Coherent Control of a Nitrogen-Vacancy Center Spin Ensemble with a Diamond Mechanical Resonator

Coherent control of the nitrogen-vacancy (NV) center in diamond's triplet spin state has traditionally been accomplished with resonant ac magnetic fields under the constraint of the magnetic dipole selection rule, which forbids direct control of the $|-1>\leftrightarrow |+1>$ spin transition. We show that high-frequency stress resonant with the spin state splitting can coherently control NV center spins within this subspace. Using a bulk-mode mechanical microresonator fabricated from single-crystal diamond, we apply intense ac stress to the diamond substrate and observe mechanically driven Rabi oscillations between the $|-1>$ and $|+1>$ states of an NV center spin ensemble. Additionally, we measure the inhomogeneous spin dephasing time ($T_{2}^{*}$) of the spin ensemble using a mechanical Ramsey sequence and compare it to the dephasing times measured with a magnetic Ramsey sequence for each of the three spin qubit combinations available within the NV center ground state. These results demonstrate coherent spin driving with a mechanical resonator and could enable the creation of a phase-sensitive $\Delta$-system within the NV center ground state.

Spin-based quantum systems typically rely on resonant magnetic fields to drive coherent transitions between different spin states. Although such magnetic driving has been effective, developing alternative modes of control opens new routes for coupling disparate quantum states to form a hybrid quantum system [1]. New techniques for manipulating a spin state also naturally extend to new sensing capabilities and an enhanced understanding of how spin systems interact with their environment.
The spin triplet ground state of the nitrogen-vacancy (NV) center in diamond represents a coherently addressable paramagnetic defect confined within a largely non-magnetic carbon lattice. This creates an excellent laboratory for studying how spin-based quantum systems interact with their environment [2] and for exploring new methods of quantum control [3]. Studies have shown that NV center spins can be controlled magnetically [4], optically [5,6], electrically [7], and mechanically [8][9][10]. The direct spin-phonon coupling that enables mechanical spin control mediated by lattice strain has prompted the experimental development of single-crystal diamond mechanical resonators [8][9][10][11] and motivated theoretical calculations showing that this interaction could enable spin squeezing [12] and mechanical resonator cooling [13]. Nonetheless, coherent Rabi driving of NV center spins with a mechanical resonator has not been previously demonstrated. Furthermore, understanding the dynamics of mechanical driving in spin ensembles could have applications in NV center-based sensing and quantum optomechanics where spin-phonon interactions can be enhanced by using a large number of spins.
Here we use a mechanical microresonator to apply a large amplitude ac stress to a single crystal diamond. Building on recent spectroscopy experiments [8], we tune the frequency of this stress wave into resonance with the |(m s =) − 1 ↔ |+1 spin transition to mechanically drive Rabi oscillations of an NV center spin ensemble. Using this capability, we measure the inhomogeneous dephasing time for an ensemble of mechanically controlled NV center spin qubits to be T * 2 = 0.45±0.05 µs and compare this result to T * 2 for magnetically driven qubits constructed from the same NV center ensemble. We find that the mechanically driven {−1, +1} qubit coherence is similar to that of a magnetically driven {−1, +1} qubit, and these {−1, +1} qubits dephase twice as quickly as magnetically driven {0, −1} or {+1, 0} qubits.
NV centers couple to mechanical stress (σ ⊥ and σ ) and magnetic fields (B ⊥ and B ) through their ground-state spin Hamiltonian (shown schematically in Fig. 1a) where D 0 /2π = 2.87 GHz is the zero-field splitting, γ N V /2π = 2.8 MHz/G is the gyromagnetic ratio, ⊥ /2π = 0.015 MHz/MPa and /2π = 0.012 MHz/MPa are the perpendicular and axial stress coupling constants [10,14], P/2π = −4.945 MHz and A /2π = −2.166 MHz are the hyperfine parameters [15][16][17], and S x , S y , S z (I x , I y , I z ) are the x, y, and z components of the electronic (nuclear) spin-1 operator. The NV center symmetry axis defines the z-axis of our coordinate system as depicted in Fig. 1b [10,14]. Non-axial stress σ ⊥ couples the |−1 and |+1 spin states, enabling coherent control of the magnetically-forbidden ∆m s = ±2 spin transition and providing direct access to the {−1, +1} spin qubit. This qubit combination has recently become a topic of interest because it is isolated from thermal fluctuations [18] and can make a more sensitive magnetometer than either the {0, −1} or {+1, 0} qubit [18,19].
