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Abstract
Wide-range high-precision velocity detection with nitrogen-vacancy (NV) color center has been realized. By treating the NV color center as a mixer, the high-precision microwave measurement is realized. Through optimization of acquisition time, the microwave frequency resolution is improved to the mHz level. Combined with the frequency-velocity conversion model, velocity detection is realized in the range of 0-100 cm/s, and the velocity resolution is up to 0.012 cm/s. The maximum deviation in repeated measurements does not exceed 1/1000. Finally, combined with the multiplexed microwave reference technique, the range of velocity can be extended to 7.4 × 105 m/s. All of the results provide reference for high-precision velocity detection and play a significant role in various domains of quantum precision measurement. This study provides a crucial technical foundation for the development of high-dynamic-range velocity detectors and novel quantum precision velocity measurement technologies.
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1. Introduction
High-precision microwave detection technology is the basis for the realization of high-precision velocity measurement. With the rapid development of quantum precision measurement technology, high-precision microwave measurement technology [1–3] has made significant strides. Ultracold atoms [4], Rydberg atoms [5], and nitrogen-vacancy (NV) color centers [6,7] have all shown the ability to perform high-precision microwave frequency measurements [8,9]. These advances lay the foundation for wide-range velocity detection.
High-precision microwave detection based on ultracold atoms has been realized, but the complex system of ultracold atoms cannot meet the demand for high-precision velocity detection. High resolution frequency detection with Rydberg atoms has been documented [10], based on the high-resolution frequency detection, velocity detection based on Rydberg atoms has also been reported. Nevertheless, the miniaturization and high-precision integration of the Rydberg atom detection system continue to pose challenges, primarily because of the necessity for multiple light sources and unavoidable factors such as optical absorption within the atomic vapor cell.
In recent years, the NV color center in diamond has received significant attention due to its robustness, ease of miniaturization, and other advantageous characteristics [11,12]. The NV color center system offers substantial benefits in terms of size, integration, solid-state characteristics, and on-chip measurements. The application of NV color centers in diamond focuses primarily on vector magnetometry [13–15], magnetic imaging, and near-field microwave detection.
Microwave frequency measurements using diamond NV color centers utilize primarily the manipulation of spin states by microwave fields. High-quality microwave antennas [16–21] are capable of providing a variety of complex microwave sequences, which in turn has led to the development of a variety of measurement protocols that allow microwave frequency measurements to be accomplished using NV color centers. The primary methods commonly employed at present include optical detection of magnetic resonance (ODMR) technology and hybrid dynamical decoupling techniques [22]. However, due to limitations imposed by sensing principles, ODMR technology typically achieves frequency measurement accuracy only at the kHz level. Within the sensitive frequency band of the NV color center, the uncertainty in velocity measurements exceeds 100 m/s. Dynamical decoupling technology holds the potential for high-precision frequency measurement, but its effective frequency range is confined to the kHz-MHz domain [23,24], velocity uncertainties are higher than 200 m/s at carrier frequencies sensitive to NV color centers and there are limitations in the range of velocity measurements. Consequently, it does not meet the testing requirements for precision and wide-range velocity measurement technology.
In this paper, we propose a high-precision velocity measurement technique with a wide dynamic range for microwave targets, employing NV color centers. High-precision microwave frequency measurement is achieved through the use of the transition between spin states induced by two electromagnetic fields [25,26]in conjunction with continuous wave optical detection of magnetic resonance (CW-ODMR) technology. Subsequently, combined with the velocity-frequency conversion model, high-precision detection of velocity is realized in a large bandwidth range. We investigate the correlation between frequency changes and the velocities of moving target at various velocities, validating the reliability of our velocity measurement method. Finally, by employing multiple reference microwaves, we extend the velocity measurement bandwidth, thereby confirming the theoretical feasibility of measuring maximum velocities.
