Everybody loves adiabatic sweeps. They are a popular way of achieving Iz → Sz polarization transfer because you can kill two birds with one stone: you usually get an impressive enhancement factor as well as automatic robustness against errors in whatever parameter you felt like sweeping, whether it was pulse strength, resonance offsets/detuning, or something else.
But what goes up must come down. Here’s the unfortunate catch: an adiabatic sweep can never give you the maximum possible enhancement factor. See my OEIS sequence A362534, and consult the Sacred Table:
In this blog post we’re going to try and deal with the adiabatic column of this table, but I should first provide a disclaimer: note that the basic observation that adiabatic sequences do not lead to the maximum possible transfer of polarization dates back (at least) to Hodgkinson and Pines’ paper from 1997 [1]. The adiabatic-associated bounds themselves were derived by Chingas, Garroway, and friends at the US Naval Research Laboratory [2, 3] even earlier.
The bounds on an adiabatic Iz → Sz transfer, in INS spin systems, are simply:
Where we have used our good friend the Catalan triangle to avoid doing more group theory than we really need to. (I shall not bear gruppenpest in this house.)
But hold on, where does the above result come from? The bound is very simply obtained by considering the much easier problem of an adiabatic Iz → Sz transfer in a two-spin subsystem (I = ℓ, S = 1/2):
All we’ve really done here is some sneaky normalization i.e. multiply by the (2ℓ+1) factor associated with the old 1-spin-ℓ basis, divide by the 2N factor associated with the new N-spin-1/2 basis, and you’re golden. Then, just consult the aforementioned Catalan triangle to account for the multiplicity of the spin-ℓ manifolds [As emphasized in that blog post, a common theme in tackling multispin problems where permutation symmetry is strictly satisfied, is solving for the irreducible subspaces and simply adding things up.]
Conveniently, it turns out that there is a closed form for the adiabatic-associated bounds:
Which I think is somewhat less intimidating than the equivalent form given in the aforementioned paper by Chingas et al.:
