1. Introduction

High quality nano (micro)-mechanical resonator is one of the best testbed for fundamental physics [1], such as the macroscopic quantum superpositions [2, 3], the gravity induced wavefunction collapse [4], the boundary between quantum and classical regimes [5, 6], and etc. It is found that the large quantum superpositions of the nano-mechanical resonator could be realized with the help of cavity modes [7], superconducting circuits [8, 9], nitrogen-vacancy centers [10–13], and etc. The quantum-classical boundaries can be tested in these systems through matter-wave interferometry [8, 14, 15]. On the other hand, the nano (micro)-mechanical resonator is also widely used in precision measurement of masses [16], torsion [17], forces and accelerations [18, 19], because of its high mechanical Q.

As we know, the interferometry firstly appeared in optics, which was used for prescise measurement. Later, the matter-wave interferometry was realized with electrons [20] and neutrons [21], then with larger particles such as atoms [22, 23] and molecules [24]. Atom interferometry has evolved from the demonstration of quantum superpositions into instruments at the cutting edge of precise measurement, including measurements of platform rotation, the Molar-Planck constant, the fine structure constant [25,26] and the gravitational acceleration [27–30]. One of the motivations to replace the light with atoms for interferometry is that the shorter atomic de Broglie wavelength could make the measured phase shift much more accurate. Therefore it is natural to anticipate that the interferometers with macroscopic object, such as the nano-mechanical resonators, could greatly increase the measurement precision of the phase shift.

In this paper, we propose a scheme to realize the high-precision gravimeter with a nano-mechanical resonator hybrid with a nitrogen-vacancy center by using matter-wave interferometry. We give a physically intuitive derivation of the interferometer phase shift. With the state-of-the-art technologies, we estimate the phase shift to be 3 orders of magnitude larger than those using atom interferometry [24, 26, 27]. We briefly analyze random and systematic noise and find that the relative measurement precision 10⁻¹⁰ for gravitational acceleration is achievable. Besides, our scheme is solid based and on-chip. Unlike the gravimeter based on atomic interferometry, here neither the complex lasers nor the big vibration isolation system is required. Therefore, our scheme is suitable for portable gravimeters with a high precision.

2. The scheme

We consider a nano-mechanical resonator hybrid with a nitrogen-vacancy (NV) center through magnetic field gradient induced coupling. There are two different setups. The first one, as shown in Fig. 1(a), consists of a nanoscale diamond bead containing a NV center levitated by an optical tweezer in ultrahigh vacuum [31–34]. A magnetic tip nearby induces a large magnetic field gradient, and couples the NV center with the center of mass (CoM) motion of the nano-diamond. The other setup uses a cantilever resonator [10, 35–39]. As shown in Fig. 1(b), a magnetic tip attached to the cantilever is used to couple the mechanical mode to an NV center embedded in bulk diamond below the cantilever. In both setups, the mechanical motion can be described by the same Hamiltonian (1), as we will discuss below. In both cases, the gravity induced dynamical phase is measured through a Ramsey scheme similar to atom interferometry [27, 40–42]. We will analyze the phase shift of an oscillator in the gravitational field coupled to a solid spin, then this phase shift is revealed by Ramsey interferometry to measure the gravitational acceleration.

 

Fig. 1 (a) An optical trap holds a nano-diamond with a build-in NV center with both the weakest confinement and the electron spin quantization along the z axis. A magnetic gradient along the z axis produces spin-dependent shifts to the center of the harmonic well. The z axis is oriented along the vertical gravitational acceleration by tuning the control system. The CoM of the nano-diamond oscillates around the two balanced points z_±, accumulating a relative gravitational phase difference Δϕ. At t₀ = 2π/ω_z this phase can be read from spin population. (b) A scheme to strongly couple an NV center with a cantilever. The NV center is embedded in bulk diamond lattice. A magnetic tip is attached to the cantilever, which provides the magnetic gradient. This setup can fulfill our requirement of large magnetic field gradient and long dephasing time T₂, thus can significantly improve gravitational measurement precision.

