Nonperturbative bounds on electron spin coherence times induced by hyperfine interactions

We address the decoherence of a localized electron spin in an external magnetic field due to the hyperfine interaction with a lattice of nuclear spins. Using a completely nonperturbative method, rigorous bounds on the $T_1$ and $T_2$ coherence times for the electron spin are provided. It is shown that for magnetic fields $B$ greater than some critical field $B_c$ ($B_c\approx 0.001\text{–}2\mathrm{T}$ for the systems studied here), the $z$ polarization of the electron spin cannot relax, and hence $T_1$ is infinite. However, even at high fields dephasing can still occur. We provide a lower bound on the $T_2$ coherence time that explicitly takes into account the effects of a spin-echo pulse sequence.