Inertial measurement with solid-state spins of nitrogen-vacancy center in diamond

ABSTRACT The nitrogen-vacancy (NV) center is one of the major platforms in the evolving field of quantum technologies. The inertial surveying technology based on NV centers in diamond is a developing field with both scientific and technological importance. Quantum measurement using the solid-state spin of the NV center has demonstrated potential in both high-precision and small-volume low-cost devices. In terms of rotation measurement, the optically detected magnetic resonance has provided a perspective of the rotation measurement mechanism via the solid-state spin of the NV center. A new type of gyroscope based on the solid-state spin in diamond according to the theory has attracted considerable attention. In addition, combined with the ingenious quantum mechanics manipulation and coupling mechanism, acceleration measurement can be achieved through an efficient quantum detection technology of the NV center. This review summarizes the recent research progress in diamond-based inertial measurement, including sensitivity optimization methods for inertial measurement systems based on the NV center. Graphical abstract


Introduction
High-precision inertial measurement plays a paramount important role in both fundamental research and applications [1]. Rotation and acceleration measurements in an inertial frame are vital for modern advanced navigation systems [2]. In recent years, quantum physics-based technologies have played an important role in high-precision measurements, and their accuracy has improved considerably compared with that of classical systems. The nitrogen-vacancy (NV) center ( Figure 1) has recently emerged as a powerful quantum system that combines the good coherence properties and control techniques of atomic systems with the small size and fabrication capabilities of solid-state devices [3]. The advantages of the NV center quantum technology are reflected in various fields, such as quantum sensing [4][5][6][7][8][9][10][11][12][13][14] and quantum computing [15][16][17]. After their rapid development in the past decades, NV center quantum inertial measurement systems have been recognized as promising candidates for inertial measurement systems [18,19].
With regard to rotation measurement, Wood et al. [21][22][23] reported a measurement method using a single NV-spin phase shift arising directly from physical rotation. The rotation is obtained by measuring the phase difference between a microwave driving field and a rotating two-energy level electron spin system, and the phase shift can accumulate nonlinear behaviors over time. The carrier's rotation information can be obtained by measuring the accumulation of the quantum geometric phase. This solidstate spin inertial sensor based on NV center is competitive with conventional gyroscopes in terms of sensitivity, cost, and size. For acceleration measurements, a gravimeter based on the spin superposition of the NV center can achieve a higher precision compared with that of the atomic interference gravimeter, because of the higher mass of the mechanical resonator in the former system [24]. In addition, compared to conventional atomic inertial sensors, inertial sensors based on the solid-state spin of NV center have faster start time. Moreover, because they can operate at normal temperature and have more relax requirements of working temperature, an air chamber is not required, so they have the potential for future miniaturization.
This review focuses on inertial measurement using solid-state spins of the NV Center to guide future research efforts. First, the concept and background of the inertial measurement with solid-state spins of the NV center in diamond are introduced in Section 1. Then, the development of the spinbased rotation inertial measurement is presented in Section 2, with Section 2.1 and Section 2.2 describing the development of the spin-based rotation measurement and spin-based acceleration measurement, respectively. Section 3 discusses the sensing mechanisms of spin-based rotation inertial measurements, namely, the solid-state spin rotation measurement (Section 3.1) and solid-state spin acceleration measurement (Section 3.2). Subsequently, the methods for improving the sensitivity of gyroscopes (Section 4.1) and accelerometers (Section 4.2) are reviewed in Section 4. Finally, the summary and outlook are presented in Section 5.

Spin-based rotation measurement
The NV center is an atomic-scale defect point color center, which has good optical [25] and spin properties [26]. The NV center can be optically addressed, and the spin state of the NV center can be observed optically by the fluorescence [25]. The long coherence time of the NV − electronic spin [26,27] at both room temperature and cryogenic condition makes it possible to realize high precise sensing utilizing the quantum resources at ambient conditions. Physical quantities that coupled with the NV spin can be sensed with high precision, including magnetic field [28][29][30], electric field [31], force/strain field [32] and thermometry [33][34][35]. The best sensitivity achieves 0.43 pT � Hz À 1=2 [36], 200 V � cm À 1 Hz À 1=2 [31] and 76 mK � Hz À 1=2 [37], respectively.
