I. INTRODUCTIONThe past few decades have witnessed remarkable developments of laser techniques setting the stage for new areas of research in molecular physics. It is now possible to interrogate molecules in the ultrafast and ultracold regimes of molecular dynamics and the measurements of molecular structure and dynamics can be made with unprecedented precision. Molecules are inherently complex quantum-mechanical systems. The complexity of molecular structure, if harnessed, can be exploited for yet another step forward in science, potentially leading to technology for quantum computing, quantum simulation, precise field sensors, and new lasers.The goal of the present article is to review the major developments that have led to the current understanding of molecule -field interactions and experimental methods for manipulating molecules with electromagnetic fields. Molecule -field interactions are at the core of several, seemingly distinct, areas of molecular physics. This is reflected in the organization of this article, which includes sections on Field control of molecular beams, External field traps for cold molecules, Control of molecular orientation and molecular alignment, Manipulation of molecules by nonconservative forces, Ultracold molecules and ultracold chemistry, Controlled many-body phenomena, Entanglement of molecules and dipole arrays, and Stability of molecular systems in high-frequency super-intense laser fields. By combining these topics in the same review, we would like to emphasize that all this work is based on the same basic Hamiltonian.This review is also intended to serve as an introduction to the excellent collection of articles appearing in this same-titled volume of Molecular Physics [1-27]. These original contributions demonstrate the latest developments exploiting control of molecules with electromagnetic fields. The reader will be treated to a colourful selection of articles on topics as diverse as Chemistry in laser fields, Quantum dynamics in helium droplets, Effects of microwave and laser fields on molecular motion, Rydberg molecules, Molecular structure in external fields, Quantum simulation with ultracold molecules, and Controlled molecular interactions, written by many of the leading protagonists of these fields.This article is concerned chiefly with the effects of electromagnetic fields on low-energy rotational, fine-structure and translational degrees of freedom. There are several important research areas that are left outside the scope of this paper, most notably the large body of work on the interaction of molecules with attosecond laser pulses and high harmonic generation [28], coherent control of molecular dynamics [29] and optimal control of molecular processes [30]. We limit the discussion of resonant interaction of light with molecules to laser cooling strategies. We do not survey spectroscopy or transfer of population between molecular states. Even with these restrictions, this is a vast area to review, as is apparent from the number of references. We did our best to in...

We develop a microscopic theory describing a quantum impurity whose rotational degree of freedom is coupled to a many-particle bath. We approach the problem by introducing the concept of an "angulon" -a quantum rotor dressed by a quantum field -and reveal its quasiparticle properties using a combination of variational and diagrammatic techniques. Our theory predicts renormalisation of the impurity rotational structure, such as observed in experiments with molecules in superfluid helium droplets, in terms of a rotational Lamb shift induced by the many-particle environment. Furthermore, we discover a rich many-body-induced fine structure, emerging in rotational spectra due to a redistribution of angular momentum within the quantum many-body system.

The angular momentum of molecules, or, equivalently, their rotation in three-dimensional space, is ideally suited for quantum control. Molecular angular momentum is naturally quantized, time evolution is governed by a well-known Hamiltonian with only a few accurately known parameters, and transitions between rotational levels can be driven by external fields from various parts of the electromagnetic spectrum. Control over the rotational motion can be exerted in one-, two-and many-body scenarios, thereby allowing to probe Anderson localization, target stereoselectivity of bimolecular reactions, or encode quantum information, to name just a few examples. The corresponding approaches to quantum control are pursued within separate, and typically disjoint, subfields of physics, including ultrafast science, cold collisions, ultracold gases, quantum information science, and condensed matter physics. It is the purpose of this review to present the various control phenomena, which all rely on the same underlying physics, within a unified framework. To this end, we recall the Hamiltonian for free rotations, assuming the rigid rotor approximation to be valid, and summarize the different ways for a rotor to interact with external electromagnetic fields. These interactions can be exploited for control -from achieving alignment, orientation, or laser cooling in a one-body framework, steering bimolecular collisions, or realizing a quantum computer or quantum simulator in the many-body setting. IntroductionMolecules, unlike atoms, are extended objects that possess a number of different types of internal motion. In particular, the geometric arrangement of their constituent atoms endows molecules with the basic capability to rotate in three-dimensional space. Rotations can couple to vibrations of the atomic nuclei as well as to the orbital and spin angular momentum of the electrons. The resulting complexity of the energy level structure [1,2,3,4] may be daunting. It offers, on the other hand, a variety of knobs for control and thus is at the core of numerous applications, from the classic example of the ammonia maser [5] all the way to recent measurements of the electron's electric dipole moment in a cryogenic molecular beam of thorium monoxide [6].A key advantage of internal degrees of freedom such as rotation is that they occupy the low-energy part of the energy spectrum. Quantization of the rotational motion has been an early hallmark of quantum mechanics due to its connection to selection rules that govern all light-matter interactions [7]. Nowadays, rotational states and rotational molecular dynamics feature prominently in all active areas of AMO physics research as well as in neighbouring fields such as physical chemistry and quantum information science. Control over the rotational motion is crucial in one-body, two-body and many-body scenarios. For example, rotational state-selective excitation could pave the way towards separating left-and right handed enantiomers of chiral molecules [8,9]. Still within the one-body scenar...

