## Introduction

The detection of electromagnetic field scattering and absorption is an indispensable tool for research from fundamental physics to industry applications, including remote sensing1,2, imaging3, communication4 and ranging5. In free space, the diffraction determines the distribution of electromagnetic field, and limits the spatial resolution of imaging and precision of position estimation. Super-resolution optical far-field microscopy has been developed to beat the diffraction limit6,7,8,9. One of the most widely studied solutions is based on the nonlinear interaction with a structured light field10,11,12. Multi-photon excitation, as a typical nonlinear process, has been proven to be a powerful tool for high resolution microscopy imaging. It can enhance the spatial resolution with a factor of approximate $$\sqrt{N}$$13,14,15, where N is the number of photons that are evolved in the optical process.

The idea of multi-photon interaction with a spatially modulated light field can be extended to the radio frequency (RF) regime, and help to develop high resolution RF imaging and ranging, which has attracted wide interest in autonomous driving16, manufacturing17, health monitoring18 and the Internet of things19,20. However, the traditional multi-photon process is usually observed with a high power excitation. It hinders applications with a free space RF field. A potential solution is to utilize the quantum properties of interaction between nanoscale system and a weak electromagnetic field21,22,23. The coherent interaction between quantum emitter and the light field is recently applied for high resolution optical imaging with a low excitation power24,25. Quantum illumination is also studied to improve the signal-to-noise ratio of imaging beyond the classical limit of $$\sqrt{N}$$26,27.

In this work, by utilizing the coherent interaction between the RF field and solid state spins, we demonstrate quantum-enhanced radio detection and ranging under ambient conditions. The spin-RF interaction is enhanced approximately four orders of magnitude through the RF field focusing and quantum sensing at the nanoscale. It enables the detection of a free space GHz radio signal with a sensitivity of 21 $${{{{{{{\rm{pT}}}}}}}}/\sqrt{{{{{{{{\rm{Hz}}}}}}}}}$$. An RF interferometer with the quantum sensor is subsequently built to detect the phase of a radio signal that is reflected by a target. The coherent multi-photon interaction improves the accuracy of radio ranging beyond the classical limit. The target’s position is estimated with a precision of 16 μm (corresponding to 1.6 × 10−4λ). The proof-of-principle results demonstrate the potential of solid spin qubits in quantum radar, microwave photonics and high sensitivity metrology in astronomy.

## Results

### Design of the experiment

Phase estimation can provide a higher spatial resolution for radio ranging in comparison with the signal strength and time of flight methods. Here, the quantum sensor detects the phase of a backscattered RF signal through an RF interferometer. The concept is shown in Fig. 1a. The signal from a RF generator is split into two individual paths, which are subsequently radiated into free space by horn antennas. One path serves as the reference. In the second path, the free space RF signal is reflected by an aluminum plate (40 × 40 cm2). The phase difference between the backscattered and reference RF is then determined by the target’s distance L as:

$$\varphi=4\pi \cdot \frac{L}{\lambda },$$
(1)

where λ is the wavelength of RF field in free space. Assuming the same magnetic amplitude (B1) of the two paths (this is not a stringent requirement), the total amplitude of the interference in free space will vary as:

$${B}_{{{{{{{{\rm{RF}}}}}}}}}=2{B}_{1}\cdot \left|\cos \frac{\varphi }{2}\right|.$$
(2)

Then, the radio ranging is realized by detecting the interaction between the quantum sensor and a spatially modulated RF field (Fig. 1b). The resolution of ranging and imaging will be improved by enhancing the RF field sensitivity and exploring the nonlinearity of coherent multi-photon RF-spin interaction.

The quantum sensor of RF field is nitrogen vacancy (NV) color center ensemble (with a density of approximately 5000/μm2) in diamond. It is produced through nitrogen implantation and subsequent annealing. The ground state of NV center is a spin triplet. The resonant frequency of the transition between ms = 0 and ms = ± 1 is ω± = D ± γBz, where D = 2.87 GHz is the zero-field splitting, γ = 2.8 MHz/G is the gyromagnetic ratio of electron spin and Bz is an external static magnetic field along the axis of NV center. Therefore, the spin state transition of NV center will provide the information of a gigahertz RF signal. Here, the frequency of the carrier wave is set to 2.885 GHz, resonant with ω+ under a bias magnetic field, as shown in Fig. 2a.