Recently, the NV-based quantum sensing has been used for inertial measurements. In 2012, Maclaurin et al. proposed a scheme for detecting the phase induced by the rotation of a system with a single NV center [38], Ledbetter et al. and Ajoy and Cappellaro et.al. proposed the NV-based gyroscopes [39,40]. The sensitivity of the proposed scheme [39] without and with the use of 14 N nuclear Figure 1. Energy level diagram of the NV center spins in diamond. The inset shows the crystal structure of the NV center in diamond. Two adjacent carbon atoms (red spheres) are substituted by a nitrogen atom (brown sphere) and a vacancy (green sphere). With the phonon sideband and singlet states, the electron spin can be initialized to m s ¼ 0 state using a 532 nm laser. The electron spin-state population is detected by monitoring the 600-800 nm fluorescence signal. The zero phonon line is at 637 nm and the zero-field splitting of the triplet ground state is at 2.87 GHz [20]. spin is 5 � 10 À 3 rads À 1 = ffi ffi ffi ffi ffi ffi Hz p and 9 � 10 À 5 rads À 1 = ffi ffi ffi ffi ffi ffi Hz p . In 2017 and 2018, Maclaurin et al. further explored the effect of rotation-induced pseudo-fields on sample spins, using a fast-spinning diamond containing either an ensemble of NV centers or a single NV center mounted on a rotor, providing a technical reference for the development of accurate NV center-based spin measurements [21,23]. In 2021, utilizing the 14 N nuclear spin within the corresponding NV center and the double-quantum pulse protocol, an integral rotating scheme was designed and demonstrated in the laboratory. The rotation sensitivity and zero bias stability are 4:7 � = ffi ffi s p and 0:4 � =s respectively [41]. The experimental progress stimulated the practicalization of gyroscopes. In 2016, Zhang et al. in Beihang University put forward a structure of an NV-based gyroscope in which a bilateral layout design of coils was adopted to guarantee the uniformity of the distribution of the MW and the radio frequency (RF) field [1]. In 2018, Song et al. in China Academy of space technology showed a 13 C-NV rotation sensing system, which is analogous to the scheme mentioned above [42]. In 2021, Soshenko et al. at Lebedev Physical Institute obtained the dynamic phase difference induced by the macroscopic rotation of the diamond via the double-quantum Ramsey spectroscopy. The validity of their measurement setup based on the 14 N nuclear spin ensemble under sub-Hz rotation was verified by a MEMS gyroscope, which further promotes the miniaturization and integration of diamond gyroscope [43].

Spin-based acceleration measurement
Acceleration measurements based on interferometry have evolved from optics to matter waves [44,45]. Matter-wave interferences have been demonstrated with neutrons [46,47], atoms [48,49], and electrons. The relatively shorter de Broglie wavelength of matter waves can increase the sensitivity of phase shift measurements [50]. Diamond particles levitating in optical tweezers act as nanoscale resonators, facilitating the creation of macroscopic superposition states, which is a remarkable feature of quantum mechanics [51,52]. Inspired by this, a superposition generation scheme of a macroscopic mechanical resonator hybrid with an NV center was proposed to improve the interferometric precision of the gravitationally induced phase shift [24,53]. Due to their broad application prospects, techniques for coupling mechanical resonators with the spin of the NV center have been developed [54,55]. A hybrid system with an NV center [56] or a magnetic tip [57] attached to the end of a cantilever, with the help of a magnetic field gradient, can sense the motion of the mechanical cantilever using a single [58] or multiple NV centers [59,60]. With the development of optomechanics [61][62][63], an optical force levitated diamond containing an NV center has emerged as a powerful scheme for realizing a high-quality factor [64,65]. Figure 2 shows the sensing principle of the NV-based rotation measurement. The measured precession frequency ω 0 , which is completely determined by the external field in a non-rotating frame, changes with different angular velocities. The rotation information can be calculated from the acquired signal and a measurable external field.