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As an example, we analyze the optically levitated nano-diamond scheme in detail. The motion of the nano-diamond in the harmonic potential of the tweezer is coupled to the S = 1 spin of the NV center by a magnetic field gradient B_g = ∂B/∂z oriented along the z direction, which can be generated by a magnetic tip. We assume the trapping frequencies satisfy ω_x, ω_y ≫ ω_z. Therefore the effects of motion on x and y directions are neglected. Then the whole system Hamiltonian, including the Earth’s gravitational field effects, reads [12]

The phase diagram of Ramsey interferometry based on π/2 − π/2 pulse sequence is shown in Fig. 2. We assume that the NV center is initialized to state |0), and CoM motion of the nano-diamond is cooled down to mK or lower. The CoM motion of the levitated nanoparticle has been cooled to around 0.1 mK [45, 46]. For the optically levitated nanodiamond, the heating induced by the NV center makes it difficult to levitated in high vacuum and to cool the CoM motion to mK. However, this could be solved by using the purified nanodiamond [47], the levitated nano-refrigerator [48], or the ion trap [49, 50]. In the first step, we apply a microwave pulse corresponding to the effective interaction Hamiltonian H_mw = ħΩ(| + 1〉〈0| + |− 1〉〈0| + H.c.), where Ω is the Rabi frequency. In the limit that Ω is much larger than any other coupling strength in Eq. (1), we can neglect any other interactions when applying the pulse. With the pulse duration $t_p=\pi /(2\sqrt{2}\Omega )$ (π/2 pulse), the NV center electron spin state becomes $(|+1〉+|-1〉)/\sqrt{2}$, which is a superposition of | + 1〉 and |− 1〉 with equal amplitudes. The two states experience a different force due to the spin-dependent coupling term 2λS_z (c + c^†), which leads to an additional spin-dependent acceleration

 

Fig. 2 Phase space diagram of the matter wave interferometer based on π/2-π/2 pulse sequence. The nano-object can either be in the internal NV spin state | + 1〉 (blue) of |− 1〉 (orange). The lines represent the classical trajectories originating from one of the space-time points comprising the initial wave packet.

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The equilibrium position of the two states is then determined by z_± = z₀ ± Δz as shown in Fig. 2, where $z_0=g/{\omega }_z^2$ and the displacement

So after the π/2 pulse the two states will oscillate around their equilibrium points z_± and the two paths will recombine after an oscillation period t₀ = 2π/ω_z. The spin states after the oscillation period is (| + 1〉 + e^iΔϕ |− 1〉). The phase shift between the two paths due to propagation can be calculated by their classical actions

The classical paths of the mechanical oscillator of the two spin states |± 1〉 are

To reveal Δϕ we apply another π/2 pulse H_mw = ħΩ(| + 1〉〈0| + e^iϕ |−1〉〈0|) + H.c. with a relative phase ϕ. After time t_p, the population of the spin state with S_z = 0 becomes

The Feynman path integral techniques [43] we apply here indicates that the phase shift acquired during the evolution only depends on the CoM motion of the mechanical system, regardless whether it is in thermal states or coherent states. Alternatively, since the thermal state can be written as superposition of coherent states ρ_th = d² βP_th(β)|β〉〈β|, where P_th(β) is the Glauber P representation of the thermal state, one can focus on the evolution of a coherent state of the mechanical oscillator. Consider the system starts in a separable state |β〉|s_z = 0〉. After the first π/2-pulse the system is in $|\beta 〉(|+1〉+|-1〉)/\sqrt{2}$. It has been shown in [12] that the state factorizes after the t₀ evolution and reads $|\beta 〉(|+1〉+e^{i\Delta \varphi }|-1〉)/\sqrt{2}$ after dropping a global phase factor. Since the phase shift Δϕ is independent to the coherent state amplitude, it is also immune to thermal effect.

