Long-range electron-electron interactions in quantum dot systems and applications in quantum chemistry

Long-range interactions play a key role in several phenomena of quantum physics and chemistry. To study these phenomena, analog quantum simulators provide an appealing alternative to classical numerical methods. Gate-defined quantum dots have been established as a platform for quantum simulation, but for those experiments the effect of long-range interactions between the electrons did not play a crucial role. Here we present a detailed experimental characterization of long-range electron-electron interactions in an array of gate-defined semiconductor quantum dots. We demonstrate significant interaction strength among electrons that are separated by up to four sites, and show that our theoretical prediction of the screening effects matches well the experimental results. Based on these findings, we investigate how long-range interactions in quantum dot arrays may be utilized for analog simulations of artificial quantum matter. We numerically show that about ten quantum dots are sufficient to observe binding for a one-dimensional $\mathrm{H}{}_2$-like molecule. These combined experimental and theoretical results pave the way for future quantum simulations with quantum dot arrays and benchmarks of numerical methods in quantum chemistry.

Figure 1

Quantum dot system with schematics of chemistry simulations. (a) Scanning electron micrograph of the device. Dot locations are indicated with dashed circles, and charge sensors are indicated by resistance meters labeled with $\mathrm{\Omega }$. Barrier gates between sites $i$ and $j$ are labeled by ${\mathrm{B}}_{\mathrm{ij}}$, and plunger gates at site $k$ are labeled by ${\mathrm{P}}_{\mathrm{k}}$. Charge sensors are controlled by gates ${\mathrm{SP}}_1$ and ${\mathrm{SP}}_2$, respectively. (b) Sketch of competing terms in Eq. (1) describing the quantum dot array: intersite interaction $V_{ij}$, tunnel coupling $t_{kl}$, and on-site interaction $U_m$. (c) Schematic of the QD array and its relation to atomic QC simulation, exemplarily shown for ten QDs. The electron-nucleus interaction potential of an artificial hydrogen atom is encoded in the local energy offsets ${\varepsilon }_i$. (d) Similar setup in the case of the molecular ion $H_2^{+}$ with two nuclei at ${\mathbf{R}}_1$ and ${\mathbf{R}}_2$, respectively. The system can be studied for different internuclear distances $|{\mathbf{R}}_1-{\mathbf{R}}_2|$ separately. Blue squares (red diamonds) indicate the local expectation values $\langle {\widehat{n}}_i\rangle$ in the ground state (first-excited state), and the blue-dashed (red-solid) line shows the fit to the ground-state (excited-state) wave function [23, 24]. The orange colorscale encodes the local potential offset.

Figure 2

[(a)–(f)] Charge-stability diagrams showing the sum of signals from both charge sensors as a function of the local energy offsets for the dots highlighted by the white dashed circles. (a) For the leftmost dot only the left sensor signal is shown, and is measured relative to the gate for the rightmost sensor. This gate here acts as a dummy, because it has negligible effect on both the leftmost dot and the signal from the left sensor. The broad vertical band is a Coulomb peak for the left sensor, which appears due to crosstalk from ${\varepsilon }_1$. (b–f) For the leftmost dot in combination with a dot (b) one, (c) two, (d) three, (e) four, and (f) five sites away. Black dashed lines are fits to the anticrossings, and are used to extract the interaction elements; the lighter dashed line in (b) is a guide to the eye to indicate the voltage of the shifted vertical addition line. The voltages are converted to energies with the lever arms $\{105,94,104,86,104,95\}\mu \mathrm{e}\mathrm{V}\mathrm{m}{\mathrm{V}}^{-1}$, which were obtained with photon-assisted tunneling experiments [27]. (g) Interaction matrix elements $V_{ij}$ vs distance between QDs $i$ and $j$, which are centered around ${\tau }_i$ and ${\tau }_j$, respectively. Blue dots indicate the experimentally obtained interaction elements and the solid line shows the numerical result based on screening due to the gate metal as described in detail in Appendix pp2. The uncertainties in the interaction elements are dominated by uncertainties in the lever arms, estimated to be below 10%, and the electron temperature ($\approx 10\mu \mathrm{e}\mathrm{V}$). The inset shows the interaction and fits on a logarithmic scale for better comparison. For simplicity $U_i$ is denoted as $V_{ii}$

Figure 3

Numerical results for the atom and the two-electron molecule. Lowest part of the artificial hydrogenlike atom spectrum as a function of $V_0$ and $\eta =t/V_0$ for system sizes (a) $N=6$, (b) $N=25$, and (c) $N=300$. The left axis (dark squares) shows the atomic binding energy $E_b$ in $\mathrm{Ry}$, while the right axis (light circles) shows the corresponding eigenenergies of Eq. (1) in $\mu \mathrm{eV}$. Shown are the ground (blue), first-excited-state (green), and second-excited-state (red) energies. The dashed lines indicate the three lowest Balmer series values. In all cases, $t=20\mu \mathrm{eV}$. (d) Molecular binding energy $\mathrm{\Delta }$ (see text) for the artificial $H_2$-like molecule with two electrons interacting via effective potential $V_{ee}$, and two nuclei separated by distance $R$. Shown is the molecular binding energy for different system sizes $N=6,10,15,...,35$. Curve fits to guide the eye (solid line for $N=35$ and dash-dotted lines for all other $N$). Other parameters: $t=40\mu \mathrm{eV}$, $V_0=200\mu \mathrm{eV}$, and $\mathrm{Ry}=1\mathrm{meV}$.

Figure 4

Full set of pairwise charge-stability diagrams. The leftmost column shows the diagrams presented in Fig. 2 in the main text. For non-nearest-neighbor pairs smaller voltage ranges were used to maintain sufficient resolution with the same number of data points.

Figure 5

Screened interaction potential. (a) Schematic illustration of the numerical procedure to calculate the screened interaction potential. The device geometry shown in Fig. 2 is taken and its surface area discretized. The interaction potential is obtained from the pairwise interaction terms of all tile charges ${\lambda }_i$ at position ${\mathbf{s}}_i$ and the two electrons at ${\mathbf{r}}_1$ and ${\mathbf{r}}_2$. (b) Interaction matrix elements $V_{ij}$, between dots located at ${\tau }_i$ and ${\tau }_j$, compared for different cases: unscreened Coulomb interactions (red, dashed), screened interactions via image-charge method for a surface completely covered by metal (orange, dash-dotted), and screened interaction via numerical discretization of real device geometry (green, solid).

Figure 6

Charge occupation numbers $\langle {\widehat{n}}_i\rangle$ for a system with $N=10$ QDs and two electrons. Upper panel: Results for ground (left) and first-excited (middle) state, respectively. The $x$ axis denotes the site index of the $i\mathrm{th}$ dot in the array, and the $y$ axis denotes the internuclear separation $R/a_{\mathrm{QD}}$. The absolute value of the difference between the first two columns is shown on the right. Lower panel: Same as upper panel, but for a model with an additional bias term $\mathrm{\Delta }{\varepsilon }_i=(i-x_c)\times 10\mu \mathrm{eV}$, where $x_c$ denotes the center of the QD array.