Started by initializing the NV center in ms = 0 with a green laser pulse, the interaction between NV center and a resonant RF field is depicted as the Rabi oscillation:

$${\rho }_{\left|0\right\rangle }=\frac{1}{2}(1+\cos (\Omega {t}_{{{{{{{{\rm{RF}}}}}}}}})),$$
(3)

where $${\rho }_{\left|0\right\rangle }$$ is the population of ms = 0 state and tRF is the duration of RF pulse. The Rabi oscillation frequency is determined by the magnetic component of local RF field Bloc, as Ω = 2πγBloc. The amplitude of local RF field is assumed to be linearly dependent on the RF field in free space as Bloc = kBRF, with a conversion efficiency of k. In practice, the spin state transition is affected by the decoherence and inhomogeneous broadening with NV center ensemble. By detecting the spin-dependent fluorescence emission, we observe that the oscillation’s amplitude exponentially decays with a time τ ≈ 460 ns, as shown in Fig. 2b. According to the results of Rabi oscillation, the free space RF field magnetic sensitivity with NV center ensemble is deduced as (Supplementary Note 1):

$${\eta }_{{{{{{{{\rm{B}}}}}}}}}\propto \frac{1}{\gamma kC\sqrt{\epsilon }}\frac{\sqrt{1+{t}_{{{{{{{{\rm{RF}}}}}}}}}/{t}_{\det }}}{{t}_{{{{{{{{\rm{RF}}}}}}}}}{e}^{-{t}_{{{{{{{{\rm{RF}}}}}}}}}/\tau }}.$$
(4)

Here, C denotes the optical contrast between different spin states. ϵ consists of the photon emission rate of NV center ensemble and the fluorescence detection efficiency of setup. $${t}_{\det }$$ is the signal’s detection time.

For the sensing of a weak radio signal, the idea is to focus the free space RF field and detect the spin state transition of NV center ensemble in a nano/micro scale volume. As shown in Fig. 2c, a nanowire-bowtie antenna on the diamond surface is adopted23 (for details, see Supplementary Note 2). The two arms of the metallic bowtie structure are connected by a conductive wire with a width of approximately 1 μm. The antenna collects the free space RF signal and transmits the RF current through the nanowire (Fig. 2d). As a result, the magnetic component of local RF field and the interaction with NV center ensemble are enhanced by 7.6 × 103 times near the nanowire. Therefore, an excessive sensing volume, which includes the NV center away from the wire, should be avoided because it will bring a high level of useless background fluorescence signal. Here, using a confocal microscope, the fluorescence signal of NV center ensemble is detected with a spatial resolution of approximate 500 nm. This enable us to separate the spin state information of NV center ensemble in the strong interaction area. In this way, we increase k to 7.6 × 103 while maintaining the properties of NV center sensor in Eq. (4).

The sensitivity of free space radio detection is experimentally estimated by comparing the noise ratio with the optical response to the RF field. In Fig. 2e, we record the fluorescence emission of NV center ensemble while changing the amplitude of RF field. With an RF-spin interaction time tRF = 265 ns, which is close to τ, the result shows that the normalized optical response is ΔIBRF ≈ 1.1%/nT. Here, the fluorescence collection efficiency ϵ has been reduced 10 times to avoid the possible saturation of optical detector (Supplementary Note 3). Under this condition, the standard deviation of the experimental data with a total measurement time of 0.27 s is 0.14%. The RF field magnetic sensitivity is 66 $${{{{{{{\rm{pT}}}}}}}}/\sqrt{{{{{{{{\rm{Hz}}}}}}}}}$$ with attenuated fluorescence signal. It corresponds to a sensitivity of 21 $${{{{{{{\rm{pT}}}}}}}}/\sqrt{{{{{{{{\rm{Hz}}}}}}}}}$$ with all the fluorescence signal. Therefore, we estimate that a free space RF field electrical sensitivity of 63 $$\mu {{{{{{{\rm{V/cm/}}}}}}}}\sqrt{{{{{{{{\rm{Hz}}}}}}}}}$$ can be realized. It worth noting that, the decay time28 and amplitude of Rabi oscillation with NV center ensemble change with the Rabi frequency (Supplementary Note 4). As a result, the RF sensitivity would be lower with a weaker RF signal. In future applications, this problem can be avoided by applying a strong bias RF field.

