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.

Figure 1

A schematic representation of the eigenspectra of $H_0$ and $H$. The eigenstates of the unperturbed Hamiltonian can be grouped into two subspaces $\Uparrow$ and $\Downarrow$ based on the $z$ polarization of the electron. In Theorems 1 and 2, we demonstrate that the eigenstates of the full Hamiltonian $H$ can be grouped into two subspaces $+$ and $-$ based on the approximate $z$ polarization of the electron. Theorem 3 further states that the $+$ and $-$ eigenenergies fall within the same range as the $\Uparrow$ and $\Downarrow$ eigenenergies, respectively.

Figure 2

Lower bound on $T_2$ versus the magnetic field for a donor impurity in a sample of natural Si. Note that this plot shows the dependence of the lower bound on the magnetic field, not the dependence of $T_2$ itself. At high magnetic fields $B⪢B_c$ the lower bound scales approximately as $B$. The dashed line shows the predicted time scale for decoherence due to nuclear spectral diffusion arising from internuclear, dipolar coupling (Ref. 11). Thus, at fields greater than 3.0 T, this nuclear spectral diffusion will dominate the hyperfine coupling as a source for decoherence.

Figure 3

Lower bound on $T_2$ versus the magnetic field $B$ for a donor impurity in GaAs and a GaAs quantum dot with radius $l_0=20\mathrm{nm}$ and thickness $z_0=10\mathrm{nm}$. The dashed line shows the approximate decoherence time of $\sim 10\mu \mathrm{s}$ for both systems, due to the dipolar-induced nuclear spectral diffusion (Refs. 11, 23). At fields above $\sim 12.5\mathrm{T}$, spectral diffusion will dominate the hyperfine coupling as a source for decoherence.