1. Introduction

Using the laws of quantum mechanics, microscopic quantum systems and their quantum states quantum sensing can realize high-precision and high-sensitivity measurement of specific physical parameters [1]. Nitrogen-vacancy (NV) center in diamond is an excellent solid-state quantum sensor for multi-physics parameters with high sensitivity and spatial resolution, such as magnetic field, electric field, temperature, and stress [2,3]. As a stable light-emitting defect, the NV center consists of a nitrogen atom replacing a carbon atom and a vacancy at the adjacent position as shown in Fig. 1(a). The NV center has a variety of charge states [4,5], and the negatively charged state NV^— (capturing an electron) is often used in quantum sensing. The NV center mentioned in this paper defaults to NV^—.

 

Fig. 1. (a) Crystal structure of NV center. (b) Level structure of NV center. (c) home-built confocal microscopy for ODMR experiment. The nanodiamonds are distributed on a 0.17-millimeter cover glass on the radiation structure.

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Figure 1(b) is the electron spin energy-level diagram of NV centers. For NV-based quantum sensing, the state-selective intersystem crossing (ISC) enables the electron spin state of the NV center to be optically initialized and readout at ambient temperatures [6,7]. The electron spin of NV centers can be excited from ground state ³A₂ to excited state ³E by a 532 nm laser. Then, the electron spin of excited state ³E can decay from the excited state ³E to ground state ³A₂ by the non-radiative ISC transition and radiative transition. The electron spin in the m_s= ±1 sublevels of excited state ³E has a significantly higher probability to undergo non-radiative ISC toward the singlet system and transit from excited state ³E to m_s= 0 of ground state ³A₂, which leads to being used to realize the initialization and readout of the NV center. The initialized electron spin in the ground state ³A₂ can be manipulated by microwave (MW) and is measured by optically detected magnetic resonance (ODMR).

NV centers are along one of [1,1,1] orientations of the diamond lattice, and the Zeeman effect is sensitive to the projection of magnetic fields along the NV’s orientation. Therefore, under an external magnetic field, electron spin levels of an NV ensemble in a single crystal diamond may be non-degenerately divided into eight [8]. This property makes it an ideal vector magnetic field measurement carrier. In addition, these spin levels will also shift with the thermal expansion of the lattice, which allows the NV center to measure temperature. Based on the response of the energy levels of the NV center to the external environment, a large number of quantum measurement schemes have been proposed and realized [1,9].

Similar to most ODMR experimental setups, as shown in Fig. 1(c), the home-built confocal microscopy with a single photon detector is used for green light excitation (532 nm laser) and fluorescence detection (640 nm to 750 nm). During microwave scanning, the fluorescence intensity is weakened when the microwave resonates with the electron spin transition. This technique is called continuous wave optically detected magnetic resonance (CW-ODMR). Using microwave switches and acousto-optic modulators (AOM), we can generate microwave and light pulses on this setup to achieve pulsed ODMR, Rabi, and other pulsed experiments.

For NV-based quantum sensing schemes, the microwave is the key to manipulating the spins of NV centers. The magnetic field component in the microwave field will couple with multiple NV electron spins to realize the control of the quantum states. To improve microwave coupling and manipulation efficiency, a variety of microwave antennas are designed to adapt to specific environments and samples [10–17]. Since our target is to manipulate ensembles or an array of NV centers within a certain area, the microwave device is also required to generate a relatively uniform microwave field on the millimeter-scale area in addition to the radiation requirements of the single spin experiment. In single NV center experiments, Omega ring and microstrip line [18–21] waveguides are widely used due to the low requirements on microwave range and magnetic field uniformity. A microstrip line can provide an extremely high magnetic field if the NV center is close enough, but the intensity decreases with the square of the distance. Omega ring can generate a uniform field inside the ring, but the ring must be small to get a strong field. The other advantage of the Omega ring and microstrip line is their available bandwidth as broad as 1 GHz or more.

