1. Introduction

The quantum spectrum aims to reconstruct the frequency spectrum of a system Hamiltonian in a high resolution [1–3]. The traditional methods focus on isolating the components of the frequencies by adjusting the Hamiltonian to an interaction period [4]. Referred to dynamic decoupling (DD), successive pulses are used to generate a spectral filter for individual frequency selections [5,6], which results in a linewidth that is inverse proportional to the interaction time [4]. It is underlined that the resolution and precision of the frequency spectrum are limited by the sensor’s linewidth [3,4,7,8]. However, the existence of the linewidth gives rise to difficulties in the high-resolution of signal frequency, which is quite essential in various applications including microwave spectroscopy [9,10], magnetic navigation [11], molecular structure determination [12], and biomagnetic field detection [13]. It is of great significance to effectively reduce the linewidth of the sensor to improve the frequency resolution.

The linewidth of the sensor is determined by the sensor’s coherence time [4,5,7]. The coherence time has been extended from many works in the literature to improve the frequency resolution. By transferring the quantum phase of the sensor to the state population, the upper bound of the coherence time to the longitudinal relaxation time T₁ is successfully extended [14]. The implementation of hybrid qubits as a more stable clock can significantly prolong the sensor’s coherence time to the lifetime of a memory qubit T_M [15]. In addition, the quantum heterodyne (Qdyne) technique with DD sequences can completely break the limitation of the coherence time. Meanwhile, the resolution is only limited by the stability of the external reference clock signal and the total measurement time [4,16,17]. Enlightening by the fact that the continue-wave optically detected magnetic resonance (CWODMR) method can be regarded as a limit case of the DD intervals setting to zero, a dissipative method with CWODMR could also achieve a similar resolution as the Qdyne technique [18].

The heterodyne readout technique possesses the unique advantage that no feedback to lock the sensor to the signal phase is required [4]. The CWODMR spectrum benefits from a wide response bandwidth [18] and simple operations [2,3]. Under this direction, two advantages are combined by the developed protocol. The frequency of the magnetic field was swept to match the unknown signal. Moreover, the heterodyne readout was applied to ensure real-time high-resolution spectroscopy of the unknown signal corresponding to frequency sweeping. The nitrogen-vacancy (NV) center ensemble was selected as the solid-state sensor due to its excellent physical properties at room temperature [19,20]. The steady state of the NV center was designed to monitor the alternating current (AC) magnetic field instantly. The resolution and precision of the frequency spectrum were enhanced by adjusting the low-pass filter parameter T_c. Furthermore, rigorous mathematical models were established to clarify the factors that determine the limits of the resolution and precision of the frequency spectrum. Finally, the developed protocol was applied to analyze the frequency spectrum of arbitrary audio signals.

2. Theory and experimental method

The NV center in diamonds is considered a delicate platform to demonstrate the mechanism of the proposed protocol. The NV center is formed by a nitrogen atom binding an adjacent vacancy and negatively charged by trapping a free electron [19,20], as schematically illustrated in Fig. 1(a). Six electrons can be found in the periphery of the NV center, two of which are unpaired [21]. As can be ascertained from the electronic energy-level diagram in Fig. 1(b), the ground state ³A₂ and the excited state ³E are degenerated triplet states under zero magnetic field. Electrons can be excited from ³A₂ to ³E by laser irradiation at 532 nm. Due to the short lifetime of the ³E state (∼ 10 ns), some electrons quickly fall back to ³A₂, and through this process, photoluminescence at 600 - 800 nm is released. The excitation and radiative decay processes are also spin-preserving. A non-radiative decay process called the “intersystem-crossing process” (ISC) [22] makes electrons decay from ³E to the |m_s = 0〉 state of ³A₂ through the metastable states (¹E and ¹A₁). This cycling process can be used for the polarization of the NV electrons, and the photoluminescence intensity represents the population of spin states. These two processes enable |m_s = 0, ± 1〉 states encoded as a sensor. The degenerated |m_s = ±1〉 states can be distinguished by the Zeeman splitting via applying a static magnetic field B₀, and the |m_s = 0, −1〉 states were selected as a two-level sensor for the sake of simplicity in this work. After the polarization of the sensor, the flips between the |m_s = 0, −1〉 states in the ³A₂ were manipulated by microwaves (MW). The dissipation process between the |m_s = 0, −1〉 states with the decay rate Γ₁ can gradually disrupt the MW manipulation, which renders the sensing time limited by the relaxation time. To break through the limitation of the relaxation time, the CWODMR technique combined with the heterodyne readout was employed to achieve the high-resolution frequency spectrum of the low-frequency signal.

