Electron–phonon processes of the silicon-vacancy centre in diamond

The spin–orbit interaction results in four optical dipole transitions, labelled A–D in order of increasing wavelength, centred around ∼737 nm at cryogenic temperatures [30–32]. At liquid helium temperatures, the optical line widths are broader for transitions A and B than for the lower energy transitions C and D [7]. This was attributed to thermal relaxation reducing the effective lifetime of the upper branch via the decay rate $\gamma _{-}^{{\rm u}}$, which is faster than $\gamma _{+}^{{\rm u}}$ by the Boltzmann factor $\gamma _{-}^{{\rm u}}=\gamma _{+}^{{\rm u}}{\rm exp} (\lambda _{{\rm so}}^{{\rm u}}/{{k}_{{\rm B}}}T)$ [30, 31].

To probe the microscopic mechanism of the orbital relaxation in the excited states, the temperature dependence of the line width of transition D was measured for individual ${\rm Si}{{{\rm V}}^{-}}$ centres. Since these ${\rm Si}{{{\rm V}}^{-}}$ centres exhibit negligible spectral diffusion [7], the measured optical line widths $\Gamma (T)$ correspond to homogeneous broadening mechanisms associated with depolarisation and dephasing: $\Gamma (T)={{\gamma }_{{\rm r}}}+{{\gamma }_{{\rm nr}}}(T)+\gamma _{+}^{{\rm u}}(T)+{{\gamma }_{{\rm d}}}(T)$. The non-radiative decay rate ${{\gamma }_{{\rm nr}}}(T)$ has a very weak temperature dependence, as discussed later in section 2.3, leading to a small contribution compared to the other rates for all temperature regions of interest. The most significant temperature dependence comes from the relaxation rates within the excited states: $\gamma _{+}^{{\rm u}}(T)$ and ${{\gamma }_{{\rm d}}}(T)$. As will be shown in section 3.2, the optical transition line width is dominated by relaxation rates in the excited states with little contribution from the ground states.

For temperatures between 4 K and 50 K, the optical transition line width was measured using photoluminescence excitation in a continuous flow cryostat, where a weak probe laser was scanned across transition D and fluorescence in the phonon-sideband (PSB) was detected. At higher temperatures, 532 nm excitation was used and emission line widths were measured with a spectrometer (Princeton Instruments Acton 2500 equipped with a Pixis 100 cooled CCD-array and a 1596 mm⁻¹ grating) giving a resolution of 16 GHz. To measure fundamental properties of the ${\rm Si}{{{\rm V}}^{-}}$ centre, a bulk diamond sample containing highly uniform defect sites and low strain was used in these experiments. It was a low strain high-pressure high-temperature (HPHT) type-IIa diamond with a {100} surface on which a 60 μm layer incorporating ${\rm Si}{{{\rm V}}^{-}}$ was created by microwave-plasma chemical-vapour-deposition (MPCVD). This sample was used in previous publications [7, 8, 23] and shows a narrow inhomogeneous distribution for the ${\rm Si}{{{\rm V}}^{-}}$ optical transitions.

Figure 2 shows the full width at half maximum line widths (determined from Lorentzian fits) measured for single ${\rm Si}{{{\rm V}}^{-}}$ sites in a 200 × 200 μm⁻² region containing $^{28}$Si. Above ∼70 K the line width scales as the cube of the temperature ($\Gamma =(103+0.12\cdot {{[T/{\rm K}]}^{3}})$ MHz). However for low temperatures ($\lt 20\;{\rm K}$ ), the behaviour deviates from T³ and is better approximated by a linear dependence on temperature ($\Gamma =(-1.05+24.26\cdot [T/{\rm K}])$ MHz) saturating at about 4 K to the lifetime limited line width. Early studies on nanodiamonds have measured the T³ dependence of the line width on temperature, but were not able to resolve this linear contribution due to a combination of inhomogeneous broadening and spectral resolution limits [33]. It is shown in section 3 that the observed T and T³ mechanisms result from first- and second-order transitions due to linear electron–phonon interactions with ${\rm E}$-symmetric phonon modes.

