Quantum technologies for information processing, networking, and sensing increasingly employ hybrid architectures leveraging the advantageous properties of multiple physical qubit types. In platforms for quantum computing, this approach enables increased system sizes, while maintaining high fidelity and low crosstalk [1][2][3][4]. For timekeeping applications, ion clocks using two species, one "clock" ion with a stable transition and a second "logic" ion for preparation and readout, have demonstrated exceptional performance [5][6][7][8].

Analogous multiqubit strategies are similarly compelling for quantum sensors. In particular, the use of additional memory qubits in support of sensing qubits presents an attractive avenue towards entanglement-enhanced performance using quantum logic [9,10]. Among leading quantum sensing platforms, defect centers in solid-state systems are naturally amenable to quantum logic assisted protocols since they commonly consist of multiple individually controllable spin degrees of freedom localized within or near a defect.

An early demonstration of such multiqubit sensing by Jiang et al. used a single-negatively charged nitrogenvacancy (NV) center spin state coupled to a nearby 13 C nuclear spin [11]. The technique overcomes the poor optical readout fidelity of NV centers by using quantum logic to map the NV electronic spin state onto the longerlived 13 C nuclear spin memory qubit. This protocol repeatedly interrogates the 13 C spin by reentangling it with the NV electronic spin between each optical readout.

Subsequent work with single NV centers improved upon this protocol by instead using the nitrogen nuclear spin inherent to the NV center as a memory qubit [12,13]. In contrast to the randomly distributed 13 C nuclear spins, the on-site nitrogen nuclear spin has a well-determined hyperfine coupling with the NV electronic spin. This homogeneous coupling makes it particularly suitable for translation to ensembles of NV centers, as reported here.

NV ensemble measurements trade nanometer-scale spatial resolution for dramatically improved sensitivity [9], enabling a wide range of applications across the physical and life sciences, including micron-scale nuclear magnetic resonance (NMR) [14], magnetic microscopy [15], crystal stress and pressure spectroscopy [16,17], and thermometry [18,19]. However in nearly all such experiments, only global control of the NV ensemble is available. Therefore, the useful realization of quantum logic protocols with NV ensembles demands each constituent multiqubit sensor system be nearly identical, for both interactions within the NV center and diamond lattice, as well as in response to external fields.

In this Letter, we demonstrate quantum logic enhanced (QLE) sensing with a macroscopic ensemble of two-qubit sensors, each consisting of a NV electronic spin and its onsite 15 N nuclear spin. Interrogating a 2.3 × 10 3 μm 3 volume of diamond containing ∼10 9 of these two-qubit sensors, with careful control of ensemble homogeneity and stability, the QLE protocol increases the effective readout fidelity of each constituent NV, achieving a 33× improvement in single shot signal-to-noise ratio (SNR) compared to measurements using only the NV electronic spins (i.e., with conventional NV readout). This observed SNR improvement translates to enhancements in ac magnetic field sensitivity for repeated (i.e., time-averaged) measurements that can exceed an order-of-magnitude over conventional, non-QLE readout. Importantly, the present approach is universally applicable (independent of the sensing protocol or target signal), and thus provides a benchmark for quantum sensing using quantum logic architectures, both for ensembles of NV centers and other platforms.

Figure 1(a) illustrates the experimental setup (further detail the in Supplemental Material [20]), which operates at ambient laboratory conditions and probes a ð2 × 2 × 0.5Þ mm 3 diamond chip. In this chip, a 13 μm layer contains an ensemble of NV centers at a concentration of ½NV ≈ 2.3 ppm (½N ≈ 14 ppm) [21]. Each near-identical two-qubit system in the ensemble consists of the NV electronic "sensor" spin (S ¼ 1) and the nuclear "memory" spin (I ¼ 1=2) of the associated 15 N nucleus. By applying a bias magnetic field along an NV symmetry axis, the NV ground state spin sublevels are nondegenerate [Fig. 1(b)], allowing them to be individually addressed using microwave fields. We employ the NV electronic spin states m s ¼ 0 and m s ¼ -1 as an effective two-level system, with representative qubit states j↓ e i and j↑ e i, respectively. The two NV nuclear spin states m I ¼ -1 2 ; 1 2 are represented by j↓ n i and j↑ n i qubit states. The eigenstates of the composite electronic-nuclear system are denoted by jf↑ e ; ↓ e g; f↑ n ; ↓ n gi.

We first demonstrate control of the NV nuclear spin (memory qubit) using the NV electronic spin (sensor qubit) via a quantum logic protocol. Figures 2(a) and 2(c) illustrate this procedure, which begins with a selective microwave (MW) π pulse that exchanges the spin populations of the states j↓ e ; ↑ n i and j↑ e ; ↑ n i. This pulse acts as a CNOT operation on the electronic spin, conditioned on the nuclear spin state (CNOT ejn ). Next, a radio frequency (rf) π pulse is applied, resulting in a CNOT nje gate that exchanges the populations of j↑ e ; ↑ n i and j↑ e ; ↓ n i. This operation encodes the information measured by the sensor spins onto the memory spins. These two CNOT gates, applied in succession, form a SWAP operation (i.e., gate) of duration T SWAP .

