Strain engineering of the silicon-vacancy center in diamond

We control the electronic structure of the silicon-vacancy (SiV) color-center in diamond by changing its static strain environment with a nano-electro-mechanical system. This allows deterministic and local tuning of SiV optical and spin transition frequencies over a wide range, an essential step towards multiqubit networks. In the process, we infer the strain Hamiltonian of the SiV revealing large strain susceptibilities of order 1 PHz/strain for the electronic orbital states. We identify regimes where the spin-orbit interaction results in a large strain susceptibility of order 100 THz/strain for spin transitions, and propose an experiment where the SiV spin is strongly coupled to a nanomechanical resonator.

Figure 1

(a) Electronic level structure of the SiV center (molecular structure shown in inset) at zero strain showing ground and excited manifolds with spin-orbit eigenstates. The four optical transitions A, B, C, and D at zero magnetic field, and splittings between orbital branches in the ground state (GS) and excited state (ES), ${\mathrm{\Delta }}_{\mathrm{gs}}$ and ${\mathrm{\Delta }}_{\mathrm{es}}$, respectively, are indicated. In the presence of a magnetic field, each orbital branch splits into two Zeeman sublevels. A long-lived qubit can be defined with the sublevels of the lower orbital branch in the GS. (b) Schematic of the diamond cantilever device and surrounding electrodes. Diamond crystal axes relative to the cantilever orientation are shown. Four possible orientations of the highest symmetry axis of an SiV are indicated by the four arrows above the cantilever. Under application of strain, these can be grouped into axial (red) and transverse (blue) orientations. The molecular structure of a transverse-orientation SiV as viewed in the plane normal to the cantilever axis is shown below, and crystal axes that define the internal co-ordinate frame of the color center are indicated. The $z$ axis is the highest symmetry axis, which defines the orientation of the SiV. (c) SEM image of diamond cantilever NEMS device.

Figure 2

Tuning of optical transitions of (a) transverse-orientation SiV [blue in Fig. 1], and (b) axial orientation SiV [red in Fig. 1]. Voltage applied to the device is indicated next to each spectrum.

Figure 3

(a) Dominant effect of $E_g$-strain on the electronic levels of the SiV. (b) Dominant effect of $A_{1g}$ strain on the electronic levels of the SiV. (c) Normalized strain-tensor components experienced by transverse-orientation SiV [red in Fig. 1] and (d) axial orientation SiV [blue in Fig. 1] in the SiV coordinate frame upon deflection of the cantilever. (e) Variation in orbital splittings within GS (green dots) and ES (blue dots) upon application of $E_g$ strain. The $x$ axis refers to the magnitude of $E_g$ strain, which is approximately given by $|{\epsilon }_{xx}-{\epsilon }_{yy}|$ for our device. Data points are extracted from the optical spectra in Fig. 2. Solid curves are fits to theory in text. (f) Tuning of mean optical wavelength with $A_{1g}$ strain due to the uniaxial component ${\epsilon }_{zz}$. Data points are extracted from the optical spectra in Fig. 2. Solid line is a linear fit as predicted by theory in text. Appendix pp3 details the fitting procedure used in (e) and (f).

Figure 4

(a) Illustration of dephasing and population decay processes for the SiV qubit. Blue arrows show spin-conserving transitions responsible for dephasing. Red arrow shows a spin-flipping transition driving decay from $|e_{g-}{\downarrow \rangle }^{\prime }$ to $|e_{g-}{\uparrow \rangle }^{\prime }$. Processes suppressed at high strain are crossed out. (b) Calculated rates for spin-conserving upward and downward phonon processes. Both rates are normalized to their values at zero strain. (c) Reduction in CPT linewidth with increasing GS splitting ${\mathrm{\Delta }}_{\mathrm{gs}}$. Inset shows an example of a CPT spectrum taken at ${\mathrm{\Delta }}_{\mathrm{gs}}=460$ GHz. The two resonances in the spectrum are due to the presence of a neighboring nuclear spin [33]. Linewidths of both are plotted and indicated as Dip 1 and Dip 2 in the main plot. (d) Reduction in spin relaxation rate ($1/T_1$) with increasing GS splitting ${\mathrm{\Delta }}_{\mathrm{gs}}$ as extracted from pump-probe measurements. Solid line is a fit to the resonant two-phonon relaxation model in Appendix pp4 for ${\mathrm{\Delta }}_{\mathrm{gs}}$ above 200 GHz where this model is valid. Dotted line is an extrapolation of the fit into the low-strain regime.