2. Experimental setup and principle
The NV color center has a triplet ground state 3A2 with $|{{m_s} = 0} \rangle$ and $|{{m_s} ={\pm} 1} \rangle$, $|{{m_s} ={\pm} 1} \rangle$ are degenerate in the absence of magnetic field B. $D \approx 2.87$ GHz is a zero-field splitting, and microwaves at the resonance frequency can induce transitions between $|{{m_s} = 0} \rangle$ and $|{{m_s} ={\pm} 1} \rangle$.When a laser with a wavelength of 532 nm is applied to the NV color center, the ground state electrons absorb energy to jump to the excited state 3E and fall back rapidly. All the $|{{m_s} = 0} \rangle$ state and part of the $|{{m_s} ={\pm} 1} \rangle$ state electrons will fall back directly to the ground state and maintain the spin state, this optical transitions between the ground (3A2) and excited (3E) triplet states have a characteristic zero-phonon line (ZPL) at 637 nm, with a broad phonon-sideband (640–800 nm) at room temperature. Another part of the $|{{m_s} ={\pm} 1} \rangle$ state of the electrons will go through an intermediate state (1A1→1E) to fall back to the ground state of the spin state of $|{{m_s} = 0} \rangle$. As a result, continuous optical excitation will eventually pump the NV center into the $|{{m_s} = 0} \rangle$ spin state. When only the laser is active, the electronic states are fully polarized to $|{{m_s} = 0} \rangle$, which is the highest fluorescence intensity; When the laser and microwave collaborate at the same time, the relaxation process caused by the two competes with each other until the equilibrium, which will cause the fluorescence intensity to decrease and stabilize; When two microwave signals break this equilibrium, the fluorescence signal will become a regular sinusoidal oscillation (as shown in Fig. 1(a)).
Fig. 1. The schematic diagram of NV color center mixer. (a)Energy level diagram of the NV color center displaying the microwave-induced transition between ms states. (b) Interaction between the stationary signal microwave and the reference microwave on the NV color center. (c) Interaction between the moving signal microwave and the reference microwave on the NV color center.
For a specific dark state with $|{{m_s} = 1} \rangle$, in addition to the intrinsic longitudinal relaxation Γ1 = 1/T1, the weak microwave field opens a new relaxation channel between $|{{m_s} = 0} \rangle$⇒ $|{{m_s} = 1} \rangle$. The relaxation rate of this channel is indicated by Γb as follows [27]:
Here, Γ2 = 1/T2 represents the transverse relaxation rate, Δ represents the detuning between the ω and the NV color center resonance frequency, γNV = 2.8 MHz/Gs represents the gyromagnetic ratio of the NV electron spin, b is the amplitude of the microwave signal. Under continuous laser excitation, the rate at which $|{{m_s} ={\pm} 1} \rangle$ is polarized to $|{{m_s} = 0} \rangle$ denoted as Γp and competes with relaxation. Eventually, the population reaches steady-state equilibrium. In the steady state, the population P0 of $|{{m_s} = 0} \rangle$ is represented by:
The relaxation process Γb induced by microwave can lead to the degradation of the fluorescence signal [28]. In the case of a weak microwave field, i.e., Γb ≪ Γ1, the decay of the fluorescence signal ΔS can be simplified as:
In other words, the NV color center exhibits a second-order response to the power of the microwave field.
Similarly, we apply both the signal microwave fsig and the reference microwave fref to interact with the NV color center [29]. The reference microwave and signal microwave taking the form of:
When both frequencies of the microwaves resonate with the NV color center, i.e., δ ≪ Γ2, and thus they are regarded a signal that varies over time, leading to a time dependent relaxation rate of ${\varGamma _{\textrm{aux}}} + 2\cos (\delta t + \phi )\sqrt {{\varGamma _{\textrm{aux}}}{\varGamma _b}}$, where ${\varGamma _{\textrm{aux}}} = \gamma _{NV}^2b_1^2{\varGamma _2}/2(\varGamma _2^2{\Delta ^2})$, which represents a fixed term, does not vary over time. The oscillation term $2\cos (\delta t + \phi )\sqrt {{\varGamma _{aux}}{\varGamma _b}}$ induces fluorescence oscillation, and the change in fluorescence is proportional to the population of $|{{m_s} = 0} \rangle$ can be described as:
According to Eq. (6), the fluorescence change of the NV color center exhibits a linear response to microwaves. In other words, the oscillation frequency of the fluorescence signal is equal to the difference between the frequencies of the signal microwave and the reference microwave.
