Spin-bath narrowing with adaptive parameter estimation

We present a measurement scheme capable of achieving the quantum limit of parameter estimation using an adaptive strategy that minimizes the parameter's variance at each step. The adaptive rule we propose makes the scheme robust against errors, in particular imperfect readouts, a critical requirement to extend adaptive schemes from quantum optics to solid-state sensors. Thanks to recent advances in single-shot readout capabilities for electronic spins in the solid state (such as nitrogen vacancy centers in diamond), this scheme can also be applied to estimate the polarization of a spin bath coupled to the sensor spin. In turns, the measurement process decreases the entropy of the spin bath resulting in longer coherence times of the sensor spin.

Figure 1

P.d.f (left) and its Fourier transform (right) after an eight-step adaptive measurement, with one (red, dashed) and two measurements per step (black). Inset: Ramsey sequence.

Figure 2

Holevo variance vs total time $T=M\tau (2^{N+1}-1)$, with $\tau =1$. ($•$) with one measurement per step ($M=1$) $V_H$ follows the SQL. ($\circ$) with two measurements per step ($M=2$) achieve the QML. ($*$) with $c=0.95$, the QML scaling is lost, but can be preserved for longer with $M=4$ ($▵$) and restored ($\square$) by setting $M=n+1$, even for lower c, e.g., $c=0.85$ ($☆$).

Figure 3

Simulated NV center spectrum from a Ramsey sequence before (dotted) and after (solid line) an eight-step adaptive measurement. The adaptive scheme narrows the bath distribution, here corresponding to a $50\times$ improvement of the coherence time. We considered $\sim 2600$ close nuclear spins, randomly positioned in the diamond lattice, in a 1 T magnetic field and initially in the maximally mixed state. We simulated full dipolar couplings among the nuclear spins and between the NV and the bath spins (approximating the hyperfine interaction) to validate the secular approximation. The narrow spectrum reveals an average nuclear field of $〈B_{\mathrm{n}}〉\approx 700$ nT (this was chosen at random among possible nuclear spin-state-dependent fields; similar results were obtained for different fields and spin positions in the lattice). In the inset, Holevo variance (circles) as a function of the total time, and signal decay (dotted line) due to intrabath couplings during the adaptive scheme interrogation time ($T_{\text{Ramsey}}$).