The Quantum Pascal Pyramid

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) coherences observed in NMR experiments.

Suppose one has an I_NS (or AX_N) spin system. Begin by considering the following cumulative tensor product:

The expansion may be organized by the spin product rank q, yielding a convenient operator basis of I-spin longitudinal (z-) spin-operators which Sorensen denoted Z_N^q. Some examples for the cases N=3 and N=4 are:

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?)

Now, suppose you wanted to observe single-quantum coherences involving a product of any of these Z_N^q operators with, say, S_x. What would the S-spin spectrum look like? Some visual examples are given here:

Intuitively, one can reason that each Z_N^q.S_x operator has to be a sum of the “pure” single-transition operators corresponding to each m=J×{-N/2,…,+N/2} line component making up the S-spin spectrum. One might notice that each Z_N^q operator changes sign exactly q times. But it’s not so clear to see how.

It turns out there is an incredible direct map between the operators describing pure populations of states with azimuthal quantum number m (which I will denote P_N^m) and Z_N^q . The relationship is given by:

The map can be represented by an (N+1)×(N+1) matrix. Some matrices illustrating this relationship are shown below:

Naturally, this type of relationship has appeared in other fields 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 on relaxation in the AX₄ spin system of ¹⁵NH₄⁺ by Nicolas Werbeck and D. Flemming Hansen, where the combinatorial structure is referred to as a “modified Pascal triangle”.

EDIT (February 23, 2024, 15:56 UK time): after writing this article, I realized something neat. Recall the de Moivre-Laplace theorem, which is the famous mathematical result that the binomial distribution (corresponding to the multiplets associated with Z_N⁰) converges to a normal (Gaussian) distribution as N→∞. I have observed a generalization of the de Moivre-Laplace theorem: that the columns of Pascal’s pyramid, i.e the multiplets associated with Z_N^q, converge to the q-th derivatives of a Gaussian distribution. The q-th derivative of a generic Gaussian function has a general form that can be written in terms of a regularized hypergeometric function:

That is to say, the Z_N^q operator may be approximated by a q-th order Gaussian derivative:

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:

1. each Z_N^q operator transforms under, say, an I-spin x-rotation of a flip angle β, with an increasing “responsiveness” to rotation that depends on the spin product rank q. The self-evolution of each Z_N^q operator would be given by:

2. each Z_N^q operator can be converted into I-spin multiple quantum coherence with a (maximum) coherence order q. This well-known property is commonly exploited in NMR experiments. Of course, by definition, an MQC of coherence order q (which I’ll dub qQC for q-quantum coherence) has the following property under an I-spin z-rotation with a flip-angle β:

EDIT (May 16, 2024, 14:20 UK time): I have written this blog post as a small paper on arXiV, which goes into a bit more detail.