J-Synchronized Echoes in the Pantheon of Symmetry-based Sequences?

I often make the point that PulsePol is expected to replace the “default” option of M2S as the go-to windowed (meaning there are gaps between pulses) or hard-pulse sequence for exciting nuclear singlet order in strongly-coupled spin systems. (Shoutout to the archetypal soft-pulse/low-power sequence SLIC.)

That being said, I religiously avoid the misleading statement “symmetry-based sequences will replace M2S” and wince uncomfortably at the frequently-asked question “do you think symmetry-based sequences will replace M2S?”

This is because the building block of the M2S sequence, the J-synchronized echo (JSE; see refs. 1, 2, and 3), is technically a symmetry-based sequence.

Figure 1: The J-synchronized spin echo pulse sequence.

The fundamental difference between the JSE and PulsePol is that the JSE is designed to selectively swap the |S₀〉and |T₀〉states (via a zero-quantum effective Hamiltonian) while PulsePol is designed to selectively swap |S₀〉and either of the |T_±〉states (via a single-quantum effective Hamiltonian):

Figure 2: illustration of transitions excited by different symmetry-based sequences.

This is painfully obvious when you phrase it the right way: the R-element (i.e. basic inversion element) of the JSE is simply a windowed 180₀ pulse, and the nominal duration of the R-element is half a “rotor” period 1/(2J). With regards to the first-order selection rules, no phase cycling is applied*. Hence, we may denote the JSE as R2₁⁰ in the notation describing symmetry-based sequences. Or you could denote it C1₁⁰ too (assuming you count a full cycle including two spin echoes).

As an aside, it is easy to see why the M2S sequence ended up appearing clunky with awkwardly placed pulses and delays interrupting the echoes: the selective transition being engineered by the JSE building block was the wrong one, assuming one was starting from thermal equilibrium magnetization. Which suggests that the JSE would’ve worked handsomely for nuclear singlet excitation if one was starting from dipolar order (an excess population of the central triplet state)…

*Much like PulsePol, “riffling” in the form of supercycles is necessary for the optimal performance of the JSE/M2S sequences vis-a-vis pulse strength and detuning errors.