We investigate the tomography of unknown unitary quantum processes within the framework of a finite-dimensional Wigner-type representation. This representation provides a rich visualization of quantum operators by depicting them as shapes assembled as a linear combination of spherical harmonics. These shapes can be experimentally tomographed using a scanning-based phase-space tomography approach. However, so far, this approach was limited to known target processes and only provided information about the controlled version of the process rather than the process itself. To overcome this limitation, we introduce a general protocol to extend Wigner tomography to unknown unitary processes. This new method enables experimental tomography by combining a set of experiments with classical post-processing algorithms introduced herein to reconstruct the unknown process. We also demonstrate the tomography approach experimentally on IBM quantum devices and present the specific calibration circuits required for quantifying undesired errors in the measurement outcomes of these demonstrations.

The SpinDrops Didactics Framework (SDDF) is a built-in framework for the development of educational contents in the free software SpinDrops. SpinDrops was specifically designed for dynamically simulating and visualizing systems of coupled spins-1/2 in the context of nuclear magnetic resonance (NMR), yet it provides a vast range of tools and applications in an unguided manner, potentially overloading users who are to learn NMR spectroscopy. Previously, we introduced the digital SpinDrops Learning Environment (SDLE), engineered with the SDDF. This learning environment targeted the most relevant topics in liquid state H-1 NMR spectroscopy, that is, precession, chemical shift, and scalar coupling. These are relevant concepts for chemists and other scientists when interpreting NMR spectra which were addressed by offering carefully designed interactive SDDF exercises. In the following, the SDDF is introduced from a detailed technological point of view. We show that the SDDF offers a fully modular toolbox which can be used to create highly interactive multifaceted exercises and learning units in a simple and easy-to-understand way. Our software particularly aims to enable teachers to design custom educational content which can be tailored to the teaching of magnetic resonance from school to expert level and can cover topics ranging from experimental applications of NMR in organic chemistry to quantum mechanical theory of magnetic resonance. The obtained exercises can be distributed to students and directly imported into SpinDrops.

The chemical shift range of many NMR‐active isotopes cannot be excited in a single experiment by classical hard pulse high‐resolution spectroscopy or even conventional broadband excitation. Such nuclei can be addressed by specifically optimized saturation pulses or xy ‐excitation, which are derived from linear frequency sweeps that are further optimized using methods derived from optimal control theory. A multi‐isotope 1D experiment covering 6 MHz as well as homonuclear COSY and heteronuclear HMBC experiments covering more than 100 kHz are demonstrated, which can be adapted to fit any needs for specific isotopes at any spectrometer field. In general, the approach is very useful for 1D and 2D absolute value overview spectra at high magnetic fields and/or wideband and low‐gamma nuclei.

We introduce a generalized phase-space representation of qubit systems called the BEADS representation which makes it possible to visualize arbitrary quantum states in an intuitive and an easy to grasp way. Our representation is exact, bijective, and general. It bridges the gap between the highly abstract mathematical description of quantum mechanical phenomena and the mission to convey them to non-specialists in terms of meaningful pictures and tangible models. Several levels of simplifications can be chosen, e.g. when using the BEADS representation in the communication of quantum mechanics to the general public. In particular, this visualization has predictive power in contrast to simple metaphors such as Schrödinger’s cat.

We derive new, fundamental insights into the dynamics of an optical qubit, revealing how this is influenced by the motion of the trapped particle. Leveraging these new insights, we show that photon-recoil heating can be suppressed at relatively high Rabi frequencies by modulating the phase of the driving laser field in time. This technique enables single-qubit gates that are up to 20 times faster than current state-of-the-art approaches while maintaining the same fidelity. Remarkably, even when photon recoil is eliminated (i.e., when occupation of Fock states is preserved), we find that the gate infidelity does not vanish, but is rather limited by a fundamental mechanism, which we identify as thermal motion-induced entanglement. To overcome this limitation and the effect of recoil, we derive motion-insensitive control pulses that enable the execution of fast, very high-fidelity gates with optical qubits.

