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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…

J-Synchronized Echoes in the Pantheon of Symmetry-based Sequences?

Posted on by Mohamed Sabba

I often make the point that PulsePol is expected to replace the “default” option of M2S as the go-to windowed (meaning there are gaps between pulses) or hard-pulse sequence for exciting nuclear singlet order in strongly-coupled spin systems. (Shoutout to the archetypal soft-pulse/low-power sequence SLIC.)

That being said, I religiously avoid the misleading statement “symmetry-based sequences will replace M2S” and wince uncomfortably at the frequently-asked question “do you think symmetry-based sequences will replace M2S?”

This is because the building block of the M2S sequence, the J-synchronized echo (JSE; see refs. 1, 2, and 3), is technically a symmetry-based sequence.

Figure 1: The J-synchronized spin echo pulse sequence.

The fundamental difference between the JSE and PulsePol is that the JSE is designed to selectively swap the |S0〉and |T0〉states (via a zero-quantum effective Hamiltonian) while PulsePol is designed to selectively swap |S0〉and either of the |T±〉states (via a single-quantum effective Hamiltonian):

Figure 2: illustration of transitions excited by different symmetry-based sequences.

This is painfully obvious when you phrase it the right way: the R-element (i.e. basic inversion element) of the JSE is simply a windowed 1800 pulse, and the nominal duration of the R-element is half a “rotor” period 1/(2J). With regards to the first-order selection rules, no phase cycling is applied*. Hence, we may denote the JSE as R210 in the notation describing symmetry-based sequences. Or you could denote it C110 too (assuming you count a full cycle including two spin echoes).

As an aside, it is easy to see why the M2S sequence ended up appearing clunky with awkwardly placed pulses and delays interrupting the echoes: the selective transition being engineered by the JSE building block was the wrong one, assuming one was starting from thermal equilibrium magnetization. Which suggests that the JSE would’ve worked handsomely for nuclear singlet excitation if one was starting from dipolar order (an excess population of the central triplet state)…

*Much like PulsePol, “riffling” in the form of supercycles is necessary for the optimal performance of the JSE/M2S sequences vis-a-vis pulse strength and detuning errors.

Symmetry-based Sequences for Singlet-Triplet Excitation (PulsePol)

Posted on by Mohamed Sabba

In 2016 a Master’s student at Ulm’s Institute of Theoretical Physics, by the name of Benedikt Tratzmiller, described a simple yet powerful control sequence for optical DNP in diamond NV centres. The pulse sequence had the attractive name of PulsePol. The sequence made further appearances in an excellent paper and Tratzmiller’s PhD thesis.

Some time ago I listened to an inspiring online talk by Nino Wili (who quite convincingly spoke about cross-pollination between different fields of magnetic resonance) and we got to talking. After a few discussions/simulations eventually we realized that the PulsePol sequence could be applied to singlet NMR, and that application forms the basis of our paper, which chose to explain PulsePol using the language of symmetry-based sequence design borrowed from solid-state NMR.

PulsePol has some advantages over the M2S sequence, which was essentially a “default” option in NMR groups for generating nuclear singlet order (alongside variations of the arguably more elegant SLIC method invented by DeVience, Walsworth, and Rosen). These advantages include superior robustness (PulsePol is generally less sensitive to rf errors than M2S), simplicity, and even a small time advantage (PulsePol is ~1.21x faster than M2S). One should expect PulsePol to replace M2S in the future.

Here I provide some ready-to-use pulse programs, written for Bruker TopSpin, that the NMR community may use to actually implement the PulsePol sequence.

The pulse programs have a number of features:

  1. The so-called “riffling” 180 phase shift modification on the central 180° pulses, which improves robustness against pulse strength (Rabi frequency) and resonance offset (detuning) errors.
  2. A T00 gradient filter for selective filtration of singlet order, and a z-filter to select longitudinal magnetization.
  3. A singlet order destruction (SOD) element before the relaxation delay to purge residual singlet order, which may interfere with experiments.
  4. Wimperis’ BB1 composite pulse (in the symmetrized implementation) replacing the pulses in the filters, improving robustness against pulse strength errors.

The pulse sequences can be downloaded here:

  1. R431
  2. R873
  3. R411
  4. R612
  5. R813
  6. R1014

I also attach a quick reference table which provides the 2 experimental parameters that are actually relevant for optimal excitation of singlet order, because it has always annoyed me that the first one was not explicitly stated in our somewhat cryptic original paper:



Note that I have given the total duration of the PulsePol in terms of the SLIC duration 1/(√2 Δ) – which is the fastest currently known way to fully excite singlet order from longitudinal magnetization. The minimum of the total duration is around n = 3, N = 4, where the total evolution is ~1.38x longer than the SLIC sequence. The (fixed) total duration of the M2S sequence (~1.67x longer than SLIC) is given for comparison.

The “default” sequence for most people working with nearly equivalent spin-1/2 pairs would be R431 – or possibly R873, which is only ~3.5% slower than R431 but provides improved robustness against resonance offset/detuning errors. However, in the common case where a spin system is at intermediate inequivalence (the chemical shift has a comparable magnitude with respect to the J coupling), one can benefit from the sequences in the series R411, R612, R813, R1014… which perform over an increasingly wider range of inequivalence angles at the expense of being increasingly slower, as I discussed in the appendix to this paper.