Spin-based rotation measurement
where ω 0 is the measured frequency, ω L ¼ γB is the Larmor precession frequency, and Ω is the rotation angular velocity [66]. The quantum phase accumulated on the spin states is then expressed as: where t is the rotation time and Ω is the uniform angular velocity of the carrier. By measuring the spin states, the angular velocity of the carrier is then obtained. Notably, the influences of external parameters, such as magnetic fields, can be examined independently during the operation of the carrier [39,40]. The carrier rotation information can be obtained by measuring the accumulation of the quantum phase. In a non-rotating state, the spin magnetic moment processes in the direction of the external magnetic field, and the precession frequency is proportional to the applied magnetic field (black arrow). In a rotating state, the precession frequency increases by a term proportional to the rotational angular velocity. The 532 nm laser is used for spin polarization, the microwave source is used for spin manipulation, and the detected fluorescence contains rotation information.
Consider a solid-state NV spin system with an electron spin S ¼ 1 and 14 N nuclear spin I ¼ 1. The ground-state Hamiltonian of the NV − center electron spin and the adjacent 14 N nuclear spin is [67]: where D ¼ 2870MHz is the zero-field splitting; γ n and γ e are the nuclear and electron gyromagnetic ratios ; respectively; B is the external magnetic field; Q ¼ 4:95MHz is the intrinsic quadrupole interaction; A k � 2:2MHz and A ? � 2:1MHz is the hyperfine constant. An external magnetic field can realize the ground-state Zeeman splitting of the NV electronic spin, thus forming a two-level system. Microwave pulses are required to achieve spinstate flipping and manipulation. An off-resonance laser (ex. 532 nm) is chosen to help electron spin polarization [68][69][70][71]. The m s ¼ 0 state will decay directly to the ground state and m s ¼ �1 state will shelve into a nonradiative metastable state preferentially. The fluorescence of m s ¼ �1 state is weaker, and thus the spin state can be readout by the photon detector. Thus, the spin-state population can be determined via fluorescence detection owing to the phonon sideband and non-radiative transition of the metastable state [72]. The long coherence time of NV center (1.8 ms at room temperature [26] and ~ 1 s at liquid nitrogen temperature [27]) allows complicated control methods [73] and high sensitive measurements of the physics quantities coupled to the NV center [74]. Although the NV center has good optical and spin properties, the coherence time is limited [29,75] and the readout efficiency [29,76] depends on the photonic structure and operation temperature, especially for the NV center ensemble. The adjacent 14 N nuclear spin can be used as a sensor, which has a longer relaxation time and can be readout with high efficiency by quantum logic [11,77]. The 14 N nuclear spin can be conveniently polarized, controlled and readout by the NV center. Both excited-state level anticrossing [78,79], dynamic nuclear polarizing techniques [80][81][82][83][84] is applied to polarize the 14 N nuclear spin. Due to the development of the NV-based quantum information, the high-fidelity controls of the adjacent 14 N nuclear spin and different quantum gates are realized in Ref [16,85]. The nuclear spin state can be readout with high fidelity [86,87] enhanced by the quantum non-demolition detection [86].
Note that the natural quantization axis of the NV is set by the host diamond crystal orientation, and rotating the crystal can effectively induce rotation of the qubit [22]. The large quantum superposition of nanomechanical resonator can be achieved with the help of NV center [57,64,88,89]. To improve the sensitivity of the mechanical measurement, diamonds hosting NV centers have been trapped both in liquid [89,90] and under vacuum [91][92][93][94] for spin readout, which are important steps towards coupling spins to macroscopic particle motion. Ramsey interferometry is a fundamental method used in atomic physics and quantum measurements, one example of which is the preparation of the ground state of motion of macroscopic oscillators [58,65,95,96]. Notably, the influences of external parameters, such as magnetic fields, can be examined independently during the operation of the carrier [39,40]. The carrier rotation information can be obtained by measuring the accumulation of the quantum geometric phase (Figure 3). Consider an adiabatic angular velocity measurement system whose Hamiltonian depends on the magnetic field and evolves in cycles. The research presented in [97] is based on the coupling characteristics of the Hamiltonian control of the NV center spin system and the quantum geometric phase in the Bloch space ( Figure 3). Depending on the angle between the intrinsic axis of the NV center, the microwave field, and the axis of rotation, this rotation-induced phase can accumulate nonlinearly and be detected using a spin-echo scheme.
