Spin dynamics in a double quantum dot: Exact diagonalization study

We study spin dynamics and singlet-triplet decoherence due to the hyperfine interaction in a parabolic double quantum dot. We use exact diagonalization and perturbation theory for calculating the time evolution of the wave function. The probabilities for the singlet state exhibit damped oscillations and converge toward a value, which depends only on the strength of the hyperfine field and the exchange energy. We derive expressions for the asymptotic singlet probability and its variance using a $2\times 2$ effective Hamiltonian matrix. The asymptotic values and variances for singlet probabilities are in agreement with previous experimental data.

Figure 1

Confinement potential $V_c\left(x,0\right)$ and $L_x=50\text{nm}$ for detuning parameters $L_T=0\text{nm}$ (dashed) and $L_T=20\text{nm}$ (solid).

Figure 2

(Color online) Exchange $J$ as a function of detuning parameter $L_T$. From top to bottom at $L_T=0\text{nm}$: $L_x=10$, 20, 30, and 50 nm. Distance between the dots is $d=2L_x$. Inset shows $L_x=50\text{nm}$ case zoomed in the region of small $L_T$ values.

Figure 3

Singlet probability $P_S$ as a function of time $t$ for $J=41\text{neV}$ and $L_x=50\text{nm}$. Dashed line is a single realization and dash-dotted line is an average over ${10}^4$ realizations. The inset is zoomed in the region of few nanoseconds and shows the singlet probability of a single realization obtained with full ED calculation (solid) until 1.2 ns and after that with the perturbation calculation (dashed).

Figure 4

(Color online) Measurements of Laird et al. (Ref. 24) with numerical results. From top to bottom: $J=133$ (red), 62 (green), 41 (blue), 22 (brown), and 12 neV (magenta). The unit time determined by $J$ is from top to bottom, respectively, 5, 11, 16, 30, and 55 ns. The dashed lines give the standard error of the mean when the number of realizations is 50. The distance of the dots $2L_x$ is 100 nm. The curves are offset by 0.05 for clarity, i.e., all cases have $P_S=1$ at $t=0$.

Figure 5

Asymptotic triplet probability ${\overline{P}}_T$ and its variance ${\sigma }^2$. Solid line is a fit ${\sigma }^2=0.56{\left({\overline{P}}_T\right)}^{1.5}$.

Figure 6

Mean (solid) and variance (dashed) of the asymptotic triplet probability $1-{\overline{P}}_S$, calculated from Eqs. (19, 20).

Figure 7

The variance of the asymptotic triplet probability ${\sigma }^2\left({\overline{P}}_T\right)$ as a function of ${\overline{P}}_T$. Dashed line is a fit ${\sigma }^2=0.56{\left({\overline{P}}_T\right)}^{1.5}$.