We now discuss the parameters necessary to obtain a high precision gravitational acceleration measurement. Specifically, we need to obtain a large relative phase shift which is proportional to gravitational acceleration. From Eq. (7), by setting a large magnetic gradient B_g = 10⁶T/m and the oscillating period t₀ = 2 ms ≲ T₁ comparable to relaxation time of the NV, we can obtain a phase shift Δϕ = 1.4 × 10⁹ which is three orders of magnitude larger than phase shift in cold atom experiment Δϕ = 3.8 × 10⁶ [27] and lead to a raise in precision by three orders of magnitude, if the errors in phase measurement is comparable.

Why our gravimeter based on nano-mechanical matter-wave interferometry is orders of magnitude more precise than the cold atom interferometry gravimeter? For a diamond sphere with radius R ∼ 200 nm, considering the density 3000 kg/cm³ for diamond, the corresponding mass is m ∼ 1 × 10⁻¹⁶kg, which is 10¹⁰ times massive than sodium atoms, and the oscillating amplitude ∼ 50 nm. The cantilever nano-mechanical resonator is at least 10¹⁶ times massive than sodium atoms frequently used in interferometry with much smaller oscillating amplitude Δz. Therefore, our device can be made on chip, and much smaller than atom interferometry based gravimeter. Rewriting phase shift Eq. (7) using Eq. (3)

Here we discuss the features of the two proposals. For the cantilever scheme, the magnetic tip can approach the NV center ∼ 100 nm and induce a magnetic field gradient > 1 × 10⁵T/m [38]. The dephasing time of the NV in bulk diamond is exceptional ∼ 2 ms. Even in the presence of a magnetic field gradient ∼ 10⁵ T/m [37, 38], as long as the distance to the magnetic tip is ∼ 1 µm, the magnetic field strength is ∼ 0.01 T and does not significantly reduce the dephasing time. Therefore the main decoherence effect is the mechanical damping. Recently, high Q ∼ 8 × 10⁸ nanomechnical resonators using elastic strain engineering [51] has been realized in experiment, leading to low thermal decoherence rate k_BT/ħQ exceeds one oscillation periods.

For the trapped nanoparticle scheme, the coherence time of NV center can be prolonged to the limit of T₁ by decoherence decoupling techniques [52]. In our proposal, no transition between |± 1〉 is needed during the propagation, therefore it is convenient to use a continuous dynamical decoupling which prolong the quantum memory to T₂ ∼ 2 ms. For example, we can use the time-dependent detuning method described in [53]. Considering only the states |± 1〉 during the propagation, the two-level system with an ambient magnetic field noise δB(t) is described by

To compensate for this noise, which causes dephasing, we use a single continuous dynamical decoupling driving field

As we can see from the above discussion, each of the two methods has its own advantages. The optically trapped nanoparticle has ultra-high quality factor Q ∼ 10¹², and its trapping frequency can be tuned to optimize the phase shift. Because of its high Q, the proposed scheme could be performed even under room temperature. The cantilever setup does not require a laser system to cool or trap the oscillator. Besides, since the coupled NV is embedded in bulk solid, the dephasing time T₂ is significantly longer than the one in nano-diamond. Therefore, the precision of the gravimeter could be greatly enhanced, compared with the other setup.

We have ignored various noise effects in the above estimation of phase shift. In the following section, we show that how to optimize the phase shift after considering random and systematic noises.

3. Noise estimation

3.1. Fringe visibility and random noise

The decoherence effect will reduce fringe visibility, leading to increase in amplitude noise terms (shot noise and detection noise). Here we explore two main decoherence effects: (1) motional decay of the optomechanical system; (2) dephasing of the NV center.