During Rabi oscillation, a single RF π pulse could pump the spin state from ms = 0 to ms = ± 1, after the electron spin is initialized in ms = 0. And a second π pulse will pump the spin back to ms = 0. This transition can be endlessly pumped with the sequential RF π pulses, as long as the spin state coherence is preserved. It indicates that, the spin state Rabi oscillation with a multi-π pulse can be seen as a coherent multi-photon process, with real intermediate states. In analogy to the multi-photon microscopy, Rabi oscillation will increase the spatial frequency of NV center ensemble’s response to the spatially modulated RF field in Eq. (2). It subsequently improves the resolution of radio ranging with NV center ensemble.

Due to the large enhancement of local spin-RF interaction strength, the coherent multi-photon process can be pumped with a weak free space RF field here. To demonstrate the high resolution radio ranging with NV center ensemble, a peak power of 0.03 W at RF A and a peak power of 0.35 W at RF B are applied to balance the amplitude of reference and back scattered RF. The information of target’s position is obtained with the pulse sequence in Fig. 3a. The spin state of NV center ensemble is initialized to ms = 0, and the fluorescence is simultaneously recorded as an optical reference I0. The RF pulse is then radiated by the horn antenna and reflected by the target. The interference between the backscattered and reference RF determines the spin state of NV center ensemble. The distance dependent fluorescence of NV center ensemble, as I(L), is finally pumped with another green laser pulse.

Moving the target at a distance of approximately 2 m by a stepper motor, the amplitude of RF field interference varies between 2B1 (78 nT here) and 0. We define the RF excitation with $${t}_{{{{{{{{\rm{RF}}}}}}}}}{B}_{1}=\frac{N}{4k\gamma }$$ as an N-π pulse. As shown in Fig. 3b, with a single π pulse (30 ns duration) excitation, the spin state of NV center changes from ms = + 1 to ms = 0 by moving the target’s position from $$L=\frac{n}{2}\lambda$$ to $$(\frac{n}{2}+\frac{1}{4})\lambda$$, where n presents the integer ambiguities of ranging.

As explained before, the increase of N enhances the nonlinearity of response to the RF field. More fringes appear with a higher N in Fig. 3b. The full width at half maximum (FWHM), which is usually used to characterize the spatial resolution of imaging, decreases with N (Fig. 3c). It matches the expectation in Eq. (2) and (3) (details in Supplementary Note 5). In comparison, with tradition optical multi-photon excitation, the FWHM of a scanning microscopy is reduced with $$\sqrt{N}$$. The reduction of FWHM can help to develop high spatial resolution RF imaging. In addition, since a perfect Rabi oscillation will show no saturation of excitation, the resolution of imaging is theoretically unlimited by increasing N.

The FWHM does not include the contrast between the maximum and minimum fluorescence signals of NV center ensemble, which is related to the amplitude of Rabi oscillation. Therefore, spin state decoherence and inhomogeneous broadening do not change the reduction in FWHM. However, they will affect the evaluation of NV center ensemble’s response to the target’s position. In Fig. 3d, we plot the derivative of the normalized NV center ensemble fluorescence to the position of aluminum plate. The response of NV changes with the target’s position. With a single π pulse excitation, the highest normalized dI/dL is found to be 0.9%/mm. The ratio increases to approximately 2.6%/mm with N = 4. It confirms that the maximum of dI/dL increases with the RF excitation above $$\sqrt{N}$$, as shown in Fig. 3e. However, the increase of optical response with a higher N will be limited by the decoherence and inhomogeneous broadening with NV center ensemble, as explained before. In the experiments, we mainly test the ranging with N≤6, as it is close to the highest expected accuracy. The RF phase sensitivity and ranging accuracy can be calculated from the optical response and the noise (σI) of fluorescence signal, as $$\frac{{\sigma }_{{{{{{{{\rm{I}}}}}}}}}}{dI/dL}$$. Since the spin-RF interaction time (30 × N ns) is much shorter than the time for spin initialization and detection (> 1μs) in the ranging experiments, the increase in N does not substantially change the duty ratio of fluorescence detection. The noise ratio of ranging signal is almost the same with varied N. Therefore, the phase sensitivity and ranging accuracy is mainly determined by the normalized optical response, and will be enhanced beyond the classical limit of $$\frac{1}{\sqrt{N}}$$.

