## Introduction

Solid state quantum bits can offer the key advantage of scalability when used to realise a quantum network. However, their coherence is often limited by the impact of fluctuations in the solid-state environment. In this context, the effects of fluctuating electric1,2,3 and magnetic fields4,5,6,7,8 on the optical and spin coherence of solid-state emitters can be mitigated by applying a variety of techniques ranging from materials engineering to dynamical decoupling. On the other hand, thermal decoherence can typically be overcome only by maintaining the quantum system at low enough temperatures to freeze out relevant phonons. Phonon-driven processes are responsible for relaxation and decoherence processes in a variety of solid-state emitters that can serve as optically accessible quantum memories9,10,11,12,13,14,15. In particular, for emitters with spin-orbit coupling, such processes can demand operation at sub-Kelvin temperatures16,17,18, or the use of magnetic fields of several Tesla19 to achieve long spin relaxation and coherence times. This requires cryogenic setups that are significantly more complex than common helium-4 cryostats employed to obtain coherent optical photons from solid-state emitters.

In this work, we show that strain engineering can be used to quench the effect of the thermal phonon bath on an electronic spin qubit without lowering the operating temperature. Our experiments are performed on the negatively charged silicon-vacancy (SiV) centre in diamond, an emerging building block for photonic quantum networks20 due to its remarkable optical properties stemming from its inversion symmetric structure21. However, unless operated at dilution refrigerator temperatures16,17, the SiV centre is subject to phonon-induced transitions between its energy levels11,22 which limit its spin coherence23. Our approach to mitigate phonon-induced decoherence takes advantage of the fact that the large electron-phonon coupling responsible for such decoherence processes fundamentally arises from a high susceptibility of the electronic orbitals to lattice strain. Through strain control, we increase the energy scale for phonon absorption by the emitter to far above the thermal energy (kBT ≈ 0.3 meV at the experimental temperature, T = 4 K). The resulting depletion of thermal phonons seen by the SiV leads to an improvement in its spin coherence time.

## Results

### Description of the nano-electro-mechanical system

The detailed level structure of the SiV centre is depicted in Fig. 1a, with the ground-state (GS) and excited-state (ES) manifolds, each containing two distinct orbital branches24. Orbital degeneracy in each manifold is lifted by spin-orbit coupling: $$\left| 1 \right\rangle ,\left| 2 \right\rangle$$ in the GS split by 46 GHz, and $$\left| 3 \right\rangle ,\left| 4 \right\rangle$$ in the ES split by 255 GHz in the absence of strain. Phonons with frequencies corresponding to these splittings can drive orbital transitions within the ground and excited manifolds11. As a first step towards controlling the electron-phonon interaction, we investigate the effect of static strain on these orbitals through strain-dependent photoluminescence excitation (PLE) of the optical transitions labelled A, B, C and D at 4 K. Static strain control at the location of the emitter is achieved with a nano-electro-mechanical system (NEMS) device, a monolithic single-crystal diamond cantilever with metal electrodes patterned above and below it (see Supplementary Fig. 1), as shown in the scanning electron microscope (SEM) image in Fig. 1b. An opening in the top electrode allows optical access to SiV centres located in an array (inset of Fig. 1b), precisely positioned by focused ion-beam (FIB) implantation of 28Si+ ions25,26. A DC voltage applied across the electrodes deflects the cantilever downwards due to electrostatic attraction and generates controllable static strain oriented predominantly along the long axis of the cantilever. The strain profile can be calculated numerically via a finite-element-method (FEM) simulation, as shown in Fig. 1c. Of the two possible orientations of SiV centres in our device, we address those with transverse orientation (labelled blue, and shown in detail in inset of Fig. 1c), which predominantly experience strain in the plane normal to their highest symmetry axis (Eg-symmetric strain27). Upon applying strain, transitions A and D shift towards shorter and longer wavelengths, respectively. These shifts indicate increasing GS and ES splittings as shown in Fig. 2a. This result is consistent with a previous experiment on a dense ensemble of SiV centres28. Complete characterisation of the strain response of the SiV electronic levels and relevant group theory analysis are detailed in ref29. The variations in GS and ES splittings shown in Fig. 2a are quadratic at low strain, and linear at high strain. This indicates that Eg-symmetric strain mixes orbitals within the GS and ES manifolds, and thus phonon modes with corresponding strain components can induce resonant transitions between these orbitals. In contrast, strain along the SiV axis (A1g-symmetric strain) is found to leave the GS and ES splittings unchanged, and therefore cannot cause electronic transitions.