 

Fig. 1. (a) Depiction of the lattice structure of NV center in the diamond, where N atoms (blue) are combined with adjacent vacancies (green). Four axis orientations exist for the NV centers. (b) Distribution of the fine level of the NV electron. The degenerated triplet states of the ground and excited states are denoted as ³A₂ and ³E, while ³A₂ consists of three sublevels |m_s = 0, ± 1〉. The states of |m_s = 0〉 and |m_s = −1〉were encoded as the sensor. The initialization and readout of the sensor were performed by laser irradiation at 532 nm. The intersystem-crossing process (ISC) through the metastable states (¹E and ¹A₁) is marked by the blue dotted line. (c) AC signal measurement by the application of the CWODMR technique. The laser and MW were continuously employed to the NV centers. The fluorescence intensity changed synchronously along the AC signal. (d) Schematic illustration of the heterodyne readout technique. The laser beam was focused on the diamond surface through a confocal system. The MW and AC signal B(t) were conducted to the diamond surface via homemade MW and radio-frequency (RF) antennas. A bias magnetic field B₀ was also enforced along the (111) crystal direction. The fluorescence of the NV was used as input to the lock-in amplifier (LIA) through the avalanche photodiode (APD). The fluorescence of the NV and the reference signal were then mixed by the multiplier, and the mixed signal was passed through the low-pass (L-P) filter to remove the high-frequency components. The heterodyne readout technique can extract the amplitude and phase information of the signal under detection. Meanwhile, it can convert the signal from the time domain to the frequency domain.

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Figure 1(c) depicts a schematic illustration of the AC signal monitored by the CWODMR technique. The 532-nm laser irradiation and the MW were always on, and the MW frequency corresponded to the transition frequency between the |m_s = 0, −1〉 states under the bias magnetic field B₀. The AC magnetic field B(t) = b_ac·cos(ω_ac + ϕ) was externally applied to the diamond surface through a radio-frequency (RF) antenna. Meanwhile, the fluorescence signal changed synchronously along the B(t), as shown in the lower part of Fig. 1(c).

The Hamiltonian of the two-level NV sensor can be described as follows: $H_{\mathrm{NV}}=H_{\mathrm{ZF}}+H_{\mathrm{ZM}}+H_{\mathrm{MW}}+H_{\mathrm{ac}}+H_{\mathrm{hyp}}$ [18]. Where, H_ZF = DS_z is the zero-field splitting Hamiltonian of the NV, with D = 2π × 2.87 GHz, and S_j (j = x, y, z) represents the spin operator. H_ZM = −γ_eB₀S_z is the Zeeman splitting Hamiltonian of the NV under the enforcement of a static magnetic field B₀ along the NV-axis, with γ_e = −2π × 28 GHz/T signifies the electron gyromagnetic ratio. H_MW = −γ_e × B₁cos(ω_MWt)·S_x stands for the MW control Hamiltonian, where B₁ and ω_MW denote the amplitude, and frequency of the MW, respectively. H_ac = −γ_eB(t)S_z refers to the Hamiltonian of the AC magnetic field, which was assumed along the NV-axis for simplicity. H_hyp = A·S_z·I_z denotes the Hamiltonian of the hyperfine interaction with the ¹⁴N, with A = −2π × 2.16 MHz and I_z being the hyperfine coupling constant and the nuclear spin operator of the ¹⁴N, respectively. In this work, the ¹⁴N was not polarized, and H_hyp was determined by nuclear spin state |m_I〉 (I = −1, 0, 1) of the ¹⁴N, namely H_hyp = A·m_I·S_z. The Hamiltonian of the NV sensor can be integrated as follows:

The evolution state of the NV sensor can be described by the Lindblad master equation [23]

