Efficient Orthogonal Control of Tunnel Couplings in a Quantum Dot Array

Electrostatically-defined semiconductor quantum dot arrays offer a promising platform for quantum computation and quantum simulation. However, crosstalk of gate voltages to dot potentials and interdot tunnel couplings complicates the tuning of the device parameters. To date, crosstalk to the dot potentials is routinely and efficiently compensated using so-called virtual gates, which are specific linear combinations of physical gate voltages. However, due to exponential dependence of tunnel couplings on gate voltages, crosstalk to the tunnel barriers is currently compensated through a slow iterative process. In this work, we show that the crosstalk on tunnel barriers can be efficiently characterized and compensated for, using the fact that the same exponential dependence applies to all gates. We demonstrate efficient calibration of crosstalk in a quadruple quantum dot array and define a set of virtual barrier gates, with which we show orthogonal control of all interdot tunnel couplings. Our method marks a key step forward in the scalability of the tuning process of large-scale quantum dot arrays.

Figure 1

(a) A scanning electron microscope image of a device nominally identical to the one used here. The dashed circles indicate the intended positions of the quadruple dot and sensing dot. (b) Schematics illustrating the influence of changes in $B_{23}^{\prime }$ and $B_{23}^{†}$ on the potential landscape of a quadruple quantum dot. The grey area denotes the original landscape, and the blue (red) dashed line indicates the landscape when $B_{23}^{\prime }$ ($B_{23}^{†}$) is changed. Here $B_{23}^{\prime }$ controls the interdot tunnel coupling, $t_{23}$, while keeping the dot potentials fixed, but also influences $t_{12}$ and $t_{34}$. In contrast, $B_{23}^{†}$ controls $t_{23}$ while affecting neither other tunnel couplings nor dot potentials.

Figure 2

(a) Charge stability diagram showing the sensing-dot signal as a function of voltages on $P_2^{\prime }$ and $P_3^{\prime }$. ($N_2$, $N_3$) indicates charge occupation of dot 2 and 3. The red dotted line indicates the interdot detuning axis. (b) Excess charge (in units of e) extracted from a fit to the sensing-dot signal as a function of detuning near the interdot transition in (a). Data (colored circles) for different $t_{23}$ (in $\mu \mathrm{eV}$) is shown together with the fitted curves (dashed lines). The model of the fit is described in Ref. [18]. Here $t_{23}$ is obtained from the fit. (c) Measured tunnel coupling $t_{23}$ as a function of barrier voltage $B_{12}^{\prime }$ and $B_{23}^{\prime }$, with an exponential fit to the data. (d) Same as (c) but with a smaller voltage variation in $B_{12}^{\prime }$ and $B_{23}^{\prime }$, plotted with a linear fit.

Figure 3

(a)–(c) Measured tunnel couplings as a function of $B^{\prime }$ for (a) $t_{23}$, (b) $t_{12}$, and (c) $t_{34}$. Dashed lines show linear fits to the data. (d)–(f) Measured tunnel couplings as a function of $B^{†}$ for (d) $t_{23}$, (e) $t_{12}$, and (f) $t_{34}$. After calibration, each $t_{ij}$ only depends on the corresponding $B_{ij}^{†}$. Dashed lines show linear fits to the data.

Figure 4

The experimentally measured tunnel coupling $t_{23}$ as a function of $\mathrm{\Delta }{B}_{23}^{†}$ for different values of $\mathrm{\Delta }{B}_{12}^{†}$ (a) and $\mathrm{\Delta }{B}_{34}^{†}$ (b) (in mV), plotted with an exponential fit to the data. Here $\mathrm{\Delta }{B}_{ij}^{†}$ is the voltage relative to $B_{ij}^{†}$ when $t_{ij}\sim 25\mu \mathrm{eV}$. The exponential fit has an offset of $13\mu \mathrm{eV}$. As observed in other works, the expression Eq. (2) is a good approximation over a finite range of gate voltages, for instance because of the presence of other tunnel barriers nearby.

Figure 5

Photon-assisted tunneling measurement showing the sensor signal as a function of frequency and detuning at the $(1,0,1,1)–(0,1,1,1)$ interdot transition, shown after background subtraction [18]. The red dashed line is the fit of the form $hf=\sqrt{{ϵ}_{12}^2+4t_{12}^2}$.

Figure 6

Tuning tunnel couplings using $B^{†}$. (a) The tunnel couplings in the initial configuration where $(\mathrm{\Delta }{B}_{12}^{†},\mathrm{\Delta }{B}_{23}^{†},\mathrm{\Delta }{B}_{34}^{†})=(0,0,0)$, the expected (“Target”) and the measured (“Result”) tunnel couplings in the target configuration where $(\mathrm{\Delta }{B}_{12}^{†},\mathrm{\Delta }{B}_{23}^{†},\mathrm{\Delta }{B}_{34}^{†})=(0,6,9)$. (b) Excess charge (in units of e) as a function of detuning at the $(1,0,1,1)$–$(0,1,1,1)$ (green), $(1,1,0,1)$–$(1,0,1,1)$ (yellow) and $(1,1,1,0)$–$(1,1,0,1)$ (blue) interdot transitions, along with the measured tunnel couplings (in $\mu \mathrm{eV}$), when $(\mathrm{\Delta }{B}_{12}^{†},\mathrm{\Delta }{B}_{23}^{†},\mathrm{\Delta }{B}_{34}^{†})=(0,6,9)$. Offset in y axis is added to the data for clarity. The dashed lines show the fit to the data.

Figure 7

Stepwise tuning and calibration of tunnel couplings: $t_{23}$ as a function of (a) $B^{\prime }$ and (b) $B^{\ast 1}$; $t_{34}$ as a function of (c) $B^{\ast 1}$ and (d) $B^{\ast 2}$; $t_{12}$ as a function of (e) $B^{\ast 2}$ and (f) $B^{†}$.