Suppression of charge noise using barrier control of a singlet-triplet qubit

It has been recently demonstrated that a singlet-triplet spin qubit in semiconductor double quantum dots can be controlled by changing the height of the potential barrier between the two dots (“barrier control”), which has led to a considerable reduction of charge noises as compared with the traditional tilt control method. In this paper we show, through a molecular-orbital-theoretic calculation of double quantum dots influenced by a charged impurity, that the relative charge noise for a system under the barrier control not only is smaller than that for the tilt control but actually decreases as a function of an increasing exchange interaction. This is understood as a combined consequence of the greatly suppressed detuning noise when the two dots are symmetrically operated, as well as an enhancement of the interdot hopping energy of an electron when the barrier is lowered which in turn reduces the relative charge noise at large exchange interaction values. We have also studied the response of the qubit to charged impurities at different locations and found that the improvement of barrier control is least for impurities equidistant from the two dots due to the small detuning noise they cause but is otherwise significant along other directions.

Figure 1

(a) Schematic diagram of the double-well confinement potential of a DQD system, each dot occupied by one electron. (b) A charged impurity located at ${\mathit{R}}_c=(x_c,y_c)$ shown as the red dot. Blue (thick) dashed lines show the $\widehat{x}$, $\widehat{y}$, and $\widehat{x}+\widehat{y}$ directions to be used later when discussing the effect of the impurity when it lies along one of these directions. The two wells are centered at $(\pm a,0)$ as shown by two gray dots. The unit charge impurity is located on ($x_c$, $y_c$). (c) The confinement potential used in the calculation as described in Eq. (5). Black solid line shows the potential neither tilt nor barrier controlled, which serves as the starting point as the comparison between the two control schemes. Red (asymmetric) dashed line shows the result of tilt control with detuning $\varepsilon$ with the barrier height being fixed. Blue (symmetric) dashed line shows the consequence of the barrier control while the two dots are kept leveled. Parameters are $a=100\mathrm{nm}$, $\hslash {\omega }_0=1.29\mathrm{meV}$.

Figure 2

Calculated energy spectra of the DQD system. Only the lowest two energy levels are shown. Black solid lines: energy spectrum with the middle barrier fixed by $\xi =1.3\text{meV}$. In this case, the exchange interaction $J\left(\varepsilon \right)$ is varied by changing the detuning $\varepsilon$. Red (gray) dashed lines show the result of the barrier control of changing $\xi$ to 1 meV (only the exchange interaction at $\varepsilon =0$ is used in the quantum computation). Parameters are $a=100\mathrm{nm}$, $\hslash {\omega }_0=100\mu \mathrm{eV}$.

Figure 3

(a) The exchange interaction $J\left(\varepsilon \right)$ under the tilt control while the barrier is fixed by $\xi =1.3$ meV. Black solid line shows the exchange interaction without any impurity, while the red (gray) dashed line shows the case with an impurity located at ${\mathit{R}}_c=(-6a,6a)$. (b) The exchange interaction $J\left(\xi \right)$ under the barrier control without detuning. Black solid line shows exchange interaction without any impurity, red (gray) dashed line shows the case with the impurity located at the same location as in panel (a). Inset shows a zoom-in of the range 0.5 meV $\le \xi \le 0.6$ meV. Parameters are $a=100\mathrm{nm}$, $\hslash {\omega }_0=100\mu \mathrm{eV}$.

Figure 4

(a) The relative charge noise $\delta J/J$ vs $J$. Black solid line shows the result for the tilt control case. Red (gray) dashed line shows the result under barrier control. Comparison of the different exchange interaction $\delta J/J$ caused by impurity with tilt and barrier control. Note the log scale of the $y$ axis. (b) The improvement ratio $\chi$, defined as the charge noise $\delta J$ for the tilt control, divided by the value for barrier control. The impurity is positioned at ${\mathit{R}}_c=(-6a,6a)$. Parameters are $a=100\mathrm{nm}$, $\hslash {\omega }_0=100\mu \mathrm{eV}$.

Figure 5

The quality factor $Q$ vs exchange interaction $J$. Black solid line shows the result for the tilt-control case. Red (gray) dashed line shows the result under barrier control. Blue (gray) dotted line shows the result of $Q$ when the relative charge noise is assumed to be independent of the exchange interaction, i.e., $\delta J/J=\mathrm{const}.$ The impurity is positioned at ${\mathit{R}}_c=(-6a,6a)$. Parameters are $a=100\mathrm{nm}$, $\hslash {\omega }_0=100\mu \mathrm{eV}$.

Figure 6

The relative charge noise $\delta J/J$ vs the distance between the impurity and the center of the DQD system, $R_c=\left|{\mathit{R}}_c\right|$. The impurity is positioned along (a) the $\widehat{x}$ direction, (b) the $\widehat{y}$ direction, and (c) the $\widehat{x}+\widehat{y}$ direction [cf. the blue (gray) dashed lines of Fig. 1]. The two control schemes are compared at $J=242$ MHz. Parameters are $a=100\mathrm{nm}$, $\hslash {\omega }_0=100\mu \mathrm{eV}$.

Figure 7

The relative charge noise $\delta J/J$ vs $J$. Black solid line shows the result for the tilt-control case. Red (gray) dashed line shows the result under barrier control. Comparison of the different exchange interaction $\delta J/J$ caused by impurity with tilt and barrier control. Note the log scale of the $y$ axis. The impurity is positioned at ${\mathit{R}}_c=(-1.5a,0.5a)$ and has charge $-0.01e$. Parameters are $a=100\mathrm{nm}$, $\hslash {\omega }_0=100\mu \mathrm{eV}$.