A critical overview of the theory of the chirality-induced spin selectivity (CISS) effect, that is, phenomena in which the chirality of molecular species imparts significant spin selectivity to various electron processes, is provided. Based on discussions in a recently held workshop, and further work published since, the status of CISS effects-in electron transmission, electron transport, and chemical reactions-is reviewed. For each, a detailed discussion of the stateof-the-art in theoretical understanding is provided and remaining challenges and research opportunities are identified.The ORCID identification number(s) for the author(s) of this article can be found under https://doi.org/10.1002/adma.202106629.

During the last 70 years, the quantum theory of angular momentum has been successfully applied to describing the properties of nuclei, atoms, and molecules, their interactions with each other as well as with external fields. Due to the properties of quantum rotations, the angular momentum algebra can be of tremendous complexity even for a few interacting particles, such as valence electrons of an atom, not to mention larger many-particle systems. In this work, we study an example of the latter: a rotating quantum impurity coupled to a many-body bosonic bath. In the regime of strong impurity-bath couplings the problem involves addition of an infinite number of angular momenta which renders it intractable using currently available techniques. Here, we introduce a novel canonical transformation which allows to eliminate the complex angular momentum algebra from such a class of many-body problems. In addition, the transformation exposes the problem's constants of motion, and renders it solvable exactly in the limit of a slowly-rotating impurity. We exemplify the technique by showing that there exists a critical rotational speed at which the impurity suddenly acquires one quantum of angular momentum from the many-particle bath. Such an instability is accompanied by the deformation of the phonon density in the frame rotating along with the impurity.

Understanding the behavior of molecules interacting with superfluid helium represents a formidable challenge and, in general, requires approaches relying on large-scale numerical simulations. Here we demonstrate that experimental data collected over the last 20 years provide evidence that molecules immersed in superfluid helium form recently-predicted angulon quasiparticles [Phys. Rev. Lett. 114, 203001 (2015)]. Most important, casting the many-body problem in terms of angulons amounts to a drastic simplification and yields effective molecular moments of inertia as straightforward analytic solutions of a simple microscopic Hamiltonian. The outcome of the angulon theory is in good agreement with experiment for a broad range of molecular impurities, from heavy to medium-mass to light species. These results pave the way to understanding molecular rotation in liquid and crystalline phases in terms of the angulon quasiparticle. Among its many peculiar properties, superfluid4 He is quite averse to mixing with impurities which could serve as a microscopic probe of the superfluid phase. As a result, for several decades after the discovery of superfluidity by Allen, Misener, and Kapitza [1,2], only macroscopic -hydrodynamic -properties of superfluid helium have been studied in the laboratory. In the 1990's, however, it was demonstrated that atoms and molecules can be trapped in superfluid helium if the latter forms little droplets containing on the order of a thousand helium atoms [3][4][5][6][7]. Over the following years, trapping atoms, molecules, and ions inside superfluid helium nanodroplets -sometimes called 'nanocryostats' -emerged as an important tool of molecular spectroscopy [6][7][8][9][10][11][12]. Such nanodroplets allow to trap single molecules in a cold environment (∼ 0.4 Kelvin), thereby isolating them from external perturbations. This allows to record spectra free of collisional and Doppler broadening, as well as to study species that are unstable in the gas phase, such as free radicals.While superfluid helium does not cause a substantial broadening of molecular spectral lines, it affects molecular rotation. In particular, molecules in superfluid helium nanodroplets acquire an effective moment of inertia, that is larger compared to its gas-phase value [6,9]. The relative magnitude of the effect increases from lighter to heavier species and is somewhat similar to renormalization of the effective mass for electrons interacting with a crystalline lattice [13][14][15][16].Semiclassically, molecular rotation in helium can be rationalized within the 'adiabatic following' model [6,7,[17][18][19][20][21][22]. There, it is assumed that the molecule induces a local density deformation (a 'non superfluid shell') of helium which co-rotates along with the molecule, thereby increasing its moment of inertia. However, such a classical approach does not allow to get insight into the intriguing aspects of the problem arising from quantum many-body physics. Helium, on the other hand, represents a dense, strongly-interactin...

