Indirect Control of the ²⁹SiV^- Nuclear Spin in Diamond

We begin by defining the optical ground state of ²⁹SiV^- Hamiltonian in the lab frame. The combined Hilbert space is a product of the ground state orbitals, and the electron and nuclear spin basis states. The total Hamiltonian is expressed in Eq. 1, and the relevant interaction terms for the electron ($\textbf{H}^{\rm e}$) & nuclear ($\textbf{H}^{\rm n}$) Zeeman, hyperfine coupling ($\textbf{H}^{\rm hf}$), spin-orbit coupling ($\textbf{H}^{\rm SO}$) and strain ($\textbf{H}^{\rm str}$ defined in the SiV coordinate frame [39]) are expressed in Eqs. (2) to (6). The z axis is defined along the main symmetry axis of the SiV defect $\langle 111\rangle$. Here, $\boldsymbol{\rm{S}}=[S_{x}~{}S_{y}~{}S_{z}]^{\intercal}$ and $\boldsymbol{\rm{I}}=[I_{x}~{}I_{y}~{}I_{z}]^{\intercal}$ are proportional to the Pauli operators for the electron and nuclear spins in the combined Hilbert space. The operators $\boldsymbol{\rm{L}}=[L_{x}~{}L_{y}~{}L_{z}]^{\intercal}$ are the orbital angular momentum operators and are expressed in the spin-orbit basis of $e_{\pm}$; the operators $L_{x}$ & $L_{y}$ vanish as they only couple the basis states to energy levels much higher in energy [40]. The static magnetic field vector is defined as $\boldsymbol{\rm{B_{0}}}=[B_{x}~{}B_{y}~{}B_{z}]^{\intercal}$. The strain Hamiltonian is also expressed in the spin-orbit basis using the transformation operator $\mathds{T}$ as shown in Eq. (6) [39]. The relevant parameters of the Hamiltonian are summarized in Table 1.

$$\textbf{H}=\textbf{H}^{\rm e}+\textbf{H}^{\rm n}+\textbf{H}^{\rm hf}+\textbf{H}^{\rm SO}+\textbf{H}^{\rm str},$$

(1)

where:

$$\textbf{H}^{\rm e}=\gamma_{e}(\boldsymbol{\rm{B_{0}}}\cdot{}\boldsymbol{\rm{S}})+\gamma_{L}qL_{z}B_{z},$$

(2)

$$\textbf{H}^{\rm n}=-\gamma_{n}(\boldsymbol{\rm{B_{0}}}\cdot{}\bf{I}),$$

(3)

$$\textbf{H}^{\rm hf}=A_{||}{S_{z}}{I_{z}}+A_{\perp}({S_{x}}{I_{x}}+{S_{y}}{I_{y}}),$$

(4)

$$\textbf{H}^{\rm SO}=-\lambda_{\rm SO}L_{z}S_{z},$$

(5)

$$\textbf{H}^{\rm str}=\mathds{T}\left(\begin{bmatrix}\alpha&\beta\\ \beta&-\alpha\end{bmatrix}\otimes\mathds{1}_{4}\right)\mathds{T}^{-1},\\ \mathds{T}=\begin{bmatrix}-1&-i\\ 1&-i\end{bmatrix}\otimes\mathds{1}_{4}$$

(6)

In this work we investigate both the unstrained ($\alpha=\beta$= 0 GHz) and strained ($\alpha=\beta$= 150 GHz) regimes of the ²⁹SiV^- Hamiltonian, as they can lead to different implementations of the spin qubit, i.e., by either driving the spin-orbit separated transition $\omega_{1}$ or the mixed orbital spin transitions $\omega_{2}$ [see Fig. 1(a)] [5]. This is a relevant consideration as it is possible to strain engineer [38] the silicon vacancy color centers, or fabricate them with naturally unstrained (or minimally strained) [40] properties. The values of strain were chosen such that there is sufficient mixing between orbitals, allowing electron spin transitions between spin states. Such values of strain allow the electron spin to undergo a full $\pi$-rotation for an optimised $\boldsymbol{\rm{B_{0}}}$ field, which need not be re-optimised for higher strain values. The relevant electron spin transitions of interest are illustrated in Fig. 1(a) for the unstrained and strained Hamiltonians, with transition labeled $\rm\omega_{1}$ and $\rm\omega_{2}$, respectively. These transitions represent electron $\pi$-rotations between eigenstates of the Hamiltonian. Lastly, the contribution of the Jahn-Teller interaction strength is not considered due a relatively negligible strength compared to the $\lambda_{SO}$ and strain [40].

