Research / Quantum optimal control

Removing the Bloch–Siegert shift by construction

A systematic error the field has lived with for decades, deleted at the design stage instead of subtracted from the data.

The problem

Irradiate one nucleus and the resonance frequency of another moves. This is the Bloch–Siegert shift, and it is not an instrumental defect — it is a real consequence of driving a spin system off-resonance. The counter-rotating component of the applied field perturbs the energy levels of spins it was never aimed at, and their apparent frequency shifts by an amount that depends on the pulse amplitude and how far off resonance it is.

In a modern multidimensional experiment there are decoupling pulses, mixing sequences and selective inversions firing on several channels. Each contributes its own shift. The result is a phase error that accumulates through the sequence and a frequency error in the indirect dimension — one that varies across the spectrum, so it cannot be removed by a single global phase correction.

The field has good remedies. You can apply a compensating off-resonance pulse of opposite sign, or measure the shift and correct the spectrum afterward, or design the sequence so the errors cancel between blocks. All of these work. All of them also cost something: extra pulses cost power and duration, post-hoc corrections need calibration, and cancellation schemes constrain how the rest of the sequence can be built.

The approach

If the shift is a deterministic consequence of the pulse, then it is a function of the same parameters the optimizer is already searching over. So put it in the cost function.

Rather than optimizing only for the intended rotation and then repairing what the pulse did to the neighboring spins, the optimization targets both simultaneously: perform the desired operation on the target nucleus, and leave the passive spins' phase unperturbed at the end of the pulse. The Bloch–Siegert contribution is not canceled by an opposing error — it is simply never introduced, because waveforms that introduce it score worse and get optimized away.

Schematic, not measured data. A passive spin's line under a conventional off-resonance pulse is displaced by Δν; under a pulse optimized with the Bloch–Siegert contribution in its cost function it stays where it belongs. The distinction that matters is that nothing is being subtracted afterward — there is no residual to calibrate.

Why it is worth doing

No compensation pulses. Nothing added to the sequence means no extra power deposited and no extra duration for relaxation to eat into.
No calibration step. Corrections that depend on measuring the shift depend on measuring it correctly, on this sample, on this day.
Composable. A pulse that leaves no residue can be dropped into a sequence without auditing what it does to every other block.
Especially useful at high field, where wider offsets in hertz make the shift larger in the first place.

This work was published in the Journal of Magnetic Resonance (2026, 386, 108030) with colleagues at the Max Planck Institute for Multidisciplinary Sciences, including solid-state NMR work from Loren Andreas's group, where the same considerations apply under magic-angle spinning.