{"id":62,"date":"2024-02-22T20:28:37","date_gmt":"2024-02-22T20:28:37","guid":{"rendered":"https:\/\/sabba.me\/nmr\/?p=62"},"modified":"2024-05-16T13:20:37","modified_gmt":"2024-05-16T13:20:37","slug":"the-quantum-pascal-pyramid","status":"publish","type":"post","link":"https:\/\/sabba.me\/nmr\/2024\/02\/22\/the-quantum-pascal-pyramid\/","title":{"rendered":"The Quantum Pascal Pyramid"},"content":{"rendered":"\n

In the previous blog post<\/a>, I discussed a close relative of Pascal’s triangle – a Catalan triangle – and briefly alluded to z1<\/sup> multiplets at the end.

In this post I will discuss another less well-known (but very useful!) combinatorial structure – the Pascal (difference) pyramid<\/a><\/em>, which has a beautiful correspondence to the multispin (single-quantum) coherences observed in NMR experiments.

Suppose one has an IN<\/sub>S (or AXN<\/sub>) spin system. Begin by considering the following cumulative tensor product: <\/p>\n\n\n\n

\"\"<\/p>\n\n\n\n

The expansion may be organized by the spin product rank q<\/em>, yielding a convenient operator basis of I-spin longitudinal (z<\/em>-) spin-operators which Sorensen denoted ZN<\/sub>q<\/sup><\/em><\/a>. Some examples for the cases N=3<\/em> and N=4<\/em> are:
<\/p>\n\n\n\n

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


In my mind, I like to call these “the binomial Z-operators” since the number of operators forms binomial patterns much like NMR spectra (1:3:3:1, 1:4:6:4:1, etc.) for reasons I hope are obvious. (If it isn’t obvious, consider the basic combinatorial problem: How many unique words with length q can be formed by combining N different letters?<\/em>)<\/p>\n\n\n\n

Now, suppose you wanted to observe single-quantum coherences involving a product of any of these ZN<\/sub>q<\/sup><\/em> operators with, say, Sx<\/sub><\/em>. What would the S-spin spectrum look like? Some visual examples are given here:
<\/p>\n\n\n\n

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



Intuitively, one can reason that each ZN<\/sub>q<\/sup><\/em>.Sx<\/sub> operator has to be a sum of the “pure” single-transition operators corresponding to each m<\/em>=J<\/em>\u00d7{-N\/2<\/em>,…,+N\/2<\/em>} line component making up the S-spin spectrum. One might notice that each ZN<\/sub>q<\/sup><\/em> operator changes sign exactly q<\/em> times. But it’s not so clear to see how<\/em>.<\/p>\n\n\n\n

It turns out there is an incredible direct map<\/em> between the operators describing pure populations of states with azimuthal quantum number m<\/em> (which I will denote PN<\/sub>m<\/sup><\/em>) and ZN<\/sub>q<\/sup><\/em> . The relationship is given by:<\/p>\n\n\n\n

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

The map can be represented by an (N+1)\u00d7(N+1)<\/em> matrix. Some matrices illustrating this relationship are shown below:

<\/p>\n\n\n\n

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

Naturally, this type of relationship has appeared in other fields<\/a> of quantum mechanics (see equation 11). But to my knowledge, the only magnetic resonance paper which describes something resembling a “Pascal’s pyramid” is the excellent paper<\/a> on relaxation in the AX4<\/sub> spin system of 15<\/sup>NH4<\/sub>+<\/sup> by Nicolas Werbeck and D. Flemming Hansen, where the combinatorial structure is referred to as a “modified Pascal triangle”.

EDIT<\/strong> (February 23, 2024, 15:56 UK time): after writing this article, I realized something neat. Recall the de Moivre-Laplace theorem<\/a>, which is the famous mathematical result that the binomial distribution (corresponding to the multiplets associated with ZN<\/sub>0<\/sup><\/em>) converges to a normal (Gaussian) distribution<\/a> as N\u2192<\/strong>\u221e<\/em>. I have observed a generalization of the de Moivre-Laplace theorem: that the columns of Pascal’s pyramid, i.e the multiplets associated with ZN<\/sub><\/em>q<\/em><\/sup>, converge to the q<\/em>-th derivatives of a Gaussian distribution. The q<\/em>-th derivative of a generic Gaussian function has a general form that can be written in terms of a regularized hypergeometric function<\/a>:
<\/p>\n\n\n\n

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




That is to say, the ZN<\/sub><\/em>q<\/em><\/sup> operator may be approximated by a q<\/em>-th order Gaussian derivative:

<\/p>\n\n\n\n

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

We know from basic mathematics that a derivative is a measure of a rate of change. We can appreciate the corresponding physical picture in more than one way: <\/p>\n\n\n\n

1. each ZN<\/sub><\/em>q<\/em><\/sup> operator transforms under, say, an I-spin x<\/em>-rotation of a flip angle \u03b2<\/em>, with an increasing “responsiveness” to rotation that depends on the spin product rank q<\/em>. The self-evolution of each ZN<\/sub><\/em>q<\/em><\/sup> operator would be given by:<\/p>\n\n\n\n

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

2. each ZN<\/sub><\/em>q<\/em><\/sup> operator can be converted into I-spin multiple quantum coherence with a (maximum) coherence order q<\/em>. This well-known property is commonly exploited<\/a> in NMR experiments. Of course, by definition, an MQC of coherence order q<\/em> (which I’ll dub qQC for q<\/em>-quantum coherence) has the following property under an I-spin z<\/em>-rotation with a flip-angle \u03b2<\/em>:
<\/p>\n\n\n\n

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

EDIT<\/strong> (May 16, 2024, 14:20 UK time): I have written this blog post as a small paper<\/a> on arXiV, which goes into a bit more detail.<\/p>\n","protected":false},"excerpt":{"rendered":"

In the previous blog post, I discussed a close relative of Pascal’s triangle – a Catalan triangle – and briefly alluded to z1 multiplets at the end. In this post I will discuss another less well-known (but very useful!) combinatorial structure – the Pascal (difference) pyramid, which has a beautiful correspondence to the multispin (single-quantum) […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[16,7],"tags":[11,12,13],"class_list":["post-62","post","type-post","status-publish","format-standard","hentry","category-combinatorial-aspects-of-magnetic-resonance","category-theory","tag-combinatorics","tag-pascal","tag-theory"],"_links":{"self":[{"href":"https:\/\/sabba.me\/nmr\/wp-json\/wp\/v2\/posts\/62","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=62"}],"version-history":[{"count":7,"href":"https:\/\/sabba.me\/nmr\/wp-json\/wp\/v2\/posts\/62\/revisions"}],"predecessor-version":[{"id":137,"href":"https:\/\/sabba.me\/nmr\/wp-json\/wp\/v2\/posts\/62\/revisions\/137"}],"wp:attachment":[{"href":"https:\/\/sabba.me\/nmr\/wp-json\/wp\/v2\/media?parent=62"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/sabba.me\/nmr\/wp-json\/wp\/v2\/categories?post=62"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/sabba.me\/nmr\/wp-json\/wp\/v2\/tags?post=62"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}