A method for robust spin relaxometry in the presence of imperfect state preparation

TL;DR

This work addresses artifacts in NV-based spin relaxometry caused by imperfect state preparation, which can bias $T_1$ (relaxation) estimates. It introduces a two-state polarization-relaxation model with polarization efficiency $\eta=e^{-t_p \Gamma_p}$ that decouples polarization and relaxation, yielding an analytic expression for $n_{\ket{0}}(\tau)$ and an effective rate $\Gamma_{\text{app}}=\Gamma_1/(1-\eta)$. The approach outperforms conventional single-exponential and stretched-exponential fits, especially under low polarization, by recovering the true $\Gamma_1$ and enabling robust, wide-field analysis across NV ensembles. This advancement improves accuracy and sensitivity of NV relaxometry for nanoscale sensing and parallel spin-dynamics studies, with practical implications for biomagnetic sensing and condensed-matter investigations.

Abstract

Spin relaxometry based on quantum spin systems has developed as a valuable tool in medical and condensed matter systems, offering the advantage of operating without the need for external DC or RF fields. Spin relaxometry with nitrogen-vacancy (NV) centers has been applied to paramagnetic sensing using both single crystal diamond and nanodiamond materials. However, these methods often suffer from artifacts and systematic uncertainties, particularly due to imperfect spin state preparation, leading to artificially fast T$_1$ relaxation times. Current analysis techniques fail to adequately account for these issues, limiting the precision of parameter estimation. In this work, we introduce a minimal fitting procedure that enables more robust parameter estimation in the presence of imperfect spin polarization. Our model improves upon existing approaches by offering more accurate fits and provides a framework for efficiently parallelizing single-spin dynamics studies.

Figures (10)

• Figure 1: a) Widefield imaging of an ensemble of NV centers is captured with a camera. Laser power can be controlled by changing the linear polarization of the laser through a polarizing beam splitter. Choice of laser power, length of laser application, $t_p$ and beam size all impact the polarization rate, $\Gamma_p$, which limits the polarization efficiency of the NVs, and therefore the PL received. b)i) A spin relaxometry sequence involves laser pulses to polarize the spin state, with increasingly long dark times, $\tau$, for spin relaxation to take place. In widefield (ii), the entire polarization pulse and dark time is typically recorded by the camera due to the length of the camera exposure time. Each dark time and initialization/readout pulse is repeated N times to fill the exposure time. iii) The polarization power limits the amount of polarization achieved, and the recorded amount of relaxation in each $\tau_i$ step. c) An example of a T$_1$ relaxometry curve under different polarization conditions, resulting in different apparent relaxation rates.
• Figure 2: a) The two state system used to model relaxometry decays in the limit that $\Gamma_{\textnormal{p}}\gg\Gamma_1$. A relaxometry sequence is made up of polarization pulses followed by increasingly long dark times. The laser power or pulse duration can both limit the maximum polarization achieved, $\eta$ denotes the effectiveness of the polarization. b) With decreasing polarization rates, $\Gamma_p$, polarization becomes more inefficient and the area under the decay curve can be seen to reduce which results in a shorter $1/ e$-time. c) Even when producing a robust fit to the data, a single exponential will only report the $1/e$-time corresponding to the true relaxation rate if the spin state is perfectly polarized. Otherwise, it will over estimate the relaxation rate as the tail of the curve deviates from a single exponential. The two-state model can account for the imperfect state preparation.
• Figure 3: a) The PL from the reference measurement of the bare diamond, used to characterize $\eta$ (b). The characterization map is then used to inform the fit to the relaxometry measurement of the paramagnetic particles. The map of the relaxation rates in the presence of the particles is shown for the (c) stretched exponential and two-state model (d) fits. e) The average fitted relaxation rates of the particles becomes highly variable for low ($1-\eta < 0.35$) polarization regimes when using a stretched exponential. The two-state model remains consistent across the FOV. f) Histograms of the fits to particles and the background on a single pixel level reveal the underlying distribution and uncertainty in stretched exponential fits in a widefield setting. The two-state model increases rate estimation accuracy with smaller standard deviations and improves sensitivity a with slower intrinsic $\Gamma_1$.
• Figure SI 1: a) Probability of occupying $|0\rangle$ can be seen to converge as N increases. b), c) Population of initial state plotted as a function of time through the sequence in Figure 2a as an analogy to PL measured in experiment with varying polarization time.
• Figure SI 2: Single exponential fitting of relaxation curves under varying polarization, $\eta$.