Environment-assisted Quantum-enhanced Sensing with Electronic Spins in Diamond

The performance of solid-state quantum sensors based on electronic spin defects is often limited by the presence of environmental spin impurities that cause decoherence. A promising approach to improve these quantum sensors is to convert environment spins into useful resources for sensing, in particular, entangled states. However, the sensitivity enhancement that can be achieved from entangled states is limited by experimental constraints, such as control errors, decoherence, and time overheads. Here we experimentally demonstrate the efficient use of an unknown electronic spin defect in the proximity of a nitrogen-vacancy center in diamond to achieve both an entangled quantum sensor and a quantum memory for readout. We show that, whereas entanglement alone does not provide an enhancement in sensitivity, combining both entanglement and repetitive readout achieves an enhancement in performance over the use of a single-spin sensor, and more broadly we discuss regimes where sensitivity could be enhanced. Our results critically highlight the challenges in improving quantum sensors using entangled states of electronic spins, while providing an important benchmark in the quest for entanglement-assisted metrology.

Figure 1

Environment-assisted quantum-enhanced sensing. (a)–(c) Circuit, pulse sequence, and experimental system for sensing magnetic fields with a mixed entangled state of two electronic spins associated with a N-$V$ center and an electron-nuclear spin defect (X) in diamond. The two spins are polarized with a dissipative channel $\chi$ acting on the N-$V$ spin (green laser pulse) and a coherent spin-exchange gate between the N-$V$ and X spins (cross-polarization sequence). A mixed entangled state ${\rho }_{\mathrm{\Phi }}$, prepared with an entangling gate (cross-polarization sequence), acquires a coherent phase ${\varphi }_{\mathrm{\Phi }}={\varphi }_{\mathrm{N}\text{−}V}+{\varphi }_{\mathrm{X}}=2\varphi$, upon interacting with a sinusoidal magnetic field $b(t)$, which is then mapped as a population difference onto both the N-$V$ and X spins (cross-polarization sequence) and readout by performing a projective measurement on the N-$V$ spin (green laser pulse) and a series of $m$ repetitive measurements on the X spin (recoupled spin-echo sequence). Resonant microwave pulses selectively drive the N-$V$ and X spins, whereas green laser pulses excite the N-$V$ spin for polarization and readout, while leaving the X spin unaffected. The phase of the pulses of the disentangling gate can be incremented at a constant rate to modulate the readout signal. (d)–(f) The spectrally mismatched spins are continuously driven at the Hartmann-Hahn matching condition, $|{\mathrm{\Omega }}_{\mathrm{N}\text{−}V}|=|{\mathrm{\Omega }}_{\mathrm{X}}|$, to induce coherent spin exchange at the dipolar coupling strength $d=58(4)\mathrm{kHz}$ with a decay time of $T_{1\rho }=132(11)\mu \mathrm{s}$.

Figure 2

Quantum control. (a) The increase in contrast of the cross-polarization signal after $N=0$ (circle), $N=1$ (triangle), and $N=3$ (square) rounds of polarization transfer indicates an increase in X-spin polarization from $14(3)\mathrm{\%}$ to $76(3)\mathrm{\%}$ and $94(6)\mathrm{\%}$, respectively. (b) The coherent oscillations of the signal measured after applying the entangling and disentangling gates indicate the preparation and detection of two-spin coherence with a contrast of $85(3)\mathrm{\%}$. The phases of the pulses of the disentangling gate are modulated at 500 and 250 kHz for the N-$V$ and X spins, such that the signal oscillates at 751(2) kHz, the sum of both frequencies. (c) The decay of the spin-echo signal (purple diamonds) for the mixed entangled state corresponds to a two-spin decoherence rate of ${\mathrm{\Gamma }}_2^{\mathrm{\Phi }}=36(4)\mathrm{kHz}$, which is consistent with the sum of the decoherence rates for the N-$V$ spin [blue triangles, ${\mathrm{\Gamma }}_2^{\mathrm{N}\text{−}V}=22(1)\mathrm{kHz}$] and X spin [not shown, ${\mathrm{\Gamma }}_2^{\mathrm{X}}=15(2)\mathrm{kHz}$]. The coherent modulation is generated by incrementing the phase of the pulses of the disentangling gate rather than applying an external magnetic field. (d) Cumulative magnetometry signal measured for a sensing time of $\tau =19\mu \mathrm{s}$ using $m=0$ (light purple diamonds) to $m=9$ (dark purple diamonds) repetitive measurements of the X spin. The cumulative signal is normalized to the maximum amplitude of the magnetometry signal measured without repetitive measurements ($m=0$). The cumulative signal contrast increases by a factor of $4.2$ after $m=9$ repetitive measurements, providing an increase in signal-to-noise ratio of 1.91(8) using optimal weights.

Figure 3

Magnetic sensing. (a) Magnetometry signal measured with a single N-$V$ spin (blue triangles) and a mixed entangled state of two spins (purple diamonds) obtained by varying the amplitude of a sinusoidal magnetic field for a sensing time of $\tau =10\mu \mathrm{s}$. The two-spin state precesses at twice the rate as the single-spin state. The sensitivity is proportional to the slope (dash lines) of the sinusoidal fit (solid curves). The magnetometry signal for the N-$V$ spin has been inverted for clarity. (b) The decrease in contrast of the magnetometry signal (normalized amplitude) for the single-spin state (blue triangles) and the two-spin state (purple diamonds) measured for a sensing time of $\tau =2$, $10$, and $19\mu \mathrm{s}$ is explained by decoherence during sensing (solid curves, which are fits to the echo decays in Fig. 2). The nominal loss in amplitude, ${\alpha }_0^{\mathrm{N}\text{−}V}=96(3)\mathrm{\%}$ and ${\alpha }_0^{\mathrm{\Phi }}=78(6)\mathrm{\%}$, is instead explained by dissipation and unitary control errors.

Figure 4

Performance analysis. (a) Gain in performance (purple diamonds) and gain in sensitivity (green squares) measured for different sensing times before and after accounting for time overheads. Solid lines are extrapolated values estimated using our decoherence model. The expected gain in performance for a fully polarized X nuclear spin (dashed purple line) is greater than unity for up to approximately $25\mu \mathrm{s}$, but is less than unity for all times when accounting for time overheads (dashed green line). (b) Using the X spin as a memory for repetitive measurements improves the performance. The measured gain in performance (purple solid diamonds) before accounting for time overheads surpasses unity after $m=6$ repetitive measurements and reaches a maximum of $g=1.06(4)$ after $m=9$ repetitive measurements. The measured gain in sensitivity maximizes to $\tilde{g}=0.55(2)$ when accounting for time overheads (green solid squares); however, the extrapolated gain in sensitivity for a fully polarized X nuclear spin achieves a maximum gain of $\tilde{g}=1.10$ after $m=6$ repetitive measurements. (c) Maximum gain in sensitivity achievable (dashed contour lines) for different values of the dipolar coupling strengths and relative decay rates of the two-spin system used for sensing. We assume a fully polarized X nuclear spin, but use the control errors and N-$V$ coherence time measured experimentally. Without repetitive measurements, our two-spin system (purple diamond) lies outside the region where it outperforms a single spin (gain in performance $g>1$, dark green region). With repetitive measurements (solid contour curves), the increase in signal-to-noise ratio, even after accounting for additional time overheads, shifts the boundary of the region of favorable performance, $g>1$ (light green region), such that our two-spin system lies within it.