Concluding remarks: The age of molecular movies

Abstract

This Faraday Discussion has demonstrated enormous progress towards using advanced light sources, together with a variety of experimental and theoretical tools and techniques, to film the motion of both electrons and nuclei in molecules undergoing photo-induced reactions. The new tools are beginning to offer reliable opportunities for achieving the required spatio-temporal resolution, all the way to sub-femtosecond and sub-angstrom scales. The age of quantum molecular movies has arrived.

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1 Introduction

A picture is worth a thousand words. A movie – even a short one – needs thousands of pictures. Thus, a (good) movie is worth at least a million words. The word “imaging” in the title of this Faraday Discussion, “Time-resolved imaging of photo-induced dynamics”, boldly implies the possibility of making such movies on the spatio-temporal scale relevant to molecules. The dream of making molecular movies has a long history; this Faraday Discussion has demonstrated remarkable progress towards making this dream a reality.

Electrons are the first to respond to light. As light absorption disturbs the electron glue that holds the atoms in a molecule, the atoms begin to move, dragging the electrons with them. This interplay of the electronic and nuclear motion, especially when multiple electronic states are both coupled by the nuclear motion and coherently populated by light, has been one of the central themes in this Faraday Discussion; it is also destined to remain so for years to come. The role of initial coherence in electronic excitations and the possible importance of such coherent electronic excitations for light-induced molecular dynamics interrogated by a delayed probe pulse (see e.g. ref. 1–6 for pioneering work on this topic) remains a central question. Addressing this question requires few-femtosecond to sub-femtosecond temporal resolution. Therefore, ultimate molecular movies imaging the full interplay of electrons and nuclei will require the combination of sub-femtosecond temporal and sub-angstrom spatial resolution.

These movies will also have to be quantum. That is, reflecting the quantum mechanical nature of the dynamics they encode, they will have to deal with the potentially very high information content implied by quantum evolution of the many entangled degrees of freedom. This Faraday Discussion has presented us with remarkable progress towards both making and analysing quantum molecular movies.

Below I describe, perhaps somewhat fragmented, impressions from the papers presented during this Faraday Discussion. The choice of impressions was dictated solely by the personal tastes of the author. This choice in no way implies that the papers reflected upon presented superior science to the papers that are not represented in this incomplete essay.

2 Impressions

Electron diffraction has historically been among the favorite ways to image structures. It is very attractive since, compared to X-rays, the relevant scattering cross sections are much higher. However, its superior imaging properties face a hurdle: electrons in the incident beam repel each other, inevitably stretching the electronic pulse that probes the system; the electron pulses remain limited in their time resolution. The paper by Yanwei Xiong et al. (DOI: 10.1039/D0FD00125B) offers an impressive example of using the combination of ultrafast electron diffraction and mass spectrometry for imaging strong-field induced fragmentation and isomerization of toluene. This paper highlights one of the important trends presented during this Faraday Discussion: using the combination of several techniques, which address a problem from different angles, greatly helps one in simplifying the “development” of the “molecular movie”. Here, mass-spectrometry data were used to select the most important fragmentation channels, theory was used to compute the diffraction images these channels should yield, and ultrafast electron diffraction experiments were used to quantify the temporal evolution of the different channels and to distinguish between isomers that cannot be separated in the mass spectra.

X-rays are another key imaging tool in the molecular movie toolbox, complementing electron diffraction. The high photon flux available at modern free electron laser facilities can successfully offset the relatively low X-ray scattering cross sections, while the lack of Coulomb repulsion between the photons in the X-ray pulse is a very welcome advantage in making these pulses short. This Discussion has demonstrated major progress in using X-rays for ultrafast imaging, both on the technological and conceptual levels – the latter level being closer to the heart of the author. In particular, this Discussion has outlined clear and significant progress towards understanding the sensitivity of X-ray scattering to the instantaneous electron density, its time-derivative, and electronic coherences, i.e. to currents. Building upon the recent results of ref. 7, Andrés Moreno Carrascosa et al. (DOI: 10.1039/D0FD00124D) and Jean Christophe Tremblay et al. (DOI: 10.1039/D0FD00116C) argue that, looking at both elastic and inelastic ultrafast X-ray scattering, one can image not just the electron densities but also the electron currents across the molecule, sensing electronic coherences. This important step in conceptual understanding of the physics encoded in elastic and inelastic ultrafast X-ray scattering allows one to formulate the major goal for X-ray molecular movies: direct imaging of electron currents.

