the excitation of the pump pulse, the evolution of the RWP created from each initially populated rotational state $|JM\rangle$ [the eigenstate of the field-free rigid rotor described by the spherical harmonic $Y_{J}^{M}(\theta,\phi)$] can be expanded as njp

[TABLE]

where $C_{JM,J^{\prime}M^{\prime}}(t)=|C_{JM,J^{\prime}M^{\prime}}(t)|e^{i\varphi_{JM,J^{\prime}M^{\prime}}(t)}$ is the time-dependent complex coefficient of each rotational eigenstate with the amplitude $|C_{JM,J^{\prime}M^{\prime}}(t)|$ and phase $\varphi_{JM,J^{\prime}M^{\prime}}(t)$. The labels $J$ (or $J^{\prime}$) and $M$ (or $M^{\prime}$) are associated with the total angular momentum and its projection onto the space-fixed z axis. Assuming a thermal distribution of the initial rotational states, the time-dependent molecular axis distribution (MAD) $\rho(\theta,\phi,t)$, which directly reflects the molecular rotational dynamics after the interaction of the molecular ensemble with the pump pulse, can be written as a weighted average of the modulus squares of the wave packet $\Psi_{JM}(\theta,\phi,t)$ ro1 ; njp ; tdse1 ; tdse2 ; tdse3 ; qrs2 , i.e.,

[TABLE]

where $\Gamma_{JM}$ is the statistical weight (i.e., the population) of the initial state $|JM\rangle$ given by the Boltzmann distribution. HHG from the impulsively excited molecular ensemble can be related to the single-molecule contribution through the time-dependent MAD $\rho(\theta,\phi,t)$. Considering the coherent nature of HHG, the time-dependent harmonic ADs will thus be given by tdse3 ; qrs2 :

[TABLE]

Here $\theta$ and $\phi$ are the polar and azimuthal angles of molecular axis with respect to the z and x axes [see the geometry in Fig. 1(d)]. $\beta$ is the angle between the molecular axis and the polarization of the probe laser, which obeys $\cos\beta=\sin\theta\sin\alpha\cos\phi+\cos\theta\cos\alpha$. $S_{q}(\beta)$ is the induced dipole moment of the $q$-th harmonic related to the single molecule response with given orientation. In our reconstruction, we assume that the single molecule dipole moment $S_{q}(\beta)$ is known from the calculations by the quantitative rescattering (QRS) theory QRS1 ; qrs2 , of which the accuracy and validity in modeling molecular HHG have been well established. Equations (1)-(3) set up a function between the harmonic ADs $I_{q}(\alpha,t)$ and the complex coefficients $C_{JM,J^{\prime}M^{\prime}}(t)$. The reconstruction of molecular rotation movie requires the retrieval of the complex coefficients (both amplitude and phase) for all the relevant initial states from the measured harmonic ADs. To this end, we introduce the machine learning to the HHS study, and employ a global optimization algorithm—the simulated annealing (SA) algorithm (see Supplementary Note 1), which is an efficient solver for approximating the global optimum of a given function in a large search space, to do the retrieval.

In Fig. 2 and 3, we have recuperated the time-dependent MAD $\rho(\theta,\phi,t)$ as well as the harmonic ADs $I_{q}(\alpha,t)$ in terms of Eq. (2) and (3) by using the retrieved coefficients $C_{JM,J^{\prime}M^{\prime}}$. As shown in Fig. 2 (b)-(c), the recuperated harmonic ADs (solid lines) are in excellent agreement with experiment data (dots), indicating a high accuracy of the SA algorithm. In Fig. 3 (a), we show the recuperated MAD $\rho(\theta,\phi,t)$. Note that imposed by the linearly polarized pump pulse, the resulting RWP has a cylindrical symmetry in space, thus $\rho(\theta,\phi,t)$ is independent on the azimuthal angle $\phi$ ro1 ; njp . Figure 3 just shows the result of $\phi=0^{\circ}$. From Fig. 3 (a), one can see clear signatures of the antialignment and alignment revivals at $t$=8.2 and 8.45 ps, respectively. To evaluate the quality of the retrievals, we have theoretically calculated the MAD $\rho(\theta,\phi,t)$ by solving the TDSE of molecular RWP (see Methods). In our calculations, the laser intensity of the pump pulse has been optimized so that the squared difference between the measured and calculated signals is minimal. The calculated results are shown in Fig. 3 (b). It’s evident that the main distribution of the experimental retrieval agrees well with the theoretical one. For better comparison, the retrieved (solid line) and calculated (dashed line) results at the time delays of 8.2 ps and 8.45 ps are also presented in Fig. 3 (c)-(d). The overall agreement between the retrievals (solid line) and calculations (dashed line) are excellent.

