Quantum control and Berry phase of electron spins in rotating levitated diamonds in high vacuum

Levitated diamond particles in high vacuum with internal spin qubits have been proposed for exploring macroscopic quantum mechanics, quantum gravity, and precision measurements. The coupling between spins and particle rotation can be utilized to study quantum geometric phase, create gyroscopes and rotational matter-wave interferometers. However, previous efforts in levitated diamonds struggled with vacuum level or spin state readouts. To address these gaps, we fabricate an integrated surface ion trap with multiple stabilization electrodes. This facilitates on-chip levitation and, for the first time, optically detected magnetic resonance measurements of a nanodiamond levitated in high vacuum. The internal temperature of our levitated nanodiamond remains moderate at pressures below 10−5 Torr. We have driven a nanodiamond to rotate up to 20 MHz (1.2 × 109 rpm), surpassing typical nitrogen-vacancy (NV) center electron spin dephasing rates. Using these NV spins, we observe the effect of the Berry phase arising from particle rotation. In addition, we demonstrate quantum control of spins in a rotating nanodiamond. These results mark an important development in interfacing mechanical rotation with spin qubits, expanding our capacity to study quantum phenomena.

In this article, we design and fabricate an integrated surface ion trap (Fig. 1(a)) that incorporates an Ωshaped stripline to deliver both a low-frequency high voltage for trapping and a microwave for NV spin control.Additionally, it comprises multiple electrodes to stabilize the trap and drive a levitated diamond to rotate.With this advanced Paul trap, we have performed optically detected magnetic resonance (ODMR) measurements of a levitated nanodiamond in high vacuum for the first time.Using NV spins, we measure the internal temperature of the levitated nanodiamond, which remains stable at approximately 350 K under pressures below 10 −5 Torr.This suggests prospects for levitation in ultra-high vacuum.With a rotating electric field, we have been able to drive a levitated nanodiamond to rotate at high speeds up to 20 MHz (1.2 × 10 9 rpm), which is about three orders of magnitudes faster than previous achievements using diamonds mounted on motor spindles [29,30].Notably, this rotation speed surpasses the typical dephasing rate of NV spins in the diamond.With embedded NV electron spins in the levitated nanodiamond, we observe the effect of Berry phase generated by the mechanical rotation, which also improves the ODMR spectrum in an external magnetic field.Moreover, we achieve quantum control of NV centers in a rotating levitated nanodiamond.Our work represents a pivotal advancement in interfacing mechanical rotation with spin qubits.

A. Levitation of a nanodiamond in high vacuum
In the experiment, we levitate a nanodiamond in vacuum using a surface ion trap (Fig. 1(a)).The surface ion trap is fabricated on a sapphire wafer, which has high transmittance for visible and near-infrared lasers.To achieve levitation of nanodiamonds and quantum control of NV spins simultaneously, we apply both an AC high voltage and a microwave on a Ω-shaped circuit.The AC high voltage has a frequency of about 20 kHz and an amplitude of about 200 V.The microwave has a frequency of a few GHz.They are combined together with a homemade bias tee.The center ring electrode is grounded to generate a trapping center above the chip surface.The four electrodes at the corners are used to compensate the static electric fields from surface charges to minimize the micro-motion of a levitated nanodiamond.Fig. 1(c) shows a simulated distribution of the electric field of the trap.The trapping center is 253 µm away from the chip surface.
The trapping potential depends on the charge to mass ratio (Q/m) of a levitated particle.Thus, it is necessary to increase the charge number of particles for stable levitation in an ion trap.In our experiment, nanodiamonds are charged and sprayed out by electrospray and delivered to the surface ion trap with an extra linear Paul trap.The charge of the sprayed nanodiamond is typically larger than 1000 e, where e is the elementary charge, enabling a large trapping depth of more than 100 eV (see Supplementary Information for more details).A 532 nm laser is used to excite diamond nitrogen-vacancy (NV) centers and a 1064 nm laser is applied to monitor the nanodiamond's motion.More details of our experimental setup are shown in Supplementary Fig. 1.
A main result of our experiment is that we can levitate a nanodiamond with the surface ion trap in high vacuum, which is a breakthrough as levitated diamond particles were lost around 0.01 Torr in previous studies using ion traps [43][44][45][46][47].The red curve in Fig. 1(d) shows the power spectrum density (PSD) of the center-of-mass (CoM) motion of a levitated nanodiamond at 9.8 × 10 −6 Torr.The radius of the levitated nanodiamond is estimated to be about 264 nm based on its PSDs at 0.01 Torr (Supplementary Fig. 2).Our surface ion trap is remarkably stable in high vacuum.We can levitate a nanodiamond in high vacuum continuously for several weeks.
The internal temperature of a levitated nanodiamond is important as it will affect the spin coherence time and trapping stability.We measure the internal temperature using NV centers.The energy levels of an NV center is shown in Fig. 1(b).We use a 532 nm laser to excite the NV centers and a single photon counting module to detect their photoluminescence.Then we sweep the frequency of a microwave to perform the ODMR measurement of a levitated nanodiamond in the absence of an external magnetic field.Fig. 1(e) shows the ODMRs measured at 10 Torr (blue circles) and 6.9 × 10 −6 Torr (red squares).Based on the fitting of the ODMRs, the corresponding zero-field splittings (blue and red dashed lines) are 2.8694 GHz and 2.8650 GHz, respectively.The internal temperature of the levitated nanodiamond can be obtained form the zero-field splitting (see Methods for details).The measured internal temperature at different pressures are shown in Fig. 1(f).The internal temperature is close to the room temperature at pressures above 0.1 Torr, and increases when we reduce the pressure from 0.1 Torr to 10 −4 Torr.Finally, it remains stable at approximately 350 K at pressures below 5×10 −5 Torr.This temperature is low enough to maintain quantum coherence of NV spins for quantum control [50].
The observed phenomena (Fig. 1(f)) arise from the balance between laser-induced heating (Supplementary Fig. 3) and the cooling effects of air molecules and blackbody radiation on the internal temperature of a levitated nanodiamond [51,52].When the air pressure is high, the cooling rate due to surrounding air molecules is large and the internal temperature of the levitated nanodiamond is close to the room temperature.However, as air pressure decreases, cooling from air molecules diminishes, leading to a rise in internal temperature.When the pressure is below 5 × 10 −5 Torr, the temperature stabilizes as the cooling is dominated by the black body radiation, which is independent of the air pressure.

