Spin dynamics during chirped pulses: applications to homonuclear decoupling and broadband excitation

Swept-frequency pulses have found applications in a wide range of areas including spectroscopic techniques where efficient control of spins is required. For many of these applications, a good understanding of the evolution of spin systems during these pulses plays a vital role, not only in describing the mechanism of techniques, but also in enabling new methodologies. In magnetic resonance spectroscopy, broadband inversion, refocusing, and excitation using these pulses are among the most used applications in NMR, ESR, MRI, and $in$ $vivo$ MRS. In the present survey, a general expression for chirped pulses will be introduced, and some numerical approaches to calculate the spin dynamics during chirped pulses via solutions of the well-known Liouville-von Neumann equation and the lesser-explored Wei-Norman Lie algebra along with comprehensive examples are presented. In both cases, spin state trajectories are calculated using the solution of differential equations. Additionally, applications of the proposed methods to study the spin dynamics during the PSYCHE pulse element for broadband homonuclear decoupling and the CHORUS sequence for broadband excitation will be presented

Optimal Control of Spins by Analytical Lie Algebraic Derivatives

Computation of derivatives (gradient and Hessian) of a fidelity function is one of the most crucial steps in many optimization algorithms. Having access to accurate methods to calculate these derivatives is even more desired where the optimization process requires propagation of these calculations over many steps, which is in particular important in optimal control of spin systems. Here we propose a novel numerical approach, ESCALADE (Efficient Spin Control using Analytical Lie Algebraic Derivatives) that offers the exact first and second derivatives of the fidelity function by taking advantage of the properties of the Lie group of $2\times 2$ Hermitian matrices, $\mathrm{SU}(2)$, and its Lie algebra, the Lie algebra of skew-Hermitian matrices, $\mathfrak{su}(2)$. A full mathematical treatment of the proposed method along with some numerical examples are presented

Adaptive optimal control of entangled qubits

Developing fast, robust, and accurate methods for optimal control of quantum systems comprising interacting particles is one of the most active areas of current science. Although a valuable repository of algorithms is available for numerical applications in quantum control, the high computational cost is somewhat overlooked. Here, we present a fast and accurate optimal control algorithm for systems of interacting qubits, QOALA (quantum optimal control by adaptive low-cost algorithm), which is predicted to offer [Formula: see text] (M(2)) speedup for an M-qubit system, compared to the state-of-the-art exact methods, without compromising overall accuracy of the optimal solution. The method is general and compatible with diverse Hamiltonian structures. The proposed approach uses inexpensive low-accuracy approximations of propagators far from the optimum, adaptively switching to higher accuracy, higher-cost propagators when approaching the optimum. In addition, the utilization of analytical Lie algebraic derivatives that do not require computationally expensive matrix exponential brings even better performance

NMR pulse sequence developments for high resolution in heteronuclear 2d experiments

Low resolution in the indirect dimension of 1H–13C heteronuclear experiments is a long-standing problem of NMR spectroscopy, which has not found a satisfactory solution. Among the numerous techniques aiming at increasing the resolution of the indirect dimension of 2D NMR spectra, we focused on spectral aliasing, which plays a unique role for its simplicity of application and efficiency. We also developed a homonuclear 13C decoupling in the F1 dimension, which enhances sensitivity and resolution by transforming the spread of multiplet patterns of carbons into singlets when samples with 13C labeling are involved

Deciphered Chemical Shifts in Aliased Spectra Recorded with two Slightly Different Narrow Windows or Differential Chemical Shift Evolution

The overlap of two HSQC spectra recorded with 10 and 9.9 ppm carbon spectral windows gives rise to highly resolved signals with a pattern providing unambiguous precise and accurate chemical shifts (see picture). Alternatively, the new DENA-HSQC pulse sequence takes advantage of a differential evolution of carbon chemical shift to do the same in a single experiment. Combined with multiplicity edition, DENA-HSQC advantageously replaces commonly used 1D DEPT-135 experiments

Reconstruction of full high-resolution HSQC using signal split in aliased spectra

Resolution enhancement is a long-sought goal in NMR spectroscopy. In conventional multidimensional NMR experiments, such as the 1H-13C HSQC, the resolution in the indirect dimensions is typically 100 times lower as in 1D spectra because it is limited by the experimental time. Reducing the spectral window can significantly increase the resolution but at the cost of ambiguities in frequencies as a result of spectral aliasing. Fortunately, this information is not completely lost and can be retrieved using methods in which chemical shifts are encoded in the aliased spectra and decoded after processing to reconstruct high-resolution 1H-13C HSQC spectrum with full spectral width and a resolution similar to that of 1D spectra. We applied a new reconstruction method, RHUMBA (reconstruction of high-resolution using multiplet built on aliased spectra), to spectra obtained from the differential evolution for non-ambiguous aliasing-HSQC and the new AMNA (additional modulation for non-ambiguous aliasing)-HSQC experiments. The reconstructed spectra significantly facilitate both manual and automated spectral analyses and structure elucidation based on heteronuclear 2D experiments. The resolution is enhanced by two orders of magnitudes without the usual complications due to spectral aliasing

Exact solutions for the time-evolution of quantum spin systems under arbitrary waveforms using algebraic graph theory

A general approach is presented that offers exact analytical solutions for the time-evolution of quantum spin systems during parametric waveforms of arbitrary functions of time. The proposed method utilises the \emph{path-sum} method that relies on the algebraic and combinatorial properties of walks on graphs. A full mathematical treatment of the proposed formalism is presented, accompanied by an implementation in \textsc{Matlab}. Using computation of the spin dynamics of monopartite, bipartite, and tripartite quantum spin systems under chirped pulses as exemplar parametric waveforms, it is demonstrated that the proposed method consistently outperforms conventional numerical methods, including ODE integrators and piecewise-constant propagator approximations

Broadband 13C-Homodecoupled Heteronuclear Single-Quantum Correlation Nuclear Magnetic Resonance

13C-homodecoupling in F1: Eliminating homonuclear 13C scalar couplings is a challenging NMR problem when enriched compounds are studied by NMR. The 13C-homodecoupled heteronuclear single-quantum correlation (HSQC) experiment (BBHD-HSQC) relies on the Zangger–Sterk pulse sequence element to combine spatial encoding of the chemical shift with the refocusing of couplings. The sequence is applied to fully enriched 13C cholesterol

Combination of Homonuclear Decoupling and Spectral Aliasing to Increase the Resolution in the ¹H Dimension of 2D NMR Experiments

Broadband homonuclear decoupling (BBHD) in the indirect 1H dimension of 2D experiments can be obtained using a modified Zangger and Sterk combination of a selective pulse with a pulsed-field gradient. The coupling structure of signals is reduced to a singlet along the F1 dimension at the cost of a sensitivity loss. With the classical sampling in F1, the full resolving power of BBHD-experiments requires very long acquisition times. Spectral aliasing can reduce the number of time increments accessing the top resolution of homodecoupled spectra of small molecules by two orders of magnitude. The TOCSY spectra of androst-4-ene-3,17-dione are shown as an example