Nuclear magnetic resonance (NMR) spectroscopy is one of the most important spectroscopy methods in modern chemistry, yet students need help learning and understanding its complex nature. Empirical investigations show that simulation and (interactive) visualizations can support students by offering new possibilities for investigating connections and direct effects of parameter changes. Therefore, this article introduces a digital and interactive learning environment that addresses students' learning difficulties to facilitate the understanding of crucial H-1 NMR background knowledge. The SpinDrops Learning Environment (SDLE) covers practically useful, basic concepts for interpreting a H-1 NMR spectrum from a theoretical point of view. It includes new interactive visualizations and a dynamic and realistically simulated ppm-spectrum. To evaluate the role and influence of interactivity by parameter control regarding students' learning processes and conceptual understanding, we designed two SDLE versions differing in the degree of interactivity and dynamics. We then asked N = 50 students to work with the learning environment in a pre-post study with questionnaires on affective constructs and a H-1 NMR knowledge test on conceptual understanding. N = 12 students additionally took part in a think-aloud study. The results showed that students benefit from learning with the SDLE as their conceptual understanding and NMR-related interest, self-efficacy, and estimated knowledge increase significantly. Although the SDLE showed significantly positive effects on students' learning results and processes, the difference in the degree of interactivity and dynamics inside the software only had a small impact, as revealed in the quantitative and qualitative data.

We propose a scheme to perform optical pulses that suppress the effect of photon recoil by three orders of magnitude compared to ordinary pulses in the Lamb-Dicke regime. We derive analytical insight about the fundamental limits to the fidelity of optical qubits for trapped atoms and ions. This paves the way towards applications in quantum computing for realizing >1000 of gates with an overall fidelity above 99%.

In trapped-atom quantum computers, high-fidelity control of optical qubits is challenging due to the motion of atoms in the trap. If not corrected, the atom motion gets entangled with the qubit degrees of freedom through two fundamental mechanisms, (i) photon recoil and (ii) thermal motion, both leading to a reduction of the gate fidelity. We develop motion-insensitive pulses that suppress both sources of infidelity by modulating the phase of the driving laser field in time. To eliminate photon recoil, we use bang-bang pulses-derived using time-optimal control-which shorten the gate duration by about 20 times compared to conventional pulses. However, even when photon recoil is eliminated, we find that the gate error does not vanish, but is rather limited by a bound arising from thermal motion-induced entanglement. Remarkably, this bound is independent of the Rabi frequency, meaning that, unlike for photon recoil, operating in the resolved sideband regime does not mitigate this source of infidelity. To overcome this bound, we derive smooth-phase pulses, which allow for a further reduction of the gate error by more than an order of magnitude for typical thermal atoms. Motion-insensitive pulses can be refined to compensate for laser inhomogeneities, enhancing the gate performance in practical situations. Our results are validated through simulations of one-qubit gates operating on the optical clock transition of ^88Sr atoms trapped in an optical tweezers array.

We present an experimental scanning-based tomography approach for near-term quantum devices. The underlying method has previously been introduced in an ensemble-based NMR setting. Here we provide a tutorial-style explanation along with suitable software tools to guide experimentalists in its adaptation to near-term pure-state quantum devices. The approach is based on a Wigner-type representation of quantum states and operators. These representations provide a rich visualization of quantum operators using shapes assembled from a linear combination of spherical harmonics. These shapes (called droplets in the following) can be experimentally tomographed by measuring the expectation values of rotated axial tensor operators. We present an experimental framework for implementing the scanning-based tomography technique for circuit-based quantum computers and showcase results from IBM quantum experience. We also present a method for estimating the density and process matrices from experimentally tomographed Wigner functions (droplets). This tomography approach can be directly implemented using the Python-based software package \texttt{DROPStomo}.

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 [A. Grossmann, Comm. Math. Phys. 48(3), 191 (1976)], 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 [L. Cohen, J. Math. Phys. 7(5), 781 (1966)] 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 [M. Born, P. Jordan, Z. Phys. 34(1), 858 (1925)]. 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.