B. Fast rotation and Berry phase
After a nanodiamond is levitated in high vacuum, we use a rotating electric field to drive the levitated nanodiamond to rotate at high speeds, which also stabilizes the orientation of the levitated nanodiamond.The four electrodes at the corners are applied with AC voltage signals (A sin (ωt + φ)) with the same frequency (ω) and amplitude (A) but different phases (φ) to generate a rotating electric field (Fig. 2(a)).The phases of neighboring signals are different by π/2.Fig. 2(b) shows the simulation of the electric potential in the xy-plane at t = 0.More information can be found in Supplementary Information (Supplementary Fig. 4 and Supplementary Fig. 5).A levitated charged object naturally has an electric dipole moment due to inhomogeneous distribution of charges.In a rotating electric field, the levitated charged particle will rotate due to the torque produced by the interaction between the rotating electric field and the electric dipole of the particle.Fig. 2(c) depicts the PSDs of the rotation at different driving frequencies (0.1 MHz -20 MHz).The maximum rotation frequency is 20 MHz in the experiment, which is limited by our phase shifters used to generate phase delays between signals on the four electrodes.This is about 3 orders of magnitudes faster than previous achievements using diamonds mounted on electric motor spindles [29,30].When the rotation frequency is 100 kHz, the linewidth of the PSD of the rotational signal is fitted to be about 9.9 × 10 −5 Hz (Supplementary Fig. 5(d)), which is limited by the measurement time.This shows that the rotation is extremely stable and is locked to the driving electric signal.With easy control and ultra-stability, this driving scheme enables us to adjust and lock the rotation of the levitated nanodiamond over a large range of frequencies (see Supplementary Information for more details).
The fast rotating diamond with embedded NV spins allows us to observe the effects of the Berry phase due to mechanical rotation.The Berry phase, also known as the geometric phase, is a fundamental aspect of quantum mechanics with applications in multiple fields, including the topological phase of matter and the quantum hall effect [53][54][55][56][57]. The Berry phase in the laboratory frame is equivalent to the pseudo-magnetic field (called the Barnett field in [57]) in the rotating frame: B ω = ω r /γ, where γ is the spin gyromagnetic ratio.In this work, the microwave source is fixed in the laboratory frame.Only the levitated diamond is rotating.So, we can observe the effect of the Berry phase due to rotation [57].
In a rotating diamond, the embedded NV centers also follow the rotation with an angular frequency of ω r (Fig. 3).The levitated nanodiamond in our experiment contains ensembles of NV centers with four groups of orientations.Fig. 3(d) shows a NV center embedded in a nanodiamond rotating around the z axis in the presence of an external magnetic field.The direction of the magnetic field is along the z axis.The angle between the NV axis and z axis is θ, and the azimuth is ϕ(t) relative to the x axis.The Hamiltonian of the rotating NV electron spin in the laboratory frame, neglecting strain effects, can be written as [34]: where D is the zero-field splitting, R (t) = R z (ϕ (t)) R y (θ) is the rotation transformation, and R j (θ) = exp(−iθn • S) for the rotation angle θ around the n direction, j = y, z, and S is the spin operators.The Stark shift for NV centers induced by the electric field is negligible and hence is not included in the equation.The Hamiltonian possesses three eigenstates |m s , t⟩ lab (m s = 0, ±1).The detailed expressions can be found in the Supplementary Information.Based on its definition, the Berry phase can be calculated as [57] (2) Here the Berry phase is calculated for an open-path and is hence gauge-dependent.The spin state of the NV center is observed through the interaction with a microwave magnetic field.In our experiment, the direction of the microwave is in the yz-plane and has a small angle θ ′ = 8.5 • relative to the z axis, resulting from the asymmetric design of the waveguide.However, the dominant transition probability arises from the longitudinal (z) component.The expected value of the transition probability of the spin states interacting with the microwave can be expressed as lab ⟨±1, t| e iH lab t/ℏ e −iγ±1 H M W,z,lab e iγ0 e −iH lab t/ℏ |0, According to Eq. 3, the transition of spin states from |m s = 0⟩ lab to |m s = ±1⟩ lab can be driven by a microwave at the resonance frequency of D ± gµ B B cos θ ∓ ω r cos θ, where the frequency shift ∓ω r cos θ is due to the Berry phase induced by the mechanical rotation.
We first investigate the effect of the Berry phase in the absence of an external magnetic field.Fig. 3(a) shows the diagram of a NV center rotating around the z axis without an external magnetic field.To observe the frequency shift due to fast rotation, ODMR measurements of the levitated nanodiamond are carried out at different rotation frequencies.Fig. 3(b) displays ODMRs at rotation frequencies of 0.1 MHz (bule circles) and 14 MHz (red squares).The full width at half maximum (FWHM) of the ODMR at ω r = 2π × 14 MHz is clearly larger than that at ω r = 2π × 0.1 MHz , which is caused by the Berry phase due to rotation.The FWHM of the ODMR at different rotation frequencies is shown in Fig. 3(c).The blue circles are the experimental results.The red and orange curves are theoretical results for θ = 0 • and θ = 20 • , respectively.Experimentally, the NV ensemble contains NV centers with four orientations.Based on Eq. 3, the broadening of the ODMR spectrum is mainly determined by NV centers with the smallest θ, which have the largest frequency shift induced by the Berry phase(Supplementary Fig. 6).The frequency shift of ∓ω r cos θ is insensitive to the angle θ for small θ.This explains why the theoretical results for θ = 0 • and θ = 20 • are similar, and both agree well with the experimental results.All data shown in Fig. 3(b),(c) are taken from one levitated diamond.
To determine the frequency shift as a function of the rotational frequency unambiguously, an external magnetic field can be applied to separate the energy levels of NV centers along four different orientations.Here we apply a static magnetic field of about 100 G along the z axis to separate energy levels (Fig. 3(d)).Data shown in Fig. 3(e),(f) are taken from one levitated diamond, which is different from the one used for Fig. 3(b),(c).In Fig. 3(e), the red squares show the ODMR spectrum measured at a rotation frequency of 0.1 MHz.The linewidths of ODMR dips for levitated diamond NV centers are broader than those for fixed NV centers due to the continuous change of NV orientations relative to the magnetic field.Compared with the ODMR spectrum of a levitated nanodiamond without stable rotation (gray circles), the linewidth of each dip for a diamond rotating at 0.1 MHz is narrower.This clearly demonstrates that fast rotation can stabilize the orientation of the levitated nanodiamond.Now we consider the NV centers with the smallest θ (largest Zeeman shift) and the transition between the state |m s = 0⟩ and the state |m s = +1⟩ as an example.The electron spin resonance frequency is 3.120 GHz at ω r = 2π × 0.1 MHz for this transition.The corresponding angle between the NV axis and the rotation axis is θ = 20.7 • , which is calculated based on the transition frequency and the magnitude of the external magnetic field.We then measure the resonance frequency at different clockwise (unless otherwise specified, all are viewed from the positive z direction) rotation frequencies, as shown in Fig. 3(f).The resonance frequency increases following the increase of the rotation frequency.The experimental data points are in between the theoretically calculated curves for θ = 20.7 • (green solid line) and θ = 24.0• (violet dashed line), indicating the orientation of the NV axes changes slightly when the rotation frequency increases.This is because the electric dipole moment of the levitated nanodiamond is not exactly perpendicular to the axis of the largest or the smallest moment of inertia.Once the rotation frequency increases, the nanodiamond tends to rotate along its stable axis and the driving torque is not large enough to keep its former orientation.The magenta dashed curve is a linear fitting of the resonance frequency.The orientation of the NV center can be calculated by the resonance frequency at the various rotation frequencies.The angle θ changes by approximately 3.3 • at ω r = 2π × 10 MHz, compared with that at ω r = 2π × 0.1 MHz.A rotating diamond can also serve as a gyroscope [58,59].
The effect of the Berry phase in a levitated nanodiamond rotating at the counterclockwise direction is shown in Supplementary Fig. 6.The resonance frequency between the state |m s = 0⟩ and the state |m s = +1⟩ decreases as the rotation frequency increases for counterclockwise rotation (Supplementary Fig. 6(c)), which is different from that of the levitated nanodiamond rotate clockwise (Fig. 3(f)).