However, the susceptibilities of electron spin energy levels with temperature and magnetic field are -74 kHz/K and 2.8 MHz/Gauss respectively [22–27], and the target we want to measure may only have a temperature change of tens of K or a magnetic field change of several Gauss. In this case, the measured energy level shift is only a few MHz usually. The inhomogeneous broadening linewidth of the NV ensemble electron spins in nanodiamonds is about 10 MHz, so we may only need an antenna with a bandwidth of 30 to 40 MHz in the experiment. Considering the randomness of nanodiamond lattice orientation, the uniformity of the magnetic field is less important, keeping it in the same order of magnitude is enough. Therefore, we put bandwidth on the back burner and adopted a double 89 split-ring resonator (DSRR) as the microwave antenna [28–30], which can effectively couple 90 the microwave magnetic field to many NV centers with different orientations in the 91 millimeter-scale area. This kind of structure has been well-developed and used in 92 metamaterials [31–33], microwave filters [34], and antennas. Generally speaking, the antenna 93 bandwidth is defined as S11 ≤ -10 dB [35]. However, due to the shortcoming of DSRR that the bandwidth is very narrow, only about 30 MHz, which is only enough to cover one or two dips on ODMR spectrum. For example, in the quantum sensing based on NV ensembles in our laboratory, the frequency range of four levels (different orientations of the same electron spin) is about 100 MHz, which is hard to cover by one DSRR antenna. So, in this article, we studied a way to tune the resonance point of the double-split ring by changing the capacitance between the outer ring and the ground. The frequency range of the DSRR can be shifted to meet the experimental requirements and cover the spin levels we care about by a piece of a moveable copper sheet, which will significantly benefit our quantum sensing experiment. In this article, we did simulations (CST Microwave Studio) and experiments (network analyzer, Rohde & Schwarz-ZNA26) on the tunable microwave device base on the analysis of the original DSRR, and the effect of our design is verified by the ODMR and Rabi experiment on nanodiamonds with an NV ensemble. This antenna is promising in the fabrication of integrated quantum detectors based on the ensemble of NV centers, nanodiamonds, or an array of nanodiamonds [36].

2. Model design and analysis

The design of DSRR starts from the closed metal ring whose electrical size at the first resonance point does not satisfy the subwavelength condition [36]. The resonant frequency of the entire unit can be gradually reduced by opening a slit on the edge of the simply closed metal ring, which is called the split resonant ring, and the structure with an approximately optimal electrical size that satisfies the subwavelength condition can be obtained. The resonant frequency can be further lowered by adding an inner ring to the closed metal ring and opening a slot at the symmetrical position of the second ring. The electromagnetic coupling between the inner and outer rings will also reduce the resonance frequency by bringing a capacitance. Then the structure with the lowest resonant frequency at this time is the DSRR unit. And the DSRR is usually etched on the dielectric substrate which is also helpful in reducing the electrical size. The resonance of DSRR mainly depends on the resonance of the outer ring because most of the energy of the electric field is concentrated in the outer ring [14,37]. The optimized DSRR structure satisfies the sub-wavelength condition and has a high symmetry. It has been so perfect and is widely used in the microwave and radio frequency area. As shown in Fig. 2(a), w is the width of the rings and wire, r₁ represents the inner radius of the inner ring, r₂ represents the inner radius of the outer ring, r₀= (r₁+ r₂+ w)/2 is the mean radius between two rings, d₀ is the splits, and d₂ is the distance between the microstrip line and outer ring. These parameters determine the resonance performance of the DSRR. The equivalent circuit model of DSRR is an LC resonant circuit, as shown in Fig. 2. (b) [37], C₁ represents the equivalent capacitance of the ring, C₁= 2πr₀C₀, C₀ represents the capacitance per unit length, C_S is the series capacitance, C_S= C₁/4, L_S represents the equivalent inductance of the ring, then the resonant frequency of the entire resonant tank is as follows:

 

Fig. 2. (a) Model design of the DSRR. (b) Equivalent Circuit of the DSRR. (c) Simulated and experimental S-parameters of loading sample and unloaded (without sample) resonator. (d) The simulated and experimental resonant frequency of these DSRRs with different radii of rings.

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Based on this simplified model, we did a series of simulations and experiment to optimize the structure. Metal copper is used as the material of the ring, and the intermediate dielectric substrate is Rogers 6010, whose dielectric constant is 10.2. The backside is covered with copper as the ground wire, and the holes at the corners are connected to the ground on the backside. The microwave is provided through a microstrip line that meets the impedance matching of 50 Ohms and is coupled to the double ring through the gap. The gap between the microstrip line and the ring determines the S₁₁ by impendence matching. Theoretically, the smaller the gap, the higher the coupled microwave power. The optimized size is 0.05 mm, but because the precision of the copper sinking process is hard to achieve, the minimum size that can be well achieved by the process is selected to be 0.1 mm. The S₁₁ of simulation and experimental measurement with or without samples is shown in Fig. 2(c). And the deviation between simulation and experimental measurement is less than 8 MHz. In addition, compared to unloaded, a small piece of glass (0.1 mm) with nanodiamonds was glued on the structure, which cause a change in permittivity and shifted the resonant frequency of the structure in the experimental measurement loading sample. So we can find some sample-related deviation from the experiment with simulation.

