Harnessing the GaAs quantum dot nuclear spin bath for quantum control

We theoretically demonstrate that nuclear spins can be harnessed to coherently control two-electron spin states in a double quantum dot. Hyperfine interactions lead to an avoided crossing between the spin singlet state and the $m_{\text{s}}=+1$ triplet state, ${\text{T}}_{+}$. We show that a coherent superposition of singlet and triplet states can be achieved using finite-time Landau-Zener-Stückelberg interferometry. In this system the coherent rotation rate is set by the Zeeman energy, resulting in $\sim 1\text{ns}$ single spin rotations. We analyze the coherence of this spin qubit by considering the coupling to the nuclear spin bath and show that $T_2^{\ast }\sim 16\text{ns}$, in good agreement with experimental data. Our analysis further demonstrates that efficient single qubit and two-qubit control can be achieved using Landau-Zener-Stückelberg interferometry.

Figure 1

(Color online) (a) Energy diagram for the relevant states in the DQD as a function of $\epsilon$. The spin states for the implementation of the qubit are the hybridized singlet S and the triplet ${\text{T}}_{+}$. (b) A coherent superposition of the qubit states is generated by LZS interferometry. (c) The initialized S(2,0) is swept through the avoided crossing by means of an applied linear gate voltage pulse ${\epsilon }_{\text{i}}\to {\epsilon }_{\text{f}}$. The final state is a coherent superposition of S and ${\text{T}}_{+}$ generated by LZS tunneling. At ${\epsilon }_{\text{f}}$, the system evolves in the external magnetic field $B$ for a time $t_{\text{w}}$ before a reverse gate voltage pulse brings the system back to ${\epsilon }_{\text{i}}$ where a quantum point contact (QPC) measurement is performed to determine the singlet state return probability, $P_{\text{S}}$.

Figure 2

(Color online) Theoretical results. (a) The singlet return probability $P_{\text{S}}$ from the semiclassical model as a function of the waiting time $t_{\text{w}}$ and final detuning ${\epsilon }_{\text{f}}$. We find nanosecond oscillation periods and the dephasing time $T_2^{\ast }\sim 16\text{ns}$, in good agreement with experiment. We used $B=100\text{mT}$, $\gamma =0.015\text{meV}/\text{ns}$, $u=4\text{meV}$, and $\tau =5\mu \text{eV}$. (b) $P_{\text{S}}$ (blue) as a function of $t_{\text{w}}$ for ${\epsilon }_{\text{f}}=3.97\text{meV}$ [horizontal line in (a)]. We plot (orange) $C\text{sin}\left(\omega t\right)\text{exp}\left[-{\left(t/T_2^{\ast }\right)}^2\right]$, where $\omega =\left|E_{\text{S}}\left({\epsilon }_{\text{f}}\right)-E_{{\text{T}}_{+}}\right|/\hslash \sim 2\pi \cdot 0.23\text{GHz}$, $C=0.95$, and $T_2^{\ast }=\left(16.0\pm 0.4\right)\text{ns}$ is extracted from a best fit.

Figure 3

(Color online) Comparison between experimental (purple, open circles) and theoretical values (blue, filled circles) of $P_{\text{S}}$ for $B=45\text{mT}$. Theoretical values are obtained by finding the detuning ${\epsilon }_{\text{f}}=3.78\text{meV}$ for which the theoretical and experimental oscillation periods match. A suitable $\gamma =0.12\text{meV}/\text{ns}$ is then chosen such that ${\epsilon }_{\text{i}}>{\epsilon }_{\text{c}}$. The experiments used passive filtering of square pulses to reduce $P_{\text{LZS}}$ and thereby increase the visibility of the oscillations. As a result, oscillations in the experimental data are delayed to longer times due to finite rise time effects. The theory points are obtained with a perfect square pulse.

Figure 4

(Color online) Bloch sphere representation of LZS transitions as rotations. (a) When the propagation times are equal, $-t_{\text{i}}=t_{\text{f}}$, the rotation axis lies in the $xy$ plane. In addition with the rotations generated around the $z$ axis by letting the qubit evolve in the external magnetic field, any rotation can be achieved by the Euler angle method. (b) If the propagation times are different the rotations axis may not lie (see Appendix ) in the $xy$ plane. In this case, a pure LZS interferometry technique can be used to generate any rotation.

Figure 5

(Color online) A conditional gate can be implemented by capacitively coupling electrons trapped in quantum dots belonging to different qubits. The crossing position ${\epsilon }_{\text{c}}$ has different values whenever the charge state of the control qubit is (a) (0,2) or (b) (1,1). The later case results in ${\epsilon }_{\text{f}}⪡{\epsilon }_{\text{c}}$ which suppresses any LZS transition.

Figure 6

(Color online) (a) Cosine of the rotation angle and components of the rotation axis (b)–(d) as a function of ${\tau }_{\text{f}}$ for the dimensionless parameters ${\tau }_{\text{i}}=10$ and $\eta =3$.