C. Quantum control of fast rotating NV centers
Quantum control of spins is important for creating superposition states [25,26,39] and performing advanced quantum sensing protocols [60].Here we apply a resonant microwave pulse to demonstrate quantum state control of fast rotating NV centers.The spin state can be read out by measuring the emission PL.Because a weak 532 nm laser is used to avoid significant heating, the initialization time should be long enough to prepare the NV spins to the |m s = 0⟩ state.When the laser intensity is 0.113 W/mm 2 , the initialization time is 1.05 ms (Supplementary Fig. 7).This is shorter than the spin relaxation time (T 1 ∼ 3.6 ms) of this levitated nanodiamond (Supplementary Fig. 7).We also measure Rabi oscillation of a nanodiamond fixed on a glass cover slip with the same 532 nm laser intensity for comparison.We get similar results for both high and low intensities of the 532 nm laser (Supplementary Fig. 8).Due to the Ω-shape of the microwave antenna, the orientation of the magnetic field of the microwave is located in the yz-plane and slightly different from the z axis with an angle of about θ ′ = 8.5 • (Fig. 4(a)).So, n M W = (− sin θ ′ , 0, cos θ ′ ).The effective microwave magnetic field acting on NV spins, with the orientation of n N V = (cos ϕ (t) sin θ, sin ϕ (t) sin θ, cos θ), changes as a function of the rotation phase ϕ(t) of the levitated nanodiamond.The Rabi frequency Ω Rabi can be written as [61,62]: Therefore, it is necessary to synchronize the microwave pulse and the rotation phase of the levitated nanodiamond.Fig. 4(b) shows the pulse sequence of the Rabi oscillation measurement.The time gap between the initialization and the readout laser pulses is twice of the rotation period, which allows us to apply the microwave pulse at an arbitrary rotation phase between 0 and 2π.
The measured Rabi oscillations between the state |m s = 0⟩ and the state |m s = +1⟩ of NV centers are shown in Fig. 4(d), (e).All these measurements are carried out at a rotation frequency of 100 kHz.The rotation period is 10 µs which is much longer than the microwave pulse.For NV centers with different orientations, the Rabi frequencies are different.The measured Rabi frequencies are 7.10 MHz, 6.57MHz and 2.80 MHz when the applied microwave frequencies are 2.936 GHz (dip 1), 3.009 GHz (dip 2) and 3.129 GHz (dip 3), respec-tively (Fig. 4(d)).Fig. 4(e) shows Rabi oscillations of the NV centers with θ = 22 • at different rotation phases.The blue circles and black squares are measured at the rotation phase of ϕ = π/2 and ϕ = π, respectively.The corresponding Rabi frequencies are 2.72 MHz and 2.23 MHz due to the different projections of the microwave magnetic field along the NV axis.We also apply microwave pulse at other rotation phases to explore how it affects the Rabi frequency.Fig. 4(f) shows the Rabi frequency Ω Rabi for NV centers with θ = 22 • (blue circles) as a function of the rotation phase.The Rabi frequency is smallest at ϕ = 3π/2.The red curve is the theoretical prediction, which agrees well with our experimental results.

D. Feedback cooling of the Center-of-Mass motion
To study quantum spin-mechanics and use a levitated diamond for precision measurements, it will be crucial The cooling efficiency can be improved in the future with backward detection by using the backward scattered light of the levitated diamond collected by the objective lens.