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Relaxation within the ground state doublet, $\gamma _{+}^{{\rm g}}$ in figure 1(b), was probed directly using pulsed excitation and time-resolved fluorescence measurements. Transitions C and D form an optical Λ-system which allows ground state populations to be optically pumped. For these experiments a second diamond sample was used in which the properties and orientation of the ${\rm Si}{{{\rm V}}^{-}}$ centres were known from earlier studies [23]. The sample is a low strain HPHT diamond observed through a {111} surface with a low density of in-grown ${\rm Si}{{{\rm V}}^{-}}$ centres and the optical properties of the ${\rm Si}{{{\rm V}}^{-}}$ centres found within are comparable to those in the {100} sample. A laser was tuned to transition D and 80-ns-pulses were generated using an electro-optical amplitude modulator with a measured extinction ratio of up to 20 dB. The signal was detected by counting the photon arrival times in relation to the laser pulses using a time-tagged data acquisition card (FAST ComTec MCS26A) giving a time resolution of up to 200 ps.

At the start of each laser pulse we observed a fluorescence peak that decayed to a steady state level. For the first laser pulse, the peak height a corresponds to the thermal population in the bright state ($|e_{+}^{{\rm g}}\rangle |\uparrow \rangle$ or $|e_{-}^{{\rm g}}\rangle |\downarrow \rangle$ ) which is ∼50% for the temperatures in our measurements. The decay of this initial peak when the laser is on corresponds to optical pumping into the dark state ($|e_{-}^{{\rm g}}\rangle |\uparrow \rangle$ or $|e_{+}^{{\rm g}}\rangle |\downarrow \rangle$). After a dark interval τ, the dark state relaxes back to the bright state, leading to a recovery of peak height h for subsequent pulses. The peak height, $h(\tau )$, exhibits a simple exponential recovery indicating a single characteristic relaxation time, ${{T}_{1}}$, as shown in figures 3(a) and (b). This measurement was repeated for a single ${\rm Si}{{{\rm V}}^{-}}$ centre at various temperatures between 4.5 K and 22 K and the relaxation rate was found to scale linearly with temperature (figure 3(c)). The longest ${{T}_{1}}$ time was measured at the lowest temperatures to be ${{T}_{1}}(5\;{\rm K})=39\pm 1\;{\rm ns}$.

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We note that the steady state fluorescence level under laser excitation, $h(0)$, is determined by a competition between the optical pumping rate and thermalization rates ($1/{{T}_{1}}$). With increasing temperature, the thermalization rate increases (figure 3(c)) while the optical pumping rate remains nearly constant at saturation. This leads to a reduced measured peak contrast ($h(0)/2a$, figure 3(d)) with increasing temperature, showing that the ground state of the ${\rm Si}{{{\rm V}}^{-}}$ centre cannot be polarised at elevated temperatures.

The results presented so far have only highlighted the processes within the ground and excited state doublets. The transition rates from excited to ground states also have a temperature dependence which can be probed by measuring the fluorescence lifetime of the excited state as a function of temperature. Previous experiments have reported excited state lifetimes in the ∼1–4 ns range along with various estimates of the quantum yield [7, 30, 32–35]. Figure 1(b) shows potential radiative (${{\gamma }_{{\rm r}}}$) and non-radiative (${{\gamma }_{{\rm nr}}}$) processes taking place at the optical energy scale that determine the excited state lifetimes and the quantum yield. The total decay rate from the excited states, ${{\gamma }_{{\rm t}}}(T)=1/{{\tau }_{0}}(T)={{\gamma }_{{\rm r}}}+{{\gamma }_{{\rm nr}}}(T)$, is a combination of a constant radiative (${{\gamma }_{{\rm r}}}$) and a temperature dependent non-radiative rate (${{\gamma }_{{\rm nr}}}(T)$).

