Abstract. It is control that turns scientific knowledge into useful technology: in physics and engineering it provides a systematic way for driving a dynamical system from a given initial state into a desired target state with minimized expenditure of energy and resources. As one of the cornerstones for enabling quantum technologies, optimal quantum control keeps evolving and expanding into areas as diverse as quantumenhanced sensing, manipulation of single spins, photons, or atoms, optical spectroscopy, photochemistry, magnetic resonance (spectroscopy as well as medical imaging), quantum information processing and quantum simulation. In this communication, state-of-the-art quantum control techniques are reviewed and put into perspective by a consortium of experts in optimal control theory and applications to spectroscopy, imaging, as well as quantum dynamics of closed and open systems. We address key challenges and sketch a roadmap for future developments. ForewordThe authors of this paper represent the QUAINT consortium, a European Coordination Action on Optimal Control of Quantum Systems, funded by the European Commission Framework Programme 7, Future Emerging Technologies FET-OPEN programme and the Virtual Facility for Quantum Control (VF-QC). This consortium has considerable expertise in optimal control theory and its applications to quantum systems, both in existing areas, such as spectroscopy and imaging, and in emerging quantum technologies, such as quantum information processing, quantum communication, quantum simulation a e-mail: fwm@lusi.uni-sb.de and quantum sensing. The list of challenges for quantum control has been gathered by a broad poll of leading researchers across the communities of general and mathematical control theory, atomic, molecular-, and chemical physics, electron and nuclear magnetic resonance spectroscopy, as well as medical imaging, quantum information, communication and simulation. 144 experts in these fields have provided feedback and specific input on the state of the art, mid-term and long-term goals. Those have been summarized in this document, which can be viewed as a perspectives paper, providing a roadmap for the future development of quantum control. Because such an endeavour can hardly ever be complete (there are many additional areas of quantum control applications, such as spintronics, nano-optomechanical technologies etc.), this roadmap

The non-linear optimization method developed by A. Konnov and V. Krotov [Autom. Remote Cont. (Engl. Transl.) 60, 1427 (1999)] has been used previously to extend the capabilities of optimal control theory from the linear to the non-linear Schrödinger equation [S. E. Sklarz and D. J. Tannor, Phys. Rev. A 66, 053619 (2002)]. Here we show that based on the Konnov-Krotov method, monotonically convergent algorithms are obtained for a large class of quantum control problems. It includes, in addition to nonlinear equations of motion, control problems that are characterized by non-unitary time evolution, nonlinear dependencies of the Hamiltonian on the control, time-dependent targets, and optimization functionals that depend to higher than second order on the time-evolving states. We furthermore show that the nonlinear (second order) contribution can be estimated either analytically or numerically, yielding readily applicable optimization algorithms. We demonstrate monotonic convergence for an optimization functional that is an eighth-degree polynomial in the states. For the "standard" quantum control problem of a convex final-time functional, linear equations of motion and linear dependency of the Hamiltonian on the field, the second-order contribution is not required for monotonic convergence but can be used to speed up convergence. We demonstrate this by comparing the performance of first- and second-order algorithms for two examples.

The advent of quantum devices, which exploit the two essential elements of quantum physics, coherence and entanglement, has sparked renewed interest in the control of open quantum systems. Successful implementations face the challenge to preserve the relevant nonclassical features at the level of device operation. A major obstacle is decoherence which is caused by interaction with the environment. Optimal control theory is a tool that can be used to identify control strategies in the presence of decoherence. We review here recent advances in optimal control methodology that allow for tackling typical tasks in device operation for open quantum systems and discuss examples of relaxation-optimized dynamics. Optimal control theory is also a useful tool to exploit the environment for control. We discuss examples and point out possible future extensions.

