Decoherence of spin qubits due to a nearby charge fluctuator in gate-defined double dots

The effects of a nearby two-level charge fluctuator on a double-dot two-spin qubit are studied theoretically. Assuming no direct tunneling between the charge fluctuator and the qubit quantum dots, the Coulomb couplings between the qubit orbital states and the fluctuator are calculated within the Hund-Mulliken framework to quadrupole-quadrupole order in a multipole expansion. We identify and quantify the coupling term that entangles the qubit to the fluctuator and analyze qubit decoherence effects that result from the decay of the fluctuator to its reservoir. Our results show that the charge environment can severely impact the performance of spin qubits, and indicate working points at which this decoherence channel is minimized. Our analysis also suggests that an ancillary double-dot can provide a convenient point for single-qubit operations and idle position, adding flexibility in the quantum control of the two-spin qubit.

Figure 1

(Color online) Orbital energy diagram for the double dot near the (1,1)-(0,2) transition vs the dimensionless bias shift $\widetilde{\Delta x}$. Shown are the Hybridized singlet states (black curves) and split (1,1) triplet states $T_{-}$ (dash-dotted green), $T_0$ (solid blue), and $T_{+}$ (dashed red). The inset shows the exchange energy dependence on bias. Here and throughout the paper $B=100\text{mT}$, $d=2.8$, and ${\omega }_0=3\text{meV}$

Figure 2

qubit-TLS system geometry.

Figure 3

(Color online) Angle-averaged Coulomb coupling terms vs qubit-TLS distance at double-dot bias corresponding to the (1,1) to (0,2) charge transition $\left(\widetilde{\Delta x}=0.405\right)$. (a) $\alpha$ terms; (b) $\beta$ terms; (c) $\gamma$ terms.

Figure 4

(Color online) Angle-averaged qubit-TLS coupling terms as functions of the double-dot bias for qubit-TLS distances of: (a) $R=30\text{nm}$, (b) $R=200\text{nm}$. The qubit exchange energy is shown for comparison. All other system parameters are the same as those used in Fig. 3.

Figure 5

(Color online) (a) Singlet probability vs time for several qubit-TLS distances, at bias $\widetilde{\Delta x}=0.382$. (b) Singlet probability vs time for several bias points at $R=100\text{nm}$. (c) End-of-pulse error relative to the $\beta$ corrected rotation about $\tilde{\vartheta }$-tilted axis vs TLS spontaneous emission rate $\Gamma$, for several $R$ values, at bias $\widetilde{\Delta x}=0.382$. (d) End-of-pulse error vs $\Gamma$ for several bias points at $R=100\text{nm}$. In figures (a) and (b) $\Gamma =0.04\mu \text{eV}$, the solid lines represent the actual time evolution, and dashed lines correspond to pure rotations around the $\tilde{\vartheta }$-tilted axes (see main text). In all the figures pulse duration is taken as $T=\pi /\delta h$ with magnetic field inhomogeneity $\delta h=1\mu \text{eV}$.

Figure 6

(Color online) Singlet probability as a function of time for original, $\beta$-corrected, and ${\beta }^{\prime }$-corrected pulse cycles, with qubit-TLS distance: (a) $R=500\text{nm}$, (b) $R=200\text{nm}$, and (c) $R=80\text{nm}$. (d) $\pi$ flip error as a function of qubit-TLS distance. In all figures $\delta h=1\mu \text{eV}$, $\Gamma =0.04\mu \text{eV}$, $D_T=5\text{nm}$, and $a_T=19.05\text{nm}$. The two bias points in the cycle are: ${\widetilde{\Delta x}}_{\vartheta }=0.357$, and ${\widetilde{\Delta x}}_z=0.414$. The kinks evident in these plots are the result of the three pulses employed in these gate operations.

Figure 7

(Color online) $\pi$ flip error of $\beta$- corrected (solid lines) and ${\beta }^{\prime }$-corrected (dashed lines) cycles for three qubit-TLS distances as a function of: (a) TLS Bohr radius $D_T$ ($a_T=4D_T$ at all points), (b) TLS centers half separation $a_T$ ($D_T=5\text{nm}$ for all points), (c) TLS spontaneous emission rate $\Gamma \left(D_T=5\text{nm},a_T=20\text{nm}\right)$, and (d) magnetic field inhomogeneity between the two dots $\left(D_T=5\text{nm},a_T=20\text{nm}\right)$. The inset in (b) shows TLS tunneling vs. $a_T$. Dotted lines in (c) depict the gate error for the ${\beta }^{\prime }$-corrected cycle with an initial TLS superposition state. (e) Number of cycles needed to complete the $\pi$ flip as a function of $\delta h$, for $\beta$-corrected cycles. (f) Singlet probability vs. time for $R=200\text{nm}$, $\delta h=0.02\mu \text{eV}$, where 7 $\beta$-corrected cycles are needed to complete the operation. In plots (a)–(c), the magnetic field inhomogeneity is $\delta h=1\mu \text{eV}$ and in all plots except (c), the TLS spontaneous emission rate is $\Gamma =0.1\mu \text{eV}$.

Figure 8

(Color online) Several qubit-TLS geometries for $\theta ={\theta }_T=90°$ (qubit and TLS lie in the $X-Y$ plane): (a) Same axis $\left(\widehat{R}={\widehat{x}}_T=\widehat{x}\right)$; (b) Parallel axes $\left(\widehat{R}=\widehat{y},{\widehat{x}}_T=\widehat{x}\right)$; (c) Perpendicular axes $\left(\widehat{R}={\widehat{x}}_T=\widehat{y}\right)$; (d) Perpendicular axes $\left(\widehat{R}=\widehat{x},{\widehat{x}}_T=\widehat{y}\right)$. For the geometries depicted in figures (a) and (b) the $\beta$ coupling is always positive, while for those in figures (c) and (d) it is always negative.

Figure 9

(Color online) Quantum control using a second double dot in same axis configuration [Fig. 8a] with $R=200\text{nm}$ (a) A $\pi$ rotation about the $X$ axis with $\delta h=1\mu \text{eV}$, obtained by a single pulse at the $\tilde{J}=0$ sweet spot, $\widetilde{\Delta x}=0.3866$ (solid-blue line) and by using a three-pulse cycle (dashed-green line). The two bias points in the three-pulse cycles are: ${\widetilde{\Delta x}}_{\vartheta }=0.357$, and ${\widetilde{\Delta x}}_z=0.414$. Here, $\Gamma =0.04\mu \text{eV}$. (b) Real part of the off-diagonal element of the qubit’s reduced density matrix vs time at the sweet spot, for several values of $\Gamma$, with $\delta h=0$. For this figure we used $D_T=10\text{nm}$, $a_T=36\text{nm}$ corresponding to TLS tunneling rate of $t_T=0.38\mu \text{eV}$.