Phase spaces as given by the Wigner distribution function provide a natural description of infinite-dimensional quantum systems. They are an important tool in quantum optics and have been widely applied in the context of time-frequency analysis and pseudo-differential operators. Phase-space distribution functions are usually specified via integral transformations or convolutions which can be averted and subsumed by (displaced) parity operators proposed in this work. Building on earlier work for Wigner distribution functions (Grossmann in Commun Math Phys 48(3):191-194, 1976. https://doi.org/10.1007/BF01617867), parity operators give rise to a general class of distribution functions in the form of quantum-mechanical expectation values. This enables us to precisely characterize the mathematical existence of general phase-space distribution functions. We then relate these distribution functions to the so-called Cohen class (Cohen in J Math Phys 7(5):781-786, 1966. https://doi.org/ 10.1063/1.1931206) and recover various quantization schemes and distribution functions from the literature. The parity operator approach is also applied to the Born-Jordan distribution which originates from the Born- Jordan quantization (Born and Jordan in Z Phys 34(1):858-888, 1925. https://doi.org/10.1007/BF01328531). The corresponding parity operator is written as a weighted average of both displacements and squeezing operators, and we determine its generalized spectral decomposition. This leads to an efficient computation of the Born-Jordan parity operator in the number-state basis, and example quantum states reveal unique features of the Born-Jordan distribution.

Quantum optimal control, a toolbox for devising and implementing the shapes of external fields that accomplish given tasks in the operation of a quantum device in the best way possible, has evolved into one of the cornerstones for enabling quantum technologies. The last few years have seen a rapid evolution and expansion of the field. We review here recent progress in our understanding of the controllability of open quantum systems and in the development and application of quantum control techniques to quantum technologies. We also address key challenges and sketch a roadmap for future developments.

Nitrogen-vacancy (N-V) diamonds have attracted keen interest for nanoscale sensing and spin manipulation. In particular, the nonequilibrium electron spin polarization after optical excitation of single N-V centers has successfully been transferred to nuclear spin baths in the surrounding of defects. However, these experiments need to be extended to N-V ensembles that have promising practical applications in the hyperpolarization of bulk sample volumes for NMR signal enhancement. Here, we use a dense, shallow ensemble of N-V centers to demonstrate polarization transfer to nuclear spins in a well-defined composite diamond sample system. This allows us to address three different types of nuclear spins in different positions with respect to the N-V polarization source: from the close proximity of C-13 inside the diamond lattice to the self-assembled molecular system consisting of H-1 and F-19 spins outside the diamond and over multiple interfaces. We show that ensemble N-V experiments face problems different from single N-V experiments. In particular, using spinlock pulses, the inhomogeneously broadened electron spin resonance line of the N-V ensemble limits the minimal resonance linewidth with which the transfer protocol can occur. Furthermore, we compare the N-V spin-polarization losses and polarization transfer rates to the different nuclear baths and discuss the role of spin diffusion as detrimentally affecting the direct observation of nuclear polarization buildup within the detection volume of nanoscale N-V-NMR experiments.

Dipolar recoupling is a central concept in the nuclear magnetic resonance spectroscopy of powdered solids and is used to establish correlations between different nuclei by magnetization transfer. The efficiency of conventional cross-polarization methods is low because of the inherent radio frequency (rf) field inhomogeneity present in the magic angle spinning (MAS) experiments and the large chemical shift anisotropies at high magnetic fields. Very high transfer efficiencies can be obtained using optimal control-derived experiments. These sequences had to be optimized individually for a particular MAS frequency. We show that by adjusting the length and the rf field amplitude of the shaped pulse synchronously with sample rotation, optimal control sequences can be successfully applied over a range of MAS frequencies without the need of reoptimization. This feature greatly enhances their applicability on spectrometers operating at differing external fields where the MAS frequency needs to be adjusted to avoid detrimental resonance effects.

We apply the small-tip-angle approximation in the toggling frame in order to analytically design robust pulses against resonance offsets for state to state transfer in two-level quantum systems. We show that a broadband or a local robustness up to an arbitrary order can be achieved. We provide different control parametrizations to satisfy experimental constraints and limitations on the amplitude or energy of the pulse. A comparison with numerical optimal solutions is made.

We study the application of Optimal Control Theory to Ion Cyclotron Resonance. We test the validity and the efficiency of this approach for the robust excitation of an ensemble of ions with a wide range of cyclotron frequencies. Optimal analytical solutions are derived in the case without any pulse constraint. A gradient-based numerical optimization algorithm is proposed to take into account limitation in the control intensity. The efficiency of optimal pulses is investigated as a function of control time, maximum amplitude and range of excited frequencies. A comparison with adiabatic and SWIFT pulses is done. On the basis of recent results in Nuclear Magnetic Resonance, this study highlights the potential usefulness of optimal control in Ion Cyclotron Resonance.