III. DISCUSSION
In conclusion, we have levitated a nanodiamond at pressures below 10 −5 Torr with a surface ion trap.We performed ODMR measurement of a levitated nanodiamond in high vacuum for the first time.The internal temperature of the levitated nanodiamond remains stable at about 350 K when the pressure is below 5 × 10 −5 Torr, which means stable levitation with an ion trap will not be limited by heating even in ultrahigh vacuum.This offers a unique platform for studying fundamental physics, such as massive quantum superposition [25,26,39].
Additionally, we apply a rotating electric field that exerts a torque on the levitated nanodiamond to drive it to rotate at high speeds up to 20 MHz.20 MHz rotation can generate a pseudo-magnetic field of 0.71 mT for an electron spin, and a pseudo-magnetic field of 6.5 T for an 14 N nuclear spin.With this method, the rotation frequency of a levitated nanodiamond is extremely stable and easily controllable.The effect of the Berry phase generated by rotation [35] is observed with the embedded NV center electron spins.This will be useful for creating a gyroscope for rotation sensing [36,58,59].We also demonstrate quantum control of rotating NV centers in high vacuum, which will be important for using spins to create nonclassical states of mechanical motion [25,26,39].Using feedback cooling, the CoM of the levitated nanodiamond is cooled in all three directions with a minimum temperature of about 1.2 K along one direction.
The maximum rotation frequency in this experiment is limited by the bandwidth of the multichannel waveform generation system for generating the phase-shifted signals on the four electrodes.The rotation frequency can be much higher with a better waveform generation system.Furthermore, in the presence of a DC external magnetic field, the NV centers within a rotating nanodiamond experience an AC magnetic field.Quantum sensing of an AC magnetic field can have a higher sensitivity compared to that of a DC magnetic field [63].Consequently, the mechanical rotation can enhance the sensitivity of a magnetometer in measuring DC magnetic fields.By using purer diamond particles, i.e.CVD diamonds, a higher excitation power of the 532 nm laser can be employed to reduce the initialization time of NV centers.

A. Experiment setup and materials
The surface ion trap is fabricated on a sapphire wafer by photolithography.The chip is fixed on a 3D stage and installed in a vacuum chamber.The AC high voltage signal used to levitate nanoparticles and the microwave used for quantum control are combined with a bias tee to be delivered to the chip.A 532 nm laser beam is incident from the bottom to excite diamond NV centers.The photoluminescence (PL) is collected by an objective lens with a numerical aperture (NA) of 0.55.A 1064 nm laser beam focused by the same objective lens is used to monitor both the center-of-mass (CoM) motion and the rotation of the levitated nanoparticle.The PL is separated with the 532 nm laser and the 1064 nm laser by dichroic mirrors.The counting rate and optical spectrum of the PL are measured by a single photon couting module and a spectrometer.The processes of particle launching and trapping are monitored by two cameras.
The diamond particles were acquired from Adamas Nano.The product model is MDNV1umHi10mg (1 mi-cron Carboxylated Red Fluorescence, 1 mg/mL in DI Water, ∼3.5 ppm NV).The experimental data shown in the main text of the manuscript are obtained from four different diamond particles.The data presented in Fig. 1, Fig. 2, and Figs.3(a-c) originate from measurements conducted on the same nanodiamond particle.Figs.3(df) show the data from a second nanodiamond particle, while the data in Fig. 4 is measured using the third nanodiamond particle.Fig. 5 uses the fourth diamond particle.

B. Internal temperature of a levitated nanodiamond
In the experiment, we measure the ODMR of levitated nanodiamond NV centers to detect the internal temperature in the absence of an external magnetic field.The zero-field Hamiltonian of NV center is: x − S 2 y /ℏ, where D is the zero-field energy splitting between the states of |m s = 0⟩ and |m s = ±1⟩, E is the splitting between the states due to the strain effect.The small splitting between two dips in the ODMR spectra (Fig. 1(e)) without an external magnetic field is due to the E term from strain in the nanodiamond.The zero-field splitting D is dependent on temperature [41,50]: where c 0 = 2.8697 GHz, c 1 = 9.7 × 10 −5 GHz/K, c 2 = −3.7 × 10 −7 GHz/K 2 , c 3 = 1.7 × 10 −10 GHz/K 3 , ∆ pressure = 1.5×10 −6 GHz/bar, and ∆ strain is caused by the internal strain effect.∆ pressure is smaller and can be neglected in vacuum.Fig. 1(e) is the ODMR measured at the pressure of 10 Torr (blue circles) and 6.9 × 10 −6 Torr (red squares).The zero-field splitting obtained by fitting can be used to calculate the temperature of the levitated nanodiamond.
The internal temperature T of a levitated nanodiamond is determined by the balance between heating and cooling effects [51,52]: where A a = λ η λ I λ V is the heating of the excitation laser (λ = 532 nm) and the detecting laser (λ = 1064 nm), η λ is the absorption coefficient of nanodiamond and I λ is the laser intensity, V is the volume of nanodiamond.The first term at the right side of the equation is the cooling rate caused by gas molecule collisions, A gas = 1 2 κπR 2 vT 0 γ ′ +1 γ ′ −1 , κ ≈ 1 is the thermal accommodation coefficient, R is the radius of nanodiamond, v is the mean thermal speed of gas molecules, γ ′ is the specific heat ratio (γ ′ = 7/5 for air near room temperature), p is the pressure, T 0 is the thermal temperature.The last term is the cooling rate of blackbody radiation.A bb = 72ζ (5) V k 5  B / π 2 c 3 ℏ 4 Im ε−1 ε+2 , where ζ (5) ≈ 1.04 is the Riemann zeta function, k B is the Boltzmann constant, c is the vacuum light speed, ℏ is the reduced Planck's constant, ε is a constant and time-independent permittivity of nanodiamond across the black-body radiation spectrum.By measuring the internal temperature as a function of the intensities of the 532 nm laser and the 1064 nm laser, the absorption coefficients of the nanodiamond are estimated to be 111 cm −1 at 532 nm and 5.87 cm −1 at 1064 nm (Supplementary Fig. 3).The dimension of the surface ion trap is designed as a = 270µm and b = 450µm.We theoretically calculate the trapping potential of a levitated nanodiamond in z axis, as shown in Supplement Fig. 2(c).The red dash-dotted curve and blue solid curve are calculated by Eq.A4 and Eq.A5, respectively.All the parameters are summarized in Table I.The theoretical trapping position z 0 is 245 µm, which is very close to the simulation result 253 µm for the current ion trap design.The difference is due to the asymmetric ion trap design.
The trapping potential is dependent on the eigenfrequency of a levitated particle, which is proportional to the charge to mass ratio (Q/m).Thus, it is necessary to increase the charge number carried on particles to achieve stable levitation in an ion trap.In our experiment, the diamond particles were purchased from Adamas Nano and the product model is MDNV1umHi10mg (1 micron Carboxylated Red Fluorescence, 1 mg/mL in DI Water, 3.5 ppm NV).These particles exhibit an average size of 750 nm.They are created by irradiating 2-3 MeV electrons on diamonds manufactured by static high-pressure, high-temperature (HPHT) synthesis and containing about 100 ppm of substitutional N. The nanodiamonds are first sprayed out by electrospray, which is supplied by a DC high voltage (∼ 2kV ).Then the nanodiamonds are delivered to the trapping region of the surface ion trap with an extra linear Paul trap.
After a nanodiamond is trapped, we apply a 1064 nm laser to measure the CoM motion of the levitated nanodiamond.Supplementary Fig. 2(d) is the PSDs of the CoM motion in x,y,z directions at the pressure of 0.01 Torr.The radius of the levitated nanodiamond is obtained to be about 264 nm based on the fitting of the PSDs.The experimental trapping frequency in z direction is about ω z /2π = 1.65 kHz.Using Eq.A3, the charge number is estimated to be about 2000 for this nanodiamond.The surface ion trap creates an extremely deep potential well of 420 eV (Supplementary Fig. 2(c)).According to our experimental results, the charge number of different levitated nanodiamonds varies from 1,000 to 10,000.nanodiamond is aligned to the direction of the electric field (E xy ) by the torque where β is the angle between the dipole moment and the electric field, z is the unit vector along z direction.
Based on the simulation as shown in Supplementary Fig. 4, the amplitude and the direction (α) of the electric field in the xy-plane can be calculated, which is displayed in Supplementary Fig. 5(a) and 5(b).α = 0 indicates that the rotating electric field points to the positive x direction.The blue circles and red squares are the simulations with and without the compensation electrodes GND 1 and GND 2. Ideally, the direction of the rotating electric field should be α (t) = π/4−ωt(orange dashed curve).The inset of Supplementary Fig. 5(b) is the asynchrony between the simulation result and an ideal rotation field with and without the compensation electrodes.The orientation of the electric field does not perfectly rotate at a constant speed in one period without the compensation electrodes (blue circles).The maximum deviation is 5.5 • .It hurts the stability of the rotational motion of the levitated nanodiamonds and expand the linewidth of nanodiamond's rotation signal.Moreover, the E z component of the rotating electrical field oscillates with a large amplitude if there are no compensation electrodes (Supplementary Fig. 5(c)).The electrical field drives a levitated nanodiamond to oscillate in the z direction, causing the loss of the levitated nanodiamond in high vacuum.The two compensation electrodes effectively solve these issues.The transmittance of a microwave through the Ωshaped circuit is simulated (Supplementary Fig. 5(f)) to ensure the microwave has low loss for frequencies from 2.6 GHz to 3.1 GHz.
Then we drive a levitated nanodiamond to rotate using the rotating electric field.The PSD of the rotational motion at the rotation frequency of 0.1 MHz is shown in Supplementary Fig. 5(d).The linewidth of the rotation signal is about 9.9 × 10 −5 Hz based on a Lorentzian fitting (inset of Supplementary Fig. 5(d)).Thus, the ratio of the center frequency to the linewidth is 2 × 10 9 , demonstrating the rotational motion is ultra-stable with easy control by this

