A kernel of resonance: Perturbative finite-time responses in spin-lock magnetic resonance

Spin-lock preparations have been proposed to detect weak low-frequency magnetic fields, including neuronal magnetic fields, by magnetic resonance. Most current analytic models are limited to sinusoidal targets or long-time effective descriptions. They are therefore not directly suitable for reconstructing arbitrary time-varying waveforms. We formulate spin-lock detection as a finite-time perturbative observation problem. From the rotating-frame Bloch equation, we separate the weak target field from the reference spin-lock dynamics and apply a first-order Dyson expansion. For unmodulated spin-lock, the longitudinal-magnetization perturbation is a linear functional of the target waveform. Equivalently, it is a finite-time, convolution-like kernel response. We extend this approach to Spin-lock Phase-modulated Active Resonance eXcitation (SPARX). The modulation-induced pseudo-magnetic field is included in the reference dynamics, while only the weak target field is treated as the perturbation. Numerical Bloch simulations validate the first-order endpoint response for a Gaussian-envelope carrier, a Hann-windowed sinc-envelope carrier, and a waveform derived from an open rapid invisible frequency tagging magnetoencephalography dataset. We also compare full Bloch and first-order solutions over 0-3ω₁ and for all three magnetization components. Amplitude sweeps identify the perturbative regime and the onset of higher-order effects. For SPARX, we compare an adjoint numerical kernel with a finite damped-trigonometric basis. We show that pseudo-field modulation shifts the response bands relative to unmodulated spin-lock. This forward model links spin-lock pulse design to time-series reconstruction of weak magnetic-field waveforms.