Qubit Protection in Nuclear-Spin Quantum Dot Memories

We present a mechanism to protect quantum information stored in an ensemble of nuclear spins in a semiconductor quantum dot. When the dot is charged the nuclei interact with the spin of the excess electron through the hyperfine coupling. If this coupling is made off-resonant, it leads to an energy gap between the collective storage states and all other states. We show that the energy gap protects the quantum memory from local spin-flip and spin-dephasing noise. Effects of nonperfect initial spin polarization and inhomogeneous hyperfine coupling are discussed.

Figure 1

Left: Charged quantum dot with a single, polarized excess electron. Right: Spectrum of the effective nuclear Hamiltonian in the presence of a polarized electron. Off-resonant hyperfine coupling results in a gap ${\Delta }_{\mathrm{gap}}$ between the storage state $|\mathbf{1}⟩$ and the nonstorage states $|1_q⟩$. ${\Delta }_{\mathrm{K}}$ denotes the Zeeman shift due to the effective magnetic field associated with the electron spin (Knight shift).

Figure 2

Hyperfine Rabi frequency ($\Omega$), protective energy gap (${\Delta }_{\mathrm{gap}}$), Knight shift difference between the logical states (${\Delta }_{\mathrm{K}}$), symmetry breaking couplings due to inhomogeneities ($\zeta$ and $\omega$), and qubit decoherence rates due to dipolar spin diffusion without (${\Gamma }_D$) and with (${\Gamma }_D^{\prime }$) protection. (a) The fully polarized (zero temperature) case shown versus the number of spin-$\frac{1}{2}$ nuclei ($N$) taking part in the storage, i.e., located within $3\sigma$ of the oblate Gaussian electron distribution with in-plane variance $\sigma$. (b) Estimated energies in dark states $|{\mathcal{D}}_{n,\beta }⟩$ with $n$ spins flipped from the fully polarized state for $N={10}^5$. Energy units are chosen to match GaAs. ${\Delta }_{\mathrm{gap}}$ is obtained by taking $\delta =10\Omega$.

Figure 3

The parameters $\Xi$ and ${\Xi }^{\prime }$ describing the effects of spatial correlations in the classical noise (${\xi }_{jk}=e^{-r_{jk}/\xi }$) for different number of nuclei. The same family of Gaussian electron densities was used as in Fig. 2. The bullets on the curves denote the linear size of the dot given by the variance $\sigma$.