The quality factor Q expresses the magnitude of the resonant frequency relative to the bandwidth, and also means the ratio of the electromagnetic energy in the resonator to the energy lost in one cycle. A larger Q results in a larger magnetic field amplitude at the center of the resonator, but usually at the expense of narrow bandwidth. The resonant frequency of the resonant ring can be tuned within a range by changing the mean radius r₀. In the simulation and experiment, we keep widths and gaps constant, then tune r₁ and r₂ together. The resonant frequency corresponding to r₁ (r₁= r₂-1.2 mm) is plotted in Fig. 2(d). As the radius increases, the frequency gradually decreases, so we chose three radius sizes with frequencies close to 2.87 GHz for casting, to test the performance experimentally. Finally, after the network analyzer test, the optimized inner diameter of the size is 1.1 mm, that is, the double split-ring structure with an average radius of 2.2 mm, resonant frequency f = 2.878 GHz, S₁₁=-31 dB, bandwidth is 30 MHz, Q = 96, it can be estimated that 99% of the microwave power at the center frequency can be directly coupled to the resonator.

The energy stored in the resonator is (1-S₁₁²)QP_in/w_r, P_in represents the input microwave power delivered to the device, and w_r represents the resonant frequency. The magnetic field energy is the strongest at the resonant point [33].

Optimizing the structural parameters of the antenna can tune the resonant frequency to meet the requirement of different experiments. But repeated antenna replacements are time-consuming and risky to damage samples, especially nanodiamond arrays and fragile ultrathin diamonds. Therefore, as theoretically suggested in the literature [33], we choose to paste a small piece of conductor on the surface of a microwave antenna to fine-tune the resonant frequency. As discussed earlier, a small piece of copper sheet is used to slightly change the capacitance of the DSRR structure, lowering the resonant frequency of the antenna.

3. Simulation and experiment of the tunable radiation structure

Frequency tuning using stubs and strips is a common practice in microwave circuits [38]. In general, stub tuning is simply the process of adding a length of the transmission line to the existing length in either series or shunt circuit configuration to match the line to the load. This is achieved by placing a specific length of stub a specific distance away from the load. stub tuners are simple to implement and inexpensive to make, they only require the same material as making transmission lines. The stub is designed to counteract the reactance component of the load to be matched, and special stubs help increase the bandwidth of the circuit. In our case, the copper tape affects the capacitive coupling between the outer ring and the ground plane (backside of the PCB) and hence shifts the resonance frequency.

The copper sheet not only changes the area of the outer ring but also changes the shape of the outer ring. The definition of the copper sheet positions on the microwave device is shown in Fig. 3(a). The materials are the same as mentioned in Sec. 2, and parameters of the structure are listed in Table 1. To study the effect of the shape of the outer ring on the resonant frequency, we simulated and measured the resonant frequency of the antenna with a 1.5mm × 1.5 mm copper sheet at different positions, shown in Fig. 3(b). During the experiment, we keep the coincidence distance d_c= 0.2 mm to balance the stability and tuning range of the antenna. Smaller coincidence distance d_c provides a larger tuning range if the copper sheet contacts the outer ring well. The relationships between coincidence distance d_c and resonant frequency at the different positions are simulated in Supplement 1.

Table 1. Optimized parameters of the radiation structure

View Table

 

Fig. 3. (a) Position of the copper sheet in tuning the resonance frequency. The overlapping of the copper sheet and the outer ring is represented by the coincidence distance d_c. In the experiment of resonance turning, we tried to keep d_c= 0.2 mm (±0.1 mm). (b) simulation and measurement of the tuned resonance frequency of the structure by moving the copper sheet. The data of S₁₁ is plotted in Fig. 2(c). (c) Resonant Rabi frequency for different microwave frequencies on the same spin level with and without tuning copper sheet on the antenna. The spin level is scanned by a magnetic field. (d) Resonant Rabi frequency of the fixed spin levels with the tuning of the microwave antenna.

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For a specific energy level of NV centers in a nanodiamond, the transition magnetic moment $\vec{\mu }$ is fixed, so the Rabi frequency can be expressed as:

The direction of the alternating magnetic field generated by the antenna is perpendicular to the plane of the device, which is also fixed. So, the resonant Rabi frequency of the specific transition is proportional to the amplitude of the alternating magnetic field $\vec{B}$. In this case, we can use the Rabi frequency of a specific energy level to characterize the performance of the antenna.

