Control of a two-electron double quantum dot with an external magnetic field

We investigate the utilization of external magnetic field for the manipulation of the quantum states in a two-electron double quantum dot. The singlet and triplet states are coupled through the hyperfine interaction between the electrons and surrounding nuclei. When the magnetic field is changed as a function of time, the singlet-triplet transition is possible for certain values of magnetic field, when the singlet state becomes degenerate with the triplet state. The transition probability depends on the sweeping speed of the magnetic field through the averaged Landau-Zener formula. We evaluate proper time scales for efficient control of the quantum dot, evaluate the singlet probabilities for different time-dependent magnetic fields, and study the average of the final singlet probability, averaged over the orientations of the nuclear spins surrounding the quantum dot.

Figure 1

The energy difference $\Delta E$ between the singlet state $S$ and triplet states $T_{-}$, $T_0$, and $T_{+}$ as a function of the external magnetic field. In the calculation of these energies, the hyperfine field is set to zero. The dark (light) gray lines correspond to dot distance 40 (50) nm.

Figure 2

(Color online) Probabilities $P_S$ of the singlet state $S$ as a function of the inverse of the time derivative $K$ of the energy difference for different values of the mean ${\delta }_0$ and variance ${\sigma }^2$ of the hyperfine field.

Figure 3

(Color online) Probabilities $P$ of the singlet state $S$ and triplet states $T_{-}$, $T_0$, and $T_{+}$ as a function of time averaged over 1000 realizations. In the left (right) figure the magnetic field is switched from 0.75 to 1.35 T in $55\left(550\right)\mu \text{s}$. Distance of the dots is 40 nm.

Figure 4

The probability distribution of the singlet probability $P_S$ for large collection of nuclear-field realizations. Panel (a) is for the case shown in Fig. 3a and (b) is for Fig. 3b. The times are 22 $\left(220\right)\mu \text{s}$ for circles, 39 $\left(390\right)\mu \text{s}$ for squares, and 55 $\left(550\right)\mu \text{s}$ for asterisks. The data points are connected with lines for clarity.

Figure 5

The asymptotic singlet probability $P_S$, indicated with markers, as a function of the switching time $\tau$ of the magnetic field. The light (dark) gray area gives the range of the asymptotic values inside the standard error of the mean of 50 (1000) realizations.