In this work, we use two devices, both fabricated from type IIa, 100 "optical grade" diamonds purchased from Element Six. These samples are specified to contain fewer than 1 ppm nitrogen impurities, and each contained a native NV ensemble as received.  reverse side of each diamond, we fabricate a loop antenna that produces gigahertz-frequency magnetic fields for conventional magnetic spin control. Fig. 1d depicts a schematic version of the resulting device.
To perform mechanically driven spin coherence measurements, we first tune the axial magnetic field B to bring the spins into resonance with a high-frequency stress wave as described in Ref. [8]. At this resonant B , we mechanically drive Rabi oscillations of the {−1, +1} qubit. Fig. 2a shows the pulse sequence used to drive Rabi oscillations in the relatively low Q modes of Sample A. To initialize the NV center spins, we first optically polarize into |0 and then transfer the spin population from |0 to |−1 with a magnetic π-pulse. Next, we apply a mechanical Rabi pulse of length τ that is resonant with the |−1 ↔ |+1 spin transition. To read out the spin signal, a second magnetic π-pulse shuttles the population in |−1 to |0 . Fluorescence measurement of the |0 state population reveals how much spin population was transferred to |+1 according to the relation P |+1 = 1 − P |0 [20]. In order to maintain a constant average power to the device, we apply a second mechanical pulse at each data point of length L − τ where L is the length of the longest Rabi pulse. This pulse comes before fluorescence read out but does not affect our measurement since the spin population we detect has left the {−1, +1} subspace. Fig. 2b shows mechanically driven Rabi oscillations as measured on Sample A for 33 dBm of input power to the HBAR.
The damping observed in Fig. 2b arises from a combination of spin dephasing from magnetic bath noise and dephasing derived from spatial variations in the amplitude of the stress standing wave within the spin ensemble. NV centers near an anti-node of the stress wave feel a larger Rabi frequency than NV centers near a node. The finite collection volume of our confocal microscope necessitates measuring a distribution of coupling strengths, which causes the measured spin signal to dephase. To account for both of these dephasing sources, we model the data in Fig. 2b with the spatially-weighted average where the factor of 1/3 arises because we drive only one of the unpolarized nuclear spin sublevels, Ω(z) = Ω mech |sin 2πz λ A | is the mechanical driving field, λ A is the wavelength of the stress standing wave, and g(z, z 0 ) represents a Gaussian approximation to the microscope point spread function (PSF) with a FWHM that grows linearly with the depth of focus inside the diamond z 0 as described in Ref. [8]. We assume resonant driving and include quasi-static spin bath noise as a random detuning δ drawn from a Gaussian distribution with a standard [21]. The mechanical Ramsey measurement presented below sets T * 2 = 0.45 µs in the {−1, +1} subspace. With the parameters Ω mech /2π = 1.0 MHz, λ A = 19.9 µm, and z 0 = 18 µm as inputs, we average 200 iterations of the simulation to produce the model curve in Fig. 2b, which is not a fit to the experimental data.
For devices with Q-factors substantially larger than Sample A, we find a standard Rabi pulse sequence is not effective. In these devices, the large bandwidth of short microwave pulses reduces their spectral precision, which in turn distorts the coupling between the mechanical resonator and its microwave drive. This becomes important in the higher Q  τ r = 2Q/ω m [22]. As before, the model -which is not a fit to the data -accounts for driving field inhomogeneities by applying a spatially-weighted average over an approximated optical PSF and includes quasi-static magnetic bath noise through a randomized detuning. The SI provides additional details on the pulse sequence and model [14].
For the measurement shown, τ mag = L+τ r = 5.41 µs where L = 3 µs. As such, the critical delay τ c = 6.03 µs corresponds to the largest mechanical pulse area enclosed between the two magnetic π-pulses. To either side of this time step, the pulse area decreases at roughly the same rate. The asymmetry in the data about this point arises because for τ 0 < τ c the mechanical pulse amplitude and thus instantaneous driving field is higher than when τ 0 > τ c .