We study the selective and robust time-optimal rotation control of several spin-1/2 particles with different offset terms. For that purpose, the Pontryagin maximum principle is applied to a model of two spins, which is simple enough for analytic computations and sufficiently complex to describe inhomogeneity effects. We find that selective and robust controls are respectively described by singular and regular trajectories. Using a geometric analysis combined with numerical simulations, we determine the optimal solutions of different control problems. Selective and robust controls can be derived analytically without numerical optimization. We show the optimality of several standard control mechanisms in Nuclear Magnetic Resonance, but new robust controls are also designed.

Quantum devices are preparing increasingly more complex entangled quantum states. How can one effectively study these states in light of their increasing dimensions? Phase spaces such as Wigner functions provide a suitable framework. We focus on spherical phase spaces for finite-dimensional quantum states of single qudits or permutationally symmetric states of multiple qubits. We present methods to efficiently compute the corresponding spherical phase-space functions which are at least an order of magnitude faster than traditional methods. Quantum many-body states in much larger dimensions can now be effectively studied by experimentalists and theorists using spherical phase-space techniques.

We study the representation and visualization of finite-dimensional, coupled quantum systems. To establish a generalizedWigner representation, multi-spin operators are decomposed into a symmetry-adapted tensor basis and are mapped to multiple spherical plots that are each assembled from linear combinations of spherical harmonics. We explicitly determine the corresponding symmetry-adapted tensor basis for up to six coupled spins 1/2 (qubits) using a first step that relies on a Clebsch-Gordan decomposition and a second step which is implemented with two different approaches based on explicit projection operators and coefficients of fractional parentage. The approach based on explicit projection operators is currently only applicable for up to four spins 1/2. The resulting generalized Wigner representation is illustrated with various examples for the cases of four to six coupled spins 1/2. We also treat the case of two coupled spins with arbitrary spin numbers (qudits) not necessarily equal to 1/2 and highlight a quantum system of a spin 1/2 coupled to a spin 1 (qutrit). Our work offers a much more detailed understanding of the symmetries appearing in coupled quantum systems.

We study the simultaneous control of an ensemble of springs with different frequencies by means of an adiabatic shortcut to adiabaticity and optimal processes. The linearity of the system allows us to derive analytical expressions for the control fields and the time evolution of the dynamics. We discuss the relative advantages of the different solutions. These results are applied in two different examples. For ion cyclotron resonance, we show how to optimally control ions by means of electric field. Using a mapping between spins and springs, we derive analytical shortcut protocols to realize robust and selective excitations of two-level quantum systems.

Efficient transfer of spin polarisation from electron to nuclear spins is emerging as a common target of several advanced spectroscopic experiments, ranging from sensitivity enhancement in nuclear magnetic resonance (NMR) and methods for the detection of single molecules based on optically detected magnetic resonance (ODMR) to hyperfine spectroscopy. Here, we examine the feasibility of electron-nuclear cross-polarisation at a modified Hartmann-Hahn condition (called eNCP) for applications in ENDOR experiments with spin-1 deuterium nuclei, which are important targets in studies of hydrogen bonds in biological systems and materials. We have investigated a two-spin model system of deuterated malonic acid radicals in a single crystal. Energy matching conditions as well as ENDOR signal intensities were determined for a spin Hamiltonian under the effect of microwave and radiofrequency irradiation. The results were compared with numerical simulations and 94-GHz ENDOR experiments. The compelling agreement between theoretical predictions and experimental results demonstrates that spin density operator formalism in conjunction with suitable approximations in regard to spin relaxation provides a satisfactory description of the polarisation transfer effect. The results establish a basis for future numerical optimizations of polarisation transfer experiments using multiple-pulse sequences or shaped pulses and for moving from model systems to real applications in disordered systems.

We introduce the concept of magic subspaces for the control of dissipative Nlevel quantum systems whose dynamics are governed by the Lindblad equation. For a given purity, these subspaces can be defined as the set of density matrices for which the rate of purity change is maximum or minimum. Adding fictitious control fields to the system so two density operators with the same purity can be connected in a very short time, we show that magic subspaces allow us to derive a purity speed limit, which only depends on the relaxation rates. We emphasize the superiority of this limit with respect to established bounds and its tightness in the case of a two-level dissipative quantum system. The link between the speed limit and the corresponding time-optimal solution is discussed in the framework of this study. Explicit examples are described for twoand three-level quantum systems.

Continuous phase spaces have become a powerful tool for describing, analyzing, and tomographically reconstructing quantum states in quantum optics and beyond. A plethora of these phase-space techniques are known, however a thorough understanding of their relations is still lacking for finite-dimensional quantum states. We present a unified approach to continuous phase-space representations which highlights their relations and tomography. The infinite-dimensional case from quantum optics is then recovered in the large-spin limit.

In high sensitivity inductive electron spin resonance spectroscopy, superconducting microwave resonators with large quality factors are employed. While they enhance the sensitivity, they also distort considerably the shape of the applied rectangular microwave control pulses, which limits the degree of control over the spin ensemble. Here, we employ shaped microwave pulses compensating the signal distortion to drive the spins faster than the resonator bandwidth. This translates into a shorter echo, with enhanced signal-to-noise ratio. The shaped pulses are also useful to minimize the dead-time of our spectrometer, which allows to reduce the wait time between successive drive pulses. (C) 2019 The Authors. Published by Elsevier Inc.