### Strain-tuning of energy levels

With our device we can tune the splitting of the orbitals in the GS manifold from 46 GHz to typically up to 500 GHz, and in the best case, up to 1.2 THz (see Supplementary Discussion). In doing so, we can probe the interaction between the colour centre and phonons of different frequencies. This is achieved by measuring the thermal relaxation rate between the orbitals with a time-resolved pump-probe technique (Fig. 2b). Measurements are performed in the frequency range Δgs = 46 to 110 GHz where this technique can be applied. The total relaxation rate is a sum of the rates of phonon absorption, γup, and emission, γdown (shown in Fig. 1a), which can be individually extracted using the theory described in Supplementary Discussion. Over the range of Δgs measured, phonon processes in both directions are observed to accelerate with increasing orbital splitting. This is because the number of acoustic modes resonant with the GS splitting, i.e. the phonon density of states (DOS) at Δgs, increases with the dependence $$\Delta _{{\mathrm{gs}}}^n$$ (n depends on the geometry of material seen by resonant phonons, see Supplementary Discussion). However, if the orbital splitting is increased far above 120 GHz (at temperature T = 4 K) as plotted in Fig. 2c, the phonon absorption rate (γup) is theoretically expected to reverse its initial trend. In this regime, the polynomial increase in phonon DOS is outweighed by the exponential decrease in thermal phonon occupation (~exp(−hΔgs/kBT))11, and consequently γup is rapidly quenched.

### Strain-enhanced spin coherence

Such a suppression of phonon absorption at high strain can improve the spin coherence of the emitter. In the presence of a magnetic field, the SiV electronic levels further split into spin sub-levels and provide an optically accessible spin qubit as shown in Fig. 3a23,30,31,32. We use coherent population trapping (CPT) through simultaneous resonant laser excitation of the optical transitions labeled C1 and C2 to pump the SiV into a dark state, a coherent superposition of the spin sub-levels $$\left| {1 \downarrow } \right\rangle$$, $$\left| {1 \uparrow } \right\rangle$$31,32. When the two-photon detuning is scanned, preparation of the dark state results in a fluorescence dip, whose linewidth is determined by the optical driving and spin dephasing rates. At low laser powers, the linewidth is limited by spin dephasing, which is dominated by phonon-mediated transitions within the GS manifold11,23. In Fig. 3b, as the dark resonance narrows down due to prolonged spin coherence with increasing strain, we reveal a fine structure not visible before. Further measurements in Supplementary Discussion suggest that the presence of two resonances is due to the interaction of the SiV electron spin with a neighbouring spin such as a 13C nuclear spin. This indicates the possibility of achieving a local register of qubits as has been demonstrated with NV centres33. Figure 3c shows the decreasing linewidths of the CPT resonances with increasing GS orbital splitting, indicating an improved spin coherence time. Beyond a GS splitting of ~400 GHz, the linewidths saturate at ~1 MHz. At the highest strain condition, we perform a power-dependent CPT measurement to eliminate the contribution of power broadening, and extract a spin coherence time of $$T_2^ \ast$$ = 0.25 ± 0.02 μs (compared with other CPT-based measurements which reported $$T_2^ \ast$$ = 40 ns without strain control31,32). This saturation of $$T_2^ \ast$$ suggests the mitigation of the primary dephasing source, single-phonon transitions between the GS orbitals, and the emergence of a secondary dephasing mechanism such as slowly varying magnetic fields from naturally abundant (1.1%) 13C nuclear spins in diamond. We note that our longest $$T_2^ \ast$$ = 0.25 ± 0.02 μs is on par with that of the NV centre without dynamical decoupling4,34 and of low-strain SiV centres operated at a much lower temperature of 100 mK16, the conventional approach to suppress phonon-mediated dephasing.

## Discussion

In conclusion, we use a nano-electro-mechanical system to probe and control the interaction between a single electronic spin and the phonon bath of its solid-state environment. In doing so, we demonstrate a six-fold prolongation of spin coherence by suppressing phonon-mediated dephasing as the dominant decoherence mechanism. As a next step, we can further improve the spin coherence by cancelling the effect of slowly varying non-Markovian noise from the environment16 using dynamical decoupling techniques that are well-studied with other spin systems6,7,33. Our strain engineering approach can be applied to overcome phonon-induced decoherence in other emitters such as emerging inversion-symmetric centres in diamond13,14,35,36 Kramers rare earth ions15,18,19, and in general, systems with spin-orbit coupling in their ground state. The high strain needed to quench phonon processes can be achieved simply by deposition of a thin film37, which passively stresses the underlying crystal. A NEMS platform such as the one demonstrated in this paper can provide the added benefit of active wavelength tuning, which can enable generation of indistinguishable photons from multiple emitters, and hence scalable photonic quantum networks20,38. Another natural extension of our work is coherent coupling of the SiV spin to phonons in a well-defined mechanical mode, which will enable the use of phonons as a quantum resource. In particular, we can combine the large strain susceptibility of the SiV electronic levels with mechanical resonators of dimensions close to the phonon wavelength, such as optomechanical crystals39 to obtain orders of magnitude larger spin-phonon interaction strengths compared with previous works40,41,42,43,44,45, leading to strong spin-phonon coupling29. In this regime, one can realise phonon-mediated two-qubit gates46,47 analogous to those implemented with trapped ions48, and achieve quantum non-linearities required to deterministically generate single phonons and non-classical mechanical states49,50,51,52,53, a long sought-after goal since phonons can be used to interface spins with other quantum systems such as superconducting qubits54.

