Validity of the single-particle description and charge noise resilience for multielectron quantum dots

We construct an optimal set of single-particle states for few-electron quantum dots (QDs) using the method of natural orbitals (NOs). The NOs include also the effects of the Coulomb repulsion between electrons. We find that they agree well with the noniteracting orbitals for GaAs QDs of realistic parameters, while the Coulomb interactions only rescale the radius of the NOs compared to the noninteracting case. We use NOs to show that four-electron QDs are less susceptible to charge noise than their two-electron counterparts.

Figure 1

Noninteracting orbital energy spectrum of a harmonic QD for the confining strength $\hslash {\omega }_0=3\text{meV}$ and out-of-plane magnetic fields $B_z$ according to Eq. (1). The eigenstates are the FD states $|n,l\rangle$ that are described by the orbital quantum number $n$ and the magnetic quantum number $l$. For $B_z=0$, the FD states are grouped into atomic shells. We label the nondegenerate lowest shell as $K$ shell and the doubly degenerate first-excited shell as $L$ shell. Out-of-plane magnetic fields lift the degeneracies. The QD parameters from the highlighted region protect a four-electron STQ from charge noise, as described in the main text.

Figure 2

NOs of the lowest energy configurations with $s_z=0$ of a two-electron QD and a four-electron QD of $\hslash {\omega }_0=3\text{meV}$ and $B_z=0.35\text{T}$. In each case, the ground state is a singlet, and the excited state is a triplet. The NOs are ordered by the energy expectation values of the single-particle Hamiltonian from Eq. (1): $E=\langle \psi ({\mathit{r}}_1,\cdots {\mathit{r}}_N)\left|h\left({\mathit{r}}_1\right)\right|\psi ({\mathit{r}}_1,\cdots {\mathit{r}}_N)\rangle$. We interpret all the NOs by rescaled FD states $|\widetilde{n,l}\rangle$. The dominant NOs (drawn with bold lines) resemble the predictions from a shell filling model: (a) the two-electron singlet has an occupation of $89\%$ in the lowest NO. Two additional NOs with $5.0\%$ occupations are needed to sufficiently describe the singlet. The two-electron triplet has nearly $99\%$ occupation in the lowest two NOs. All NOs are summarized in Table 1. (b) The four-electron configurations have nearly $50\%$ occupations in the ground state orbital. We interpret this configuration as a frozen core. The singlet has the dominant remaining occupation $\left(43.1\%\right)$ in the next excited NO; the triplet requires two additional NOs (each with $24.4\%$ occupation). In every case, some additional NOs have occupations of a few percent or less; all NOs are tabulated in Table 2.

Figure 3

Probability densities $\rho \left(\mathit{r}\right)$, according to Eq. (4), of the lowest energy states of a two-electron QD and a four-electron QD with the confining strength $\hslash {\omega }_0=3\text{meV}$ and $B_z=0.35\text{T}$. In both cases, the ground state is in a singlet configuration and the excited state is in a triplet configuration. The dashed lines show radii where the probability densities drop to $1/e$ of their maximal values. The two-electron singlet has a smaller spread of the charge distribution compared to the triplet configuration. In the four-electron configuration, the probability densities are very similar for the singlet and the triplet. All properties can be explained in shell-filling description using FD states, as explained in the text.

Figure 4

Radial dependency of the singlet's $\mathrm{[}{\rho }_S\left(\right|\mathit{r}\left|\right)$, solid lines] and the triplet's $\mathrm{[}{\rho }_T\left(\right|\mathit{r}\left|\right)$, dashed lines] probability densities. The curves are shifted relatively to each other for different out-of-plane magnetic fields $B_z$. (a) The probability densities of the two-electron QDs are always distinct from each other. (b) For a four-electron QD, the probability densities are similar at small magnetic fields, but they differ for larger $B_z$.

Figure 5

Shift $\delta E_{ST}$ of the energy difference between the singlet and the triplet states of a two-electron STQ and a four-electron STQ which is caused by a charge trap that is the distance $d$ away from the QD center, as approximated by Eq. (10). $\delta E_{ST}$ is nearly one order of magnitude larger for the four-electron STQ than for the two-electron STQ at $\hslash {\omega }_0=3\text{meV}$ and $B_z=0.35\text{T}$. Recall that $\delta E_{ST}=0$ for four-electron STQs in a noninteracting theory.