method.
Meanwhile, the rotational motion of the levitated nanodiamond is damped by the interaction with the remaining gas molecules in vacuum chamber.The damping torque of a sphere is [20,66] where I is the moment of inertia of the nanodiamond, ω r is the angular velocity, γ d = 40η ′ pR 2 /3mv is the damping rate of rotational motion, η ′ ≈ 1 is the accommodation factor accounting for the efficiency of the angular momentum transferred onto the nanodiamond by gas molecule collisions.Thus, the rotational motion equation can be written as: The maximum rotation frequency of the levitated nanodiamonds is obtained at M electric = −M gas and β = π/2, which is limited by the pressure in the vacuum chamber.The maximum rotation frequency at a certain pressure is: We measure the upper limit of the rotation frequency at different pressures (Supplementary Fig. 5(e)).The PSDs as functions of air pressure are measured at the rotation frequencies of 0.05 MHz, 0.1 MHz, 0.2 MHz, 0.5 MHz, 1 MHz, 2 MHz, 5 MHz and 10 MHz.The levitated nanodiamond stops rotating when the pressure is too large for that rotation frequency.The maximum rotation frequency is inversely proportional to the pressure (white dashed curve).The dipole moment of the nanodiamond (R = 264 nm) is estimated to be about |p| = 3.13 × 10 −25 C•m (1.96 e•µm).We can adjust and lock the rotation of the levitated nanodiamond at arbitrary frequency and pressure in the region below the white dashed curve.The maximum rotation frequency is ω r = 2π × 20 MHz in this experiment, which is limited by the π-phase shifter (Mini-Circuits, ZFSCJ-2-2-S) used to generate the signals on the four electrodes.Under favorable conditions, the rotational motion can achieve a frequency exceeding 10 GHz at the pressure of 10 −6 Torr based on the dashed line in Supplementary Fig. 5(e).In a rotating diamond, the embedded NV centers follow the rotation of the particle with an angular frequency of ω r .Considering an arbitrary NV center in a diamond at the time of t, the angle between the NV axis and z axis is θ, and the azimuth angle is ϕ(t) = ω r t relative to x axis.In the absence of an external magnetic field and neglecting strain effects, the Hamiltonian of the rotating NV center in the laboratory frame can be written as [34] H where D is the zero-field splitting, R (t) = R z (ϕ (t)) R y (θ) is the rotation transformation, and R y (θ) = exp (−iθS y ) (R z (ϕ) = exp (−iϕS z )) expresses the rotation of spin around the y (z) axis in terms of θ (ϕ), S is the spin operator.The Hamiltonian possesses three eigenstates |m s , t⟩ lab = R (t) |m s , 0⟩ lab (m s = 0, ±1), For a quantum system in an eigenstate, the system remains in the eigenstate and acquires a phase factor during an adiabatic evolution of the Hamiltonian.This factor arises from both the state's time evolution and the variation of the eigenstate with the changing Hamiltonian.The second term specifically corresponds to the Berry phase.Hence, the expression for the time-dependent spin state is [57] e iγm s e −iH 0,lab t/ℏ |m s , t⟩ lab = e iγm s e −iH 0,lab t/ℏ e −iϕSz e −iθSy |m s , 0⟩ lab , where γ ms is the Berry phase.Here, the diamond particle rotates around the z axis with a constant θ, the Berry phase can be calculated as [57] γ ms = i The Berry phase of Eq.D4 is calculated for an open-path, which is gauge-dependent.However, for a closed loop, the Berry phase is gauge-invariant and can be expressed as m s [−2π (1 − cos θ)].The result is equivalent to Eq. D4 of m s (2π cos θ).
The spin state of the NV center is observed through the interaction with a microwave magnetic field.In our experiment, the direction of the microwave is in the yz-plane and forms a slight angle θ ′ relative to the z axis, resulting from the asymmetric design of the waveguide.The Hamiltonian of the microwave in the laboratory frame can be written as which contains two components, the longitudinal term H M W,z,lab = gµ B B M W cos (ω M W t) S z cos θ ′ and the transverse term H M W,y,lab = gµ B B M W cos (ω M W t) S y sin θ ′ .First, we consider the longitudinal term H M W,z,lab .The expected value of the spin states can be expressed as lab ⟨±1, t| e iH 0,lab t/ℏ e −iγ±1 H M W,z,lab e iγ0 e −iH 0,lab t/ℏ |0, t⟩ lab = gµ B B M W cos (ω M W t) cos θ ′ lab ⟨±1, 0| e iθSy e iϕSz e iH 0,lab t/ℏ e −iγ±1 S z e iγ0 e −iH 0,lab t/ℏ e −iϕSz e −iθSy |0, 0⟩ lab = gµ B B M W cos (ω M W t) cos θ ′ e −i(γ±1−γ0) e i(E±1−E0)t/ℏ lab ⟨±1, 0| e iθSy e iϕSz S z e −iϕSz e −iθSy |0, 0⟩ lab = 1  2 gµ B B M W cos θ ′ e iω M W t + e −iω M W t e ∓iωrt cos θ e iDt lab ⟨±1, 0| e iθSy S z e −iθSy |0, 0⟩ lab = 1  2 gµ B B M W cos θ ′ e i(ω M W ∓ωr cos θ+D)t + e i(−ω M W ∓ωr cos θ+D)t lab ⟨±1, 0| e iθSy S z e −iθSy |0, 0⟩ lab ≈ 1  2 gµ B B M W cos θ ′ e i(−ω M W +D∓ωr cos θ)t lab ⟨±1, 0| e iθSy S z e −iθSy |0, 0⟩ lab , (D6) where the E ms is the corresponding eigenvalue of the Hamiltonian H 0,lab for the spin state |m s , t⟩.According to Eq. D6, the transformation of spin states from |m s = 0⟩ lab to |m s = ±1⟩ lab can be driven by a microwave operating at the resonance frequency of D ∓ ω r cos θ, where the frequency shift ∓ω r cos θ is attributed to the Berry phase.

(D7)
The expected values are written as lab ⟨+1, t| e iH 0,lab t/ℏ e −iγ+1 H M W,y,lab e iγ0 e −iH 0,lab t/ℏ |0, t⟩ lab = 1 4i gµ B B M W sin θ ′ e i(ω M W −ωr cos θ+D+ωr)t + e i(−ω M W −ωr cos θ+D+ωr)t × lab ⟨+1, 0| e iθSy S + e −iθSy |0, 0⟩ lab ≈ 1 4i gµ B B M W sin θ ′ e i(−ω M W +D+ωr−ωr cos θ)t lab ⟨+1, 0| e iθSy S + e −iθSy |0, 0⟩ lab , (D8) lab ⟨−1, t| e iH 0,lab t/ℏ e −iγ−1 H M W,y,lab e iγ0 e −iH 0,lab t/ℏ |0, Utilizing Eq.D8 and Eq.D9, the resonance frequency of microwave for transforming the spin state from |m s = 0⟩ lab to |m s = ±1⟩ lab is D ± ω r (1 − cos θ).In addition to the frequency shift of ∓ω r cos θ caused by the Berry phase, there is another term of ω r coming from the rotational Doppler effect [57].In our experiment, the angle θ ′ of the microwave relate to the z axis is approximately 8.5 • .Consequently, the dominant transition probability arises from the longitudinal component, characterized by a frequency shift of ∓ω r cos θ due to the Berry phase.The energy levels of NV centers with four orientations are degenerate in the absence of an external magnetic field.When the nanodiamond undergoes rotation, the electron spin resonance frequency experiences a shift due to the Berry phase, and this shift depends on the angle θ between the NV axis and the rotation axis.The electron spin resonance frequencies of NV spins along different orientations become non-degenerate.The ODMR of NV at different orientations are theoretically calculated by Eq.D6 at the rotation frequency of 20 MHz (Supplementary Fig. 6(a)).The orientations corresponding to the blue solid curve and the red dashed curve are θ = 0 • and θ = 45 • , respectively.The intrinsic linewidth is 2π × 19 MHz, and the strain effect splitting E is 2π × 6.7 MHz.The eight dips are not separated in the ODMR spectrum at a rotation frequency of a few MHz because of the large linewidth.Here we use the FWHM parameter of the ODMR spectrum to indicate the frequency shift by the Berry phase of a rotating NV center.The FWHM of the ODMR is mainly determined by the splitting of the NV centers that have the smallest θ.Supplementary Fig. 6(b) shows the FWHM of the ODMR as a function of rotation frequency.The NV centers, which have the smallest θ, show the highest sensitivity of the frequency shift due to the Berry phase .