In accordance with the above-mentioned sensing mechanism, more scholars have engaged in research on gyroscopes based on the NV center. Maclaurin [38] proposed a method using a single NV center in a spinning diamond to measure the geometric phase accumulated by the electron spin. The Ramsey pulse sequence method and spin-echo pulse sequence method were suggested for geometric phase measurement. As illustrated in Figure 4, Wood et al. [21][22][23] utilized a rotor to spin the NV center in a magnetic field. The macroscopic rotation information can be obtained by observing Figure 3. Schematic of rotation measurement. In the external magnetic field, the physical rotation of the carrier causes a change in the spin precession frequency, and the angular velocity information can be obtained via the quantum geometric phase [38,101]. the phase difference of the spin states. In the experiment of Soshenko [43], rotation of the platform is measured using the nitrogen nuclear spins of an ensemble of NV centers; in particular, no external reference is used.

Spin-based acceleration measurement
Spin-oscillator coupling schemes have been proposed as accelerometers, whereby a spatial superposition is created through the interaction of an embedded two-level system such as an NV center with an external magnetic field gradient [54,98]. A hybrid system coupling a nanomechanical resonator to the electron spin of an NV center can achieve high-precision gravity measurement via matter-wave interferometry. For acceleration applications, large objects provide significant advantages over small objects. A 10 À 10 relative detection accuracy for gravitational acceleration measurement has been estimated in [24], and the measured relative phase difference is three orders of magnitude larger than that using the atom interference scheme. The proposed hybrid schemes are exhibited in Figure 5.
As shown in Figure 5(a), the center of mass (CoM) motion of the nanodiamond, which is trapped by a light force tweezer in a vacuum environment, couples to the spin of the built-in NV center of the diamond via a magnetic field gradient. A magnetic tip is placed in the vicinity of the diamond to induce the required magnetic field gradient along the z direction. Suppose that the trapping frequency ω z is sufficiently small compared NV centers are optically prepared and addressed by a scanning confocal microscope. MW (produced by the micro coil), magnetic field, and laser are coupled to the NV center. The fluorescence signal is collected by the detector, which contains the angular velocity information [22].
with ω x and ω y to neglect the motion effect along the x and y directions. Considering the Earth's gravitational field, the corresponding Hamiltonian [24] is is the spin-motion coupling strength, and Δλ ¼ is the gravity-induced displacement, g NV ¼ 2 is the electron Landé g-factor, m is the mass of the nanoparticle, μ B is the Bohr magneton, and B z is the magnetic field gradient along the z acceleration-sensitive direction. c and c y are annihilate operator and produce operator, respectively. It can be seen that the CoM motion of the oscillator varies with different eigenvalues of S z . Ramsey interferometry is utilized to detect the gravitational accelerationinduced phase shift of the resonator-spin hybrid system. By applying the first microwave π 2 pulse (t p duration) after initialization of the solid-spin state, superposition of j þ 1i and j À 1i states with the same amplitude is produced. The equilibrium position separation of the two states due to the spin-dependent acceleration is � Δz ¼ � g NV μ B B z 2mω z 2 . After recombination of the two classic paths at t 0 ¼ 2π ω z , the states become j þ 1i þ e iΔϕ j À 1i, and a relative phase shift [24] appears between the two states.  [24,65,98]. (c) The free-flight acceleration sensing scheme using a Ramsey interferometer facilitates the possibility of realizing a larger superposition. The spin superposition creation, the spin states flipping (at times t 1 and t 2 ) and wave packet mergence (at time t 3 ) are controlled by microwave pulses [99,100].
The phase shift is proportional to the acceleration and can be determined through the application of the second microwave pulse to read the population of the spin state . Another spin-oscillator hybrid system is depicted in Figure 5(b), where the magnetic tip is fixed on a cantilever resonator to couple the mechanical vibration mode with the NV center in diamond. It creates a magnetic gradient near the electronic spin of the NV center. Similar to the aforementioned scheme, a laser is used to initialize and measure the spin states, and the Ramsey sequence is adopted to manipulate and probe the spin states.