Next, the Heterodyne readout was applied to achieve a high-resolution frequency measurement of the low-frequency signal through the lock-in amplifier (LIA). As can be observed from Fig. 1(d), the signal under detection was used as input to the LIA and played the role of the reference signal, whereas the LIA will normalize the signal under detection as cos(ω_reft). Subsequently, the fluorescence signal $f{l_{|{{m_s} = 0} \rangle }}$ and the reference signal cos(ω_reft) were mixed by the multiplier, and then the mixed signal was passed through the low-pass (L-P) filter to remove the high-frequency components. The transfer function of the first-order RC low-pass filter in the time domain can be expressed as follows: H(t) = kexp(kt), where k = 1/2πT_c is the signal bandwidth of the L-P filter. The first output X of the LIA can be calculated through the convolution operation that is described by the following expression:

Similarly, the second output Y with the reference signal shifting π/2 can be expressed as follows:

The final output of the LIA is X + iY, and R = (X² + Y²)^1/2 and ϕ = arctan(Y/X) refer to the amplitude information, and phase information of the signal, respectively. (more details can be found in Supplement 1).

As far as the frequency measurement is concerned, the measurement time t of each data point was assumed adequately long, and the frequency ω_ac of the B(t) was swept. The R curve can be estimated by the following expression:

An experiment was also set up to demonstrate the improvement of the frequency resolution by our protocol, shown in Fig. 1(d). A laser (OXXIUS, LCX-532L-500) with 400-mW power was focused on the diamond surface through a confocal lens. The MW source (Agilent, N5181B) generated continuous microwaves of 0 dBm to the diamond surface through the MW antenna. A bias magnetic field B₀ along the (111) crystal direction of the diamond was created through the permanent magnet. The produced AC signal B(t) by a wave generator (Agilent, 33500B) was emitted to the diamond surface via a radio frequency (RF) antenna. The photodiode (Thorlabs, APD120A2/M) converted the fluorescence into an electrical signal, which was employed as input into the LIA (Zurich Instruments, HF2LI). The LIA performed the heterodyne calculation with the fluorescence and the reference signal. Finally, the outputs of the LIA were used to analyze the frequency spectrum of the signal under detection (the experimental setup and parameters can be found in Supplement 1).

3. Results and discussions

3.1 Resolution Improvement of the AC spectrum

The measurements of the AC spectrum were performed by our developed method based on the NV ensemble, and the measurement time t of each data point was set sufficiently long to ensure that the system was stable at each time. Technically, the measurement time t was prolonged until the FWHM of R(ω_ref) reached the minimum at each T_c. The signal under detection was input to the LIA as the reference signal with a frequency value of ω_ref ≈ 10 kHz, and then the frequency ω_ac of B(t) was swept. When ω_ref = ω_ac matched, the peak of the R curve from the LIA emerged the maximum, as is shown in Fig. 2(a). The R curve was fitted according to Eq. (7), and all the extracted r-squares (goodness of fit) were better than 0.995, whose fit results were more accurate than the Lorentz fit, whose r-squares values were about 0.98. By increasing T_c from the value of 3.992 s to 116.6 s, the FWHM of the R curve was intensely decreased from 142.2 millihertz to 4.7 millihertz (theoretically, the FWHMs can be infinitely reduced). As T_c increased, the L-P filter required more time to stabilize the signal, which needed to gradually extend the sweep time of the B(t) frequency (or reduce the sweeping velocity). For instance, each point for T_c = 116.6 s took more than 1 hour to be stabilized, and the total sweeping time lasted more than 24 hours. Note that if the sweeping interval of B(t) was small enough, slight sinusoidal fluctuations were observed from the R curve. However, these fluctuations can be explained by considering the trigonometric function terms of ω_ref and ω_ac in Eq. (7). Due to these sinusoidal fluctuations, the fit of Eq. (7) was better than the Lorentz fit. However, these fluctuations can be neglected as macroscopically stable.

 

Fig. 2. Depiction of the frequency resolution and precision results. (a) R curve of the LIA for T_c = 3.992 s (blue dots) and T_c = 116.6 s (black forks). The data were fitted according to Eq. (7) (red lines). The full width half height (FWHM) values for T_c = 3.992 s and T_c = 116.6 s were 142.2 mHz and 4.7 mHz, respectively. The inner left diagram is a section of T_c = 3.992 s within the range of 0.1 Hz from 9994.08 to 9994.18 Hz. (b) Distribution of the frequency resolution (represented by the FWHM) as a function proportional to T_c⁻¹. (c) Distribution of frequency precision (represented by the peak uncertainty) as a function proportional to T_c^−1.5.