In the trapped particle scheme, the motional decay is associated with photon scattering from the trapping laser and heating due to random momentum kick with residual gas particles [2, 3, 54]. The background gas collision leads to heating with a damping rate γ_g/2 = (8π)(P/vr ρ) [2], where ρ is the material density, P and v are the background gas pressure and mean speed, respectively. For a sphere of radius R = 200 nm, ω_z = 2π × 0.5 kHz and a room-temperature gas with P = 1 × 10⁻⁹ Torr, the damping rate γ_g/ω_z ∼ 4 × 10⁻¹⁰. We define γ_sc as the photon scattering induced decay rate. For the diamonds with permittivity ϵ = 1.5, radius R = 200nm and trapping wavelength λ₀ ∼ 10 µm (CO₂ laser), we have ${\gamma }_{sc}/{\omega }_z=(16{\pi }^3/15)[(ϵ-1)/(ϵ+2)]R^3/{\lambda }_0^3\simeq 3.8\times {10}^{-5}$ [2], which is much larger than γ_g. Therefore the main motional decoherence comes from photon scattering with the maximum decoherence rate Γ = γ_sc |2λ/ħω_z |². With parameters in the previous section, λ/ħω_z ≃ 90, we have Γ/ω_z = 0.3 less than 1. To further reduce the scattering noise, we could lower the maximum separation λ/ħω_z by either reducing the magnetic gradient or increasing trap frequency. It is also possible to use other trap, e.g. ion trap [49, 55], where no photon scattering noise exists.

The other detrimental effect on the trapped nano-diamond scheme is due to the dephasing of the NV center. The noise is induced by magnetic field fluctuation of the diamond lattice and coupling to torsional motion [56]. In the levitated nanoparticle setup, the spin-torsional(rotational)-motion coupling term reads µS_z (b + b^†) [17,57], where µ is the torsional coupling strength and b/b^† are the annihilation/creation operator of the torsional motion. The coupling may affect the translational decoherence via spin dephasing. Its strength could be estimated by replacing b/b^† with c-number $\sqrt{{\overline{n}}_{th}}$, which is the average phonon number of the torsional motion. The torsional coupling strength µ can approached to zero if the angle between the magnetic field and axis of the NV center is zero [13]. The torsional degree of freedom could be cooled down to reduce the coupling induced dephasing even more. Both the dephasing effect can be suppressed by the dynamical decoupling scheme [53] which prolong the quantum memory to T₂ ∼ 2 ms. For t₀ ∼ T₂, the fringe contrast is reduced to 1/e ∼ 0.36.

In the cantilever scheme, the decoherence effect can be significantly reduced. The effect of motional decay of the mechanical oscillator can be estimated by the damping rate γ_sc/ω_z = 1/Q ∼ 1 × 10⁻⁸ [51], and the maximum decoherence in one oscillation period γ_sc |2λ/ħω_z |²2π/ω_z ∼ 0.001, which can be ignored. When the oscillator is cooled to sub-millikelvin temperature [10, 58, 59], the Brownian motion amplitude ∼ 7 nm is much smaller than the maximum separation between the spin up and down states, thus the motional decay caused by thermal motion is negligible. The pure dephasing time of the NV center in bulk diamond could be ∼ 10 ms, which is much longer than the one in the trapped nano-diamond.

The visibility due to mechanical motion decoherence and dephasing under different mechanical quality factor Q = ω_z /γ_sc and the pure dephasing time T₂ is plotted in Fig. 3, where ω_z = 2π × 0.5 kHz and λ/ħωz ≃ 90. Here we use a simple formula $\exp (-\frac{2\pi }{Q}|2\lambda /\hslash \omega {|}^2)\exp (-t_0/T_2)$ to estimate the visibility. To retain a large visibility, the quality factor Q should be larger than 1 × 10⁵ and the dephasing time T₂ should be longer than 2 ms.

 

Fig. 3 The visibility due to mechanical motion decoherence and dephasing under different mechanical quality factor Q = ω_z /γ_sc and the pure dephasing time T₂, where ω_z = 2π × 0.5 kHz and λ/ħω_z ≃ 90. The CoM motion temperature of the oscillator is cooled to T = 0.1 mK by feedback cooling, therefore the heating effect can be ignored.