In an attempt to identify the non-radiative process, we measured the lifetime of the $^{2}{{{\rm E}}_{{\rm u}}}$ excited states as a function of temperature from 5 K to 350 K. At each temperature, 10 separate single ${\rm Si}{{{\rm V}}^{-}}$ centres were excited using a pulsed 532 nm laser with a 80 MHz repetition rate and the measured time traces were fitted using a single exponential decay. The measured temperature dependence of ${{\tau }_{0}}(T)=1/{{\gamma }_{{\rm t}}}(T)$ is shown in figure 4(a). The excited state lifetime was found to increase as temperature was decreased down to 50 K, where it saturated to a constant level. These results suggest there is a finite non-radiative rate ${{\gamma }_{{\rm nr}}}$ at room temperature, while the saturation below 50 K does not necessarily imply ${{\gamma }_{{\rm nr}}}(T\lt 50\;{\rm K})=0$ as there might still be a finite spontaneous non-radiative rate at zero temperature. The observed temperature dependence in figure 4(a) can be described by the Mott–Seitz model for non-radiative relaxation, ${{\tau }_{0}}(T)={{\tau }_{0}}(T=0\;{\rm K}){{(1+\alpha {{{\rm e}}^{-\frac{\Delta E}{{{k}_{{\rm B}}}T}}})}^{-1}}$, with an activation energy of $\Delta E=55\pm \;2\;{\rm meV}$ and $\alpha =3.3\pm 0.3$ [36]. Our measurements do not, however, provide enough information to distinguish whether the system decays from $^{2}{{{\rm E}}_{{\rm u}}}$ directly to a higher vibrational state of $^{2}{{{\rm E}}_{{\rm g}}}$, or to an unidentified electronic level closer to $^{2}{{{\rm E}}_{{\rm u}}}$ in energy. Whilst there exists some ab initio [25] and experimental [37] evidence of an additional electronic level below the excited $^{2}{{{\rm E}}_{{\rm u}}}$ level, this evidence conflicts with the simple molecular orbital model of the centres electronic structure [21–24], which predicts no such additional level. Future studies involving spectroscopy of the $^{2}{{{\rm E}}_{{\rm g}}}{{\to }^{2}}{{{\rm E}}_{{\rm u}}}$ absorption PSB and single-shot readout capability of ${\rm Si}{{{\rm V}}^{-}}$ electronic states might help identify the relaxation paths from the $^{2}{{{\rm E}}_{{\rm u}}}$ and dark states of ${\rm Si}{{{\rm V}}^{-}}$ centres [38].

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The line positions of all four optical transitions were determined using the Lorentzian fits to the measurements described in section 2.1. The spectrometer was calibrated with respect to a wavemeter, allowing us to consistently reproduce transition wavelengths across the entire temperature range. For simplicity, only transition C is shown in figure 4(b). Fitting with a free temperature exponent results in $\Delta \lambda =19.2\cdot {{(T/{\rm K})}^{2.78\pm 0.05}}$, in close agreement with a cubic temperature dependence. Figure 4(b) compares fits of the form $\Delta \lambda \sim {{T}^{\beta }}$ for $\beta =2,3,4$ as well as a model based on thermal expansion described below. Our observation of T³ scaling differs marginally from earlier measurements made on nanodiamonds [33]. For temperatures at which the linewidth was narrow enough to resolve individual transitions, the ground and excited state splittings could also be obtained from the spectrum. We observe that the measured splittings, which correspond to the spin–orbit interaction at low temperature, are reduced with increasing temperature for both the excited (figure 4(c)) and ground (figure 4(c)) states.

In diamond, the temperature shifts of optical lines have two distinct origins: thermal expansion and electron–phonon interactions [39, 40]. The shift of the transition energy due to thermal expansion has the form $\delta {{E}_{{\rm exp} .}}(T)=A\cdot P(T)$, where A is the hydrostatic pressure shift of the transition energy, $P(T)$ $=-B\int _{0}^{T}e(x){\rm d}x$ is the negative pressure of thermal expansion, B is the diamond bulk modulus and e(T) is the bulk thermal expansion coefficient [39, 40]. Whilst B and e(T) are well-known for diamond [41], the pressure shift A of the ${\rm Si}{{{\rm V}}^{-}}$ optical transition has not been measured. The dashed green line in figure 4(b) is the best fit of the line shift obtained using the single fit parameter A of the thermal expansion mechanism, and it is clear that it does not account for the observed shift. The shift of the transition energy due to electron–phonon interactions typically arises from quadratic interactions with ${{{\rm A}}_{1}}$-symmetric phonon modes and produces a T⁴ dependence [42], which is also inconsistent with our observations. Furthermore, a linear combination of shifts caused by these two mechanisms is not able to produce a good fit to our observations. It is shown in section 3 that the atypical T³ shift arises from second-order linear interactions with ${\rm E}$-symmetric phonon modes. This fits well to the observed data and therefore the shift due to thermal expansion is negligible.