The fidelity of the CNOT gates is limited by the spin transition linewidths, with contributions from both the intrinsic linewidth of the NV ensemble δν ∼ 1=πT Ã 2 and dephasing from inhomogeneities in the bias and control fields across the interrogation volume [9]. Here, the external inhomogeneities over the ensemble are ∼0.2 kHz for the bias field, ∼10 kHz for the lattice strain, and ∼4% in the applied MW and rf control fields [20]. (a) A 532 nm optical beam enters the diamond chip through its side (45°angle cut), using total internal reflection to illuminate a spot diameter of ∼15 μm on the top surface of the diamond chip, where the 13 μm thick NV layer resides. A 1 mm diameter, single-loop antenna at ∼0.5 mm above the NV layer drives both the microwave transition of the NV electronic spin states and the radio frequency transition of the 15 N nuclear spins [20]. The magnified view depicts the multi-qubit NV spin sensors comprising an ensemble of ∼10 9 NVs in the detection region. A multiloop test coil provides the synthetic ac magnetic field measured in this study, emulating those originating from, e.g., NMR, integrated circuits, materials science, and thermometry applications. (b) NV energy levels allow optical initialization and readout of the electronic spin states, with splitting of the ground state triplet energy levels from Zeeman and hyperfine interactions.

Thus, the CNOT gate fidelities used here limited by the intrinsic broadening, with linewidth δν ≈ 0.5 MHz (T Ã 2 ¼ 600 ns). With this linewidth limited broadening, the estimated fidelity of spin population transfer by the SWAP operation ≈ 93%, as shown in Fig. 2(b).

After encoding the sensor spin population onto the memory spins using the SWAP gate, the electronic spin states are reset using an optical polarization pulse. With successive N applications of an entangling CNOT ejn gate followed by an optical readout pulse, the information stored in the nuclear memory spins is then repeatedly mapped back onto the electronic spins and measured optically. This procedure provides many readouts within a duration limited by the nuclear spin lifetime T 1 , thereby enhancing the overall readout fidelity. The large number of NVs probed allows a high-precision, ensemble-averaged measurement of the sensor spin state with one execution of this repeated readout procedure.

The nuclear spin T 1 increases with external bias magnetic field due to suppression of flip-flop transitions of the electronic and nuclear spins. We measure T 1 using the pulse sequence in Fig. 2(c) for magnetic fields up to 3700 G, see Fig. 2(d). The inset of Fig. 2(d) shows example T 1 values extracted from fits of the NV ensemble fluorescence contrast to a à exp ½-ðT op =T 1 Þ c þ d. The T 1 values follow the expected quadratic dependence on magnetic field due to anisotropic hyperfine spin state mixing [12]. Full magnetic field and laser power dependent T 1 measurements are shown in the Supplemental Material [20].

The following measurements use a bias magnetic field of 3700 G, with a corresponding 15 N nuclear spin T 1 of 3.44 (12) ms.

We demonstrate quantum logic enhanced (QLE) ac magnetic field sensing by measuring a three-tone test signal centered at 1 MHz, applied to the NV ensemble via a multiloop coil (see Fig. 1, and Supplemental Material [20]). First we use correlation spectroscopy, a popular NV T 1 -limited technique for ac magnetometry, where the time delay between two dynamical decoupling sequences (T corr ) is varied, see Fig. 3, with optical readout applied only after the second decoupling sequence [33][34][35][36][37][38]. Our correlation spectroscopy measurement employs the dynamical decoupling sequence XY8:6 (six repetitions of an XY8 sequence) [39,40]. As shown in Fig. 3, the sensing interval T sense is followed by a SWAP operation and optical reinitialization pulse (with overall duration T SWAP ) and N quantum logic readout (QLR) cycles (each with duration T QLR ). The Applying the QLE sensing protocol to measurements of the three-tone test signal, we determine a series of power spectra from the correlation time series acquired for each of the N readout cycles. As apparent in the inset of Fig. 4(c), the power spectrum signal amplitudes, A n , decay with increasing readout cycle index n due to 15 N nuclear spin T 1 relaxation. To optimize the signal-to-noise ratio SNRðNÞ after N readouts [11], the signal amplitude for the nth readout is weighted by A n =σ 2 n , where σ n is the standard deviation of the noise at the nth readout. Figure 4(b) compares the power spectrum of a weighted signal, after N ¼ 2000 QLR cycles, to the reference signal obtained using conventional readout.

The resulting QLE signal to noise ratio SNRðNÞ is given by ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi P N n¼1 A 2 n =σ 2 n p . We normalize SNRðNÞ by SNR(Ref), the SNR of conventional NV electronic spin readout (i.e., single readout), measured under the same experimental conditions. The increase in SNR realized with quantum logic is shown in Fig. 4(c). For example, with 2000 QLR cycles, we increase the SNR by 33.3ð9Þ×, larger than the best value (∼20×) previously reported for QLR applied to a single NV [13]. Similar SNR increases with the QLE protocol are found for the other ac magnetic field sensing sequences used in this study (see the Supplemental Material [20]).