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The Ramsey fringe visibility is directly related to the signal-to-noise ratio, and will appear in the amplitude noise terms. To reduce the shot noise limit to the measurement precision Δg/g below 1 × 10⁻¹⁰, we need at least 1 × 10⁵ data points for each measurement. For the atom interferometry, there are N ∼ 1 × 10⁶ atoms in the atom fountain simultaneously contribute to the signal. As for our scheme, we should fabricate M mechanical resonators with the same frequency on-chip, which can perform the measurement at the same time. With the measurement repeating frequency 1 kHz and M = 100, we can achieve the precision goal within 2s. We note that modern technology makes it possible to fabricate such solid-based gravimeter on chip.

We assume that the phase shift can be measured with precision 10 mrad, comparable to atom interferometry. Based on Eq. (7), we can plot the precision of our gravimeter under different experiment parameters, as shown in Fig. 4. In order to retain a high fringe visibility, the choice of parameters is limited to the region below the red line. This is because under the external magnetic gradient, the thermal motion of the oscillator leads to a magnetic field fluctuation for the NV center and the dephasing of the NV center electron spin. The fluctuation in magnetic field can be estimated by the root-mean-square of the mechanical motion times the magnetic gradient, which reads

 

Fig. 4 Precision of the gravity acceleration measurement under different magnetic field gradient B_g and oscillation period t₀. We chose t ≤ 2 ms and assume the visibility are high enough by choosing the parameters to be in the upper right region in Fig. 3. We take the accuracy of the phase shift to be 10 mrad in the estimation. In addition, we require the magnetic field fluctuation to be Δ < 10 MHz, which corresponds to the region below the red line. Within the region, up to 1 × 10⁻¹⁰ relative precision can be achieved at the lower right corner.

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If the condition Δ ≪ Ω₁ fulfills, the extra dephasing can be suppressed by dynamical decoupling drive with Eq. (11). This condition limit our choice of the parameters. We assume that the center of mass motion temperature of the trapped nano-diamond is cooled to T = 0.1 mK by feedback cooling, and the mass of the nanoparticle is m = 1 × 10⁻¹⁶ kg. For typical Rabi frequency Ω₁ ≲ 100 MHz, we require the magnetic field fluctuation to be Δ < 10 MHz, which corresponds to the region below the red line in Fig. 4. If the CoM temperature could be cooled to T < 1 µK, e.g., by using cold atoms [60], the measurement precision would be further improved by one to two orders of magnitudes.

3.2. Systematic noise

Thanks to the short matter-wave wave length of both the nanoparticle and cantilever, our scheme has the advantage of on-chip compared with the atomic gravimeter. Therefore, the laser system, vibration isolator and other auxiliary devices should be relatively easy to reach a high precision. We assume that in our setup the systematic error is on the same order of magnitude as the one in atom interferometry method. The resulting interferometer is accurate enough to allow phase shifts 10 mrad to be detected, leading to a value of g accurate to one part in 1 × 10¹⁰ in our method.

The second order magnetic field gradient induces extra systematic error. The additional term in Lagrangian is $\pm \frac{1}{2}{g}_{NV}{\mu }_B\frac{{\partial }^{2}B}{\partial z^2}{z}^2$, which is different for spin up and down. We can treat it as a shift in trapping frequency $\Delta {\omega }_{\pm }/\omega =\pm \frac{1}{8m}{g}_{NV}{\mu }_b\frac{{\partial }^{2}B}{\partial z^2}$. The frequency error contribute to the measured gravitational acceleration $\frac{\Delta g}{g}~\frac{\Delta \omega }{\omega }$. By this relation, we get the maximum second derivative of the magnetic field $\frac{{\partial }^{2}B}{\partial z^2}\lesssim 1.7\times {10}^5\mathrm{T}/{\mathrm{m}}^2$ for $\frac{\Delta g}{g}=1\times {10}^{-10}$. In the case of cantilever oscillator, we can embed another magnetic tip under the NV in the bulk diamond to reduce $\frac{{\partial }^{2}B}{\partial z^2}$. For trapped nanoparticle scheme, we can eliminate this effect by simultaneously rotating the trapping direction 180° and apply a microwave π-pulse to flip the spin states after one evolution period t₀ and measure the phase shift at time 2t₀. Because the trapping axis is rotated, the phase shift of the two evolution period accumulates, while the second order magnetic field induced phases will cancel out.