The temperature reductions of the ground and excited state splittings can also arise from thermal expansion and electron–phonon interactions [40]. The dashed green lines in figures 4(c) and (d) are the best thermal expansion fits obtained by introducing pressure shift parameters of the spin–orbit splittings, and as above it is clear, at least for figure 4(c), that another mechanism must be involved in the reduction of the splittings. Similarly, the T⁴ dependence of the quadratic interactions with ${{{\rm A}}_{1}}$-symmetric phonon modes and its linear combination with the thermal expansion shift do not satisfactorily fit the observations. We will show in the next section that the T² dependence of the spin–orbit splittings are also consequences of second-order linear interactions with ${\rm E}$-symmetric phonon modes as for the shift of the transition energy.

For the ${\rm Si}{{{\rm V}}^{-}}$ centre the interaction between the ${\rm E}$-symmetric orbital states $\{|{{e}_{+}}\rangle ,|{{e}_{-}}\rangle \}$ and phonon modes of ${\rm E}$ symmetry is described most conveniently if the ${\rm E}$ modes are linearly transformed to be circularly polarised. With this transformation, the phonon Hamiltonian and the linear electron–phonon interaction are

$$\begin{eqnarray}&&{{\hat{H}}_{E}}=\mathop{\sum }\limits_{p,k}\hbar \;{{\omega }_{k}}\;a_{p,k}^{\dagger }{{a}_{p,k}},\end{eqnarray} \tag{ 2 }$$

$$\begin{eqnarray}&&{{\hat{V}}_{E}}=\mathop{\sum }\limits_{k}\hbar \;{{\chi }_{k}}[{{\sigma }_{+}}({{a}_{-,k}}+a_{-,k}^{\dagger })+{{\sigma }_{-}}({{a}_{+,k}}+a_{+,k}^{\dagger })],\end{eqnarray} \tag{ 3 }$$

where ${{\chi }_{k}}$ is the interaction frequency for a single phonon, ${{\sigma }_{+}}$ (${{\sigma }_{-}}$) is the raising (lowering) operator for orbital states, and $a_{p,k}^{\dagger }$ (${{a}_{p,k}}$) is the creation (annihilation) operator for phonons with polarisation $p=\{-,+\}$ and wavevector k. Long wavelength acoustic phonons in diamond give rise to the collective translation and relative displacement of the ions of the center. Phonon modes resulting in the relative displacement have even parity and are responsible for the electron–phonon coupling. The interaction frequency and density of modes are approximately $\overline{|{{\chi }_{k}}(\omega ){{|}^{2}}}\approx \chi \omega$ and $\rho (\omega )=\rho {{\omega }^{2}}$, respectively, where the overbar denotes the average over all modes with frequency ${{\omega }_{k}}=\omega$ and χ and ρ are proportionality constants [28, 45]. Note that interactions with ${\rm A}$-symmetric modes have not been included, as they do not couple the states within the ground and excited electronic levels.

Treating ${{\hat{V}}_{{\rm E}}}$ as a time-dependent perturbation, the first-order transitions between the orbital states involve the absorption or emission of a single ${\rm E}$ phonon whose frequency is resonant with the splitting Δ (see figure 5(b)). The corresponding transition rates are

$$\begin{eqnarray}\begin{array}{rcl} {{\gamma }_{+}} & = & 2\pi \mathop{\sum }\limits_{k}{{n}_{-,k}}|{{\chi }_{k}}{{|}^{2}}\delta (\Delta -{{\omega }_{k}}), \\ {{\gamma }_{-}} & = & 2\pi \mathop{\sum }\limits_{k}({{n}_{+,k}}+1)|{{\chi }_{k}}{{|}^{2}}\delta (\Delta -{{\omega }_{k}}), \\ \end{array}\end{eqnarray} \tag{ 4 }$$

where ${{n}_{p,k}}$ is the occupation of the phonon mode with polarisation p and wavector k. Assuming acoustic phonons, performing the thermal average over initial states and the sum over all final states leads to

$$\begin{eqnarray}\begin{array}{rcl} {{\gamma }_{+}} & = & 2\pi \chi \;\rho \;{{\Delta }^{3}}n(\Delta ,T), \\ {{\gamma }_{-}} & = & 2\pi \chi \;\rho \;{{\Delta }^{3}}[n(\Delta ,T)+1]. \\ \end{array}\end{eqnarray} \tag{ 5 }$$

