Phonon-mediated decay of singlet-triplet qubits in double quantum dots

We study theoretically the phonon-induced relaxation ($T_1$) and decoherence times ($T_2$) of singlet-triplet qubits in lateral GaAs double quantum dots (DQDs). When the DQD is biased, Pauli exclusion enables strong dephasing via two-phonon processes. This mechanism requires neither hyperfine nor spin-orbit interaction and yields $T_2\ll T_1$, in contrast to previous calculations of phonon-limited lifetimes. When the DQD is unbiased, we find $T_2\simeq 2T_1$ and much longer lifetimes than in the biased DQD. For typical setups, the decoherence and relaxation rates due to one-phonon processes are proportional to the temperature $T$, whereas the rates due to two-phonon processes reveal a transition from $T^2$ to higher powers as $T$ is decreased. Remarkably, both $T_1$ and $T_2$ exhibit a maximum when the external magnetic field is applied along a certain axis within the plane of the two-dimensional electron gas. We compare our results with recent experiments and analyze the dependence of $T_1$ and $T_2$ on system properties such as the detuning, the spin-orbit parameters, the hyperfine coupling, and the orientation of the DQD and the applied magnetic field with respect to the main crystallographic axes.

Figure 1

The energy spectrum of the DQD calculated for the parameters described in the text. The $S$-$T_0$ qubit is formed by the eigenstates of type $\left|\right(1,1)S\rangle$ and $|(1,1)T_0\rangle$.

Figure 2

(a) Temperature dependence of the decoherence time ($T_2$, blue) and relaxation time ($T_1$, red) for the parameters in the text. The solid line corresponds to a power-law fit to $T_2$ for $\text{0.1}\text{K}\le T\le \text{0.2}\text{K}$, which yields $T_2\propto T^{-3}$ and good agreement with recent experiments [16]. We note that $T_2\ll T_1$. (b) The decoherence time due to one-phonon ($1/{\Gamma }_2^{1\mathrm{p}}$) and two-phonon processes ($1/{\Gamma }_2^{2\mathrm{p}}$) and the full decoherence time $T_2=1/{\Gamma }_2=1/({\Gamma }_2^{1\mathrm{p}}+{\Gamma }_2^{2\mathrm{p}})$ as a function of temperature. We note that $1/{\Gamma }_2^{2\mathrm{p}}$ changes its behavior from $\propto C_1+C_2{T}^{-5}$ to $\propto T^{-2}$, where $C_1$ and $C_2$ are constants, whereas $1/{\Gamma }_2^{1\mathrm{p}}\propto T^{-1}$ for the range of $T$ considered here.

Figure 3

Dependence of the decoherence time $T_2$ on the temperature for the parameters in the text and different spin-orbit lengths. Keeping the splitting $J_{\mathrm{tot}}$ between the qubit states constant, the values chosen for the detuning $\epsilon$ are 0.896 meV (black), 0.912 meV (blue), 0.918 meV (green), and 0.933 meV (red), increasing with increasing SOI. Within the range $T=100–200\text{mK}$, $T_2\propto T^{-3}$ in all cases. We note that the best quantitative agreement with the experiment [16] is obtained for the strongest SOI (red), where $l_R=1\mu \mathrm{m}$ and $l_D=0.5\mu \mathrm{m}$.

Figure 4

Decoherence time $T_2$ as a function of temperature from two different models. The dotted line is also shown in Fig. 3 and was calculated via Eq. (12), using the parameters in the text with $\Omega =0$ (no SOI) and $\epsilon =0.896\text{meV}$. The crosses result from Eq. (25), using exactly the same parameters. We note that the associated $J_{\mathrm{tot}}$ differ only slightly. The remarkable agreement demonstrates that the simple model of Sec. 3d accounts for the dominant decay mechanism. At $T\lesssim 50\text{mK}$, the curves start to deviate because relaxation is no longer negligible. When the hyperfine coupling in Eq. (23) is not omitted, excellent agreement is obtained also at low temperatures.

Figure 5

Dependence of the relaxation $\left(T_1\right)$ and decoherence time $\left(T_2\right)$ on the angle ${\theta }_B$ between the in-plane magnetic field $\mathit{B}$ and the $x$ axis that connects the QDs. When $\mathit{B}\perp x$ (${\theta }_B=\pi /2$), both $T_1$ and $T_2$ exhibit a maximum. Red (black) corresponds to the spin-orbit lengths $l_R=2\mu \mathrm{m}$ and $l_D=1\mu \mathrm{m}$ ($l_R=1\mu \mathrm{m}$ and $l_D=0.5\mu \mathrm{m}$). For the stronger SOI, the lifetimes increase by almost two orders of magnitude. For details, see text.

Figure 6

Temperature dependence of the decoherence time $\left(T_2\right)$ and its one-phonon ($1/{\Gamma }_2^{1\mathrm{p}}$) and two-phonon ($1/{\Gamma }_2^{2\mathrm{p}}$) parts for the detuning $\epsilon \simeq 0$, where excited states are taken into account. For this plot, $U=1\text{meV}$, $V_{+}=50\mu \text{eV}$, $V_{-}=49.5\mu \text{eV}$, $t=24\mu \text{eV}$, $J_{\mathrm{tot}}=1.41\mu \text{eV}$, and the other parameters as described in the text. We note that $T_2\simeq 2T_1$.