Hyperfine induced spin and entanglement dynamics in double quantum dots: A homogeneous coupling approach

We investigate hyperfine induced electron spin and entanglement dynamics in a system of two quantum dot spin qubits. We focus on the situation of zero external magnetic field and concentrate on approximation-free theoretical methods. We give an exact solution of the model for homogeneous hyperfine coupling constants (with all coupling coefficients being equal) and varying exchange coupling, and we derive the dynamics therefrom. After describing and explaining the basic dynamical properties, the decoherence time is calculated from the results of a detailed investigation of the short-time electron-spin dynamics. The result turns out to be in good agreement with experimental data.

Figure 1

(Color online) Spin dynamics for $|{\alpha }_1⟩=|\Uparrow \Downarrow ⟩,|T_{+}⟩,|T_0⟩$ and an even number of spins. The number of down spins in the bath is $N_D=20$ in all plots, yielding polarizations $p_b\approx 5–30\mathrm{\%}$. Note that the time unit is rescaled according to the number of bath spins. We see periodicity with $\pi$. For $|{\alpha }_1⟩=|T_0⟩$ and $N=58$ we count the number of local extrema on one period and find $N-2N_D+1=58-40+1=19$ as expected.

Figure 2

(Color online) Spin dynamics for $|{\alpha }_1⟩=|\Uparrow \Downarrow ⟩,|T_{+}⟩,|T_0⟩$ and an odd number of spins. The number of down spins in the bath is $N_D=20$ in all plots, giving polarizations $p_b\approx 2–30\mathrm{\%}$. In contrast to the case of an even number of spins we see periodicity with $2\pi$. For $|{\alpha }_1⟩=|\Uparrow \Downarrow ⟩$ and $N=45$ we count the number of local extrema on half the period and find $N-2N_D+1=45-40+1=6$ as expected.

Figure 3

(Color online) Spin dynamics for $|{\alpha }_1⟩=|\Uparrow \Downarrow ⟩$ and $N_D=20$, resulting in $p_b\approx 6–30\mathrm{\%}$. If $J_{ex}$ is an odd multiple of $A/2N$ we see periodicity with $2\pi$.

Figure 4

(Color online) Spin dynamics for $|{\alpha }_1⟩=\left(1/\sqrt{13}\right)\left(2|\Uparrow \Downarrow ⟩+3|\Downarrow \Uparrow ⟩\right)$ and $N_D=20$, resulting in $p_b\approx 6–30\mathrm{\%}$.

Figure 5

(Color online) Entanglement dynamics for $|{\alpha }_1⟩=|\Uparrow \Downarrow ⟩$ and $N_D=20$, resulting in $p_b\approx 6–30\mathrm{\%}$. In the completely homogeneous case the amplitude is small even for high polarization. Generation of entanglement benefits from high polarization.

Figure 6

(Color online) Entanglement dynamics for $|{\alpha }_1⟩=|T_{+}⟩$ and $N_D=20$, resulting in $p_b\approx 6–30\mathrm{\%}$. Instead of an oscillating function we see discrete peaks. Variation of the exchange coupling has no influence because $|T_{+}⟩$ is an eigenstate of the central spin coupling term.

Figure 7

(Color online) Spin dynamics on short time scales for $J_{ex}\lessgtr 0$, $p_b=2/N$, and $|{\alpha }_1⟩=|\Uparrow \Downarrow ⟩$. The thick solid lines mark the zero level $⟨S_1^z⟩=0$ while the thick dashed line (lower panel) represents the threshold level $⟨S_1^z⟩=0.2$ as appropriate for $J_{ex}<0$ and small spin baths.

Figure 8

(Color online) Position of the first zero of $⟨S_1^z\left(t\right)⟩$ for $J_{ex}\ge 0$, and the first intersection with the threshold level $⟨S_1^z⟩=0.2$ for $J_{ex}<0$, on a double-logarithmic scale. We choose $|{\alpha }_1⟩=|\Uparrow \Downarrow ⟩$ and a polarization of $p_b=2/N\iff N=2N_D+2$. The curves are fitted to a power law $\propto N^{\nu }$ with $\nu =-0.52$ $\left[J_{ex}=\left(A/N\right)\right]$, $\nu =-0.51$ $\left[J_{ex}=1.85\left(A/N\right)\right]$, $\nu =-0.53$ $\left(J_{ex}=0\right)$, $\nu =-0.51$ $\left[J_{ex}=-1.5\left(A/N\right)\right]$, and $\nu =-0.50$ $\left[J_{ex}=-1.85\left(A/N\right)\right]$. Note that the parallel offset between the plots for $J_{ex}\ge 0$ and $J_{ex}<0$ results from the fact that the intersection with the higher threshold level happens closer to zero.