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Next, the FWHM of the R curve was significantly reduced by increasing T_c to optimize the resolution of the R-peak further. Figure 2(b) shows the frequency resolution (representing the FWHM) as a function proportional to T_c⁻¹. As can be described from Eq. (7), when the states of the NV were steady (t sufficiently long) and ω_ref ≈ ω_ac around the R-peak, the first term F₁ (∼ I₁k/ω_ref) will be much smaller than the second term (∼ I₂k), and R(ω_ac) can be simplified as follows:

It can be deduced that ω_ac ≈ kR(I₂² − 2R²)^-1/2, and the FWHM ∝ k ∝ T_c⁻¹. These results illustrated that the frequency resolution from the proposed method could break the limitation of the coherence time, and the resolution can be infinitely reduced by prolonging T_c.

Figure 2(c) displays the frequency precision as a function proportional to T_c^−1.5, which can be described by the 95% confidence intervals of the fitting ω_ref according to Eq. (7). In the experiments, the uncertainty sources of the AC signal originated from the fluorescence readout process. As a result, the uncertainty of the reference signal Δω_ref was equivalent to the uncertainty of the fluorescence Δω_ac. The Fisher information (FI) was also introduced to analyze the uncertainty of the fluorescence. By assuming that the noises of the system (such as the drifts of laser, MW and B₀, the noise of electronic components, the temperature disturbance, etc.) were common white Gaussian noises (WGN), the fluorescence was re-described as follows: fl(n) = I₁ − I₂·cos(ω_acn+ϕ) + G(n) = s(n) + G(n), where, n = 1, 2, 3…N is the number of the experimental data, and S(n) signifies the accurate value of the fluorescence. The WGN satisfies the following expression: G(n) = exp[−(n − µ)²/(2σ²)](2πσ²)^−1/2, where µ, σ are the average and standard variance of the noise, respectively. Since the amplitude I₂, the frequency ω_ac, and the phase ϕ were all embedded in the WGN as unknown constants, the FI was introduced as follows: $I(\vec{\theta }) = E[\partial \ln P(n|\vec{\theta })/\partial \vec{\theta }]$, where $\vec{\theta }\textrm{ = [}{I_2},{\omega _{ac}}\textrm{,}\phi {\textrm{]}^T}$ refers to the vector containing all the unknown parameters, $P(n|\vec{\theta })$ stands for the probability density function (PDF) of the data under the condition of $\vec{\theta }$, and E represents the expected value. The PDF can be described by the following equation:

The matrix elements of the FI can be written as follows: ${[I(\vec{\theta })]_{i,j}} = \frac{1}{{{\sigma _2}}}\sum\limits_{n = 1}^N {\frac{{{\partial ^2}P(n|\vec{\theta })}}{{\partial {{\vec{\theta }}_i}\partial {{\vec{\theta }}_j}}}} $ (${\vec{\theta }_{1,2,3}}$ represented I₂, ω_ac, ϕ, respectively), and the FI can be described by the following expression:

The Cramer-Rao lower bounds (CRLB) of the three parameters in $\vec{\theta }\textrm{ = [}{I_2},{\omega _{\textrm{ac}}}\textrm{,}\phi {\textrm{]}^T}$ were used to evaluate their uncertainties, which can be expressed as follows:

3.2 Noise floor and sensitivity of the AC magnetic field measurement

The investigations from Eq. (11) illustrated that the T_c and SNR η were the determinants of the frequency resolution and precision. The SNR η was greatly determined by the noise floor and the sensitivity of the CWODMR method. In this direction, the limitation of our method was analyzed via the protocol of phase-sensitive detection [26]and the Allan variance.