A deep insight of this rich rotational dynamics is further revealed by analyzing the amplitude and phase of the RWP created from each initial rotational states. The amplitude is associated with the populations of the rotational states. Excited by the pump pulse, the populations of the contributing rotational states are redistributed due to the transitions between different states. The rotational transition induced by the pump pulse can be quantified by analyzing the population change $\Delta N_{JM}=N_{JM}-\Gamma_{JM}$ of each rotational state. Here, $N_{JM}$=$\sum\limits_{J^{\prime}M^{\prime}}\Gamma_{J^{\prime}M^{\prime}}|C_{J^{\prime}M^{\prime},JM}|^{2}$ represents the population of the rotational state $|JM\rangle$ after the excitation of the pump pulse. Figure 4(a) depicts the $\Delta N$ retrieved from the experiment. Note that after the impulsive excitation of the pump pulse, the created RWP evolves field-freely. The population $N_{JM}$ is almost time-delay-independent. The result in Fig 4(a) is an average of the retrievals at different time delays. It’s clear that excited by the pump pulse, the lower-lying rotational states are transferred to much higher rotational levels. Owing to the linear polarization of the pump pulse, the magnetic quantum number $M$ is conserved in the transitions, i.e., $\Delta M=0$ ro1 ; njp . Since the lower-lying rotational states possess smaller magnetic quantum number $M$, the populations of high rotational states are mainly concentrated at small $M$ states. Moreover, the distribution over the magnetic quantum number $M$ after the interaction is symmetric, agreeing with the cylindrical symmetry of the excitation. These results are in consistent with the theoretical calculations shown in Fig 4(b). For a direct comparison with the theory, we count up the populations of the sub-$M$ states for each $J$ state. As shown in Fig. 4(c), regardless of the experimental error, the agreement between the experiment (dots) and theory (bars) is excellent.

In Fig. 5, we show the retrieved phase $\varphi_{JM,J^{\prime}M^{\prime}}$ of the relevant rotational states as a function of the pump-probe delay. Figure 5(a) depicts the results of the rotational eigenstates Y${}_{0}^{0}$, Y${}_{2}^{0}$, and Y${}_{4}^{0}$ created from the initial $|00\rangle$ state. It’s obvious that the retrieved phase of each rotational eigenstate (dots) exhibits a nearly linear change with the time delay. Dotted curves in Fig. 5 show the linear fitting of these retrievals. The slopes (as indicated) of the fitting curves are very close to $-\frac{E_{J}}{\hbar}$, where $E_{J}=2\pi\hbar B_{0}cJ(J+1)=\pi\hbar J(J+1)/T_{rev}$ is the eigen energies of the considered rotational eigenstates. This result is consistent with the field-free nature of the evolution of the RWP after the interaction, which is described by a phase factor $e^{-i\frac{E_{J}}{\hbar}t}$ ro1 ; njp . With the complex coefficients $C_{JM,J^{\prime}M^{\prime}}$ retrieved, we can then reproduce the dynamics of the RWP created from each initial rotational state in terms of Eq. (1). Figure 6 (a)-(d) display some selected snapshots of the RWP created from the $|00\rangle$ , $|10\rangle$, $|20\rangle$, and $|30\rangle$ states, respectively. One can see that, owing to the time-dependent phase beating of the coherently populated rotational eigenstates, the RWPs created from different initial rotational states at the rotation revivals are synchronously condensed/stretched along the z axis, thus leading to the oblate/prolat angular distribution of molecules at the antialignment/alignment revival. Through a thermal average over these created RWPs, a complete movie of molecular rotational dynamics then is achieved (see the movie in the Supplemental Material).