However, improvements in SNR do not necessarily translate into enhanced sensitivity, since SNR does not consider the impact on the measurement timescale (and hence bandwidth) of an extended readout interval. Accounting for the overhead time associated with the SWAP operation (T SWAP ¼ 20 μs) and each readout cycle (T QLR ¼ 3 μs), the sensitivity enhancement obtained using the QLE protocol can be estimated using ηQLE ≈ SNRðNÞ SNRðRefÞ

where we have assumed the duration of a conventional readout is approximately T QLR .

As shown in Fig. 5(a) for an XY8:6 dynamical decoupling sequence with an optimal T sense ¼ 24 μs, the quantum logic protocol achieves a sensitivity enhancement of up to ηQLE ¼ 2.4ð3Þ for N ≈ 150, compared to the same sensing sequence with conventional readout. For the diamond sample used here, the optimal T sense under XY8 decoupling is constrained by the NV electronic spin XY8 coherence time T 2 ≈ 31 μs, which, in turn, is limited by NV-NV dipolar interactions. The resulting non-QLE ac sensitivity was measured to be 34ð1Þ pT= ffiffiffiffiffiffi Hz p [20]. To surpass the coherence time limit from NV-NV interactions, we employ DROID-60 decoupling introduced in Refs. [22,23] and thereby extend optimal T sense to about 144 μs [20] 25]; and providing more than an orderof-magnitude better volume-normalized non-QLE ac sensitivity than these other works. See Ref. [20] for further sensitivity discussion. Applying the QLE protocol and comparing it to conventional readout, we find ηQLE ¼ 5.7ð3Þ for DROID-60:6 at N ≈ 1400, as shown in Fig. 5(a). With this enhancement factor, our calculated QLE ac sensitivity is 3.4ð1Þ pT= ffiffiffiffiffiffi Hz p . For ac magnetometry using correlation spectroscopy, T sense varies with T corr . To account for this variation, we compare measurements using quantum logic with N readouts to conventional measurements repeated M times, maintaining a constant total acquisition time:

For measurements with T corr ranging from 0-1.5 ms, we find up to ηQLE ¼ 11.3ð3Þ for N ≈ 1000 and above. More intuitively, if we use the average value of T corr (0.75 ms) when calculating ηQLE via Eq. ( 1), we estimate a similar QLE sensitivity factor of ηQLE ≈ 11. Finally for correlation spectroscopy, ac sensitivities using N ¼ 2000 QLR cycles were measured to be 47ð1Þ pT= ffiffiffiffiffiffi Hz p with no QLE, and 4.2ð1Þ pT= ffiffiffiffiffiffi Hz p with QLE, again yielding a quantum logic enhancement of ∼11.

To highlight the versatility of the QLE technique, Fig. 5(b) provides estimates of quantum logic enhancement for a range of sensing durations T sense , given our experimental conditions. For moderate nuclear spin lifetimes (∼T 1 > 100 μs), an improvement in sensitivity (i.e., ηQLE > 1) is readily achieved when both T sense exceeds T SWAP , and T SWAP is much larger than T QLR . These conditions for T QLR and T SWAP are, respectively, obtained with typical optical intensities ∼0.1-1.0 mW=μm 2 and with commercially available rf amplifiers [20]. Thus, the QLE protocol should be applicable to a wide range of sensing sequences (and diamond materials) commonly used in NV ensemble measurements and, in principle, for other solid-state spin systems.

In summary, we leveraged quantum logic using a macroscopic ensemble of solid-state, hybrid two-qubit sensors-each consisting of an NV electronic spin and on-site 15 N nuclear spin in diamond-to realize a factor of about 30 improvement in spin state readout SNR, which in turn enables significant improvement in ac magnetic field sensitivity. The observed sensitivity enhancements can exceed an order of magnitude under favorable conditions (i.e, sensing interval ∼1 ms) using only global control of the NV ensemble.

Furthermore, the current approach is agnostic to the target signal and, therefore, broadly applicable to sensing a variety of physical quantities. In particular, QLE sensing benefits from a strong, uniform bias magnetic field, making the technique well suited for use in NV-NMR spectroscopy [14]. We expect existing NV-NMR systems can readily implement the QLE protocol. Additionally, the sensitivity improvements realized here are compatible with the growing collection of techniques for NV-NMR sample hyperpolarization [26,27].

Beyond magnetometry, NV-diamond dynamical decoupling protocols sensitive to crystal stress, pressure, and temperature have attained sensing durations of tens of microseconds or longer. For example, such sequences were recently employed in path-finding experiments for diamond-based dark matter searches [16]. With further development, the key metric of T sense > T SWAP may be realized for these alternative sensing applications, enabling quantum logic enhanced sensitivity.

Integrating additional quantum degrees of freedom, e.g., defects with couplings to multiple nuclear spins, is another promising direction for further progress in QLE sensing [41][42][43][44]. Similarly, solutions to address the random distribution of host lattice nuclear spins, such as manipulating the collective modes of a spin bath [45], may prove advantageous when exploring more advanced quantum logic or error correction algorithms [46] for further sensing enhancements.