As the schematic of the phase-sensitive detection demonstrated in Fig. 3(a), the AC signal was applied at the maximum slope of the CWODMR spectrum. When the amplitude of the AC magnetic field changed by ΔB, the fluorescence signal was altered accordingly. The output amplitude of the LIA was also shifted by γ_eΔB synchronously to the induced magnetic field change ΔB. As the sensitivity measured under the same experimental parameters in section 2, the frequency of the AC signal, the power of the MW, and the power of the laser were set as 10 kHz, 0 dBm, and 400 mW, respectively. In this case, the scale factor of the AC signal reached 2.03 × 10⁻⁴mV/nT. The noise of the system was air-mined at a sampling rate R_s of 225 sample/s for 2 hours, and then the system noise was analyzed by the Allan variance method (details see Supplement 1) [ 27]. The results in Fig. 3(b) showed that the sensitivity of the magnetic field measurement was 7.32 nT·Hz^−1/2@10 kHz, and the noise floor was 0.34 nT (comparisons of this work with the state-of-art results can be found in the Supplement 1). It has been well discussed that the higher pumping rate (higher laser power), the longer relaxation time T₁, $T{_2^\ast }$ (optimal ¹²C sample, etc.), and the longer coherence time T₂ (DD technique, etc.) could significantly reduce the FWHM of the ODMR and optimize the sensitivity. A lower noise floor can be realized by two methods, namely noise shielding and error suppression. The main noise sources of the system were from the laser, electric current, and magnetic field. Laser noises can be effectively reduced by stabilizing laser power and reducing wavelength bandwidth. Electromagnetic noises can be optimized by electromagnetic shielding, noise filter, and PID control. Moreover, it was also reported that optimized data acquisition methods combined with machine learning [28] and Bayesian estimation [29] could suppress data errors and guarantee better noise floors.

 

Fig. 3. The sensitivity measurement of the AC magnetic field is based on the implementation of the heterodyne readout CWODMR technique. (a) Schematic illustration of the phase-sensitive detection based on the heterodyne readout CWODMR technique. The AC signal was applied at the maximum slope of the ODMR spectrum, and the fluorescence signal was changed synchronously to the AC field. As a result, the amplitudes of the outputs from the heterodyne readout were shifted by γ_eΔB. (b) Distribution of the Allan variance result of the AC magnetic field sensitivity. The sensitivity of the AC field was 7.32 nT·Hz^−1/2@10 kHz, and the noise floor was 0.34 nT (green dashed line).

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It is interesting to notice that the curve of the Allan variance was upturned after 1000 s. The reason is the long-term drifts caused by the low-frequency system noise, which include the slow temperature perturbation, the laser power fluctuation, and the electronic noise from the pre-and post-amplifier circuits of the detector. The solutions such as PID temperature control, electromagnetic shielding, and low-frequency noise filter can effectively improve the situation.

3.3 Frequency sensing of arbitrary audio signals

The literature has confirmed that the response bandwidth of CWODMR is within the frequency range of DC-140 kHz [18]. The CWODMR magnetometer combining the heterodyne readout is considered a convenient approach for sensing the arbitrary audio signals audible to humans, within the range of 20-20000 Hz [30]. The arbitrary audio signal is referred to as F_sig(ω_i, ϕ_i), where ω_i represents all the frequencies contained in the signal, and ϕ_i refers to all the phases. The original F_sig(ω_i, ϕ_i) was mixed by a high frequency (ω_hf) signal for convenience of the heterodyne operation that will be performed later, and the mixed signal was described as follows: F_sig(ω_i, ϕ_i)·sin(ω_hft), with ω_hf >> ω_i. The mixed audio signal was written into the read-only memory (ROM) of the field programmable gate array (FPGA, Xilinx ZYNQ7020). In ROM, the Hashtable characterized the function between audio signal and time. The programmed FPGA converted the digital signal in the ROM into the analog signal by employing a digital-to-analog (D/A) converter under the trigger of a counter. The analog signal was amplified by an RF amplifier and then emitted to the diamond surface through a homemade RF antenna. According to the theory that was presented in section 2, the fluorescence of the NV was in proportion to the analog signal. By the same heterodyne CWODMR method described in section 3.1, the frequency of the reference signal was set as ω_ref = ω_hf. The heterodyne result of the fluorescence and reference signals can be characterized in proportion to the following equation: F_sig(ω_i, ϕ_i)·[cos(ϕ_hf) − cos(2ω_hf + ϕ_hf)]. The high-frequency components were removed by the L-P filter, and the output of the LIA was synchronized with the original signal. In this case, the frequency information of the original signal can be obtained by analyzing the measured result of the LIA. The increase of T_c could effectively improve the frequency resolution of the arbitrary audio signal.