For temperatures $T\gt \hbar \Delta /{{k}_{{\rm B}}}$, equation (5) can be approximated by a single relaxation rate with a linear temperature dependence

$$\begin{eqnarray}&&{{\gamma }_{+}}\approx {{\gamma }_{-}}\approx \frac{2\pi }{\hbar }\chi \rho {{\Delta }^{2}}{{k}_{{\rm B}}}T.\end{eqnarray} \tag{ 6 }$$

Hence, the one-phonon transitions lead to the relaxation of population between the orbital states as well as the dephasing of the states that are linearly dependent on temperature. The measurements presented in figures 2 and 3 demonstrated a clear linear dependence of broadening for temperatures below 20 K, but greater than the spin–orbit splitting ($T\gt \hbar \Delta /{{k}_{{\rm B}}}\sim 2.4\;{\rm K}$). We therefore conclude that the relaxation mechanisms are dominated by a resonant single phonon process at liquid helium temperatures for both the ground and the excited states. Equation (6) also shows that the relaxation rate is $\sim {{\Delta }^{2}}$, where Δ is the spin–orbit splitting in the zero-field limit. The ${{\Delta }^{2}}$ scaling explains why the phonon relaxation processes are much faster in the excited levels for which the splittings are larger compared with the ground states.

It was seen in figure 2 that the line broadening deviated from its linear temperature dependence above about 20 K ($T\gg \hbar \Delta /{{k}_{{\rm B}}}$), suggesting that higher order processes involving two phonons start dominating the relaxation rates. Given the form of the electron–phonon interaction in equation (3), the only allowed two-phonon processes are those where the initial and final orbital states are identical. Therefore, the inelastic Raman-type scattering processes (figure 5(c)) that are dominant for ${\rm N}{{{\rm V}}^{-}}$ centres [28], are suppressed in ${\rm Si}{{{\rm V}}^{-}}$ and the elastic Raman-type scattering processes (figure 5(d)) dominate instead. The elastic scattering rate for $|{{e}_{-}}\rangle$ is

$$\begin{eqnarray}\begin{array}{rcl} {{\gamma }_{{\rm d}-}} & = & 2\pi {{\hbar }^{2}}\mathop{\sum }\limits_{k,q}{{n}_{-,k}}({{n}_{+,q}}+1)|{{\chi }_{k}}{{|}^{2}}|{{\chi }_{q}}{{|}^{2}} \\ {} & {} & {{\left| \frac{1}{\Delta -{{\omega }_{k}}}+\frac{1}{\Delta +{{\omega }_{k}}} \right|}^{2}}\delta (\Delta -{{\omega }_{k}}+{{\omega }_{q}}). \\ \end{array}\end{eqnarray} \tag{ 7 }$$

Performing the thermal average over the initial states and the sum over all final states leads to

$$\begin{eqnarray}\begin{array}{rcl} {{\gamma }_{{\rm d}-}} & = & 2\pi {{\hbar }^{2}}\int _{0}^{\scr{\Omega} }n(\Delta +\omega ,T)(n(\omega ,T)+1)\overline{{{\left| {{\chi }_{k}}(\Delta +\omega ) \right|}^{2}}}\overline{{{\left| {{\chi }_{q}}(\omega ) \right|}^{2}}} \\ {} & {} & {{\left| \frac{1}{-\omega }+\frac{1}{\Delta +\omega } \right|}^{2}}\rho (\Delta +\omega )\rho (\omega ){\rm d}\omega , \\ \end{array}\end{eqnarray} \tag{ 8 }$$

where Ω is the Debye frequency of diamond. Assuming acoustic modes and that the temperatures are such that only modes with frequencies $\scr{\Omega} \gg \omega \gg \Delta$ contribute significantly to the integral, to lowest order in Δ, the rates become

$$\begin{eqnarray}\begin{array}{rcl} {{\gamma }_{{\rm d}-}}\approx {{\gamma }_{{\rm d}+}} & \approx & 2\pi {{\hbar }^{2}}{{\Delta }^{2}}{{\chi }^{2}}{{\rho }^{2}}\int _{0}^{\infty }n(\omega ,T)(n(\omega ,T)+1){{\omega }^{2}}{\rm d}\omega \\ {} & = & \frac{2{{\pi }^{3}}}{3\hbar }{{\Delta }^{2}}{{\chi }^{2}}{{\rho }^{2}}k_{{\rm B}}^{3}\;{{T}^{3}}. \\ \end{array}\end{eqnarray} \tag{ 9 }$$

