Quantum Metrology with Strongly Interacting Spin Systems

Quantum metrology is a powerful tool for explorations of fundamental physical phenomena and applications in material science and biochemical analysis. While in principle the sensitivity can be improved by increasing the density of sensing particles, in practice this improvement is severely hindered by interactions between them. Here, using a dense ensemble of interacting electronic spins in diamond, we demonstrate a novel approach to quantum metrology to surpass such limitations. It is based on a new method of robust quantum control, which allows us to simultaneously suppress the undesired effects associated with spin-spin interactions, disorder, and control imperfections, enabling a fivefold enhancement in coherence time compared to state-of-the-art control sequences. Combined with optimal spin state initialization and readout directions, this allows us to achieve an ac magnetic field sensitivity well beyond the previous limit imposed by interactions, opening a new regime of high-sensitivity solid-state ensemble magnetometers.

Popular Summary

Quantum metrology makes use of quantum-mechanical effects, such as superposition and entanglement, to detect weak signals with ultrahigh precision. To enhance sensitivity, it is desirable to increase the density of quantum sensors in the sensing volume. However, in practice, this sensitivity improvement is severely hindered by unwanted interactions between the sensors at a close distance. We demonstrate a novel approach to quantum metrology that overcomes this challenge by employing a robust control-pulse sequence, which decouples the sensor-sensor interactions while detecting a target signal with high sensitivity.

Our approach is based on a new method of robust quantum control, which allows us to simultaneously eliminate the undesired effects associated with sensor-sensor interactions, disorder, and control errors. Specifically, we develop and implement a novel control sequence consisting of short periodic pulses that is designed to respond sensitively to an oscillating ac signal while suppressing the interactions and disorder in a robust fashion, insensitive to experimental imperfections. Combined with optimal initialization and readout protocols, we apply this new sequence to an interacting sensor ensemble in diamond.

Our work provides the first demonstration of a solid-state ensemble quantum sensor surpassing the interaction limit, opening up a promising avenue for the development of magnetometers with unprecedented sensitivity. Looking forward, our approach will have an immediate impact on a wide range of quantum-sensing applications on the nanoscale and also offer new opportunities for engineering of many-body quantum dynamics.

Figure 1

Interaction limit to spin ensemble quantum sensing. (a) Volume-normalized magnetic field sensitivity as a function of total spin density. The dashed line denotes the standard quantum limit scaling and the solid curve shows the behavior when interactions between spins are taken into account for the typical readout efficiency factor $C\approx 0.0028$ [27]. The sensitivity plateaus beyond a critical density due to a coherence time reduction. Robust interaction decoupling (red arrow) allows us to break the interaction limit. (b) Illustration of the black diamond nanobeam used as a spin ensemble quantum sensor. Microwave and optical excitation are delivered to the NV spins to control and read out their spin states, and an ac magnetic field is used as a target sensing signal. The inset shows the three magnetic sublevels, $|0⟩$ and $|\pm 1⟩$, in the ground state of NV centers, where two levels $|0,-1⟩$ are addressed using resonant microwave driving. All measurements are performed at room temperature under ambient conditions.

