Analysis of quantum coherence in bismuth-doped silicon: A system of strongly coupled spin qubits

There is a growing interest in bismuth-doped silicon (Si:Bi) as an alternative to the well-studied proposals for silicon-based quantum information processing (QIP) using phosphorus-doped silicon (Si:P). We focus here on the implications of its anomalously strong hyperfine coupling. In particular, we analyze in detail the regime where recent pulsed magnetic resonance experiments have demonstrated the potential for orders of magnitude speedup in quantum gates by exploiting transitions that are electron paramagnetic resonance (EPR) forbidden at high fields. We also present calculations using a phenomenological Markovian master equation, which models the decoherence of the electron spin due to Gaussian temporal magnetic field perturbations. The model quantifies the advantages of certain “optimal working points” identified as the $df/dB=0$ regions, where $f$ is the transition frequency, which come in the form of frequency minima and maxima. We show that at such regions, dephasing due to the interaction of the electron spin with a fluctuating magnetic field in the $z$ direction (usually adiabatic) is completely removed.

Figure 1

The 20 spin energy levels of Si:Bi may be labeled in order of increasing energy $|1\rangle ,|2\rangle ,...,|20\rangle$. States $|10\rangle$ and $|20\rangle$ are not mixed. State $|10\rangle$ is of especial significance since it, rather than the ground state, is a favorable state to initialize the system in. Experimental hyperpolarization studies[6] concentrate the system in this state. Thus, in our coupled two-qubit scheme, state $|10\rangle$ corresponds to our $|0_e{0}_n\rangle$ state; in the same scheme, states $|9\rangle \equiv |0_e{1}_n\rangle$ and $|11\rangle \equiv |1_e{0}_n\rangle$ are related to state $|10\rangle$ by a single qubit flip, while for $|12\rangle \equiv |1_e{1}_n\rangle$, both qubits are flipped.

Figure 2

(a) Comparison between theory [see Eqs. (10), (11), and (12)] (black dots) and experimental CW EPR signal (grey/red online) at 9.7 GHz. Resonances without black dots above them are not due to Si:Bi; the large sharp resonance at $0.35$ T is due to silicon dangling bonds, while the remainder are due to defects in the sapphire ring used as a dielectric microwave resonator. The variation in relative intensities is mainly due to the mixing of states as in Eq. (8). The variability is not too high but the calculated intensities are consistent with experiment and there is excellent agreement for the line positions. (b) Calculated EPR spectra (convolved with a 0.42 mT measured linewidth); they are seen to line up with the experimental spectra at $f=\omega /2\pi =9.7$ GHz). The type I cancellation resonances are indicated by integers $-m=0,1,2,3,4,5$. The first four of these are associated with $df/dB=0$ points. The type II cancelation resonance at ${\tilde{\omega }}_0\simeq 7$ also coincides with a $df/dB=0$ point, and corresponds to that shown in the $\lesssim 2$ GHz electron nuclear double resonance (ENDOR) spectra of Ref. 5. Figure reproduced with permission from Ref. 7.

Figure 3

Shows that near the ${\tilde{\omega }}_0=7$ frequency maximum, the transition rates $|12\rangle \leftrightarrow |11\rangle$ and $|8\rangle \leftrightarrow |9\rangle$ equalize, and we may transfer the coherences between the former to the latter with a relative phase shift of $\pi$. We use ${\omega }_1/2\pi =200$ MHz.

Figure 4

Shows that at the ${\tilde{\omega }}_0=5$ resonance, second-order two-photon transitions may be exploited since $f^{10\leftrightarrow 9}\simeq f^{11\leftrightarrow 10}$. A linear oscillating microwave field of strength ${\omega }_1/2\pi =200$ MHz is used. (a) shows that driving at resonance prepares $|10\rangle \to \frac{1}{\sqrt{2}}\left(\right|11\rangle -|9\rangle )$. The process is very sensitive to detuning from resonance. (b) and (c) illustrate how slight detuning of the microwave frequency may be used to prepare other superpositions such as $|9\rangle \to \frac{1}{\sqrt{2}}\left(i|10\rangle +|11\rangle \right)$ and $|11\rangle \to \frac{1}{\sqrt{2}}\left(-i|10\rangle +|9\rangle \right)$.

Figure 5

(a) and (b) show Rabi oscillations utilizing a linearly oscillating microwave field of strength ${\omega }_1/2\pi =200$ MHz. At $B=0.21$ T, the time taken for the electron qubit flip $|0_e{0}_n\rangle \to |1_e{0}_n\rangle$ is identical to that for the nuclear qubit flip $|0_e{0}_n\rangle \to |0_e{1}_n\rangle$. (c) and (d) show selective nuclear qubit flips and have microwave fields of strength ${\omega }_1/2\pi =100$ MHz. This is because the energy difference between the eigenstates is smaller than in the previous two cases and ${\omega }_1$ must remain perturbative. (c) Utilizes a linearly oscillating microwave field, which is nonselective for short pulses. At $B=0.22$ T, the rotation speed ratio is $|{\eta }_{12\to 11}|/|{\eta }_{9\to 8}|=5/4$. A $5\pi$ rotation of the nuclear qubit corresponds to a $4\pi$ rotation of the unwanted $|9\rangle \to |8\rangle$ transition. In (d), we use a RH circularly polarized microwave field, which selects for the desired conditional nuclear qubit rotation and is much more efficient than the linearly polarized case.