Figure 5

(a) Splitting of the C transition into the four transitions C1, C2, C3, and C4 in the presence of a magnetic field. Spin transition frequencies on the lower orbital branches of the GS and ES are ${\omega }_s$, ${\omega }_s^{\prime }$ respectively. (b) Response of transitions C1, C2, C3, and C4 upon tuning GS splitting ${\mathrm{\Delta }}_{\mathrm{gs}}$ with $E_g$ strain. (c) Calculated response of optical transitions C1, C2, C3, and C4 to $E_g$ strain in presence of 0.17-T $B$ field aligned along the [001] direction. Shaded regions on the left and right ends indicate the regimes in which the GS orbitals are determined by SO coupling and strain respectively. (d) Strain response of spin transition frequencies upon tuning of ground-state orbital splitting ${\mathrm{\Delta }}_{\mathrm{gs}}$ with $E_g$ strain. SO regime data points are extracted from the optical spectra in Fig. 5. High-strain regime data points are obtained from CPT measurements on the SiV studied in Fig. 4 (error bars for these points are smaller than data markers). Solid (dashed) line is calculated spin transition frequency on the lower orbital branch of GS (ES) from Fig. 5.

Figure 6

(a) Illustration of mixing terms introduced by $E_g$ strain and an off-axis magnetic field in the GS manifold. (b) Calculated susceptibility of the qubit for interaction with ac $E_g$-strain resonant with the transition frequency ${\omega }_s$ (interaction shown in inset). This ac strain susceptibility is maximum at zero strain for the pure SO eigenstates. At high strain, it falls off as $1/{\mathrm{\Delta }}_{\mathrm{gs}}$. Color variation along the curve shows the GS splitting ${\mathrm{\Delta }}_{\mathrm{gs}}$ corresponding to the value of static $E_g$-strain at the SiV. Both the static and ac strain are assumed to be entirely in the $E_{gx}$ component. (c) SEM image of an optomechanical crystal nanobeam cavity [43] along with an FEM simulation of its 5-GHz flapping resonance. Displacement profile and a cross-sectional strain profile of the mode are shown with arbitrary normalization.

Figure 7

Histogram of C transition frequencies for SiV centers in a nanophotonic cavity. Mean frequency of 406.933 THz is indicated. Standard deviation is estimated to be 31 GHz.

Figure 8

Various pathways for a phonon-mediated spin flip (a) direct relaxation via a single phonon resonant with the $|e_{g-}{\downarrow \rangle }^{\prime }\to |e_{g+}\uparrow \rangle \prime$ spin transition. (b) Two possible channels for a resonant two-phonon process involving the upper orbital branch. (c) Off-resonant two-phonon processes.

Figure 9

Rates of all three spin-relaxation mechanisms indicating their magnitudes and scaling with strain $\beta$.

Figure 10

Calculated susceptibility of the qubit levels for interaction with off-resonant ac $E_g$-strain that modulates the transition frequency ${\omega }_s$ (interaction shown in inset). Color variation along the curve shows the GS splitting corresponding to the value of static $E_g$ strain at the SiV. Both the dc and ac strain are assumed to be entirely in the $E_{gx}$ component.

Figure 11

Variation of $g$ factor for transverse magnetic field (orthogonal to the SiV internal $z$ axis) as a function of pre-existing static $E_g$ strain. Color variation along the curve shows the GS splitting corresponding to the value of static $E_g$ strain at the SiV. Static magnetic field that splits the spin sublevels is applied along the [001] direction, while microwave magnetic field resonantly driving the spin transition is applied along the SiV $x$ axis (interaction shown in inset). Static strain is assumed to be entirely of $E_{gx}$ character.