Hence, the two-phonon elastic scattering process contributes to the dephasing of the orbital states and has rates that are proportional to $\sim {{T}^{3}}$, matching the observed line width behaviour in figure 2. Therefore our microscopic model shows perfect agreement with the measurements and we can understand the orbital relaxation process as a combination of a single phonon mixing between the orbital states and a two-phonon dephasing process.

For ${\rm Si}{{{\rm V}}^{-}}$ centres under high strain (larger than the spin–orbit interaction), the orbital eigenstates $\{|{{e}_{x}}\rangle ,|{{e}_{y}}\rangle \}$ no longer have well defined angular momentum. Under such conditions, the inelastic Raman process shown in figure 5(c) becomes allowed, which results in a competing orbital relaxation rate that scales as $\sim {{T}^{5}}$.

The electron–phonon interactions also perturb the energies of the orbital states at second-order. The second-order energy shifts $\delta {{E}_{-}}$ ( $\delta {{E}_{+}}$ ) for states $|{{e}_{-}}\rangle$ ( $|{{e}_{+}}\rangle$ ) can be expressed in a simple form using the linear phonon $\{x,y\}$ polarisation basis. The energy shift due to phonon modes with wavevector k and occupation ${{n}_{x(y),k}}$ are

$$\begin{eqnarray}\begin{array}{rcl} \delta {{E}_{-}}(x(y),k) & = & {{\hbar }^{2}}\chi _{k}^{2}\left( \frac{{{n}_{x(y),k}}}{\omega -\Delta }-\frac{{{n}_{x(y),k}}+1}{\omega +\Delta } \right), \\ \delta {{E}_{+}}(x(y),k) & = & {{\hbar }^{2}}\chi _{k}^{2}\left( \frac{{{n}_{x(y),k}}}{\omega +\Delta }-\frac{{{n}_{x(y),k}}+1}{\omega -\Delta } \right), \\ \end{array}\end{eqnarray} \tag{ 10 }$$

where each polarisation contributes independently. Assuming acoustic modes and that the temperatures are such that only modes with frequencies $\scr{\Omega} \gg \omega \gg \Delta$ contribute significantly to the integral, then correct to lowest order in Δ, the thermal averages of the shifts in the orbital energies over all (acoustic) vibrational levels are

$$\begin{eqnarray}\begin{array}{rcl} \overline{\delta {{E}_{-}}} & = & {{\hbar }^{2}}\chi \rho \left( -\frac{1}{3}{{\scr{\Omega} }^{3}}+\frac{\Delta }{2}{{\scr{\Omega} }^{2}}+\frac{{{\pi }^{2}}k_{{\rm B}}^{2}}{3{{\hbar }^{2}}}{{T}^{2}} \right), \\ \overline{\delta {{E}_{+}}} & = & -{{\hbar }^{2}}\chi \rho \left( \frac{1}{3}{{\scr{\Omega} }^{3}}+\frac{\Delta }{2}{{\scr{\Omega} }^{2}}+\frac{{{\pi }^{2}}k_{B}^{2}}{3{{\hbar }^{2}}}{{T}^{2}} \right). \\ \end{array}\end{eqnarray} \tag{ 11 }$$

This yields a temperature shift in the spin–orbit splitting

$$\begin{eqnarray}&&\delta \Delta =\overline{\delta {{E}_{+}}}-\overline{\delta {{E}_{-}}}=-{{\hbar }^{2}}\chi \rho \Delta \left( {{\scr{\Omega} }^{2}}+\frac{2{{\pi }^{2}}k_{{\rm B}}^{2}}{3{{\hbar }^{2}}}{{T}^{2}} \right),\end{eqnarray} \tag{ 12 }$$

that is proportional to T² and a temperature independent mean energy of the orbital states $(\overline{\delta {{E}_{+}}}+\overline{\delta {{E}_{-}}})/2=-{{\hbar }^{2}}\chi \rho {{\scr{\Omega} }^{3}}/3$. This correctly predicts the observed T² dependence of the fine structure splittings in figures 4 (c) and (d) , but it fails to predict the T³ dependence of the optical line position in figure 4(b).