The quantization axis of the electron spin depends primarily on the net magnetic field vector $\boldsymbol{\rm{B_{net}}}$[41] – the sum of the applied field $\boldsymbol{\rm{B_{0}}}$, the field induced by spin-orbit coupling $\boldsymbol{\rm{B_{\rm SO}}}$, and the hyperfine field via coupling to the ²⁹Si nucleus. Thus, with $|$$\boldsymbol{\rm{B_{0}}}$$|=0$ T or with $\boldsymbol{\rm{B_{0}}}$ $\parallel$ [1 1 1], the effective quantization field for the electron and nucleus remains parallel to the defect axis. However, when misalignment of the $\boldsymbol{\rm{B_{0}}}$ field is introduced, the effective quantization field is no longer parallel to $\boldsymbol{\rm{B_{0}}}$ due to the competition between the $\boldsymbol{\rm{B_{0}}}$ and $\boldsymbol{\rm{B_{\rm SO}}}$ vectors [40, 5]. The quantization field of the electron with respect to the defect symmetry axis is thus expected to vary as a function of the applied $\boldsymbol{\rm{B_{0}}}$ field. In comparison, the electron is not significantly affected by the magnetic moment of the nucleus as it results in a negligible hyperfine field of $\sim 3$ mT, as felt by the electron.

On the other hand, the electron spin orientation has a significant effect on the nuclear spin orientation due to the electron’s larger magnetic moment, resulting in a hyperfine field of $|$$\boldsymbol{\rm{B_{\rm hf}}}$$|\approx$ 4.14 T, as felt by the ²⁹Si nucleus. Thus, any change in the orientation of the electron spin also significantly affects the quantization field of the nuclear spin. Moreover, as the spin-orbit coupling is a relativistic effect experienced by the orbiting electron, the nuclear spin only experiences the applied $\boldsymbol{\rm{B_{0}}}$ field and the hyperfine field $\boldsymbol{\rm{B_{\rm hf}}}$. In summary, the main effect is that the nuclear spin eigenstate orientations are no longer strictly parallel to each other when the electron spin is flipped. This is the origin of the anisotropy-like effect of the nuclear spin quantization fields, captured by the angle $\Delta\theta$, which enables indirect control to become possible even in the absence of explicit hyperfine anisotropy. The relevant static field vectors of interest are illustrated in Fig. 1(b).

We investigate the change in orientation of the quantization fields of the nuclear spin represented by $\Delta\theta$ as a result of the relevant electron spin flipping transitions $\ket{\uparrow}\leftrightarrow\ket{\downarrow}$, illustrated in Fig. 1(a). We plot $\Delta\theta$ as a function of the $\boldsymbol{\rm{B_{0}}}$ field intensity and orientation for these transitions in Fig. 1(c) & (d), for the unstrained and strained Hamiltonians, respectively. Closer inspection of these plots shows that there are ideal parameters for the $\boldsymbol{\rm{B_{0}}}$ field that result in $\Delta\theta$ close to $90^{\circ}$. This is favourable as the efficiency of the indirect control technique is optimised by maximising this angle [28].

As an example we illustrate the exact spin eigenstates of the Hamiltonian, showing the quantization axes orientations in Fig. 2, with respect to the ²⁹SiV^- main symmetry axis. These orientations are plotted with respect to the defect axis, for both cases of strain and with the $\boldsymbol{\rm{B_{0}}}$ field orientated at $54.7^{\circ{}}$ (corresponding to [1 0 0] vector orientation). Lastly, the $\Delta\theta$ values for the nuclear spin corresponding to the relevant electron transitions [illustrated in Fig. 1(a)] are plotted in Fig. 3 as a function of $|$$\boldsymbol{\rm{B_{0}}}$$|$ applied at an angle $54^{\circ{}}$ away from the ²⁹SiV^- symmetry axis.

Two-axes control of a qubit requires control of both the zenith and azimuthal angles over the Bloch sphere. Conventionally, phase control of a spin qubit is performed using IQ modulation of the MW or RF pulses [43]– which is not possible in the absence of such oscillating fields. Importantly, simple precession of the nuclear spin around the primary quantization axis merely results in a global phase accumulation which is not relevant for quantum information processing. Fortunately, as we show here, phase control of the nuclear spin does not impose any further requirements on the electron spin control apart from simple $\pi$-rotations. In this way, the phase is accumulated as the spin precesses around the secondary quantization axis which detunes the spin from the resonance frequency.

We consider the application of a standard set of single qubit gates on the nuclear spin which include phase gates, implemented in simulation as instantaneous electron $\pi$-rotations. The nuclear spin IC gates are numerically optimised by enforcing that two seperate qubits each initialised in a different state correctly evolve to their respective final states in their rotating frames. Examples of gate actions on a pair of initial states are $\ket{+X}\leftrightarrow\ket{+X}$ and $\ket{\Uparrow}\leftrightarrow\ket{\Downarrow}$ for the X gate, where $\ket{\pm{}X}=(\ket{\Uparrow}\pm\ket{\Downarrow})/\sqrt{2}$.

