{"id":346,"date":"2024-08-09T18:20:13","date_gmt":"2024-08-09T18:20:13","guid":{"rendered":"https:\/\/sabba.me\/nmr\/?p=346"},"modified":"2024-08-09T18:26:15","modified_gmt":"2024-08-09T18:26:15","slug":"zero-quantum-hamiltonian-engineering-made-simple-time-shifted-spin-echoes","status":"publish","type":"post","link":"https:\/\/sabba.me\/nmr\/2024\/08\/09\/zero-quantum-hamiltonian-engineering-made-simple-time-shifted-spin-echoes\/","title":{"rendered":"Zero-quantum Hamiltonian engineering made simple: time-shifted spin echoes"},"content":{"rendered":"\n

I think one of the crowning achievements of our paper on double-quantum excitation in strongly-coupled spin systems via the geometric quantum phase<\/a> was the development of a general way of generating a zero-quantum effective Hamiltonian with any phase, so I’d like to talk about that.<\/p>\n\n\n\n

Introduction <\/strong>(skip if you know what singlet NMR is)<\/p>\n\n\n\n

Suppose you had a coupled 2-spin-1\/2 system where the two nuclei can be said to be identical <\/em>i.e. in the sense of, they experience the exact same spin interactions. As you may know from introductory courses (or expect from the good ol’ Catalan triangle<\/a>) the spin system would have a set of eigenstates described by a spin-0 (singlet) and a spin-1 (triplet) manifold, where the eigenkets can be expressed in a number of languages:<\/p>\n\n\n\n

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

The Hamiltonian of this spin system would look something like this in the product operator formalism<\/a>, in which “J” denotes the isotropic part of the J-coupling tensor:<\/p>\n\n\n\n

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

And something like this in the matrix representation:<\/p>\n\n\n\n

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

Now, introduce a small chemical shift difference between the spins, which we will call \u0394<\/em>, with the corresponding Hamiltonian:<\/p>\n\n\n\n

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

It is pretty useful to express the previous Hamiltonians in terms of the single-transition operators for the zero-quantum subspace<\/strong> (the subspace of |T0<\/sub>> and |S0<\/sub>>):<\/p>\n\n\n\n

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

I’ll also include the cheeky unity operator of the double quantum subspace:<\/p>\n\n\n\n

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

To make it clear that I1z<\/sub>I2z<\/sub> consists of unity operators:<\/p>\n\n\n\n

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

We have:<\/p>\n\n\n\n

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

And we really only need to consider the truncated version of HJ<\/sub> (denoted HJ<\/sub>\u03b8<\/sup>) when it comes to the overall dynamics. <\/p>\n\n\n\n

In the strong-coupling limit, we may express the perturbation H\u0394<\/sub><\/em> in the interaction frame<\/em> of HJ<\/sub>, and it shouldn’t be rocket science to see that the time-dependent interaction-frame Hamiltonian is:<\/p>\n\n\n\n

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

That is to say, the net effect of the J-coupling Hamiltonian is to modulate the chemical shift difference term in the xy<\/em>-plane of the ZQ subspace. <\/p>\n\n\n\n

Zero-quantum effective Hamiltonians<\/strong><\/p>\n\n\n\n

Consider the traditional J-CPMG<\/a> building block of the old-school M2S<\/a> experiment, which involves a basic spin echo sequence [1\/(4J) – 180 – 1\/(4J)]N<\/sup>. Its average Hamiltonian is trivial to calculate using a first-order (numbered per the modern Nielsen-Levitt convention<\/a>; this would be zeroth-order<\/em> in the traditional convention<\/a> favored by the prominent American groups i.e. Warren<\/a>) Magnus expansion:<\/p>\n\n\n\n

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

Which works out like this:<\/p>\n\n\n\n

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

In a nutshell, we’ve generated an effective Hamiltonian that generates a pure x-rotation in the ZQ subspace! What if we explored the alternative sequence {180-[1\/(4J) – 180 – 1\/(4J)]N<\/sup>-180} ? We would instead have a rotation about the (-x) axis:<\/p>\n\n\n\n

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

Now, a little thought reveals you can generate a zero-quantum effective Hamiltonian of any phase<\/em>. For a phase \u03d5 in the interval [-\u03c0\/2,+\u03c0\/2]:<\/p>\n\n\n\n

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

And for a phase \u03d5 in the interval [+\u03c0\/2,+3\u03c0\/2]:<\/p>\n\n\n\n

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

Realized via an actual pulse sequence, we have the following pair of sequences which correspond to a rotation of phase \u03d5 and flip angle \u03b2 in the zero-quantum subspace:<\/p>\n\n\n\n

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

You can use this generalized control protocol to generate many fancy trajectories<\/a> in the ZQ subspace, but I originally invented this for another reason: composite pulses. You can see what I’m talking about below:<\/p>\n\n\n\n

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

This is a simulation for a hypothetical spin system with J<\/em> = 100 and \u0394<\/em> = 1, using a naive train of J-CPMG echoes (the ideal number Nideal<\/sub><\/em> is 156) vs. a composite rotation employing 5 ZQ-phase-shifted J-CPMG trains (the composite pulse itself is nothing a special and part of a family of composite pulses I came up with a while ago while generalizing the work of Tycko and Pines<\/a>). You can see that you can achieve excellent compensation against flip-angle errors just like you would do with a normal pulse, and you could try off-resonance errors too (see Pages 82-85 of Tayler’s excellent thesis<\/a>). But for now, I leave it at this, and leave you with the suggestion that there are many more tricks to be played in the zero-quantum world…<\/p>\n","protected":false},"excerpt":{"rendered":"

I think one of the crowning achievements of our paper on double-quantum excitation in strongly-coupled spin systems via the geometric quantum phase was the development of a general way of generating a zero-quantum effective Hamiltonian with any phase, so I’d like to talk about that. Introduction (skip if you know what singlet NMR is) Suppose […]<\/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":[10,3,13],"class_list":["post-346","post","type-post","status-publish","format-standard","hentry","category-theory","tag-sequences","tag-singlet","tag-theory"],"_links":{"self":[{"href":"https:\/\/sabba.me\/nmr\/wp-json\/wp\/v2\/posts\/346","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=346"}],"version-history":[{"count":2,"href":"https:\/\/sabba.me\/nmr\/wp-json\/wp\/v2\/posts\/346\/revisions"}],"predecessor-version":[{"id":350,"href":"https:\/\/sabba.me\/nmr\/wp-json\/wp\/v2\/posts\/346\/revisions\/350"}],"wp:attachment":[{"href":"https:\/\/sabba.me\/nmr\/wp-json\/wp\/v2\/media?parent=346"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/sabba.me\/nmr\/wp-json\/wp\/v2\/categories?post=346"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/sabba.me\/nmr\/wp-json\/wp\/v2\/tags?post=346"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}