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In this paper, we study the design of pulse sequences for NMR spectroscopy as a problem of time optimal control of the unitary propagator. Radio frequency pulses are used in coherent spectroscopy to implement a unitary transfer of state. Pulse sequences that accomplish a desired transfer should be as short as possible in order to minimize the effects of relaxation and to optimize the sensitivity of the experiments. Here, we give an analytical characterization of such time optimal pulse sequences applicable to coherence transfer experiments in multiple-spin systems. We have adopted a general mathematical formulation, and present many of our results in this setting, mindful of the fact that new structures in optimal pulse design are constantly arising. Moreover, the general proofs are no more difficult than the specific problems of current interest. From a general control theory perspective, the problems we want to study have the following character. Suppose we are given a controllable right invariant system on a compact Lie group, what is the minimum time required to steer the system from some initial point to a specified final point? In NMR spectroscopy and quantum computing, this translates to, what is the minimum time required to produce a unitary propagator? We also give an analytical characterization of maximum achievable transfer in a given time for the two-spin systems.

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

General pulse sequence elements that achieve sensitivity-enhanced coherence transfer from a heteronucleus to protons of arbitrary multiplicity are introduced. The building blocks are derived from the sensitivity-enhancement scheme introduced by Cavanagh et al. ((1991) J. Magn. Reson., 91, 429-436), which was used in conjunction with gradient coherence selection by Kay et al. ((1992) J. Am. Chem. Soc., 114, 10663-10665), as well as from a multiple-pulse sequence effecting a heteronuclear planar coupling Hamiltonian. The building blocks are incorporated into heteronuclear correlation experiments, in conjunction with coherence selection by the formation of a heteronuclear gradient echo. This allows for efficient water suppression without the need for water presaturation. The methods are demonstrated in HSQC-type experiments on a sample of a decapeptide in H2O. The novel pulse sequence elements can be incorporated into multidimensional experiments.

Oxidized and reduced hen lysozyme denatured in 8 M urea at low pH have been studied in detail by NMR methods. 15N correlated NOESY and TOCSY experiments have provided near complete sequential assignment for both 1H and 15N resonances. Over 900 NOEs, including 130 (i, i + 2) and 23 (i, i + 3) NOEs, could be identified by analysis of the NOESY spectra of the denatured states, and 3J(HN, Halpha) coupling constants and 15N relaxation rates have been measured. The coupling constant and NOE data were analyzed by comparisons with theoretical predictions from a random coil polypeptide model based on amino acid specific phi,psi distributions extracted from the protein data bank. There is significant agreement between predicted and experimental NMR parameters suggesting that local conformations of the denatured states are largely determined by short-range interactions within the polypeptide chain. This result is supported by the observation that the chemical shift, coupling constant, and NOE data are little affected by whether or not the four disulfide bridge cross-links are formed in the denatured protein. The relaxation data, however, show significant differences between the oxidized and reduced protein. Analysis of the relaxation data in terms of simple dynamics models provides evidence for weak clustering of hydrophobic groups near tryptophan residues and increased barriers to motion in the more compact conformers formed when the polypeptide chain is cross-linked by the disulfide bridges. Using this information, a structural description of these denatured states is given in terms of an ensemble of conformers, which have a complex relationship between their local and global characteristics.