In Fig. 4a, we continuously record the RF ranging signal of NV center ensemble with N = 4. The normalized noise ratio is $$\frac{{\sigma }_{{{{{{{{\rm{I}}}}}}}}}}{{I}_{0}}\, \approx \, 0.077\%$$ with a one second measurement time. Compared with the optical response dI/dL, the accuracy of ranging with N = 4 is estimated to be approximately 30 μm with a one second measurement. It corresponds to a phase sensitivity of $$4\times 1{0}^{-3}{{{{{{{\rm{rad}}}}}}}}/\sqrt{{{{{{{{\rm{Hz}}}}}}}}}$$. In comparison, a phase sensitivity of $$0.095{{{{{{{\rm{rad}}}}}}}}/\sqrt{{{{{{{{\rm{Hz}}}}}}}}}$$ was previously reported by performing high frequency quantum heterodyne with NV center29.

To characterize the stability of the setup, the Allan deviation of RF ranging signal is analyzed in Fig. 4b. It shows that the highest ranging accuracy of 16 μm can be obtained with a 10 s measurement time. The instability of experimental setup mainly results from the vibration of the building and shift of the sensor. Since the fluorescence collection of NV center ensemble has been reduced approximately 10 times here, this accuracy is expected to be achieved with a much higher sampling rate when collecting all the fluorescence signals.

The integer ambiguities of ranging based on phase estimation can be solved by applying frequency modulation. For example, an external magnetic field is applied to split the spin state transition frequencies of ω±. According to Eq. (1), the maximum unambiguous range ($${R}_{\max }$$) with single RF (ω+) satisfies $$\Delta \varphi=4\pi \frac{{R}_{\max }{\omega }_{+}}{c}=2\pi$$, where c is the speed of light. It is deduced that $${R}_{\max }=\frac{c}{2{\omega }_{+}}$$. In contrast, simultaneously detecting the NV center spin state transition with ω+ and ω, the maximum unambiguous range will be extended as $$4\pi \frac{{R}_{\max }}{c}({\omega }_{+}-{\omega }_{-})=2\pi$$. It indicates that $${R}_{\max }$$ can be extended by a factor of $$\frac{{\omega }_{+}}{{\omega }_{+}-{\omega }_{-}}$$. In addition, the time of flight ranging can be performed with the quantum sensor(Supplementary Note 6). It provides the information on the target’s position with a longer dynamic range, though a lower ranging accuracy.

For the ranging with a moving object, the Doppler shift of reflected signal is fd = 2v/λ, where v is the speed of object. Considering a subsonic speed, fd will be smaller than 10 kHz with RF pulse (λ ≈ 0.1 m) in this experiments. In contrast, the width of ODMR peak (Fig. 1a) is greater than 10 MHz. Therefore, the Doppler shift of RF frequency will not change the amplitude of Rabi oscillation. We expect that the change of sensitivity can be ignored with normal moving objects.

## Discussion

In recent decade, quantum illumination30,31,32,33 and quantum detector34,35,36,37,38 have been proposed to improve the performance of a radar system. Comparing with other quantum RF receivers, such as Rydberg atoms and superconductors (where sophisticated setups or low temperature are needed), the NV center in diamond shows the advantages of robustness, high spatial resolution and miniaturization39,40,41,42,43. It is operated with simple setup under ambient condition. The working frequency band can be adjusted through Zeeman effect to tens of GHz44,45. And the CMOS-compatibility of diamond is favored for the development of a compact and integrated quantum sensor46,47. Recently, heterodyne sensing with NV center has been demonstrated for the high resolution spectroscopy of GHz radio signal29,48,49. It can be used to analyze the Doppler shift of RF scattering with moving objects. Combining these techniques, we expect that the NV center can be used to develop a practical high resolution quantum radar.

