Multi-spin control from one-spin pulses

TL;DR

RF pulses optimised for an isolated spin-1/2 can be used to control ensembles of weakly coupled spins by pulsing a single spin at a time, via band-schematic pulses with fixed pre- and post-evolution delays $aT$ and $bT$, expressed as $V_S = Z(b\Omega T) U Z(a\Omega T)$. The authors develop a method to extract the schematic parameters from arbitrary pulses using $p = \frac{i}{T} V^{\dagger} \frac{dV}{d\Omega}$ and show that many common pulses have or can be made band-schematic, enabling robust multi-spin control without full multi-spin optimisation. They demonstrate a band-schematic joint INEPT (JINEPT) element and show that continuous irradiation can replace free evolution delays in INEPT sequences; the framework and tools are implemented in the Seedless software, allowing rapid, hardware-tailored pulse generation. The work provides a practical route to high-sensitivity multi-spin NMR experiments by exploiting single-spin optimal control, with limitations in strong coupling and neglected relaxation.

Abstract

Controlling ensembles of weakly coupled spins typically requires computationally expensive multispin optimisations. We present a compact framework that enables control of weakly coupled spin systems (of any spin), but using RF pulses optimised for a single spin-1/2. We do this by explicitly creating a GRAPE pulse with fixed 'active' evolution times using single spin-1/2 methods, and pulsing on one spin at a time. By enforcing this form uniformly across offsets ('band-schematic' pulses),chemical shift and scalar coupling evolution of the entire system can be precisely controlled. We demonstrate the approach by constructing band-schematic pulses and a continuously irradiated joint INEPT (JINEPT) that achieves band-selective transfer $I_z \rightarrow 2I_zS_z$. The framework is implemented in the software Seedless, which both rapidly generates such pulses and analyses the schematic form of arbitrary pulses, enabling robust multi-spin control, without multi-spin optimisation.

Figures (5)