In summary, when a microwave with a frequency within the resonance region of the ODMR spectrum reaches the NV color center, the center emits a fluorescence signal with constant intensity due to a stable relaxation process. When two microwave signals reach the NV color center, they interfere with each other as described in Eq. (5). At this point, the NV color center can be considered as a mixer, and the frequency difference between the two microwave signals is represented in the fluorescence signal. As illustrated in Fig. 1(b), when the frequency difference Δf between the signal microwave and the reference microwave is fixed, the fluorescence oscillates fNV at a constant frequency, i.e., fNV = Δf. As shown in Fig. 1(c), when the frequency difference changes due to external factors (such as velocity variations), the corresponding oscillation frequency also changes. When the object carrying the signal microwave is at rest, the frequency difference Δf between the signal microwave and the reference microwave remains constant. When it interacts with the NV color center, it emits fluorescence oscillating at frequency Δf. However, when the object carrying the signal microwave approaches or moves away from the NV color center at a velocity, the frequency change resulting from its motion modifies the frequency difference Δf, subsequently causing a change in the oscillation frequency of the fluorescence signal.
Consider a microwave target moving towards the NV color center at a velocity v along a fixed direction. which is emitting a microwave with frequency fsig. Due to the Doppler effect, the relative velocity causes a change in the frequency of the received NV color center, and for a fixed receiver NV color center, the frequency shift Δfv caused by the target velocity v is given by:
The Δfv is superimposed on fsig, resulting in a change in the Δf and a corresponding variation in fNV. The velocity v of the target can be expressed as:
Here, c represents the velocity of light in a stable environment.
As shown in Fig. 2, the optical path section comprises a 532 nm laser (Changchun Institute of Optics, MLL-FN-532-1W), a beam expander, an objective lens, and other optical components. The measurement unit section includes a diamond sample containing NV color centers, a near-field flat antenna, a spatial horn antenna, and the microwave source (Keysight N5183B). The part related to the moving object under study consists of an electric slide rail (with a velocity control range of 0 m/s to 1.5 m/s, velocity accuracy of 0.01 mm per second.) and a mass block that carries the horn-shaped antenna. For signal collection, an avalanche photodiode detector (Thorlabs, APD130A2/M) is used, and experimental data are recorded using an oscilloscope (Tektronix, Mso640).
The diamond (4.5mm × 4.5mm × 0.5 mm, single crystal of type I b, [100] orientation, Element Six) was synthesized using the high-pressure high-temperature (HPHT) method with a nitrogen concentration of approximately 500 ppm and a natural abundance of 1.1% 13C. After annealing at 850°C for 1.5 hours in a vacuum, the diamond was irradiated with 10 MeV electrons for 4 hours, resulting in a total dose of 1.8 × 1018 cm2, which led to the formation of an ensemble of NV color centers with a concentration of approximately 3 ppm.
The 532 nm laser is emitted from the laser, the beam is adjusted to a suitable diameter by the lens, and then reflected by the dichroic mirror (long wavelength pass, cut-off wavelength: 565 nm) and focused onto the diamond surface through the objective lens. The fluorescence signal emitted from the diamond is also collected by the objective lens and passes through the dichroic mirror to enter the APD, where the signal is read out by an oscilloscope and analyzed and computed by an electronic computer. The reference microwave signal for the diamond is provided by a flat panel antenna placed under the diamond, and the signal carried by the moving target is provided by a horn antenna, which moves on a slide toward the diamond. The panel antenna and the horn antenna are driven by a microwave source.
