Sensitivity threshold defines the optimal spin subset for ensemble quantum sensing

Abstract

Finite drive power leaves unavoidable spatial gradients in control fields, preventing spin ensembles from reaching the standard-quantum-limit sensitivity. We derive an analytic expression of ensemble sensitivity for inhomogeneous spin sensors and introduce sensitivity thresholds that reveal the optimal spin subset. Applied to both pulsed and continuous-wave magnetometry, the optimal subsets deliver up to a tenfold improvement over conventional schemes relying on nominally uniform regions of the ensembles. We demonstrate phase-only digital holography to implement the optimal subsets and show that residual aberrations add less than 1 dB of sensitivity loss. Our framework imposes no fundamental trade-offs and extends quantum sensing to heterogeneous sensing environments.

Figures (4)

• Figure 1: Sensitivity threshold for ensemble quantum sensing(a) Control error $\{\epsilon_i\}$ in $U_p$ and $U_r$ leads to an inhomogeneous sensitivity distribution across the ensemble; (b) Sensors with individual sensitivity below the threshold $\eta_\text{th}$ form an optimal sensor subset that minimizes ensemble sensitivity. The dark gray line denotes the standard $1/\sqrt{N}$ scaling; (c) and (d)$U_p(\epsilon)$ and $U_r(\epsilon)$ reduce the phase accumulation for $\delta B _\text{ext}$ and its corresponding projection contrast, respectively.
• Figure 2: Optimal sensor sets for the pulsed magnetometry.(a) The circular loop antenna (left) produces the normalized Rabi-frequency map $\Omega(\mathbf{x})/\Omega_0$ (right), calculated using HFSS. Additional details of the HFSS simulations are provided in the supplemental material supp. The dashed circle refers to the antenna's inner boundary. The field uniformity inside each colored solid boundary is given in the legend. (The uniformity of the enclosed area $A$ is defined as $1-\int_A\abs{\Omega(\mathbf{x})-\Omega_0}d\mathbf{x}/\int_A \Omega_0 d\mathbf{x}$).; (b) Pulse sequences for the DC (Ramsey) and AC (Spin-echo) magnetometry; (c) Local sensitivity $\eta(\mathbf{x})$ of the DC (left) and AC magnetometry (right); (d) Optimal sensor sets for the DC (left) and AC (right) magnetometry.
• Figure 3: Optimal sensing configuration for CW magnetometry(a) Magnetic field sensitivity of a nitrogen-vacancy center under continuous microwave and laser illumination: $\Omega$ ($s$) is the Rabi frequency of the microwave (the laser saturation parameter). The dashed line plots the optimal $s$ for the minimum sensitivity. (b) Sensitivity $\eta'_{\text{CW}}(\mathbf{x})$ with the circular loop antenna: The dashed line indicates the sensitivity threshold for the optimal ensemble sensing.
• Figure 4: Structured illumination to implement optimal spin subsets(a) Schematics to address the optimal sensors using digital holography (SLM: spatial light modulator). (b) A phase-only hologram $\phi_\text{DC}(u,v)$ that produces the structured illumination $I_\text{DC}(x,y)$ shown in (c). The dashed white line indicates the $1/e^2$ diameter of the incident laser beam ($\lambda=532~\text{nm}$). The scale bar refers to 1 mm.