III Discussion

In summary, we demonstrate a full experimental reconstruction of molecular rotational dynamics by using an angle-resolved HHS method in combination with the machine learning algorithm. Under the excitation of a pump pulse, a coherent molecular RWP is created in experiment, which afterwards is probed by measuring the time-dependent ADs of HHG through the interaction the molecular ensemble with a subsequent probe pulse. By utilising a machine learning algorithm—the SA algorithm, the time-dependent angular probability distribution of the molecular RWP is successfully retrieved from the measured harmonic ADs. A full movie of the evolution of molecular RWP in space and time is then achieved. The rich dynamics of molecular rotation is further illuminated by analyzing the populations and phases of the excited rotational states. All the experiment results are consistent with the TDSE simulations. The present results afford a thoroughgoing knowledge of the spatiotemporal evolution of the field-free molecular RWP, which is extremely useful for future researches on ultrafast imaging of strong-field molecular frame dynamics.

On the other hand, machine learning provides a practical and robust method for extracting information from the HHG spectra. It’s general and can be effectively extended to other molecules, like CO2 (see Supplementary Note 2) and even some more complex polyatomic molecules. Moreover, HHG from molecules also contains abundant information on the electronic structure and dynamics. The exploitation of machine learning in HHS could also open access to the ultrafast charge migration in molecules. In the future, the machine learning is expected to be an extremely effective technique in attosecond physics for actually “watching” a chemical reaction.

IV Methods

Experimental methods. Our experiment is carried out by using a commercial Ti: sapphire laser system (Astrella-USP-1K, Coherent, Inc.), which delivers 35 fs, 800 nm pulses at a repetition rate of 1 kHz. The output laser is spilt by a beam splitter to produce the pump and probe pulses. The pump pulse polarized along the z-axis is first used to create molecular RWP. Subsequently, the intense probe pulse with adjustable time delay is applied to interact with the molecule ensemble to generate high-order harmonics. A motorized delay lines and a half-wave plate are installed in the arm of the probe pulse to adjust its time delay and polarization with respect to the pump one. These two beams are focused to a gas jet (500 $\mu$m diameter) by a 300-mm focal-length lens. The background gas pressure is maintained at 20 Torr and the rotational temperature is estimated to be about 100 K temp . The generated harmonic spectrum is detected by a homemade flat-field soft x-ray spectrometer he1 ; he2 .

Theoretical methods. To compare with the experiment, we have also simulated the molecular rotational dynamics by solving the time-dependent Schrödinger equation (TDSE) of the molecular RWP. For each initial rotational state $|JM\rangle$, the TDSE reads tdse1 ; tdse2 ; tdse3 ; qrs2

[TABLE]

Here, J is the rotation operator, $B_{0}$ is the rotational constant of N2 molecule, $\alpha_{\parallel}$ and $\alpha_{\perp}$ are the the anisotropic polarizabilities in parallel and perpendicular directions with respect to the molecular axis, respectively. $E(t)$ is the envelope of the field vector of the pump pulse. Eq. (4) is solved with the split-operator method for each initial rotational state $|JM\rangle$ split . Assuming a Boltzmann distribution of the rotational levels at the initial time, the MAD at each time delay can be obtained in terms of Eq. (2). The harmonic spectrum is then calculated according to Eq. (3) in the main text. In our simulations, the single molecule response is calculated based on the QRS theory QRS1 ; qrs2 . The intensity of the probe pulse is estimated from the harmonic cutoff in the experiment, which is about $2.3\times 10^{14}\ \mathrm{W/cm}^{2}$. The intensity of the pump pulse is determined by minimizing the squared difference between the measured and calculated HHG signals, which is about $1.1\times 10^{13}\ \mathrm{W/cm}^{2}$. Other parameters are the same as the experimental conditions.

V Acknowledgements

The authors gratefully acknowledge Xu Wang for helpful discussions. This work was supported by National Natural Science Foundation (NSFC) of China (Grants No. 11627809, No. 11874165, No. 11704137, and No. 11774109), Fundamental Research Funds for the Central Universities (2017KFXKJC002), Program for HUST Academic Frontier Youth Team.

VI Author contributions

P.X.L. and P.F.L. conceived and designed the experiment. L.X.H., Y.Q.H., B.N.W., and P.F.L. performed the experiments. Y.Q.H., L.X.H., and P.F.L. performed the simulations. L.L., X.S.Z., and W.C. participated in the discussions.