The reference microwave fref (fref = 2 628 000 002 Hz) is generated by a flat microstrip antenna and irradiated on the sample. The horn antenna, installed on the velocity measurement device, emits the signal microwave fsig (fsig = 2 628 000 000 Hz = 2.628 GHz), precisely aligned with the NV color centers. The device moves along a fixed path with a predetermined velocity on the velocity measurement track.
3. Results and discussion
Utilizing the external bias magnetic field provided by a pair of permanent magnets, we achieve a centrally symmetric alignment between the magnetic field direction and the NV color center orientation. As a result, a pair of resonant spectra is obtained within the framework of CW-ODMR, and the positions of these resonance peaks are adjusted to be in proximity to the signal microwave frequency by manipulating the magnitude of the magnetic field. Figure 3(a) illustrates that the left resonance is located at a frequency of 2.628 GHz, corresponding to the frequency of the signal under measurement.
Firstly, the signal-to-noise ratio (SNR) affected by optical power is improved, the SNR of the signal exhibits an optimal solution with respect to the variation in optical power, as:
The SNR [30] can be calculated using the following equation:
Here, Ps and Pn represent the effective power of the signal and noise, respectively, xi and ni represent the amplitude of the signal and noise, Ns and Nn are the length of the signal and noise.
To attain the optimal SNR and ensure the highest measurement accuracy, SNR testing of the signal was conducted at various laser power levels, as illustrated in Fig. 3(b). When the laser power of the system is set to 500 mW, the SNR essentially reaches its maximum level. Consequently, the laser power was configured to 500 mW to capture the signal (At this point, after attenuation by the lens set and objective lens, the actual laser power finally focused on the diamond sample is 50 mW.).
Fig. 3. Basic parameter settings and results. (a) CW-ODMR. (b) Relationship between SNR and optical power in the system. (c) Initial frequency difference Δf0 Set to 2 Hz.
Following the optimization of the laser power, in order to visualize the direction of the object when it first starts moving, even though Δf can be configured to any value within the bandwidth, the system's Δf, denoted as Δf0, was ultimately set to 2 Hz, i.e., fsig = 2 628 000 000 Hz, fref = 2 628 000 002 Hz, the fluorescence signals obtained after the interaction of two microwaves are illustrated in Fig. 3(c).
To validate the feasibility of the velocity measurement system based on the NV color center mixer, we constructed a measurement setup under laboratory conditions. In this measuring system, the target carrying the signal microwave fsig was affixed to a rail controlled by a motor for velocity control. The entire motion process consisted of five stages: stationary, acceleration, constant velocity, deceleration, and returning to a stationary state, as illustrated in Fig. 4(a). Figure 4(b) displays the fluorescence signals received by the system. The blue curve represents the fluorescence signal when the target is stationary, the red curve represents the fluorescence signal when the target moves at a constant velocity (set as 10 cm/s) in the direction of the NV color center, and the black curve represents the fluorescence signal during the acceleration and deceleration stages, which show that the velocity information can be observed with our system. Figure 4(c) presents the frequency information of stationary, constant velocity, and stationary stages. When the target is at a standstill, the frequency of the fluorescence signal matches the initial set frequency difference of 2 Hz, i.e., fNV = Δf0 = 2 Hz. When the target moves with fixed velocity, the frequency difference Δf is no longer fixed at 2 Hz, which is caused by the Doppler effect. From Fig. 4(c), we can see that the frequency difference is changed to 2.88 Hz. The characteristic peaks with the highest amplitude are extracted from the obtained spectrum for calculation, and using Eq. (8), the motion speed of the measured target can be calculated. The corresponding velocity is 10.03 cm/s, which is basically consistent with the velocity value we set. The results show that velocity detection is realized in our system.
Fig. 4. Velocity measurement setup and results. (a) Stages of target motion. (b) Fluorescence signals: blue for stationary, red for constant velocity, and balck for acceleration/deceleration. (c) Frequency changes with target motion.
We then performed error analysis and optimization. According to Eq. (8), the primary sources of error in the NV color center mixer-based velocity measurement system can be attributed to two factors. First, there are errors in the measurement results due to spectral resolution. Secondly, there are equipment self-output errors that result from aging of the internal clock of the microwave source.
