In situ Learning-Based Spin Engineering of Pulsed Dynamic Nuclear Polarization

Abstract

Pulsed Dynamic Nuclear Polarization (DNP) is currently receiving substantial interest as a means to enhance the sensitivity of nuclear magnetic resonance (NMR) and magnetic resonance imaging (MRI) by orders of magnitude. It has also received much attention as a central ingredient in many modalities of electron spin-involved quantum sensing. Relative to spin engineering associated with NMR, the design of efficient pulsed DNP experiments with a broad experimental scope are challenged by large electron-nuclear spin systems, large electron spin-involved interactions, and instrumental non-idealities and limitations. All of this may challenge traditional NMR-like theoretical and numerical pulse sequence engineering. Exploiting state-of-the-art instrumentation and taking advantage of the high sensitivity of DNP relative to NMR, we here demonstrate the use of combinations of Bayesian machine learning methods and constrained random walk procedures to design pulse sequences \textit{in situ}, by experiments, directly on the spin systems responding to spectrometer instructions. For trityl and nitroxide samples, it is demonstrated that efficient broadband DNP pulse sequences can be designed in situ with experimental protocols benchmarked against in silico analogs.

Figures (21)

• Figure 1: Schematic representation of the experimental setup used for in situ Bayesian development of DNP pulse sequences. a) The generation of amplitude and phase modulated pulses for an initial flip (or excitation) pulse and a periodic DNP transfer element in combination mediating electron to proton polarization transfer. The pulse sequences are generated using an AWG. The overall pulse program contains the initial flip pulse and the DNP transfer element at the MW channel for polarization transfer and an RF channel containing initial saturation pulses and a solid echo for proton signal detection. b) Part of the spectrometer magnet and the ENDOR/EPR probe for MW/RF pulsing and detection of signals from an inserted frozen radical-containing sample. The spectrometer is controlled by a Bayesian optimization algorithm in a feedback-loop setup receiving response from the experiment and outputs the control variables for the next experiment. c) The sample is a solution containing in our example trityl OX063 or 4-Oxo-TEMPO radicals.
• Figure 2: in situ optimization of the DNP transfer element in Fig. \ref{['fig:Setup']}a with 6 (a-c) and 24 (e,f) pulses (modulation time $t_m$=120 ns; 49 MHz maximal MW field strength; initial flip pulse of amplitude 47 MHz and duration 5 ns, see Fig. \ref{['fig:Setup']}a). The measurements were performed at 80 K using a trityl OX063 in glycerol-d$_5$ (a-e) and glycerol-d$_8$ (f). The DNP transfer element was repeated 1-30 (a-c) and 6 (e-f) times. The buildup time was 2 (a-c,e) and 64 (f) s. (a-c) Results for 3 optimization methods: a) Monte Carlo, b) Bayesian, and c) constrained Bayesian (1000 evaluations each, 51 h optimization). (a1-c1) Absolute average enhancement $|\overline{\epsilon}|$ for the offset profile; red/blue dots represent match to ZQ/DQ resonances. (a2-c2) Accumulated nutation angle during each of the evaluated sequences. Colored numbers reflects $k_I$ values for ZQ (red) and DQ (blue) resonances according to Eq. (\ref{['Eq1']}). (a3-c3) Electron-spin offset profile. In columns 2) and 3) the color is weighted by the absolute average enhancement. The best sequences are marked in black. d) Different electron-spin offsets were realized by placing the MW carrier at different frequencies with respect to the EPR resonance of trityl. e) Constrained Bayesian optimization for a 24-pulse DNP transfer element (black circle marks the sequence with best offset profile when repeated with higher resolution in electron-spin offset). f) Offset profiles for the 24-pulse BayesOpt sequence marked in e) compared to NOVEL, PLATO, and cRW-OPT1. Offset profiles for trityl in glycerol-d$_5$ can be found in Supplementary Information Fig. S3. Settings for Bayesian optimizations are provided in Supplementary Information Tables S1 and S2.
• Figure 3: a) Bayesian in situ optimization of a 4-pulse excitation sequence with flexible total duration of 5-20 ns (all pulses equal length, amplitude up to 45 MHz, free phase) followed by cRW-OPT1 DNP transfer (5 repetitions; 5 s buildup) using trityl OX063 in glycerol-d$_5$. b) The best 4-pulse excitation sequence (marked by circle in a); Supplementary Information Table S7) followed by cRW-OPT1 DNP transfer. c) Comparison of simulated (horizontal) and experimental (vertical) enhancements for all sequences in a). c) Offset profile for the best excitation sequence. Blue crosses show the enhancements found during optimization, green dots show repetition of the experiment for a finer grid of offsets, and the dashed line shows the average. e) Simulated profile of the best excitation sequence. Crosses show simulated transfer at the optimized offsets, and the dashed line shows the average. Note that the dashed lines in c) correspond to the best sequence as shown in d) and e) f) Offset profiles for cRW-OPT1 using the best excitation sequence and a hard excitation pulse (amplitude 32 MHz). The carrier of the hard excitation pulse was either following the offset for the DNP transfer sequence or fixed at the trityl resonance. These experiments used trityl OX063 in glycerol-d$_8$ and a buildup time of 64 s. Settings for the Bayesian optimizations are given in Supplementary Information Table S8.
• Figure 4: In situ experimental Monte Carlo and Bayesian optimization of excitation pulse and polarization transfer elements of pulsed DNP experiments for 2 mM sample of 4-Oxo-TEMPO. a) Evaluated enhancements for Monte Carlo optimization of control variables. b) Evaluated enhancements for an unconstrained Bayesian optimization of control variables. c) The best pulse sequence is marked by the circle in b). The duration of the excitation pulses was 4 ns each, and transfer pulses were 1250 ns each (the time axis is not to scale). The pulse sequence is given in Supplementary Information Table S11. d) Buildup curve for the best sequence found using Bayesian optimization compared to the buildup for a hard excitation pulse of 52 MHz for 5.5 ns followed by a spin-lock pulse of constant amplitude of 31.2 MHz for 10 $\mu s$. The cross showed the enhancement measured during the optimization with a buildup time of 8 s. Settings for the Bayesian optimizations are given in Supplementary Information Table S12.
• Figure 5: Diagram showing the procedure used for situ Bayesian learning-based optimization. First, a set of random input vectors is given to the objective function. Each of these vectors is then used to construct a pulsed DNP sequence, which is executed on the experimental setup. Each sequence and the resulting DNP enhancements are stored in an experiment log for future reference. After the initial input vectors have been evaluated, a Gaussian process model is initialized from the experiment log, and the chosen type of acquisition function is optimized based on this model. The (approximate) optimizer of the acquisition function is then used as the input vector for the next evaluation of the objective function, completing the feedback loop. When certain user-defined conditions, such as number of iterations, are met, the loop is set to stop, and, subsequently, the best sequence can be extracted from the experiment log.