Inhomogeneous nuclear spin flips: Feedback mechanism between electronic states in a double quantum dot and the underlying nuclear spin bath

We discuss a feedback mechanism between electronic states in a two-electron double quantum dot and the underlying nuclear spin bath. We analyze two pumping cycles for which this feedback provides a force for the Overhauser fields of the two dots to either equilibrate or diverge. Which of these effects is favored depends on the $g$ factor and Overhauser coupling constant of the material. The strength of the effect increases with the ratio of Overhauser coupling to electron exchange energy and also increases as the external magnetic field decreases.

Figure 1

(Color online) Top: electronic states near the (1,1) to (0,2) stability diagram transition. Crossing of $\Psi$ and $|{\text{T}}^{+}⟩$ at $\tilde{\epsilon }$ becomes avoided crossing in presence of transverse Overhauser field gradient. Bottom: overlap of the $\Psi$ state with (a) $S\left(0,2\right)$, with (b) $|{\text{L}}_{\uparrow }{\text{R}}_{\downarrow }⟩$, and with (c) $|{\text{L}}_{\downarrow }{\text{R}}_{\uparrow }⟩$. Parameters: $B_{ext}=0.2T$, $\gamma =1.2\mu \text{eV}$, $\Delta =1000$, $E_C=0.6\text{meV}$, and $V_x=1\mu \text{eV}$.

Figure 2

The wave-function ratio $r\equiv c/b$ evaluated at the $\Psi \text{−}|{\text{T}}^{+}⟩$ crossing point, $\tilde{\epsilon }$, as a function of $B_{ext}$ for various values of the nuclear spin $z$ component difference $\Delta$. $r\left(\tilde{\epsilon }\right)$ is monotonically increasing with $\Delta$ and decreasing with $B_{ext}$.

Figure 3

(main) Reduced distribution $W\left(\Delta ,s\right)$, calculated from Eq. (7) (solid lines), for $V_x=1\mu \text{eV}$ as a function of $\Delta$ for various $B_{ext}=0.05,0.10,\dots ,0.75\text{T}$ (lower fields have narrower $W$); and thermal $W\left(\Delta \right)$ (dashed), averaged over $s$, all with $I_L=I_R={10}^3$. Inset (a) narrowing factor ${\sigma }_T/\sigma \left(B_{ext}\right)$ versus $B_{ext}$. Inset (b) Illustration of $I_{Lz}-I_{Rz}$ plane. $\Delta$ and $s$ are the diagonal coordinates, with $\Delta \equiv I_{Lz}-I_{Rz}$.