Anisotropy is a fundamental property of particle interactions. It occupies a central role in cold and ultra-cold molecular processes, where long-range forces have been found to significantly depend on orientation in ultra-cold polar molecule collisions 1,2 . Recent experiments have demonstrated the emergence of quantum phenomena such as scattering resonances in the cold collisions regime due to quantization of the intermolecular degrees of freedom 3-8 . Although these states have been shown to be sensitive to interaction details, the effect of anisotropy on quantum resonances has eluded experimental observation so far. Here, we directly measure the anisotropy in atom-molecule interactions via quantum resonances by changing the quantum state of the internal molecular rotor. We observe that a quantum scattering resonance at a collision energy of appears in the Penning ionization of molecular hydrogen with metastable helium only if the molecule is rotationally excited. We use state of the art ab initio and multichannel quantum molecular dynamics calculations to show that the anisotropy contributes to the effective interaction only for molecules in the first excited rotational state, whereas rotationally ground state interacts purely isotropically with metastable helium. Control over the quantum state of the internal molecular rotation allows us to switch the anisotropy on or off and thus disentangle the isotropic and anisotropic parts of the interaction. These quantum phenomena provide a challenging benchmark for even the most advanced theoretical descriptions, highlighting the advantage of using cold collisions to advance the microscopic understanding of particle interactions.

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...

The electronic structure of the (LiYb) + molecular ion is investigated with two variants of the coupled cluster method restricted to single, double, and noniterative or linear triple excitations. Potential energy curves for the ground and excited states, permanent and transition electric dipole moments, and long-range interaction coefficients C4 and C6 are reported. The data is subsequently employed in scattering calculations and photoassociation studies. Feshbach resonances are shown to be measurable, despite the ion's micromotion in the Paul trap. Molecular ions can be formed in their singlet electronic ground state by one-photon photoassociation and in triplet states by two-photon photoassociation; and control of cold atom-ion chemistry based on Feshbach resonances should be feasible. Conditions for sympathetic cooling of an Yb + ion by an ultracold gas of Li atoms are found to be favorable in the temperature range of 10 nK to 10 mK; and further improvements using Feshbach resonances should be possible. Overall, these results suggest excellent prospects for building a quantum simulator with ultracold Yb + ions and Li atoms.

The role of internal molecular degrees of freedom, such as rotation, has scarcely been explored experimentally in low-energy collisions despite their significance to cold and ultracold chemistry. Particularly important to astrochemistry is the case of the most abundant molecule in interstellar space, hydrogen, for which two spin isomers have been detected, one of which exists in its rotational ground state whereas the other is rotationally excited. Here we demonstrate that quantization of molecular rotation plays a key role in cold reaction dynamics, where rotationally excited ortho-hydrogen reacts faster due to a stronger long-range attraction. We observe rotational state-dependent non-Arrhenius universal scaling laws in chemi-ionization reactions of para-H2 and ortho-H2 by He(2(3)P2), spanning three orders of magnitude in temperature. Different scaling laws serve as a sensitive gauge that enables us to directly determine the exact nature of the long-range intermolecular interactions. Our results show that the quantum state of the molecular rotor determines whether or not anisotropic long-range interactions dominate cold collisions.

Three-wave mixing spectroscopy of chiral molecules, which exist in left-handed and right-handed conformations, allows for enantio-selective population transfer despite random orientation of the molecules. This is based on constructive interference of the three-photon pathways for one enantiomer and destructive one for the other. We prove here that three mutually orthogonal polarization directions are required to this end. Two different dynamical regimes exist to realize enantioselective population transfer, and we show that they correspond to different phase conditions in the three-wave mixing. We find the excitation scheme used in current rotational three-wave mixing experiments of chiral molecules with C 1 symmetry to be close to optimal and discuss prospects for ro-vibrational three-wave mixing experiments of axially chiral molecules. Our comprehensive study allows us to clarify earlier misconceptions in the literature. arXiv:1904.02208v2 [quant-ph] 8 Jul 2019 , (6) (a) (b) (c) J J J J J+1 J+1 J J J+1 |1⟩ |2⟩ |3⟩ 12 23 13 FIG. 1. Scheme for cyclic population transfer between three rotational states. The possible combinations of J-states for such three-level cycles are denoted by (a), (b), and (c).2. Proof that enantio-selective cyclic electric dipole excitation of rotational states requires three mutually orthogonal polarization directionsWe consider the transition matrix elements between two rotational states of an asymmetric top, J , τ , M |H int |J , τ , M =