Rotation of molecules embedded in He nanodroplets is explored by a combination of fs laserinduced alignment experiments and angulon quasiparticle theory. We demonstrate that at low fluence of the fs alignment pulse, the molecule and its solvation shell can be set into coherent collective rotation lasting long enough to form revivals. With increasing fluence, however, the revivals disappear -instead, rotational dynamics as rapid as for an isolated molecule is observed during the first few picoseconds. Classical calculations trace this phenomenon to transient decoupling of the molecule from its He shell. Our results open novel opportunities for studying non-equilibrium solute-solvent dynamics and quantum thermalization.

In several settings of physics and chemistry one has to deal with molecules interacting with some kind of an external environment, be it a gas, a solution, or a crystal surface. Understanding molecular processes in the presence of such a many-particle bath is inherently challenging, and usually requires large-scale numerical computations. Here, we present an alternative approach to the problem -that based on the notion of the angulon quasiparticle. We show that molecules rotating inside superfluid helium nanodroplets and Bose-Einstein Condensates form angulons, and therefore can be described by straightforward solutions of a simple microscopic Hamiltonian. Casting the problem in the language of angulons allows not only to tremendously simplify it, but also to gain insights into the origins of the observed phenomena and to make predictions for future experimental studies. A. Second-order perturbation theory 25 B. Nonperturbative analysis in the weak-coupling regime 28 C. The canonical transformation 34 D. The limit of a slowly rotating impurity 35 VI. Conclusions and outlook 37 VII. Acknowledgements 38 A. Angular momentum operators 38 References 41 I. INTRODUCTIONThe properties of polyatomic systems we encounter in physics and chemistry can be extremely challenging to understand. First of all, many of these systems are strongly correlated, in the sense that their complex behavior cannot be easily deduced from the properties of their individual constituents -isolated atoms and molecules. Second,in realistic experiments these systems are usually found far from their thermal equilibrium, as they are perturbed by the surrounding environment, be it a solution, a gas, or lattice vibrations in a crystal. Quite often, however, insight into the behavior of such complex many-body systems can be obtained from studying the simplified problem of a single quantum particle coupled to an environment. These so-called 'impurity problems' represent an important part of modern condensed matter physics [1,2].Interest in quantum impurities goes back to the classic works of Landau, Pekar, Fröhlich, and Feynman, who studied motion of electrons in crystals [3][4][5][6][7][8]. In its most general formulation, such a problem involves the coordinates and momenta of all the electrons and nuclei in the crystal -some 10 23 degrees of freedom -and is therefore intractable by any existing numerical technique. The problem, however, can be drastically simplified by using a trick very common among condensed matter physicists -that of introducing 'quasiparticles.' A quasiparticle is a collective object, whoseproperties are qualitatively similar to those of free particles, however they quantitatively depend on the coupling between the particle and the environment. Fig. 1 shows a few examples of quasiparticles. For example, the behavior of an electron interacting with a crystalline lattice can be understood in terms of a so-called polaron quasiparticle, composed of an electron dressed by a coat of lattice excitations [9,10]. A polaron effectively behav...