In the experiments, the original audio signal was randomly selected within the DC-600 Hz range, with a bandwidth of about 100 Hz, and the amplitude was about 1.5 V_pp. The frequency of the mixing signal was set as ω_hf = 9500 Hz. The result of the LIA is shown in Fig. 4(a). The length of the original signal during one period was 3.5 s, and the sample rate was set as 143.9 k sample/s, which needed to be larger than the frequency resolution of the analog signal from the FPGA (100 k sample/s in our experiment) since an insufficient sample rate will lead to a significant loss of the signal information. As can be seen from the fast Fourier transform (FFT) result of the signal from the LIA shown in Fig. 4(b), the frequency spectrum consisted of three frequency values of 523.4 Hz, 588.2 Hz, and 660.1 Hz, which were the anastomosis with the original audio signal. Compared to the original signal, the frequency shifts were less than 0.2 Hz, and the errors of frequency were less than 0.95 ‰. The FFT results revealed that the information of the audio signal could be effectively transmitted and preserved through the applied CWODMR method combined with the heterodyne readout. Notably, some low-frequency noises were introduced by the FFT spectrum compared to the original signal. The origin of noises could be ascribed to the photon shot noise and the electromagnetic fluctuations. Several methods can effectively suppress these two noise sources, such as increasing the laser power, increasing the fluorescence collection efficiency, and suppressing the external magnetic noise. From the frequency analysis through the CWODMR method combined with the heterodyne readout, it can be concluded that a new path to the frequency sensing of arbitrary audio signals is provided.

 

Fig. 4. Frequency sensing of arbitrary audio signals. (a) The audio signal results from the LIA. The sound signal period was 3.5 s, and the sampling rate was 143.9 k sample/s. (b) Depiction of the FFT analysis of the audio signal. The FFT spectrum contained three frequency values of 523.4 Hz, 588.2 Hz, and 660.1 Hz.

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4. Conclusion

In conclusion, it was experimentally demonstrated that a high-resolution frequency spectrum was performed by the CWODMR method combined with the heterodyne readout based on the NV center ensemble. The frequency of the magnetic field was swept to match the unknown signal, and the signal can be transformed into a real-time frequency-domain curve via the heterodyne readout. The acquired results proved that the frequency resolution and precision could break the limitation induced by the sensor’s coherence time, and a frequency resolution of 4.7 millihertz was achieved. The experimental results were mathematically explained based on the Lindblad master equation and the heterodyne operation. The frequency resolution and precision were certificated as functions of the T_c parameter of the L-P filter proportional to T_c⁻¹ and T_c^−1.5, respectively. Moreover, the frequency precision was also proportional to the SNR η⁻¹. The noise floor and sensitivity of our method were tested by the phase-sensitive detection and the Allan variance. The results illustrated that the AC magnetic sensitivity was 7.32 nT·Hz^−1/2@10 kHz, and the noise floor was 0.34 nT. The sensitivity can be further improved by optimizing several parameters in the experiment (optimized ¹²C samples [31], higher pumping rate, longer relaxation time, etc.). The CWODMR method combined with the heterodyne readout also performed the frequency sensing of arbitrary audio signals. More specifically, the original audio signal was mixed with a high-frequency signal, and the final mixed signal was converted into an analog signal using an FPGA. The heterodyne technique and L-P filter removed the high-frequency components of the mixed signal and successfully restored it to its original form. From the FFT result, it was proved that the information of the audio signal could be effectively transmitted and preserved through the introduced method, and a new path to the frequency sensing of arbitrary audio signals was provided. The fidelity of the recorded audio signal can be further improved by enlarging the corresponding frequency bandwidth [18], increasing the fluorescence collection efficiency [24,32], and improving the SNR [33,34]. The proposed protocols for AC signal sensing possess the comparative advantages of wide frequency bandwidth, enhanced resolution, and precision. The states of the sensor in the process can also be monitored synchronously. Our work paves the way for developing a universal idea for quantum sensing of AC signals, and could be enlightening to other fields, such as quantum imaging and chemical analysis.