##### Application of the small-tip-angle approximation in the toggling frame for the design of analytic robust pulses in quantum control

L. Van Damme, D. Sugny, S.J. Glaser

Physical Review A 104, 042226 (2021).

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.

##### Maximizing efficiency of dipolar recoupling in solid-state NMR using optimal control sequences

Z. Tosner, M.J. Brandl, J. Blahut, S.J. Glaser, B. Reif

Science Advances 7 (42), (2021).

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.

##### Application of Optimal Control Theory to Fourier Transform Ion Cyclotron Resonance

V. Martikyan, C. Beluffi, S.J. Glaser, M.A. Delsuc, D. Sugny

Molecules 26 (10), 2860 (2021).

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.

##### Selective and robust time-optimal rotations of spin systems

Q. Ansel, S.J. Glaser, D. Sugny

Journal of Physics A-Mathematical and Theoretical 54 (8), 085204 (2021).

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.

##### Fast Computation of Spherical Phase-Space Functions of Quantum Many-Body States

B. Koczor, R. Zeier, S.J. Glaser,

Physical Review A 102 (6), 62421 (2020).

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.

##### Symmetry-adapted decomposition of tensor operators and the visualization of coupled spin systems

D. Leiner, R. Zeier, S.J. Glaser

Journal of Physics A - Mathematical and Theoretical 53 (49), 495301 (2020).

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.

##### Robust control of an ensemble of springs: Application to ion cyclotron resonance and two-level quantum systems

V. Martikyan, A. Devra, D. Guery-Odelin, S.J. Glaser, D. Sugny

Physical Review A 102 (5), 053104 (2020).

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.

##### Cross-polarisation ENDOR for spin-1 deuterium nuclei

I. Bejenke, R. Zeier, R. Rizzato, S.J. Glaser, M. Bennati

Molecular Physics 118 (18), e1763490 (2020).

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.

##### Purity speed limit of open quantum systems from magic subspaces

V.A.A. Diaz, V. Martikyan, S.J. Glaser, D. Sugny

Physical Review A 102 (3), 033104 (2020).

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-Space Representations for Finite-Dimensional Quantum States and their Tomography

B. Koczor, R. Zeier, S.J. Glaser

Physical Review A 101 (2), 22318 (2020).

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 was 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.

##### Shaped pulses for transient compensation in quantum-limited electron spin resonance spectroscopy

S. Probst, V. Ranjan, Q. Ansel, R. Heeres, B. Albanese, E. Albertinale, D. Vion, D. Esteve, S.J. Glaser, D. Sugny, P. Bertet

Journal of Magnetic Resonance 303, 42-47 (2019).

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.