The reading error of Δfv is limited to 0.01 Hz, primarily as a result of frequency resolution following the Fast Fourier Transform (FFT). The fsig output error is equipment-dependentand the calculated output error is 0.001 Hz. The absolute error limit [31] of the final measured velocity can be expressed as:
Following Eq. (11), the absolute error limit can be calculated. Ultimately, the absolute error limit for the measurement is better than 0.0011 m/s.
To improve measurement accuracy and minimize errors, we have performed further optimization of the measurement methodology.
As shown in Eq. (11) that spectral resolution is one of the key factors influencing the error results. In this work, by applying FFT, the time-domain signal is transformed into the frequency domain, enabling information conversion. The calculation formula for spectral resolution, given by δf = Fs /N = 1/t, includes the parameter δf, representing the minimum spectrum resolution, the sampling rate Fs, the total number of collected samples N, and the acquisition time t under a fixed sampling rate. When the sampling rate Fs satisfies the Nyquist sampling theorem, decreasing δf by extending the measurement time effectively improves the accuracy of the velocity measurement.
Since the accuracy of velocity measurement is determined by the accuracy of frequency measurement, we first conducted experiments to verify the limiting resolution accuracy of the system and to determine the reliability of the sensor by checking its resolution of the frequency response of the microwave signal. Due to the limitations inherent in the minimum frequency resolution provided by the microwave source, Δf was set to 0.001 Hz (fsig = 2.628 GHz, fref = 2.628 GHz + 0.001 Hz). As illustrated in Fig. 5(a), the experimental results indicate that the fluorescence signal exhibited 5 cycles during the 5000-second measurement, with the maximum peak position frequency, after data processing, recorded at 0.001 Hz. These findings demonstrate that the NV color center ensemble mixer in this study exhibits an ultra-high frequency resolution capability, reaching mHz.
Fig. 5. Relationship between measurement time and velocity accuracy. (a) The mHz-scale frequency resolution of the NV color center mixer. (b) Fluorescence signals at different measurement time. (c)1 s measurement time. (d)10 s measurement time. (e)100 s measurement time. (f)1000 s measurement time.
In order to investigate the relationship between measurement time and velocity accuracy, as shown in Fig. 5(b), we investigated the processing results at different measurement times when the target was moving at a fixed velocity of 100 cm/s. Figure 5(c)-(f) displays spectra with deviations attributed to the minimum frequency resolution, along with the corresponding velocity calculations in the context of multiple measurement repetitions, we computed velocity measurements for measurement times of 1s, 10s, 100s, and 1000s. The results indicate a significant improvement in velocity resolution with extended measurement time, this is consistent with the prediction of Eq. (11).
In this study, the measurement time was extended from 100 s to 1000 s. According to δf = 1/t, the frequency resolution was improved to 0.001 Hz, which corresponds to a velocity resolution of 1.12 × 10−4 m/s. A longer measurement time can theoretically result in higher measurement accuracy. However, due to various factors such as the minimum frequency resolution of the microwave source, the maximum operating time of the equipment, the memory limitations of the oscilloscope, and the speed of data processing, tests were conducted only for target motion with a 1000 s measurement time in this study.
In Fig. 6, it is evident that prior to optimization, the system exhibited an absolute error limit of 0.11 cm/s, an average standard deviation of 0.0565 cm/s, and a ten-test average deviation of 0.0273 cm/s. However, after optimization, the system showed a remarkable improvement, achieving an absolute error limit of 0.012 cm/s, an average standard deviation of 0.0056 cm/s, and a ten-test average deviation of 0.0048 cm/s. Employing this approach, as outlined in Eq. (11), the post-optimization absolute error limit is less than 0.012 cm/s, representing nearly a tenfold enhancement compared to the pre-optimization state.