Further, it is enforced that gate operations may only be performed when the lab frame is in phase with the rotating frame. This is to account for the fact that the quantization fields remain stationary in the lab frame. To achieve this, an appropriate delay is padded to the end of the optimised gate sequence, such that the rotating frame returns to phase with the lab frame at the end of the sequence. This padding delay is given as $\tau_{4}=\frac{\left\lceil T_{\rm gate}f_{\rm RF}\right\rceil}{f_{\rm RF}}-T_{\rm gate}$, where $f_{\rm RF}$ is the nuclear spin precession frequency, $T_{\rm gate}$ is the total optimised gate time and the $\left\lceil\right\rceil$ brackets represent a ceiling function. This ensures that there is a unique optimised gate sequence for a given gate operation.

We demonstrate the following gates: X, Y, Z, H, S & T, with the gate durations summarised in Table 2 for the unstrained (‘A’) and strained (‘B’) Hamiltonian parameters, also indicated in Fig. 3. The gates are simulated with high fidelity (F $\geq 0.98$), with diminishing returns expected for longer optimisation routines.

We find the total gate times to be at the order of hundreds of nanoseconds, significantly faster than those that can be achieved with NMR techniques. For example, the length of the IC Y gate for the unstrained Hamiltonian (‘A’) is equivalent to that achieved via an NMR Rabi frequency of $\sim 1.6$ MHz. Such high speed control would typically require an applied radio-frequency $B_{1}$ field of $\sim 400$ mT, which is impossible to achieve in millikelvin cryostats. Note that the gate times are generally higher for the strained case despite $\Delta\theta=90^{\circ{}}$. This is because the gate durations are limited by lower precession frequencies of the nuclear spin as discussed in Sec. IV.2.

Lastly, we can find the inverse gates S^-1 & T^-1 for uncomputation operations [44] using the same method described in Sec. III.1.

In this work we have shown simulations wherein instantaneous electron $\pi$-rotations are used to actuate on the nuclear spin orientation. Fast electron switching is required to minimally disturb both the electron and nuclear spins during the switching period, and to keep the electron switching fast compared to the nuclear spin timescales. Practical realisations of such high-speed electron Rabi frequencies involve high microwave frequency $B_{1}$ fields, which though achievable, would require high-Q factor resonators with contradictorily long ring-down times (order of 100 ns). This limits the suitability of magnetic resonance techniques for coherent electron spin control. Alternatively, high electron Rabi frequencies can be achieved using coherent acoustic control via surface acoustic waves [26], or using coherent all-optical control of the electron spin state [27, 45].

All-optical control is particularly favourable, as it does not involve the need for fabrication of an on-chip antenna or acoustic transducer. This would allow for precise, targeted nuclear spin qubit control, taking advantage of the state-of-the-art optical confocal microscopy methods [46]. The use of nanowatt scale optical power renders this method readily compatible with ultra low temperature cryostats such as dilution refrigerators with a cooling power of 10-500 $\mu{}$W [47]. Lastly, optical manipulation of the electron spin offers the potential of picosecond timescale $\pi$-rotations [48], which justifies the instantaneous spin rotations employed in our simulations.

The efficiency of indirect control directly depends on the delay times between switching of the quantization axes, which are determined by the precession frequency of the nuclear spin under the influence of its quantization field. A $\boldsymbol{\rm{B_{0}}}$ field in a non-optimal orientation may negate the benefits of indirect control. We plot the nuclear spin precession period as a function of magnetic field orientation, for the primary and secondary quantization axes as well as for the unstrained and strained cases in Fig. 5. These plots are to be used as reference for optimising the $\boldsymbol{\rm{B_{0}}}$ field parameters while also considering the resulting change in quantization axes as shown in Fig. 1(c)&(d). For the unstrained ²⁹SiV^- the precession times can range up to few microseconds for both the primary and secondary quantization fields [cf. Fig. 5(a)]. Whereas, for the strained ²⁹SiV^- the precession period is increased up to tens of microseconds with a $\boldsymbol{\rm{B_{0}}}$ field aligned either parallel or perpendicular to the ²⁹SiV^- defect axis [cf. Fig. 5(b)]. For both the strain and unstrained cases, it is comparatively favourable to align the $\boldsymbol{\rm{B_{0}}}$ field at $54.7^{\circ{}}$ as used in this work, given that an appropriate $|$$\boldsymbol{\rm{B_{0}}}$$|$ is chosen.