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…