In order to measure the absolute gravitational acceleration, the magnetic field gradient should be determined with high accuracy. We note that the present technology cannot achieve the magnetic field gradient accuracy around 10⁻⁹ or higher. Therefore, we cannot achieve the absolute gravimeter with accuracy higher than 10⁻⁹. However, by using NV center as a magnetic field detector, it could achieve the magnetic field gradient accuracy higher than 10⁻⁹ in future [38].

We propose the following method to measure the magnetic field gradient. We can measure the NV center electrons’ spin energy splitting change with respect to position of the NV center. The magnetic field gradient is obtained by $B_g=\frac{\hslash }{g_{NV}{\mu }_B}\frac{\Delta \omega }{\Delta z}$, where Δω is the change in energy splitting and Δz is the position displacement. The position measurement precision is on the order of $3\text{fm}\sqrt{\text{Hz}}$ [61], i.e. 1 × 10⁻¹⁶m precision within 1 × 10² s. We can measure the magnetic field in the range of 0.1 µm, therefore the relative precision of Δz can be achieved above 1 × 10⁻⁹. The decoherence time of the NV center can approach 1 s at low temperature with dynamical decoupling described in section 3. When the magnetic gradient shift the energy level on the order of GHz, the relative precision of Δω around 1 × 10⁻⁹ can be achieved. Taking into account the error in both Δω and Δz, the magnetic field gradient could be determined with precision more than 1 × 10⁻⁹. Therefore an absolute gravimeter with high sensitivity could be built.

Other systematic noise includes magnetic field drift, perturbative terms related to ω_x, ω_y, anharmonic effects of the trapping potential, random orientation and Doppler effect. The magnetic field drift in a timescale of hours is much larger than the system evolution time of 1 ms. So in principle, the magnetic gradient can be determined with relative accuracy $e^{-t/t_{drift}}~1\times {10}^9$ or higher if we consider its time dependence. Perturbative analysis shows that for ω_x = ω_y = 10ω_z, the fidelity of the evolution stays above 99% even when the initial state is thermal with an average thermal occupation number is up to 600 [62]. The anharmonic effects of the trapping potential will be avoided by feedback cooling of our oscillator to sub-millikelvin temperatures. The random orientation effect can be corrected by methods shown in [53]. The Doppler effect may appear in the spin preparation when the nanoparticle is oscillating in the trap. The corresponding frequency error could be estimated by δf = f₀v/c = f₀Δzω_z /c, where Δz is the oscillation amplitude, f₀ = 2.88 GHz is the microwave frequency in use, and c is the speed of light. With δz ≈ 100 nm and ω_z ≈ 2π × 1 kHz, we eventually have δf ≈ 6 × 10⁻³Hz, which is much smaller than the typical linewidth of an NV center of ∼ 10 MHz. For a thermal state of the CoM, with a temperature cooled to about 1 mK in the trap, the root-mean-square velocity is about $v_1=\sqrt{2kT/m}~0.002\mathrm{m}/\mathrm{s}$. Therefore the Doppler shift would not be a concern in our scheme.

4. Conclusion

In conclusion, we have proposed a solid-base on-chip gravimeter which makes use of the matter-waver interference of a mechanical resonator to significantly increase the precision. We have proposed two equivalent schemes to couple an NV center to a mechanincal oscillator in gravitational field. In order to measure the gravitational acceleration, we have proposed the method to achieve Ramsey interferometry in this system, where the inference pattern is depend on the gravitational induced phase shift. Under the experimental feasible parameters, we found that the phase shift is three order of magnitudes greater than the atomic interferometry method. We then analyze the noise effects, including motional decay of the oscillator and the dephasing of the NV center, on the precision of the gravimeter. It is found that the relative precision 10⁻¹⁰ is possible under the current experimental conditions. Finally, we have analyzed the effect of the second derivative of the magnetic field and provided methods to compensate it.