The untrapped nanodiamond interferometric scheme [51,99,100] ( Figure  5(c) is proposed to measure the mass-independent dynamic phase induced by the spin-dependent gravitational potentials. The magnetic field gradient is still used to couple the CoM motion and spin states of the object. The spin superposition creation and spin state flipping (at times t 1 and t 2 ) are controlled by microwave pulses. After splitting, acceleration, deceleration, and mergence of the wave packet, a phase factor [99] proportional to gravity can be acquired via fluorescence detection. 3 3 cosðθÞ (6) where g NV , μ B , and B z are as previously defined, θ is the angle between the applied magnetic field B z and the direction of acceleration a, and t 3 is the total free-fall time. Compared with the levitated method, this free-flight scheme facilitates the possibility of realizing a larger superposition because of its mass independence [51].

Sensitivity optimization of spin-based rotation measurement
When the magnetic field B is perturbated, its correction factor is [29]: where E 1 , E 2 and E 3 are the first-, second-and third-level mini-disturbance items, respectively. The error σ brought by the external magnetic field interference is as follows [38]: Although a single NV center in diamond has a long coherence time [27], the measurement sensitivity is limited by the single-photon signal. In addition, it is difficult to integrate a single-photon counting system into a compact diamond gyroscope. Therefore, spin ensembles have been used to replace the single NV center in diamond. According to the fluorescence signal, the sensitivity limit of the rotation measurement is [40]: where N is the number of the sensing spins, T 2 is the coherence time, C is the fluorescence collection efficiency, t d is the total of the preparation time t pol and the readout dead time t ro . The comparation in sensitivity and size of some representative gyroscopes and the NV-based gyroscope [39,40,[101][102][103][104][105][106][107][108][109][110] are shown in Table  1, including ring laser gyroscope (RLG), fiber optical gyroscope (FOG), spin relaxation free (SERF) gyroscope, atom interferometer gyroscope (AIG), micro-electro-mechanical systems (MEMS) gyroscope, and NV-based gyroscope. As a new type of atomic gyroscopes, the NV-based gyroscope has the potential for medium-sensitivity and miniaturization in the future.
Efforts have been made to enhance the sensitivity of gyroscopes. One of the keys to reach this goal is to increase the coherence of the NV center, which is a challenge when the diamond is in an optical trap and the spin angle becomes unstable. This will offer prospects for experiments, such as matter-wave interferometry [89,102], quantum gravity sensing [111], strong coupling [56,57], and cat state preparation [112], which rely on the ability to maintain a long coherence time for spin-state superpositions in a trapped object. The 14 N nuclear spin has a much longer coherence time than the electron spin, so the fundamental sensitivity can be significantly improved by using the 14 N nuclear spin. Meanwhile, increasing the NV center density also improves the sensitivity. Considering these facts, Ledbetter et al. proposed the idea of developing a gyroscope based on nuclear spin ensembles in diamond [39]. At the same time, in the proposal of Cappellaro [40], the nitrogen nuclear spins controlled by the NV center are used as a 'threeaxis diamond gyroscope'. The relative phase of the spin state is measured and the carrier rotation is estimated from it with a theoretical sensitivity of 5 � 10 À 4� = s � ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi Hz � mm 3 p � � . Optimized spin coherent control can also increase the sensitivity. The electronic spin resonance width can be determined mainly by coupling to the diamond strain and coupling to impurities [113,114]. Delord [115,116] applied a magnetic field that lifts the degeneracy between the state jm s ¼ �1. The Rabi envelope decay is a characteristic of the environmental noise spectrum [117]. However, the observed damping time does not provide direct access to timeT 2 . In fact, the decay time can be longer than T 2 and is determined largely by the employed microwave power [118][119][120]. Jaskula [121] used NV centers to sense local magnetic field fluctuations, estimated with a stable rotation measurement of several days, allowing navigation with loose or no requirement for geolocalization.