In order to reflect the amplitude of the magnetic field from the antenna, we did Rabi oscillation before and after tuning the antenna with the copper sheet. In the two cases, we choose the same energy level of NV centers in the same nanodiamond, fine-tuning the external magnetic field to scan the frequency of the electron spin levels and measure the resonant Rabi frequency at each frequency. Then the Rabi frequency, shown in Fig. 3(c), can reflect the amplitude of the AC magnetic field from the microwave antenna. The variations of Rabi frequencies show the full width at half maximum of the antenna as about 40 MHz. Tuning the antenna frequency with a copper sheet, it is observed that the peak of Rabi frequencies shifted clearly, which is consistent with the previous simulation and network analyzer test results.

Under a fixed static magnetic field (fixed energy level), the antenna frequency is tuned by changing the position of the copper sheet. Within the tunable range of about 80 MHz, the Rabi frequency changes by about two times, which means the magnetic field amplitude at the transition frequency is doubled by tuning the antenna by the copper sheet. We also did the Rabi test with two different microwave powers by scan the resonance frequency of a NV level with fine-tuning the external static magnetic field, and the results are consistent with our expectations (Fig. 3(d)). The Rabi frequency changed from 6 MHz to 13.3 MHz when the input microwave power is 10W. Since the lattice orientation of nanodiamonds is completely random, we did not calculate the magnetic field precisely. The Rabi frequency can be further increased by using bulk diamonds with the [1,1,1] direction perpendicular to the microwave device.

4. ODMR with the tunable double split-ring resonator

In quantum sensing based on NV centers, the amplitude of the microwave magnetic field at the sample directly affects the signal contrast, which determines the measurement sensitivity. We used ODMR and Rabi experiments to test and verify the performance of the microwave device.

Using CW-ODMR, as shown in Fig. 4(a), we can find that the four pairs energy levels of the NV ensembles in different orientations are separated by the Zeeman effect from the magnetic field, and the levels located near the resonant frequency of the antenna (${f_r}$=2878 MHz) has better signal contrast. The eight inversed peaks also mean the nanodiamond is an ideal single crystal. After the 1.5mm × 1.5 mm copper sheet is pasted at 30° outside the outer ring to adjust the microwave device, as shown in Fig. 3(a), the resonance frequency of the antenna is tuned down to 2830 MHz. With this copper sheet, the resonant frequency of the antenna can completely cover one of the electron spin energy levels, and at the same time, it is relatively close to other energy levels with the spin m_s= -1at the lower frequency side. The CW-ODMR result with the tuned antenna is plotted in Fig. 4(b), the contrasts of the four levels at left (m_s= -1) are much better than the right four (m_s=+1). Compared with the ODMR spectrum of the untuned antenna in Fig. 4(a), the contrast of the three energy levels on the left is significantly improved, and the fourth energy level at 2844 MHz doesn’t change much because the detuning magnitude is close to the previous one. In this case, by measuring the one-sided CW-ODMR spectrum at a specific temperature, we have been able to carry out the vector measurement of a small-scale variation magnetic field [39,40].

 

Fig. 4. (a) CW-ODMR spectrum of a nanodiamond with a magnetic field when the antenna is untuned, ${f_r}$=2878 MHz. The red crive is the fitted line. (b) CW-ODMR spectrum when the resonance of the antenna ${f_r}$ is tuned to 2830 MHz by the copper sheet. The red curve is the fitted line. (c) Rabi oscillation of the third spin level (2819 MHz) with an untuned antenna, Ω_untuned= 2π×7.1 MHz. (d) Rabi oscillation of the same spin level close to the resonance of the antenna tuned by the copper sheet, Ω_tuned= 2π×13.3 MHz.

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5. Conclusion

In this work, we analyzed and optimized the classical DSRR antenna for a millimeter-scale area first. based on the DSRR structure, we tune the antenna via a moveable copper sheet. Compared with the work of K. Bayat et al. [33], the bandwidth is increased. Besides simulation and experiment, the tunable antenna is applied to an NV ensemble in a nanodiamond. The performance of the microwave device is verified by CW-ODMR and Rabi on the NV ensemble. In more than 50 MHz range, we can tune the antenna to cover the resonance of the level and reach 13.3 MHz Rabi frequency at 10 W microwave input. For different experimental purposes and environments, when the spin level is shifted by a magnetic field, we can quickly change the resonant frequency of the antenna without moving the sample, which provides great convenience for many experiments, for example the detection of nano magnetic particles or two-dimensional magnetic materials. This tunable antenna has considerable application prospects in nanodiamond arrays and wide-field experiments of NV ensembles.