This larger instantaneous driving field offers the spins better protection from magnetic bath noise as evinced by the larger amplitude Rabi oscillations. Our model correctly reproduces this asymmetry, demonstrating the possibility of using a mechanical driving field to achieve continuous dynamical decoupling of an NV center spin from a spin bath [23].
By modeling the resonator ringing as described above, we can convert the mechanical pulse area between the two magnetic pulses into the "square-pulse" units typically used in magnetic Rabi measurements. Fig. 3c shows mechanical Rabi oscillations plotted as a function of this normalized Rabi interval for measurements taken at several depths inside the diamond substrate. As expected, the oscillations dephase faster near a node in the stress wave due to driving field inhomogeneities within the ensemble. Near the antinode, however, the relative uniformity of the stress wave mitigates this depth-dependence and, thus, the dephasing from driving field inhomogeneities.
The more traditional Rabi pulse protocol used for Sample A provides a direct means to implement conventional pulse sequences. From the data in Fig. 2b, we extract the π/2-pulse time and proceed to measure T * 2 of Sample A with a mechanical Ramsey pulse sequence. Fig. 4 shows the result of this measurement along with Ramsey measurements of T * 2 for magnetically driven {−1, +1}; {0, −1}; and {+1, 0} qubits. Details on the pulse sequences used for each of these measurements are provided in the SI [14]. Although selection rules forbid direct magnetic control of the |−1 ↔ |+1 transition, magnetic control of the {−1, +1} qubit can be accomplished indirectly by using either double-quantum pulses [19] or multi-pulse sequences [24]. Both of these alternatives use the |0 state as a waypoint in the |−1 ↔ |+1 transition. To control the {−1, +1} qubit magnetically, we employ the multipulse sequence described in the SI [14]. We fit the three magnetically driven Ramsey measurements to the function where δ represents a detuning in the driving field, the amplitudes (C 1 , spin sublevels. Therefore, we fit our mechanical Ramsey data to the function where ω rot /2π = 3.5 MHz describes an experimentally introduced phase that accumulates at ω rot t to visualize the decay envelope [14]. Our fitting procedure varies δ, T * 2 , C i , and φ i as free parameters. Since we measure the coherence of a spin ensemble, we extract T * 2 from an exponentially decaying envelope rather than from the Gaussian decay expected for a single NV center [25]. Fig. 4 displays the values of T * 2 extracted from these fits, and the figure caption lists the measured detunings δ.
The inset within each plot depicts a Fourier power spectrum of the corresponding data.
A number of engineering improvements can improve the performance of our devices. First, we expect additional refinements in device fabrication to increase the Q of our devices, which could provide at least a factor of 5 enhancement in the mechanical driving field [26]. Also, working in higher electronic purity diamond will dramatically reduce spin bath induced dephasing, and working with either a single spin or a plane of NV centers would remove dephasing from driving field inhomogeneities. Taken together, these advances can unlock high fidelity quantum control of a mechanically driven qubit.
Our results demonstrate coherent control of all three ground state spin transitions. By simultaneously driving the |0 ↔ |−1 and |+1 ↔ |0 transitions magnetically and the |−1 ↔ |+1 transition mechanically, a ∆-system in which all three states are coupled by a closed-loop interaction contour can be created within the NV center ground state.
Such a system requires at least one parity non-conserving driving field, making ∆-systems an uncommon extension of the more typical Λ-system, which has been well explored in NV centers [5,6,[27][28][29][30]. In a Λ-system, driving field amplitudes and detunings balance to enable phenomena such as coherent population trapping [28,29] and electromagnetically induced transparency [27,30]. In a ∆-system, similar phenomena occur but with an additional sensitivity to the relative phases of the driving fields [31][32][33]. Implementing an NV center ∆-system could, for instance, create a phase induced transparency where the phase of a magnetic driving field tunes the absorption of the mechanical driving field. Such a system could have value in NV center optomechanics experiments as a phase-controlled switch to rapidly gate spin-phonon interactions. Another application could be measuring the relative phase of a resonating mechanical proof mass in an inertial sensor.
In summary, we use a high-frequency mechanical resonator to drive coherent Rabi oscillations of an NV center spin ensemble with driving fields up to Ω mech /2π = 3.8 MHz. This    [35].