Such effort would require rapid and reliable inversion of the X-ray diffraction data to generate the structures these images encode. What is the optimal way to do it? Haiwang Yong et al. (DOI: 10.1039/D0FD00118J) argue that, with the advent of computational power, one can take advantage of direct quantum chemistry computation of (many) physically relevant structures, generating the corresponding diffraction images and comparing them to the measured ones. This should prevent the black-box image reconstruction algorithms, which blindly invert diffraction images, from converging to unphysical structures. The results of their simulations make a very convincing case that such direct comparison of calculated and measured diffraction patterns may be an excellent way to go. At the very least, the combination of direct simulations and image inversion algorithms will surely become a key tool in the molecular movie toolbox.

A perspective often reflects the personal tastes of its author. The personal tastes of this author were shaped decades ago, when the competition of, and the distinction between, direct and sequential multiphoton processes were a plat du jour. This is, perhaps, one of the reasons why the papers by Phay Ho et al. (DOI: 10.1039/D0FD00106F) on separating coherent and incoherent multi-photon pathways, and by Adi Natan et al. (DOI: 10.1039/D0FD00126K) on using X-ray imaging to distinguish various multi-photon channels, have struck such a cord. What could be more exciting than visualizing transient structures associated with different orders of multiphoton processes? When, not if, one reaches the regime where the X-ray probe and the IR pump are locked in phase to within one cycle, one can start to think about making a movie of how multiphoton excitations build up inside the laser pulse. And, once the X-ray probe is brought inside the strong pump pulse that induces multiphoton processes, perhaps direct and sequential absorption pathways could finally be distinguished? One should not be surprised if such distinction could be found in time delays associated with these different channels – after all, the sequential process implies such time delays in its very definition, while the direct multiphoton absorption implies superior speed. For the author of these notes, filming the race between direct and sequential multiphoton processes would make a sure contender for the Oscars.

A molecular quantum movie aiming to capture ultrafast quantum dynamics needs short laser pulses. These, however, come with a lot of bandwidth, which excites a lot of states and makes the response highly complex to interpret. As Linda Young put it during this Discussion, the broad spectrum of the probe probes too many unwanted channels. How can one solve the problem of getting a nice image of the dynamics that one wants to image, when the probe pulse can excite all kinds of auxiliary dynamics one does not want see? This is an important fundamental question one has to address.

In this context, one may recall that spectral domain measurements can sometimes offer temporal resolution far better than the time-domain duration of the pulses probing the dynamics. The text-book example is SPIDER,⁸ where virtually unlimited temporal resolution of the structure of the laser pulse can be achieved as long as one has unlimited control over the accuracy of the time-delay between the original pulse and its spectrally sheared replica. The lesson to take home is that one can often measure with far better resolution than the duration of the probe pulse if one has full command over the pump–probe time delay. Using extremely broad-bandwidth probes is, therefore, not always mandatory.

Another way to circumvent the background of unwanted channels is by using correlated measurements. Such measurements, when feasible, may also give attosecond resolution without ever using attosecond probe pulses. An example of using correlated two-electron dynamics to unravel complex bi-exponential Auger decay with attosecond precision, without using attosecond pulses, has already been discussed in ref. 9. In addition to ref. 9 and its beautiful time-domain counterpart, the self-referenced attosecond streaking,¹⁰ examples of achieving attosecond temporal resolution without using attosecond pulses include several recollision based imaging techniques such as laser-induced electron diffraction (see e.g. ref. 11–16), high harmonic generation spectroscopy (see e.g. ref. 17–23 for early work), or recollision-induced fragmentation.²⁴ The power of strong-field techniques to circumvent the limitations of the pulse duration by using highly nonlinear processes to look inside a single laser cycle should not be underestimated. At the same time, the price one has to pay for using such methods – the need to deal with complex, highly non-perturbative laser-driven dynamics – should not be overlooked either (see e.g. ref. 16 and 25 for illustrative examples of how to deal with such complexity, for laser-induced electron diffraction and high harmonic spectroscopy, respectively).