With an external magnetic field
To precisely measure the frequency shift induced by the Berry phase of a rotating NV center, an external magnetic field B along the z direction can be applied.This serves to distinguish the energy levels of NV centers in four different orientations.The Hamiltonian of a NV center in the laboratory frame with the external magnetic field can be expressed as In the rotating frame, the Hamiltonian of the NV center can be calculated by a unitary transformation, where the unitary operator is defined as U = e iθSy e iϕSz .The second term on the right side of Eq.D11 represents Zeeman interaction arising from the pseudo-magnetic field due to the rotation of the NV center.In the case of an adiabatic process, ω r ≪ D − gµ B B cos θ, the second term is significantly weaker than the first term, and can be treated as a perturbation.We neglect the off-diagonal terms in the first component since gµ B B ≪ D, which are too small to induce significant mixing of the NV spin states.
Therefore, the Hamiltonian in the rotating frame possesses three eigenstates, |m s ⟩ rot (m s = 0, ±1): and the corresponding eigenvalues are ℏ (D + gµ B B cos θ), 0, ℏ (D + gµ B B cos θ), respectively.The new Hamiltonian in the laboratory frame can be transformed by applying the rotation transformation R (t) = e −iϕSz e −iθSy , The eigenstates of the Hamiltonian in the laboratory frame also can be calculated through the rotation transformation, which are same as the eigenstates of the Hamiltonian without the external magnetic field (Eq.D2).Thus, the Berry phase of the rotating NV center is: .
The transverse component can be expressed as lab ⟨±1, t| e iH lab t/ℏ e −iγ±1 H M W,y,lab e iγ0 e −iH lab t/ℏ |0, t⟩ lab = gµ B B M W cos (ω M W t) sin θ ′ lab ⟨±1, 0| e iθSy e iϕSz e iH lab t/ℏ e −iγ±1 S y e iγ0 e −iH lab t/ℏ e −iϕSz e −iθSy |0, 0⟩ lab = gµ B B M W cos (ω M W t) sin θ ′ e −i(γ±1−γ0) e i(E B,±1 −E B,0 )t/ℏ lab ⟨±1, 0| e iθSy e iϕSz S y e −iϕSz e −iθSy |0, 0⟩ lab = 1  2 gµ B B M W sin θ ′ e iω M W t + e −iω M W t e ∓iωrt cos θ e i(D±gµ B B cos θ)t × lab ⟨±1, 0| e The transformation resonance frequency is D ± gµ B B cos θ ± ω r (1 − cos θ) between the |m s = 0⟩ lab state and |m s = ±1⟩ lab state.The corresponding frequency shift duo to the Berry phase also is ±ω r cos θ, and the frequency shift induced by the rotational Doppler effect is ω r .Similar to the case of zero external magnetic field, the predominant transition probability arises from the longitudinal component, characterized by a frequency shift of ∓ω r cos θ due to the Berry phase.Supplementary Fig. 6(c) shows the frequency shift induced by the Berry phase in a levitated nanodiamond rotating counterclockwisely (viewed from the positive z direction unless otherwise specified).The external magnetic field along the z direction is about 100 G.The resonance frequency transition between |m s = 0⟩ lab state and |m s = +1⟩ lab state decreases with an increasing of the rotation frequency, in contrast to the behavior observed in the levitated nanodiamond rotating clockwise.The red curve is the theoretical calculation for the angle of θ = 21.5 • between the NV axis and the rotating axis.The experimental data is in excellent agreement with the theoretical calculation, suggesting a consistent orientation of the NV centers at various rotation frequencies.

Pseudo-magnetic field due to rotation
The Berry phase observed in the laboratory frame is equivalent to the pseudo-magnetic field (called the Barnett field in [57]) in the rotational frame.In our experiment, the mircowave source is fixed in the laboratory frame.Only the levitated nanodiamond is rotating.Thus, we observe the effect of the Berry phase [57].It will be beneficial to also consider this system in the rotational frame.The electron spin resonance frequency shift of the rotating NV center involves the combination of the pseudo-magnetic field and the rotational Doppler effect in the rotating frame.As expressed in Eq.D11, the Hamiltonian of the pseudo-magnetic field in the rotating frame, induced by the rotation of a diamond particle, can be given by

Figure 1 .
Figure 1.Stable levitation of a nanodiamond in high vacuum.(a) Schematic of a levitated nanodiamond in a surface ion trap.The center ring electrode is grounded (GND).It has a hole at its center for sending a 1064 nm laser to monitor the nanodiamond's motion.A combination of a low-frequency high voltage (HV) and a high-frequency microwave (MW) is applied to the Ω-shaped circuit to trap the nanodiamond and control the NV centers.(b) Energy level diagram of a diamond NV center.A 532 nm laser (green arrow) excites the NV center.The red solid arrows and gray dashed arrows represent radiative decays and nonradiative decays, respectively.(c) Simulation of the electric field of the ion trap in the xy-plane (top) and in the xz-plane (bottom) when a voltage of 200 V is applied to the Ω-shaped circuit.The trap center is 253 µm away from the chip surface.(d) Power spectrum densities (PSDs) of the center-of-mass (CoM) motion of the levitated nanodiamond at the pressure of 0.1 Torr (blue) and 9.8 × 10 −6 Torr (red).(e) Optically detected magnetic resonances (ODMRs) of the levitated nanodiamond measured at 10 Torr (blue circles) and 6.9 × 10 −6 Torr (red squares).The blue and red dashed lines are the corresponding zero-field splittings.The intensities of the 532 nm laser and the 1064 nm laser are 0.030 W/mm 2 and 0.520 W/mm 2 , respectively.(f) Internal temperature of the levitated nanodiamond as a function of pressure with the same laser intensities as shown in (e).

