Spectrum of an Electron Spin Coupled to an Unpolarized Bath of Nuclear Spins

The main source of decoherence for an electron spin confined to a quantum dot is the hyperfine interaction with nuclear spins. To analyze this process theoretically we diagonalize the central spin Hamiltonian in the high magnetic $B$-field limit. Then we project the eigenstates onto an unpolarized state of the nuclear bath and find that the resulting density of states has Gaussian tails. The level spacing of the nuclear sublevels is exponentially small in the middle of each of the two electron Zeeman levels but increases superexponentially away from the center. This suggests to select states from the wings of the distribution when the system is projected on a single eigenstate by a measurement to reduce the noise of the nuclear spin bath. This theory is valid when the external magnetic field is larger than a typical Overhauser field at high nuclear spin temperature.

Figure 1

Numerical evaluation of $\nu (E)$ using Eq. (2) on a course scale—thick line and Eq. (5)—thin line ($S_j=1$, $r_0=8$, $N=18$, a fixed external $B$ field), $A$ is a maximum Overhauser field, $E_0$ is a shift from Eq. (5). Insets show $\nu (E)$ on a fine scale in the middle of the upper electron Zeeman line and at a finite detuning, the average level spacing $d$ was evaluated using Eq. (6).