Sensitivity optimization of acceleration measurement
Preparing the quantum superpositions of a large object is challenging because of the interaction between the system and the environment [122]. To achieve this goal, many methods have been investigated. In 2010, Romero-Isart et al. [52] proposed a method that used an optical cavity levitating a nano-dielectric object. The resonance frequency dependence on the position of the trapped object yields an optomechanical coupling. Because the object was not attached to other mechanical pieces, this optomechanical scheme could avoid the main source of heating, thus facilitating the realization of ground-state cooling. Romero-Isart et al. [122] further presented a cavity quantum optomechanical scheme to implement a matterwave interference experiment. This scheme can be utilized to realize spatially separated superposition preparation of a large object and test of quantum mechanics, including wave function collapse models. In 2013, Yin et al. [64] proposed a method for generating and detecting quantum superposition states of the CoM oscillation of a light force-trapped diamond based on magnetic field gradient-induced spin-optomechanical coupling. Their scheme entails the possibility of creating large quantum superpositions under feasible conditions. In the same year, Bose et al. [98] presented a Ramsey interferometry scheme that satisfies five requirements (i.e. modest cooling, no cavities, no ensembles, no spatially resolved measurements, and controllable phase). The motion of a diamond bead, which is trapped by an optical tweezer, couples to the spin of the NV center via a static magnetic field gradient. The proposed method can measure the gravitational acceleration-induced relative phase difference of the spin state. An appropriate Ramsey scheme can attain good robustness against initial conditions, such as thermal fluctuations, yet places great demands on magnetic field control [63].
In terms of acceleration measurement, considering the interferometric accelerometers, matter-wave interferometry may replace light interferometry to obtain an improved accuracy due to the shorter de Broglie wavelength of the former. The gravimeter based on a diamond resonator has higher precision than the cold atom interferometry gravimeter with suitable parameter settings, which can be explained by the mass difference. For the optical spin-oscillator sensing system and the cantilever spin-oscillator system mentioned before, the resonators are 10 10 and 10 16 times more massive than sodium atoms, respectively [24]. Eq. (5) can be rewritten as Δϕ ¼ 16πmgΔz � hω z . Given a fixed precision target and a fixed integration time, the size of the accelerometer can be considered to be inversely proportional to the mass of the resonator [24]. Therefore, the NV center accelerometer has a higher miniaturization potential and is easier to integrate into a chip.
In the future, the realization and performance improvement of the accelerometer based on the NV center can be promoted in many aspects. First, as a basic system of light force accelerometers, optical tweezers have been widely investigated [62,123]. As a promising tool for creating macroscopic superposition states of the NV center spin, levitated optomechanics needs further studies to realize precise position detection of the nanodiamond [124], reliable cooling control of the spin state [125], and stable fluorescence collection [63,93]. Second, the sensitivity can be improved by using better fluorescence detection techniques to achieve shot noise performance [29,126]. The efficiency of photoconversion can be improved with the total internal reflection light guiding method [127]. Additionally, the material properties [128] can be optimized toward longer dephasing and coherence times of spin states. Finally, to improve the long-term performance of the sensor, further exploration could focus on the sensing and stabilization of temperature and magnetic fields through passive and active methods. Schemes that implement magnetic field feedback control [121,129] would result in better robustness of the device against field fluctuations in the environment.

Summary and outlook
The application of inertial surveying technology based on NV centers in diamond is a rapidly developing field with both scientific and technological importances. Over the past decades, quantum measurement using the solid-state spin of the NV center has demonstrated its potential in both highprecision applications and small-volume low-cost devices. In terms of rotation measurement, the optically detected magnetic resonance has provided a perspective on the rotation measurement mechanism via the solid-state spin of the NV center. The theory has been used to describe a gyroscope based on the solid-state spin in a diamond. In addition, with the ingenious 'spinoscillator' coupling mechanism, acceleration measurement can be achieved via superposition generation. In summary, acceleration and angular velocity measurements based on the solid-state spin are in the experimental stage. The development of inertial measurement systems based on this principle is still facing huge challenges. Nevertheless, the NV center spin-based inertial measurement is an innovative technology at its early stage with rapid development. It has scientific exploratory significance and theoretical application value in terms of research and development of new inertial devices.