The failure of the above analysis to predict the temperature shift of the optical line position is due to a well-known problem in the treatment of the linear Jahn–Teller interaction [46]. The problem arises from the implicit choice of rectangular mode coordinates for the zero-order vibrational wavefunctions of the perturbative analysis. In rectangular coordinates, the vibrational wavefunction of a pair (${{Q}_{x}}$,Q_y) of degenerate ${\rm E}$ modes is of the form ${{\psi }_{i}}({{Q}_{x}}){{\psi }_{j}}({{Q}_{y}})$, where i and j are the independent vibrational quantum numbers of the modes. Since the rectangular coordinates do not match the cylindrical symmetry of the linear Jahn–Teller vibrational potential, the rectangular vibrational wavefunctions are a poor choice of zero-order basis [47]. As a consequence, much higher perturbative expansions are required to correctly predict a shift in the optical line position.

A superior choice of basis is obtained by transforming to polar coordinates (${{Q}_{x}}$,Q_y)$\to$(ρ,ϕ), within which the vibrational wavefunctions take the form ${{\psi }_{\nu ,l}}(\rho ,\phi )$, where $\nu =1,2,\ldots$ is the principal vibrational quantum number and $l=-\nu +1,-\nu +2,\ldots ,\nu -1$ is the vibrational angular momentum quantum number, such that the vibrational energies of modes with frequency ω are ${{E}_{\nu }}=\nu \hbar \omega$ [47]. Using the formalism of the linear Jahn–Teller effect in the polar vibrational basis [47], we obtained the second-order shifts of the vibrational energies as per section 3.4. Performing the thermal average of the shifts in the orbital energies over all (acoustic) vibrational levels, the corrected expression for the temperature shift of the optical line position is

$$\begin{eqnarray}&&\frac{1}{2}(\overline{\delta {{E}_{+}}}+\overline{\delta {{E}_{-}}})=-2\hbar \chi \rho \int _{0}^{\scr{\Omega} }\frac{2{{{\rm e}}^{\hbar \omega /{{k}_{{\rm B}}}T}}\left( {{{\rm e}}^{2\hbar \omega /{{k}_{{\rm B}}}T}}+3 \right)}{\left( {{{\rm e}}^{\hbar \omega /{{k}_{{\rm B}}}T}}-1 \right){{\left( {{{\rm e}}^{\hbar \omega /{{k}_{{\rm B}}}T}}+1 \right)}^{2}}}{{\omega }^{2}}{\rm d}\omega \propto {{T}^{3}},\end{eqnarray} \tag{ 13 }$$

which correctly predicts the T³ dependence of the optical line position. Note that this corrected approach is consistent with the previous subsection and also predicts a T² dependence of the fine structure splittings. Hence, we can conclude that the electron–phonon processes of the linear Jahn–Teller interactions within the ground and excited electronic levels are responsible for the observed temperature variations of the optical line width, position and fine structure splittings.

The ${\rm Si}{{{\rm V}}^{-}}$ ground states have spin and orbital degrees of freedom which can be used as qubit states. Figure 6(a) shows the electronic states under a magnetic field applied along the ${\rm Si}{{{\rm V}}^{-}}$ symmetry axis. The orbital relaxation rates $\gamma _{+,-}^{{\rm g},{\rm u}}$ discussed in this manuscript are spin conserving, consistent with the long spin ${{T}_{1}}$ times that were recently measured [19]. Even though such orbital relaxations are spin conserving, the detuning between $|\uparrow \rangle$ and $|\downarrow \rangle$ spin states varies depending on which orbital state is occupied. This means that the phase evolution of any coherent spin state changes after a phonon-induced orbital quantum jump, leading to the accumulation of random phase which is measured as decoherence. Hence coherences formed between any of the four states shown in figure 6 are expected to be limited by the ${{T}_{1}}$ of the orbital degree of freedom.

