Qubit coherence control in a nuclear spin bath

Coherent dynamics of localized spins in semiconductors is limited by spectral diffusion arising from dipolar fluctuation of lattice nuclear spins. Here we extend the semiclassical theory of spectral diffusion for nuclear spins $I=1∕2$ to the high nuclear spins relevant to the III-V materials and show that applying successive qubit $\pi$ rotations at a rate approximately proportional to the nuclear spin quantum number squared $\left(I^2\right)$ provides an efficient method for coherence enhancement. Hence robust coherent manipulation in the large spin environments characteristic of the III-V compounds is possible without resorting to nuclear spin polarization, provided that the $\pi$ pulses can be generated at intervals scaling as $I^{-2}$.

Figure 1

Behavior of $1∕e$ decay times for the first CPMG echo in pseudospin subspaces $I^{\prime }⩽5∕2$ as a function of flip-flop rate $\Gamma$. Circles: calculation with $M^{\prime }$-dependent rates ${\Gamma }_{M^{\prime }}$. Solid lines: calculation with average rate $⟨{\Gamma }_{M^{\prime }}⟩$. Each curve achieves minimum coherence at $\Gamma ={\Gamma }_{\mathrm{c}}$, which is identified as a critical correlation rate. Small $\Gamma$ (or strong coupling $\Delta$) imply low-frequency noise with time correlation [regime (a)], while large $\Gamma$ (or weak coupling $\Delta$) imply motional narrowing or white noise behavior [regime (c)]. Note the strong $I^{\prime }$ dependence in the weak coupling region. Here we have $\mathrm{Min}\left(T_2\right)\approx 6∕\Delta (2I^{\prime }+1)$, and ${\Gamma }_{\mathrm{c}}\approx {(2I^{\prime }+1)}^3∕12$.

Figure 2

Coherence times $T_2$ and inter-pulse times ${\tau }_{100}$ required for $100\times$ coherence enhancement as a function of nuclear spin quantum number $I$. We find that both $T_2$ and ${\tau }_{100}$ scale as $\sim 1∕I^{1.5}$.