NV-diamonds have attracted keen interest for nanoscale sensing and spin manipulation. In particular, the non-equilibrium electron spin polarization after optical excitation of single NV centers has successfully been transferred to nuclear spin baths in the surrounding of the defects. However, these experiments need to be extended to NV-ensembles which have promising practical applications in the hyperpolarization of bulk sample volumes for NMR signal enhancement. Here, we use a dense, shallow ensemble of NV-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 NV polarization source: from the close proximity of $^{13}$C inside the diamond lattice to the self-assembled molecular system consisting of $^{1}$H and $^{19}$F spins outside the diamond and over multiple interfaces. We show that ensemble NV experiments face problems different from single NV experiments. In particular, using spinlock pulses, the inhomogeneously broadened ESR line of the NV ensemble limits the minimal resonance linewidth with which the transfer protocol can occur. Furthermore, we compare the NV 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 build-up within the detection volume of nanoscale NV-NMR experiments.

Purpose To demonstrate that the concept of “universal pTx pulses” is applicable to local excitation applications. Methods A database of B 0 / maps from eight different subjects was acquired at 9.4T. Based on these maps, universal pulses that aim at local excitation of the visual cortex area in the human brain (with a flip angle of 90° or 7°) were calculated. The remaining brain regions should not experience any excitation. The pulses were designed with an extension of the “spatial domain method.” A 2D and a 3D target excitation pattern were tested, respectively. The pulse performance was examined on non‐database subjects by Bloch simulations and in vivo at 9.4T using a GRE anatomical MRI and a presaturated TurboFLASH mapping sequence. Results The calculated universal pulses show excellent performance in simulations and in vivo on subjects that were not contained in the design database. The visual cortex region is excited, while the desired non‐excitation areas produce the only minimal signal. In simulations, the pulses with 3D target pattern show a lack of excitation uniformity in the visual cortex region; however, in vivo, this inhomogeneity can be deemed acceptable. A reduced field of view application of the universal pulse design concept was performed successfully. Conclusions The proposed design approach creates universal local excitation pulses for a flip angle of 7° and 90°, respectively. Providing universal pTx pulses for local excitation applications prospectively abandons the need for time‐consuming subject‐specific B 0 / mapping and pTx‐pulse calculation during the scan session.

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.

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 parameterizations to satisfy experimental constraints and limitations on the amplitude or energy of the pulse. A comparison with numerical optimal solutions is made.

PURPOSE:The paper introduces a classical model to describe the dynamics of large spin-1/2 ensembles associated with nuclei bound in large molecule structures, commonly referred to as the semi-solid spin pool, and their magnetization transfer (MT) to spins of nuclei in water. THEORY AND METHODS:Like quantum-mechanical descriptions of spin dynamics and like the original Bloch equations, but unlike existing MT models, the proposed model is based on the algebra of angular momentum in the sense that it explicitly models the rotations induced by radiofrequency (RF) pulses. It generalizes the original Bloch model to non-exponential decays, which are, for example, observed for semi-solid spin pools. The combination of rotations with non-exponential decays is facilitated by describing the latter as Green's functions, comprised in an integro-differential equation. RESULTS:Our model describes the data of an inversion-recovery magnetization-transfer experiment with varying durations of the inversion pulse substantially better than established models. We made this observation for all measured data, but in particular for pulse durations smaller than 300 μs. Furthermore, we provide a linear approximation of the generalized Bloch model that reduces the simulation time by approximately a factor 15,000, enabling simulation of the spin dynamics caused by a rectangular RF-pulse in roughly 2 μs. CONCLUSION:The proposed theory unifies the original Bloch model, Henkelman's steady-state theory for MT, and the commonly assumed rotation induced by hard pulses (i.e., strong and infinitesimally short applications of RF-fields) and describes experimental data better than previous models.

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 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.

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 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 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 these phase-space techniques.

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.