• Figure 1: The schematic form of selected excitation/de-excitation pulses (Table \ref{['tab1']}). A The normalised amplitude (A) and phase ($\phi$) of a i) rectangular $90^\circ$ (1 ms, $\omega$ 0.25 KHz), ii) an EBURP1 burp1991 (2 ms, $\omega$ 1.86 KHz, giving a 3 ppm bandwidth) that has approximately consistent schematic parameters over its range of activity, and four pulses generated using Seedless (2 ms, $\omega$ 5 KHz, optimised between $-1.5$ and 1.5 ppm, indicated by vertical bars, with a $B_1$ field distribution of 0.95, 1, 1.03 with constant amplitude seedless, Appendix \ref{['app:SeedScript']}), iii) a $Z\rightarrow-Y$ state to state showing 'semi' evolution control ($a$ is uncontrolled and $b=0$), iv) a 90x (where $a=b=0$), v) an evolution controlled a90xb (where $a=0.95$, $b=0$) and vi) an XYcite where $a$ and $b$ are uncontrolled. The infidelities of the optimised Seedless pulses were below 10$^{-4}$ indicating that the pulses well perform their intended function. The schematic pulse parameters of each were computed versus offset (computed for a Larmor frequency of 600 MHz). B The pre ($a$, purple) and post ($b$, green) evolution delays of each are indicated. C The angle ($\psi$, orange) and unit vector ($x,y,z$) describing the central rotation versus $\Omega$. Only the 90x and a90x are band-schematic in the range optimised with $a$ and $b$ staying constant over the region. In all plots, the performance of the pulses are shown at $B_1$ valuse of 0.95, 1, 1.03 of the central value. In the general case, a new pulse can be rapidly constructed using Seedless seedless for any of these tasks, as described in the text (Appendix \ref{['app:SeedScript']}).
• Figure 2: The schematic form of selected $180^\circ$ pulses (Table \ref{['tab1']}). A The normalised amplitude (A) and phase ($\phi$) of i) a rectangular $180^\circ$ (2 ms, $\omega$ 0.25 KHz), ii) a REBURP1 burp1991 (2 ms, $\omega$ 3.13 KHz giving a 3 ppm bandwidth), and four pulses generated using Seedless ($\omega$ 5 KHz, optimised between $-1.5$ and 1.5 ppm, indicated by vertical bars sampled at 0.95, 1.0 and 1.03 $B_1$ at constant amplitude seedless, Appendix \ref{['app:SeedScript']}), iii) a $Z\rightarrow-Z$ state to state (0.5 ms), iv) a 180x (2 ms), and v) an evolution controlled a180xa (2 ms). The infidelities of the Seedless pulses were below 10$^{-4}$ indicating that the pulses well perform their intended function. The schematic pulse parameters of each were computed versus offset (computed for a Larmor frequency of 600 MHz). The pre- and post- evolution delays of the schematic description are unreliable for the case of a $180^\circ$ rotation as described in the text, and the pulses are analysed from breaking them into two halves. B The angle ($\psi$, orange) and axis ($x,y,z$) of the central rotation of the pulse overall. C The pre- ($a$, purple) middle ($b$, green) and post ($c$, blue) evolution delays derived from analysing the pulses in two halves. D The angle ($\psi$, orange) and axis ($x,y,z$) of the central rotation in the first half of the pulse. The overall unitary transformation reveals all pulses perform a $180^o$ rotation in their active band, but the axis varies. Only the REBURP and the a180xa pulses are band-schematic and suitable for refocusing, with pre and post $a$ and $c$ delays constant versus ppm, the mid delay $b=0$ (or close to zero for the REBURP1) and a $90_x$ for the central rotation of the first half of the pulse. The $Z\rightarrow -Z$ has uncontrolled pre- and post- evolution delays but the central evolution period $b=0$. This isn't important if the pulse is used for inversion as magnetisation will be longitudinal during the pre- and post- evolution delays. The 180x is uncontrolled for all evolution periods and so is unsuitable for refocusing (unless it is very short). In all plots, the performance of the pulses are shown at $B_1$ values of 0.95, 1, 1.03 of the central value. In the general case, a new pulse can be rapidly constructed using Seedless seedless for any of these tasks, as described in the text (Appendix \ref{['app:SeedScript']}).
• Figure 3: A A 1D HSQC experiment was adapted so that the first INEPT consisted either of rectangular pulses (HARD) or a continuously pulsed JINEPT element (Equation \ref{['eq:JINEPT']}). The time that scalar coupling time is active $\Delta$ could be varied by changing $\tau$ (HARD) or by altering $aT$ in the a90x pulses (JINEPT). Pulse phases are on $x$ unless otherwise indicated with $\phi_1 = (0,2)$, $\phi_2=(0,0,2,2)$ and $\phi_{\text{rec}}=(2,0,0,2)$. Spectra were recorded with $\tau_2=1.75$ ms and gradients 1-4 were applied with durations (1, 1, 1, 0.15) ms at (20, 80, 5, 20.1)% of the maximum. Rectangular pulses were calibrated to have a $90^\circ$ rotation time of 7.84 and 21.46 $\mu$s on $^1$H and $^{13}$C respectively at maximum power. They were applied on-resonance to maximise their performance, which will deteriorate moving off-resonance seedless. 41 different JINEPT sequences were tested using a90x seedless pulses with varying $T$ and $a$. These were all applied at a field equal to half of the maximum possible (15.7 KHz, see Appendix \ref{['app:expmeth']}), ensuring that the probe was not overloaded. A rectangular 180$^\circ$ pulse was used on the $^{13}$C in the JINEPT element. Switching all pulses to GRAPE pulses is expected to boost overall sensitivity by ca. 30% at 750 MHz seedless. B Spectra of $^{13}$CH$_3$ methyl labelled methionine were acquired, revealing a doublet. Both HARD and JINEPT spectra are shown with $\Delta=\Delta_{max}=\frac{1}{2J}$ leading to maximum sensitivity (the JINEPT spectrum has been offset for clarity). C Peak intensities of both spectra varied with $\Delta$. For HARD, $\tau$ was varied. For JINEPT, four total durations ($T =$ 946, 1285, 1799 and 2998 $\mu$s) were tested with $a$ ranging from 0.1 to 0.95. The datasets were combined and fitted to $I=A\sin(\Delta J\pi)$, where the fitted value of $J$ was 138.6 Hz and $\Delta_{max}$ =3.6 ms. This demonstrates that varying $\tau$ (HARD) and adjusting $aT$ (JINEPT) leads to equivalent variations in signal intensity, and that scalar coupling can be controlled with these pulses. All spectra are shown in Figure \ref{['fig:processed_spectra']}.
• Figure 4: Spectra of $^{13}$CH$_3$ L-methionine using a modified HSQC experiment (Figure \ref{['fig:JINEPT']}) where the effective scalar coupling time $\Delta$ was systematically varied. Spectra have been offset in a manner that is proportional to $\Delta$ for clarity. A Spectra acquired using rectangular pulses, where the evolution delay $\tau$ was varied. B Spectra acquired using the JINEPT element with evolution fraction $a$ varied as indicated and i) $T=946\;\mu$s , ii) $T=1285\,\mu$s, iii) $T=1799\,\mu$s, and iv) $T=2998\,\mu$s.
• Figure 5: A summary report generated automatically by Seedless showing the computation of an a180xa pulse. Schematic pulse analysis is turned on by specifying EVOLVE. Because this is a 180$^\circ$ pulse, the evolution parameters are unreliable as discussed in the text. Specifying HALF analyses both halves of the pulse, allowing determination of the pre, mid and post evolution delays. The pulse was not constrained to have order/phase reversal symmetry, this symmetry in the final pulse emerged from optimisation.