To verify the reliability of our method after verifying the principles and discussing methods to improve accuracy, we performed a series of measurements at various velocities. Each measurement had a duration of 100 seconds, and we conducted 20 sets of measurements in differential calibration mode.
These sets included 20 different values of movement toward the NV color center, encompassing velocity values ranging from -100 cm/s to 100 cm/s (the direction towards the NV color center receiver is denoted as positive, while the opposite direction is designated as negative.). The partial test results, as depicted in Fig. 7(a), reveal that when the target is stationary (i.e., v = 0 m/s), the fluorescence oscillation frequency aligns with the pre-established Δf0. As the value of the target's velocity of motion is raised, the oscillation frequency of the curve gradually ascends. The spectra, corresponding to the curves presented in Fig. 7(a), are displayed in Figs. 7(c)-7(h), respectively. Additionally, the figures indicate the positions of the most prominent characteristic peaks found in the spectra.
Fig. 7. Investigation of fluorescence curves at various velocities of the measurement target. (a) The fluorescence oscillation of NV color center. (b) Velocity calculated from the maximum characteristic peak of the fluorescence frequency. (c)-(h) Spectra extracted from fluorescence curves at velocities of 0 cm/s, 20 cm/s, 40 cm/s, 60 cm/s, 80, and 100 cm/s.
As shown in Fig. 7(b) using the fluorescence data obtained from the measurements, combined with the FFT transformation process, the result obtained by Eq. (8) also matches the velocity value.
Upon achieving velocity measurements, to broaden the velocity measurement range, a scheme involving multiple simultaneous reference microwaves is applied (as shown in Fig. 8(a)). Multiple reference microwaves, equidistant in frequency difference, are positioned within the resonance region of the ODMR. When the signal microwave to be measured surpasses the response bandwidth of the initial reference frequency fref, it enters the response range of the next reference microwave. Although this approach is constrained by the resonance region of the ODMR spectral line, it significantly expands the measurement bandwidth.
Fig. 8. Expansion of measure range. (a) schematic diagram. (b) Single reference microwave measure bandwidth. (c) Multi-reference microwave measure bandwidth.
Next, the system response bandwidth (at 3 dB attenuation) is measured under single-channel reference microwave and multi-channel reference microwave schemes, and the measurement results are shown in Fig. 8(b), (c). The system's velocity measurement range is calculated. When only a single reference microwave is used, the effective response range of the signal microwave is 1.5 kHz, and the corresponding velocity measurement range is 170 m/s. When multiple reference microwaves are introduced, the measurement effective response range is extended to 13 MHz, this value is essentially the same as the linewidth of the ODMR resonance peak we chose in this experiment, and the corresponding velocity measurement range is 7.4 × 105 m/s. In addition, this technology can be combined with the ODMR frequency adjustment and linewidth control technology [32], which is expected to further increase the velocity measurement range significantly.
4. Conclusion
In this paper, high-precision velocity detection based on continuous heterodyne detection of the NV color center ensemble mixer has been achieved. By combining with CW-ODMR control methods, we adjusted the resonant point to the signal microwave frequency band and successfully tested the motion velocity of a signal microwave source in space. Through velocity measurements in the range of 0-100 cm/s, the uncertainty of velocity measurement is less than 0.012 cm/s. The maximum deviation in repeated measurements does not exceed 1/1000. Using multichannel local microwave, the effective measurement bandwidth of this method can be enhanced to the order of 13 MHz, corresponding to a velocity measurement limit of 7.4 × 105 m/s. The research results demonstrate that this work has important reference value for the construction of quantum radar systems on future quantum platforms, high-precision measurement of electromagnetic waves in the far field, accurate measurement of moving objects, and tracking and positioning of signals. This paves the way for the potential application of this technology in integrated quantum devices for ultra-high-velocity target velocity measurement, with broad prospects in fields such as quantum radar and quantum communication.
Funding
Joint Funds of the National Natural Science Foundation of China (U21A20141); National Natural Science Foundation of China (51821003, 62103385, 62175219); Key Laboratory of Shanxi Province, China (201905D121001).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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