The discussion of X-ray and electron diffraction imaging was complemented by the discussion of a broad variety of other movie making tools: femtosecond X-ray emission spectroscopy (Camila Bacellar et al., DOI: 10.1039/D0FD00131G), nonlinear-optical spectroscopy, from attosecond wave-mixing (Yen-Cheng Lin et al., DOI: 10.1039/D0FD00113A; Christina Boemer et al., DOI: 10.1039/D0FD00130A) to transient absorption (Thomas Ding et al., DOI: 10.1039/D0FD00107D), transient resonant Auger spectroscopy (Thomas J. A. Wolf et al., DOI: 10.1039/D0FD00112K), time-resolved photo-electron spectroscopy (e.g. Varun Makhija et al., DOI: 10.1039/D0FD00128G; Evgenii Titov et al., DOI: 10.1039/D0FD00111B; S. Mandal et al., DOI: 10.1039/D0FD00120A; Sajal Kumar Giri et al., DOI: 10.1039/D0FD00117A; Roger Y. Bello et al., DOI: 10.1039/D0FD00114G; A. S. Maxwell et al., DOI: 10.1039/D0FD00105H), and strong-field techniques such as laser-induced Coulomb explosion imaging (Evangelos T. Karamatskos et al., DOI: 10.1039/D0FD00119H). All of them offer viable and exciting alternatives with complementary perspectives. Strong-field laser-induced electron diffraction and high harmonic generation spectroscopy, not represented at this Faraday Discussion, have also been mentioned above. Making use of multiple cameras with different camera angles is standard in the movie business – the same is true in quantum movies, where the complementary techniques provide complementary views and perspectives.

One of the key challenges inherent in making molecular movies is the quantum nature of the system. On the one hand, quantum mechanics of many coupled degrees of freedom deals with exponentially growing information content (as long as the coupling remains coherent), reflecting the entanglement developing in the system. On the other hand, the quantum nature of the system opens new opportunities for imaging, e.g. using post-selection. Correlation-based post-selection is particularly useful when dealing with inevitable shot-to-shot fluctuations of the X-ray pulses generated from self-amplified spontaneous emission. Such post-selection has been used to great success in developing spectral ghost imaging by Siqi Li et al. (DOI: 10.1039/D0FD00122H).

How well can one represent the quantum dynamics of a wave-packet evolving along some effective electronic potential energy surface? To what extent and how quickly do things become entangled, i.e. no longer representable as a simple wave-packet evolving on some effective electronic surface? And with many nodes inside the full wave-function, what is the best way to visualize the dynamics?

This is an existential question in all systems with several coupled degrees of freedom. Sooner or later, a classically chaotic system – and several coupled degrees of freedom are almost inevitably chaotic in the classical world – may render the structures of the corresponding quantum wave-packets too complex, even though the quantum mechanics steps in to arrest the chaos. In this context, Varun Makhija et al. (DOI: 10.1039/D0FD00128G) suggest that two coupled degrees of freedom is already one degree of freedom too many for a clear movie.

But again, isn’t it interesting to look at how entanglement between the coupled degrees of freedom develops? And how can the evolving entanglement be best recorded, imaged, and visualized?

One cannot help but think of a transition from classicism (early stage dynamics) to the abstract art of the early twenties of the previous century (entangled stage), with the movie frames rapidly evolving from the art of Leonardo through Picasso to Pavel Filonov’s “Formula of Spring”. Stripping the quantum images of their full complexity, while likely necessary, will have to be done with care and rigour to avoid disastrous information loss (as in transitioning from the rich images of nature to something reminiscent of the “Voice of Fire” disaster, hanging proudly upside-down in the National Art Gallery of Canada).

In the view of this author, development of rigorous approaches towards distilling useful information content from the complex movies of the quantum dynamics of several coupled degrees of freedom is a major challenge, both technically and conceptually. In this context, a useful measure characterizing the degree of correlation is the Schmidt decomposition. Perhaps one should use the time-dependent Schmidt decomposition of the evolving vibronic wave packet to (i) quantify the transition from the classicism of the initial excitation to the post-modernism of the fully quantum performance art during the later stages of the dynamics and (ii) to develop a minimal and optimal set of filters – the evolving basis functions – to analyze the dynamics. Schmidt decomposition may help one find the most compact representation of the developing entanglement, trace out certain degrees of freedom while minimizing information loss, and possibly identify the key set of useful entangled structures as the projective filters for the quantum movie. The key information we often want to grasp is not only the complex structures themselves, but also the flow of the quantum system between them. A quantum molecular movie might then be well advised to represent the structures themselves as simple basic blocks, Minecraft style.