Figure 6

Shows the change in density operator elements of $|\psi \rangle =\frac{2}{\sqrt{3}}|9\rangle +\frac{1}{\sqrt{3}}|12\rangle$ after a period of $20/{\alpha }^2$, given diabatic Gaussian noise of variance ${\alpha }^2$, at two field regimes: the frequency minimum of $B=0.188$ T and $B=6$ T. (a) Population of $\rho$ in the eigenbasis of ${\widehat{H}}_0$ at time $t_0$. (b) and (c) show the effect of $B$ on $X$ noise. (b) Near the frequency minima, $X$ noise couples every part of the full Hilbert space, and depolarizes the full system, ultimately resulting in $\frac{1}{20}\mathbb{1}$. (c) At $B=6$ T, $X$ noise decouples states $|+,m\rangle ,|-,m-1\rangle$ from the rest of the Hilbert space, and effects a depolarizing channel in that subspace only. (d) and (e) show the effect of $B$ on $Z$ noise. Z noise conserves angular momentum and hence keeps to the four-dimensional Hilbert space of $m=-3,m-1=-4$. (d) shows that at the frequency minimum, $Z$ noise effects independent depolarizing channels for each $m$ subspace. As a result, the population of states $|12\rangle$ and $|9\rangle$ equalize with those of states $|8\rangle$ and $|11\rangle$, respectively.(e) shows that at $B=6$ T, $\langle +,m\left|{\widehat{S}}_z\right|-,m\rangle \sim 0$ and we simply get a dephasing channel for the $|+,m\rangle ,|-,m-1\rangle$ subspace.

Figure 7

The exponential decay rate given by $\Gamma$ in units of ${\alpha }^2/2$ for diabatic $Z$ noise driven (a) depolarization and (b) dephasing in Si:Bi. This is done in the four-dimensional subspace of $m=-3,m-1=-4$. (a) shows that in each subspace, the depolarization rate maximizes when ${\theta }_m=\pi /2$, or at the avoided crossing cancellation resonances. (b) shows that at the high-field limit, the dephasing rate of $a|\pm ,m\rangle +b|\mp ,m-1\rangle$ is maximal, whilst that of $a|\pm ,m\rangle +b|\pm ,m-1\rangle$ becomes vanishingly small. It should be noted, however, that the $a|-,m\rangle +b|+,m-1\rangle$ superposition cannot be made by either EPR or NMR in the high-field limit. These rates both approximately reach the value of 1/2 at the frequency minima.

Figure 8

Simulated dephasing times in units of $2/{\alpha }^2$ for diabatic (a) $Z$ and (b) $X$ noises, calculated with ${\alpha }^2/2\pi =9$ MHz. In (a), the superpositions $a|\pm ,m\rangle +b|\mp ,m-1\rangle$ have $T_2$ times of $2/{\alpha }^2$ at $B\gtrsim 0.6$ T and approximately $4/{\alpha }^2$ at the frequency minima. Superpositions $a|\pm ,m\rangle +b|\pm ,m-1\rangle$ also have $T_2$ times of $4/{\alpha }^2$ at the frequency minima. However, as $B$ increases, these become NMR transitions and will have $T_2$ times of $2/\left({\alpha }^2\delta \right)$. The color bar has been truncated after three to aid visibility but the maximum value reaches as high as $\sim 100$. In (b), the $T_2$ time does not vary by much and reaches its maximal points at fields less than the frequency minima.

Figure 9

Simulated depolarizing times for diabatic $Z$ and $X$ noise with ${\alpha }^2/2\pi =9$ MHz. (a) Given $Z$ noise, the decay of $\text{tr}\left[{\sigma }_z\tilde{\rho }\left(t\right)\right]$ within each $m$ subspace is always exponential. For $m⩽0$, the $T_1$ time reaches a minimum at the avoided crossing cancellation resonances. The $T_1$ times for subspaces $m$ and $m-1$ become identical at the frequency minima. (b) Given $X$ noise, the decay in the $\left\{\right|+,m\rangle ,|-,m-1\rangle \}$ and $\left\{\right|-,m\rangle ,|+,m-1\rangle \}$ subspaces follows an exponential curve at high magnetic fields, but at low magnetic fields such as the frequency minima, it follows a double exponential fit.