Figure 2

Robust dynamical decoupling. (a) Measurement protocol. NV centers are initialized using pulsed laser excitation at 532 nm (green trace) and read out through emitted photons detected by a single photon counting module (red trace). We perform $N$ repetitions of a sensing sequence unit of length $T$ (blue trace) and repeat the same measurement with an additional $\pi$ pulse (yellow trace) acting on the NV centers for differential readout of the spin state [17]. The box illustrates the details of different pulse sequences, XY-8, Seq. A, and Seq. B (DROID-60), composed of $\pi /2$ and $\pi$ rotations along $\widehat{x}$ and $\widehat{y}$ axes. Bars above (below) the line indicate driving along positive (negative) axis directions. (b) Key concepts for sequence design. The sequence is described by pulses $P_i$ and the time-dependent frame transformation of the system between the pulses. We highlight the orientation of each rotated frame by the axis that points along the $\widehat{z}$ axis of the fixed external reference frame. Decoupling sequences are designed by imposing average Hamiltonian conditions on the evolution of the highlighted axis. For example, the effects of disorder can be canceled by implementing an echolike evolution, $+\widehat{\mu }\to -\widehat{\mu }$, where $\mu =x$, $y$, $z$ (top row), and interactions are symmetrized by equal evolution in each of the $\widehat{x}$, $\widehat{y}$, and $\widehat{z}$ axes in the transformed frames (bottom row). Additionally, the pulse sequence is designed to mutually correct rotation angle errors and finite pulse duration effects [17]. (c) Experimental performance of different sequences with their respective decoupling features (inset). We fit the decoherence profile with a stretched exponential $e^{-(t/T_2{)}^{\alpha }}$ (solid curves) to extract the coherence time $T_2$ for each sequence [17]. A simple spin echo (gray crosses), XY-8 (blue circles), and Seq. A (green diamonds) show $T_2=0.98(2)$, 1.6(1), and $2.8(1)\mu \mathrm{s}$ with $\alpha =1.5(1)$, 0.66(2), and 0.61(3), respectively. DROID-60 (squares), designed to correct for all leading-order effects of interactions, disorder, and control imperfections, gives $T_2=7.9(2)\mu \mathrm{s}$ with $\alpha =0.75(2)$. We confirm that its coherence time is independent of the initial state prepared along $\widehat{x}$, $\widehat{y}$, and $\widehat{z}$ axes, as shown in red, yellow, and purple, respectively. All sequences have pulse spacing $\tau =25\mathrm{ns}$ and $\pi$-pulse width ${\tau }_{\pi }=20\mathrm{ns}$.

Figure 3

Optimal sensing with unconventional spin state preparation. (a) Pulse sequence (top row) and three-axis time-domain modulation functions (blue solid curves) for the first 14 free-evolution times of DROID-60. Red and blue bars in DROID-60 indicate rotation pulses as defined in Fig. 2. The modulation period along each axis is synchronized to an ac sensing signal (green curve). (b) Frequency-domain modulation function $|{\tilde{F}}_{x,y,z}(f){|}^2$ for DROID-60, with pulse spacing $\tau =25\mathrm{ns}$ and $\pi$-pulse width ${\tau }_{\pi }=20\mathrm{ns}$. The total strength $|{\tilde{F}}_t(f){|}^2$ is obtained from individual axis amplitudes ${\tilde{F}}_{x,y,z}(f)$ considering their relative phases in the frequency domain [17]. The principal resonance is highlighted by a red arrow, yielding maximum sensitivity. (c) Illustration of the effective magnetic field created by DROID-60 at the principal resonance. In the average Hamiltonian picture, the $\widehat{z}$-direction sensing field in the external reference frame transforms to the [1,1,1]-direction field ${\overrightarrow{B}}_{\mathrm{eff}}$ in the effective spin frame with a reduced strength [17]. Optimal sensitivity is achieved by initializing the spins into the plane perpendicular to the effective magnetic field direction. This optimal state preparation allows the spins to precess along the trajectory with the largest contrast (red dashed line). For comparison, the precession evolution for initialization to the conventional $\widehat{x}$ axis is shown as a blue dashed line. (d) Sensing resonance spectra near the principal resonance. The optimal initialization (red) shows greater contrast than the $\widehat{x}$-axis initialization (blue). Markers indicate experimental data and solid lines denote theoretical predictions calculated from the frequency-domain modulation functions [17].

Figure 4

Demonstration of sensitivity enhancement. (a) Observed spin contrast as a function of ac magnetic field strength, for the XY-8 sequence (blue) and for DROID-60 (red). The fit is a sinusoidal oscillation with an exponentially decaying profile [17]. Sequence B shows a steeper slope at zero field, indicating that it is more sensitive than XY-8 to the external field. The interrogation times, $t=2.16\mu \mathrm{s}$ for XY-8 and $t=6.52\mu \mathrm{s}$ for DROID-60, are independently optimized to achieve maximal sensitivity [17]. (b) Extracted absolute sensitivity $\eta$ and volume-normalized sensitivity ${\eta }_V=\eta \sqrt{V}$ with sensing volume $V=8.1(9)\times {10}^{-3}\mu {\mathrm{m}}^3$ as a function of phase accumulation time $t$ (error bars are given for $\eta$). The pulse spacing $\tau$ is fixed at 25 ns to detect the ac signal oscillating at frequency 11 MHz [see Fig. 3]. A comparison of the two sequences at their respective optimal sensing times reveals that DROID-60 outperforms XY-8 by $\sim 40\%$. The solid lines indicate the theoretical sensitivity scaling using the independently estimated sensor characteristics [17].