## Methods

### Fabrication procedure

We use $$\left\langle {100} \right\rangle$$-cut, ultra-high purity (nitrogen concentration less than 5 ppb), type IIa, single-crystal diamond synthesized by chemical vapour deposition (CVD) from Element Six Corporation. The cantilever arrays are patterned with electron-beam lithography, and first vertically etched with oxygen plasma. These vertically etched structures are then made free-standing by etching the sample at a tilted angle with an oxygen-plasma assisted ion-milling process. After cantilever fabrication, silicon ions (Si+) are implanted at target spots on the cantilevers using a custom focused-ion-beam (FIB) system at Sandia National Labs. SiV centres are then generated by a high-temperature (1100 °C), high-vacuum annealing procedure followed by a tri-acid clean (1:1:1 sulfuric, perchloric, and nitric acids). Subsequently, electrode patterns are made by a conventional bi-layer PMMA process followed by metal evaporation. We use as a 10 nm thick tantalum (Ta) layer for the cantilever electrodes in order to reliably apply high voltages, and a 200 nm thick gold layer for the bonding pads. Detailed schematics for the above fabrication steps are shown in Supplementary Methods.

### Strain dependent photoluminescence measurements

The sample is cooled down to a nominal temperature of 6 K inside a Janis ST-500 continuous helium-flow cryostat. The cryostat is mounted under a home-built scanning confocal microscope with a 0.9 NA ×100, 1 mm working distance objective (Olympus MPLFLN 100X) housed inside the cryostat. SiV centers are identified via non-resonant excitation with a 703 nm laser diode (Thorlabs LP705-SF15), and collection of zero-phonon-line (ZPL) fluorescence in a narrow bandwidth of 10 nm around 737 nm. For resonant photoluminescence excitation (PLE) of ZPL transitions, we use a tunable continuous-wave Ti-sapphire laser (M-Squared Solstis), and collect the resulting fluorescence in the phonon-sideband (above 750 nm). Mode-hop-free tuning of the laser is achieved over the scan range of interest by using feedback from a wavemeter (High Finesse WS7). A CW 532 nm laser is periodically pulsed using an acousto-optic modulator (Crystal Technology 3080) to maintain the negative charge state of SiV(−) centers. Collected fluorescence (gated off during green excitation) is sent to an avalanche-photodiode to measure the photon-count rate. DC voltage for cantilever-deflection is supplied from a Stanford Research Systems PS300 high-voltage source. As an added precautionary measure, the weak leakage-current in the circuit (typically below 100 nA) discussed in Supplementary Methods is monitored via a Keithley 2400 source-meter.

### Orbital thermalization measurements

The pump-probe pulse sequence to measure the orbital thermalization rate is implemented by pulsing our resonant-excitation laser with a Mach-Zehnder intensity electro-optic modulator (EOM) (EO Space AZ-AV5-5-PFA-PFA-737). The EOM is driven by a digital-delay generator with rise- and fall-times of 2 ns (SRS DG645). Over the course of the measurements, the modulation index of the EOM is stabilized against long-term drifts with continuous feedback on the DC-bias voltage. The feedback loop is implemented with a lock-in amplifier (SRS SR830) generating a low-frequency (1 KHz) modulation of the DC-bias voltage. Photon-count pulses from the single-photon-detector are time-tagged on a PicoHarp 300 module triggered by the delay-generator. The laser frequency itself is stabilized by continuous feedback with a wavemeter (High Finesse WS7).

### SiV spin measurements

The sample is cooled down to a temperature of 3.8 K inside a closed-cycle liquid helium cryostat (Attodry 1000). It is placed in a dip stick, in which helium gas (pressure ~1 mbar) acts as an exchange gas. Two superconducting coils surrounding the sample chamber can be used to apply a magnetic field along two orthogonal axes, up to 8 T vertically and up to 2 T horizontally. The optical part of the setup consists of a home-built confocal microscope mounted on top of the cryostat, and a microscope objective (NA = 0.82) inside the sample chamber. The sample is mounted on piezoelectric stages (Attocube ANPx101 and ANPz101) allowing to position the sample with respect to the objective. Non-resonant excitation of SiVs is performed using a diode laser at 660 nm (Laser Quantum Ventus), while resonant excitation is achieved with a tunable diode laser around 737 nm (Toptica DLpro). The frequency of the latter is stabilised through continuous feedback from a wavemeter (High Finesse WSU). For CPT measurements, sidebands are generated on the resonant excitation laser using an EOM (Photline NIR-MX800) connected to a tunable microwave source (Rhode&Schwarz SMF 100A). Fluorescence from the emitters is collected through the microscope objective. A 750 nm long-pass filter in the confocal microscope allows collection of the phonon-sideband emission from SiV centres, filtering out the laser excitation. This emission is then sent to an avalanche photodiode (APD) (Excellitas).

### Data availability

The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.