The protocol of radio detection in this work improves the sensitivity of an ultra high frequency signal with NV center by approximately three orders29,48. It enables the prototype of NV-based high resolution radio ranging (with 0.35 W RF peak power) to realize an accuracy that is comparable with the harmonic radio-frequency identification system19, which is operated with an RF power lower than 1 W. Meanwhile, Rydberg atoms RF receiver was recently demonstrated with electrical sensitivity from tens of μV/cm/$$\sqrt{{{{{{{{\rm{Hz}}}}}}}}}$$ to sub-hundred nV/cm/$$\sqrt{{{{{{{{\rm{Hz}}}}}}}}}$$37,38,50, though it has not been applied for high resolution radio ranging. Considering the advantages of NV center RF detector, further enhancing the sensitivity of RF detection 1–2 orders would be helpful for improving the competitiveness of NV-based quantum radar in future applications. Potential approaches include optimizing the NV center production, spin manipulation and increasing the fluorescence collection and RF focusing efficiency. For example, the inhomogeneous broadening with NV center ensemble, which limits the optical contrast C and the decay time τ, can be reduced by optimizing the diamond engineering40. Using high density NV center ensemble in 12C isotopic purified diamond may further improve the RF field sensitivity by three orders51,52. In addition, optimizing the RF antenna53 and utilizing spin-wave in ferromagnetic materials54,55 will provide a solution for further enhancing of local RF-spin interaction, increasing k in Eq. (4).

In conclusion, we have studied high resolution radio ranging with a solid state quantum sensor under ambient conditions. An RF field magnetic sensitivity of 21 pT/$$\sqrt{{{{{{{{\rm{Hz}}}}}}}}}$$, and a radio ranging accuracy of 16 μm is demonstrated with quantum sensing. With efforts to further enhance the sensitivity and dynamic range, we hope that quantum enhanced radio detection and ranging can be used for research on autonomous driving, navigation, health monitoring and wireless communication.

## Methods

### Sample preparation

The diamond plate is a 2 × 2 × 0.5 mm3 size single crystal with (100) surface from Element 6. High density NV center ensemble is produced through nitrogen ion implanting (15 keV energy, 1013/cm2 dosage) and subsequent annealing (850 C for 2 h). After the production of NV center ensemble, a nanowire-bowtie structure of chromium/gold (5/200 nm thickness) film is produced on the diamond surface through lift-off. And it is subsequently connected to a cm size in-plane bowtie antenna, which is made with copper foil tape.

### Experimental setup

The NV center ensemble is optically detected with a home-built confocal microscope, as shown in Fig. 5. A 532 nm laser (MGL-DS-532nm, New Industries Optoelectronics) is used to excite the spin-dependent emission of NV center, and is modulated by an acousto-optic modulator (MT200-0.5-VIS, AA). An objective with 0.7 NA (Leica) focuses the laser onto the diamond and collects the fluorescence of NV center ensemble. The fluorescence is detected by a single-photon-counting-module (SPCM-AQRH-15-FC, Excelitas) after passing through a long-pass filter (edge wavelength 633 nm, Semrock).The diamond quantum sensor is mounted on a glass plate, which is moved by a three-dimensional manual stage. The distance between diamond and the objective is fine tuned by a piezo stage (NFL5DP20/M, Thorlabs), while the direction of laser beam is scanned by a Galvo mirror system (GVS012/M, Thorlabs).

The RF from a signal generator (SMA100A, Rhode&Schwartz) is split into two paths by a power splitter (ZFRSC-42-S, MiniCircuits). RF pulse of each path is subsequently controlled by switches. The two paths are separately amplified by two amplifiers (60S1G4A, Amplifier Research, and ZHL-16W-43+, MiniCircuits). The RF of two paths is radiated into free space by two horn antennas (LB-2080-SF, Chengdu AINFO Inc). One path of RF is reflected by the target, the position of which is moved by a stepped motor. The other path directly pumps the NV center.

### Reporting summary

Further information on research design is available in the Nature Portfolio Reporting Summary linked to this article.