Figure 2 .
Figure 2. Fast rotation of a levitated nanodiamond.(a) Optical image of the surface ion trap.AC voltage signals (A sin (ωt + φ)) with the same frequency (ω) and amplitude (A) but different phases (φ) are applied to the four corner electrodes to generate a rotating electric field.The phase is different by π/2 between neighboring electrodes.DC1, DC2, DC3 and DC4 are compensation voltages that minimize the micromotion to stabilize the trap.(b) Simulation of the electric potential in the z = 253 µm plane at t = 0.The amplitude is A = 10 V. (c) PSDs of the rotational motion of the levitated nanodiamond at the rotation frequencies from 0.1 MHz to 20 MHz.The pressure is 1.0 × 10 −4 Torr.

Figure 3 .
Figure 3. Effects of the Berry phase generated by a rotating nanodiamond.(a) Schematic of an NV center in the nanodiamond rotating around the z axis in the absence of an external magnetic field.The small angle between BMW and the z axis is due to the asymmetric design of the waveguide.(b) ODMRs of the levitated nanodiamond at rotation frequencies of 0.1 MHz (blue circles) and 14 MHz (red squares).(c) Experimentally measured FWHM of the ODMR spectrum as a function of rotation frequency (blue circles).The red solid curve and orange dashed curve are theoretically calculated FWHMs at θ = 0 • and θ = 20 • , respectively.(d) Schematic of an NV center in the nanodiamond rotating around the z axis in an external magnetic field.The magnetic field is along the z axis and is about 100 G. (e) The upper panel (red squares) shows the ODMR of the levitated nanodiamond at a rotation frequency of 0.1 MHz and a pressure of 1.0 × 10 −4 Torr.The bottom panel (gray circles) shows the ODMR of a nanodiamond without a stable rotation at the pressure of 10 Torr.The corresponding solid curves are the fittings with eight Lorentzian dips.(f) Experimentally measured frequency of the right-most dip of the ODMR spectrum of a NV center as a function of rotation frequency (blue circles).The green solid curve and violet dashed curve are theoretical calculations at θ = 20.7 • and θ = 24.0• , respectively.The magenta dashed curve is a linear fitting of the resonance frequency.

Figure 4 .
Figure 4.Quantum control of NV centers in a levitated nanodiamond in high vacuum with a rotation frequency of 100 kHz.(a) Schematic of the Rabi oscillation measurement at different rotation phase ϕ(t).The angle θ ′ between the magnetic component of microwave and the z axis is 8.5 • .(b) Pulse sequence of the Rabi oscillation measurement.(c) ODMR of the levitated nanodiamond.(d) Measured Rabi oscillations of NV centers at three different orientations.The Rabi frequencies are 7.10 MHz, 6.57MHz and 2.80 MHz at the ODMR frequencies of 2.935 GHz, 3.009 GHz and 3.129 GHz, respectively.(e) Rabi oscillation of NV centers with θ = 22 • corresponding to the resonance frequency of 3.129 GHz (dip 3).The blue circles and black squares are measured at rotation phase of ϕ = π/2 and ϕ = π, respectively.(f) Rabi frequency at θ = 22 • (blue circles) as a function of the rotation phase ϕ.The red curve is the theoretical prediction.

Figure 5 .
Figure 5. Feedback cooling of the CoM motion of a levitated nanodiamond in the ion trap.(a) Schematic diagram of the feedback cooling method.(b), (c), (d): PSDs of the CoM of the levitated nanodiamond along the (b) x, (c) y and (d) z directions at the pressure of 0.02 Torr without cooling (blue curves) and at the pressure of 2.0 × 10 −5 Torr with feedback cooling (red curves).The orange curves are the noise floors.Based on the fitting, the effective temperature of the CoM motion with feedback cooling are 1.2 ± 0.3 K, 3.5 ± 0.4 K and 86 ± 26 K along the x, y and z directions, respectively.

Supplemental Fig. 3 .
Internal temperature of a levitated nanodiamond.(a) ODMRs of the levitated nanodiamond at different intensities of the 532 nm laser and the 1064 nm laser at the pressure of 1.3 × 10 −5 Torr.(b) Internal temperature as a function of the intensity of the 532 nm laser.The intensity of the 1064 nm laser is 0.520 W/mm 2 .(c) Internal temperature as a function of the intensity of the 1064 nm laser.The intensity of the 532 nm laser is 0.030 W/mm 2 .

SupplementalFig. 5 .
Simulation of the rotating electric field and the rotational motion of a levitated nanodiamond.Time evolution of the xy-plane component (a) and direction (b), z component (c) of the electric field with (blue circles) and without (red squares) the compensation electrodes of GND1 and GND2.α describes the orientation of the electric field (α = 0 indicates the electric field points to positive x direction).(d) PSD of the rotational motion at the rotation frequency of 0.1 MHz.The linewidth is 9.9 × 10 −5 Hz by the fitting (inset).The ratio of the center frequency to the linewidth is 2 × 10 9 .(e) PSDs of the rotational motion of the levitated nanodiamond at different pressures, showing the maximum rotation frequency at different pressures.The rotation frequencies are 0.05 MHz, 0.1 MHz, 0.2 MHz, 0.5 MHz, 1 MHz, 2 MHz, 5 MHz and 10 MHz, respectively.The upper limit of the rotation frequency is inversely proportional to pressure with the electric field driving.Particles stop rotating above the dashed white line.(f) Simulation of microwave transmittance of the surface ion trap.

Supplemental Fig. 6 .
Berry phase induced by a rotating nanodiamond.(a) Theoretically calculated ODMR of NV centers with different orientations at the rotation frequency of 20 MHz.The blue solid, red dashed and orange dash-dotted curves are calculated at θ = 0 and θ = 45 • , respectively.(b) Theoretically calculated FWHM of the ODMR as a function of rotation frequency.The blue solid curve and the red dashed curve are calculated for θ = 0 and θ = 45 • , respectively.(c) Experimental results of the frequency shift due to the Berry phase induced by counterclockwise rotation (blue circles), and theoretical calculated resonance frequency as a function of the rotation frequency at θ = 21.5 • (red curve).

Table I .
Parameters for the pseudopotential calculation of equivalent surface ion trap.qz satisfies the condition of stable 3D trapping.