{"id":181,"date":"2024-06-10T16:28:46","date_gmt":"2024-06-10T16:28:46","guid":{"rendered":"https:\/\/sabba.me\/nmr\/?p=181"},"modified":"2024-06-10T16:35:45","modified_gmt":"2024-06-10T16:35:45","slug":"evolution-constrained-bounds-on-polarization-transfer-in-i_ns-spin-systems-part-ii-adiabatic-sweeps","status":"publish","type":"post","link":"https:\/\/sabba.me\/nmr\/2024\/06\/10\/evolution-constrained-bounds-on-polarization-transfer-in-i_ns-spin-systems-part-ii-adiabatic-sweeps\/","title":{"rendered":"Evolution-Constrained Bounds on Polarization Transfer in I_{N}S Spin Systems Part II: Adiabatic Sweeps"},"content":{"rendered":"\n

Everybody loves adiabatic sweeps. They are a popular way of achieving Iz<\/sub> \u2192 Sz<\/sub> 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.<\/p>\n\n\n\n

But what goes up must come down. Here’s the unfortunate catch: an adiabatic sweep can never give you the maximum <\/em>possible enhancement factor. See my OEIS sequence A362534<\/a>, and consult the Sacred Table<\/a>:<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

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<\/a>]. The adiabatic-associated bounds themselves were derived by Chingas, Garroway, and friends at the US Naval Research Laboratory [2<\/a>, 3<\/a>] even earlier.<\/p>\n\n\n\n

The bounds on an adiabatic Iz<\/sub> \u2192 Sz<\/sub> transfer, in IN<\/sub>S spin systems, are simply:<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Where we have used our good friend the Catalan triangle<\/a> to avoid doing more group theory than we really need to. (I shall not bear gruppenpest<\/em><\/a> in this house.)<\/p>\n\n\n\n

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<\/sub> \u2192 Sz<\/sub> transfer in a two-spin subsystem (I = \u2113, S = 1\/2):<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

All we’ve really done here is some sneaky normalization i.e. multiply by the (2\u2113+1) factor associated with the old 1-spin-\u2113 basis, divide by the 2N<\/sup> 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-\u2113 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.]<\/p>\n\n\n\n

Conveniently, it turns out that there is a closed form for the adiabatic-associated bounds:<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

Which I think is somewhat less intimidating than the equivalent form given in the aforementioned paper by Chingas et al.:<\/p>\n\n\n\n

\"\"<\/figure>\n","protected":false},"excerpt":{"rendered":"

Everybody loves adiabatic sweeps. They are a popular way of achieving Iz \u2192 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 […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[7],"tags":[],"class_list":["post-181","post","type-post","status-publish","format-standard","hentry","category-theory"],"_links":{"self":[{"href":"https:\/\/sabba.me\/nmr\/wp-json\/wp\/v2\/posts\/181","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/sabba.me\/nmr\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/sabba.me\/nmr\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/sabba.me\/nmr\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/sabba.me\/nmr\/wp-json\/wp\/v2\/comments?post=181"}],"version-history":[{"count":2,"href":"https:\/\/sabba.me\/nmr\/wp-json\/wp\/v2\/posts\/181\/revisions"}],"predecessor-version":[{"id":193,"href":"https:\/\/sabba.me\/nmr\/wp-json\/wp\/v2\/posts\/181\/revisions\/193"}],"wp:attachment":[{"href":"https:\/\/sabba.me\/nmr\/wp-json\/wp\/v2\/media?parent=181"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/sabba.me\/nmr\/wp-json\/wp\/v2\/categories?post=181"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/sabba.me\/nmr\/wp-json\/wp\/v2\/tags?post=181"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}