Tag: theory

About theory

Subscribe Feeds

Zero-quantum Hamiltonian engineering made simple: time-shifted spin echoes

Posted on by Mohamed Sabba

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 you had a coupled 2-spin-1/2 system where the two nuclei can be said to be identical 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) 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:

The Hamiltonian of this spin system would look something like this in the product operator formalism, in which “J” denotes the isotropic part of the J-coupling tensor:

And something like this in the matrix representation:

Now, introduce a small chemical shift difference between the spins, which we will call Δ, with the corresponding Hamiltonian:

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

I’ll also include the cheeky unity operator of the double quantum subspace:

To make it clear that I1zI2z consists of unity operators:

We have:

And we really only need to consider the truncated version of HJ (denoted HJθ) when it comes to the overall dynamics.

In the strong-coupling limit, we may express the perturbation HΔ in the interaction frame of HJ, and it shouldn’t be rocket science to see that the time-dependent interaction-frame Hamiltonian is:

That is to say, the net effect of the J-coupling Hamiltonian is to modulate the chemical shift difference term in the xy-plane of the ZQ subspace.

Zero-quantum effective Hamiltonians

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

Which works out like this:

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-180} ? We would instead have a rotation about the (-x) axis:

Now, a little thought reveals you can generate a zero-quantum effective Hamiltonian of any phase. For a phase ϕ in the interval [-π/2,+π/2]:

And for a phase ϕ in the interval [+π/2,+3π/2]:

Realized via an actual pulse sequence, we have the following pair of sequences which correspond to a rotation of phase ϕ and flip angle β in the zero-quantum subspace:

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

This is a simulation for a hypothetical spin system with J = 100 and Δ = 1, using a naive train of J-CPMG echoes (the ideal number Nideal 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). 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). 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…

The Quantum Pascal Pyramid

Posted on by Mohamed Sabba

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 INS (or AXN) 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 ZNq. 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 ZNq operators with, say, Sx. What would the S-spin spectrum look like? Some visual examples are given here:



Intuitively, one can reason that each ZNq.Sx 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 ZNq 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 PNm) and ZNq . 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 ZN0) 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 ZNq, 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 ZNq 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 ZNq 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 ZNq operator would be given by:

2. each ZNq 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.