SUPPLEMENTARY INFORMATION NV CENTER STRESS COUPLING
Ovartchaiyapong, et al measured the NV center strain coupling to be d ⊥ /2π = 21.5 GHz/strain and d /2π = 13.3 GHz/strain for perpendicular and axial strain, respectively [10]. Since our mechanical resonator generates acoustic waves by applying a pressure to one face of the diamond crystal, we choose to work in units of stress. To convert the measured constants from strain to stress, we first rotate the measured couplings from the coordinate system defined by the NV center to the lattice coordinates. We then use the stiffness matrix for to convert strain/GHz into GPa/GHz (stress/GHz). The elastic constants C ij are given in Table I. Finally, we rotate back into the coordinates of the NV center to find the stress coupling constants ⊥ /2π = 0.015 MHz/MPa and /2π = 0.012 MHz/MPa used in the main text.

MECHANICAL RABI MEASUREMENTS
Readout Through |+1 As a control, we performed a second type of Rabi measurement. In this alternative pulse sequence, after optically pumping the NV center into |0 we once again apply a magnetic π-pulse to resonantly move the population from |0 to |−1 . We then pulse the resonant mechanical driving field for a length τ to drive the |−1 ↔ |+1 transition. Finally, we use a magnetic adiabatic passage to robustly transfer the population that was driven into |+1 to |0 where we read out the spin state optically. This differs from the Rabi measurement presented in the main text in that we extract population from |+1 , not |−1 , for optical readout.

Mechanical Rabi Model for Sample B
To fit the mechanical Rabi data shown in Fig. 3b of the main text, we solve the Schrödinger equation to find the population in |+1 after applying the relevant portion of an L = 3 µs mechanical pulse. We use the Hamiltonian when the resonator is ringing up and the Hamiltonian when the resonator is ringing down. Quasi-static magnetic bath noise takes the form of a randomized detuning δ drawn from a Gaussian distribution with a standard deviation σ = √ 2/T * 2 [21]. The magnetic Ramsey measurement shown in Fig. 7 sets T * 2 = 0.68 µs. Defining the result of this computation as the function f (τ 0 , Ω(z)), we then perform a spatially-weighted average over the point spread function (PSF) of our confocal microscope to account for spatial inhomogeneities in our mechanical driving field. The resulting signal takes the form

Sample B
where C accounts for partial polarization of the nuclear spin sublevel, Ω(z) = Ω mech |sin 2πz λ B | is the mechanical driving field, λ B is the wavelength of the stress wave, and g(z, z 0 ) describes a Gaussian approximation to a PSF centered at the focal depth z 0 with a depth dependent FWHM as described in Ref. [8]. To produce the model curve in Fig. 3b of the main text, we used the parameters Ω mech /2π = 3.8 MHz, z 0 = 5.9 µm, C = 0.414 (as measured via mechanically driven spin resonance), and λ B = 29.6 µm. The simulation was repeated 200 times, and these results were averaged to produce the final curve.

RAMSEY MEASUREMENTS
Ramsey Pulse Sequences Fig. 8 shows the pulse sequences used for the Ramsey measurements presented in the main text. To eliminate experimental artifacts, we modified the typical Ramsey measurement to include a second measurement for each data point. We first execute the typical π/2-τπ/2 Ramsey sequence. Immediately afterward, we perform a π/2-τ -(−π/2) sequence.
The difference of these two measurements equals twice the imaginary portion of the qubit's coherence Im[ρ i,j ] (i, j ∈ {(m s =) + 1, 0, −1}, i = j). We further modify the Ramsey sequence for the mechanically driven qubit by advancing the phase of the second π/2-pulse by ω rot (τ + τ π/2 ). This extra phase shift introduces a known periodicity to the measurement that aids visualization of the decay envelope.

Ramsey Measurement Normalization
Two measurements were used to normalize the spin contrast for the magnetic Ramsey measurements in the {+1, 0} and {0, −1} subspaces. The maximum spin signal y N P is measured by optically pumping the NV center into |0 , shuttering the laser for the fixed dark time in which no pulses were applied, and then reading out the NV center fluorescence.
For the magnetic {−1, +1} qubit Ramsey measurement, the same "no pulse" measurement gives the maximum spin signal y N P . We define the minimum spin signal y π as the average of the signal from a single magnetic π-pulse on the {+1, 0} qubit and the signal from a single magnetic π-pulse on the {0, −1} qubit.

HAHN ECHO MEASUREMENTS
We performed magnetic Hahn echo measurements of the homogeneous dephasing time T 2 in Sample A. We were unable to perform a mechanical Hahn echo experiment as intrinsic spin dephasing in our device limited the spin contrast after a mechanically driven 2π nutation to the prohibitive value of ≈ 1%. Fig. 9 shows the Hahn echo data for each magnetically driven qubit examined in the main text. Once again, we measure roughly twice the coherence for the {+1, 0} and {0, −1} qubits when compared to the {−1, +1} qubit. * gdf9@cornell.edu