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Recent experiments [19, 20] that probed ground state coherences using coherent population trapping have reported $T_{2}^{*}$ values that are in good agreement with the orbital ${{T}_{1}}$ reported in our work. We note that the model used in [20] considered thermal relaxation mechanisms only between the two lowest energy ground states shown in figure 6. The authors concluded thermal relaxation rates between these two states to be suppressed owing to small spin overlap at low magnetic fields, and the $T_{2}^{*}$ to be limited by magnetic field noise from the ${\rm Si}{{{\rm V}}^{-}}$ environment. While the former agrees with our model (no relaxation between $|{{e}_{-}}\rangle |\downarrow \rangle \leftrightarrow |{{e}_{+}}\rangle |\uparrow \rangle$), all four ground states need to be considered to relate orbital relaxation rates to coherences. Based on the close experimental agreement between $T_{2}^{*}$ and orbital ${{T}_{1}},$ we conclude that coherence times of ${\rm Si}{{{\rm V}}^{-}}$ ground states are limited by phonon processes even at liquid helium temperatures.

We have shown that ground state coherences are limited by a single-phonon orbital relaxation process with a rate determined by a combination of phonon density of states and occupation $\left( \gamma _{\pm }^{{\rm g}}\sim \rho (\Delta )(2\;n(\Delta ,T)+1\mp 1) \right)$ at the energy of the spin–orbit splitting with $\Delta =\lambda _{{\rm so}}^{{\rm g}}$ ∼ 50 GHz. Since the interaction with the phonon bath is a Markovian process, dynamical decoupling sequences cannot be applied to extend coherences. To extend $T_{2}^{*}$, we will therefore focus on approaches that reduce the orbital relaxation rates $\gamma _{\pm }^{{\rm g}}$.

The first two approaches focus on reducing phonon occupation to decrease $\gamma _{+}^{{\rm g}}$. The occupation depends on the ratio, $T/\Delta$ , of the temperature and the energy splitting between the coupled orbital states. Substantial improvements can be achieved by minimizing this ratio in cooling the sample to lower temperatures ($T\ll \Delta \;\sim$ 2.4 K, figure 6(b)). Based on our fits in section 2.2, the expected orbital relaxation timescale is given by $1/\gamma _{+}^{{\rm g}}=101({{{\rm e}}^{2.4/T}}\;-1)$ ns which correspond to 1 μs at 1 K and 1 ms at 0.26 K. A second approach is to increase Δ by using emitters subjected to high strain. At the limit of $\Delta \gg T$, similar reductions in phonon occupation can be used to suppress relaxation rates, as shown in figure 6(c). Based on equation (5), we find that $1/\gamma _{+}^{{\rm g}}$ equals 1 ms (1 μs) for a strain shift of 1.6 THz (0.9 THz) at 4 K. We note that in both cases, only the two lowest energy states constitute a subspace that does not couple to phonons. The lowest two energy states are therefore expected to have long coherence times and could be used as a long-lived spin qubit.

The linear interaction Hamiltonian of section 3.1 and the resulting single-phonon orbital relaxation process are analogous to the Jaynes–Cummings Hamiltonian and Wigner–Weisskopf model of spontaneous emission used in quantum optics [48]. One can therefore use ideas developed in the context of cavity QED to engineer relaxation rates $\gamma _{\pm }^{{\rm g}}$. In particular, the phonon density of states can be reduced to suppress the orbital relaxation rates. This is analogous to inhibited spontaneous emission of photons [49] which has been observed for microwave and optical photons in atomic and solid-state systems [50–52]. Acoustic phonons in diamond offer an exciting new platform to probe this effect in a new regime owing to the highly broadband and reflective boundary conditions at the diamond-vacuum interface. To suppress orbital relaxation rates due to phonons at $\Delta$  ∼  50 GHz, small nano diamonds ($d\;\lt$ 120 nm) can be used to realize a complete phononic band gap for $\nu \;\lt \;$ 50 GHz phonons owing to the strong confinement [53]. An alternative approach would utilize recent advances in diamond nanofabrication [54, 55] to create 1D-optomechanical structures engineered to inhibit phonon and enhance optical transitions by modifying the density of states [56–59]. Using this approach both $\gamma _{+}^{{\rm g}}$ and $\gamma _{-}^{{\rm g}}$ are inhibited, therefore all four ground states can be used as long-lived qubits. We expect both approaches that modify phonon occupation and density of states to result in substantial improvements for the ground state coherences of ${\rm Si}{{{\rm V}}^{-}}$ centres.