Another interesting and potentially very useful tool in processing and interpreting molecular movies are deep neural networks, discussed by Sajal Kumar Giri et al. (DOI: 10.1039/D0FD00117A). This paper shows that deep neural networks can not only be a very powerful tool in analyzing complex images, but that they can also be used to extract the parameters of fluctuating X-ray pulses delivered by a free electron laser. Application of neural networks allows one to “purify” the photo-electron spectra, relieving them of the “noisy” components associated with the imperfections of the incident pulses and generating the “ideal” spectra corresponding to a theorist’s dream – a transform-limited Gaussian pulse. Moreover, this paper also shows that one can train a neural network on simple model systems before letting it out into the real world, where it can then successfully deal with complex systems. This “nursery school” concept is extremely attractive for a theorist, as it allows one to generate large training sets with little computational effort in the hope that the experience acquired by the neural network while dealing with simple systems can be used to make sense of a complex system’s response.

Making movies requires a lot of computer power. This is just as true in Hollywood as it is in making molecular movies: computational analysis of quantum dynamics is essential. Modelling charge migration in its full complexity, triggered on the sub-femtosecond time-scale and leading to the nuclear response evolving on the femtosecond to picosecond time scale, is extremely challenging due to the number of electronic surfaces involved, their intersections explored by the nuclei, and the sheer range of relevant time-scales covering several orders of magnitude. The initial excitation can be triggered either by a high-frequency attosecond pulse or by an intense laser pulse, where transitions happen on the sub-cycle time scale, but either way the multiple electronic states will inevitably get involved and become entangled with the ejected electron. Accurate modelling of even this excitation step is very challenging, let alone the follow-up dynamics.

The papers by M. Ruberti (DOI: 10.1039/D0FD00104J) and by Jorge Delgado et al. (DOI: 10.1039/D0FD00121J) demonstrate dramatic progress in addressing this challenge. The first of these focuses on the coherence of the generated ionic excitation and the separability of the full many-body wavefunction into those for the ejected electron and for the remaining hole. Such separability is crucial to even begin discussing the ionization-triggered charge migration. Here, the Schmidt decomposition of the reduced density matrix finds is natural use, helping one to unravel the multi-electron dynamics triggered by the pump. The paper then proceeds to probe the generated ionic excitation, providing a full description of the highly nonlinear, non-perturbative, many-body response of the electronic degrees of freedom.

The paper by Jorge Delgado et al. (DOI: 10.1039/D0FD00121J) is even more ambitious, reporting first attempts at complete modelling of both the multi-electron and nuclear dynamics that would arise in an attosecond two-color XUV-pump/XUV-probe experiment in a polyatomic molecule such as glycine, one of the primary candidates for charge-directed reactivity. This contribution truly pushes the absolute limits of what is computationally feasible today. The author of these notes admires the courage of Jorge Delgado et al. and is looking forward to the inevitable success of this theoretical tour de force.

3 Conclusions

The power of a movie should not be underestimated. In Hollywood, big budget productions, when combined with an exciting and meaningful plot, often generate much bigger returns. Why should quantum molecular movies be any different? This Faraday Discussion has demonstrated both important technical advances and the ability to address deep conceptual questions. These questions include the role of charge currents in molecules, driven not only by the nuclear motion but also by the initial electronic coherence, the interplay of these mechanisms, and the ability to steer this interplay with light.

Particularly attractive for time-resolved imaging are enantio-sensitive currents induced by coherent electronic or vibronic excitations of chiral molecules;²⁶ imaging such currents will teach us a lot about ultrafast chiral dynamics. Filming currents inside molecules is a goal worthy of both intellectual and technical effort.

Another key challenge is making quantum movies accessible to an audience. The importance of this task should not be underestimated – the information content encoded in molecular images is vast, reflecting the quantum nature of the dynamics. The entanglement developing between different degrees of freedom implies exponentially growing complexity of the full analysis. Finding optimal ways of performing such analysis, processing and presentation of data is one of the keys to success.

A journey of a thousand miles begins with a single step. This Faraday Discussion has documented many such steps, and the final destination no longer looks as remote as even a decade ago; for this author, that decade has taken no longer than a blink of an eye.

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

The author wishes to thank G. Dixit, A. Kirrander, J. Küpper, A. Stolow, F. Martín, A. Palacios, L. Young, K. Lopata, P. Weber, J. Marangos, N. Rohringer, and many other participants of this meeting for fruitful and stimulating discussions. Support from the DFG grant IV 152/6-2 and from the European Union’s Horizon 2020 Research and Innovation Programme under grant agreement no. 899794 is gratefully acknowledged.

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