Figure 1

(a) Schematic section of a vertical double-dot device also showing, using our bias convention, the direction of electron flow in forward and reverse bias, as well as the orientation of the applied $B$ field. (b) Cartoon of the measurement principle. Here, in reverse bias, with $V_{\text{SD}}$ applied between the emitter and collector, the ground ($1s$-like) state of dot 1 probes three mixed states $a$, $b$, and $c$ in dot 2 (appropriate, for example, for the three-level mixing to be discussed in Sec. 4 of the main text). The $1s\to b$ resonance condition is shown and all empty nonresonant states at lower energy than $a$, $b$, and $c$ are omitted.Reuse & Permissions

Figure 2

(Color online) (a) Fock-Darwin spectrum calculated with $\hslash {\omega }_0=6.1\text{meV}$. The states originating from the first five shells at 0 T are labeled with their familiar atomic-orbital-like notation. (b) Single-particle energy spectrum of an elliptical parabolic dot calculated with $\hslash {\omega }_x=6.1\text{meV}$ and $\hslash {\omega }_y=4.6\text{meV}$. Here, the states are labeled using the $n_x$ and $n_y$ quantum numbers. In both (a) and (b), energy levels which merge into the same Landau level at high $B$ field are colored the same and energy of all states is with respect to the energy of the ground state, i.e., the diamagnetic shift of the ground state has been subtracted from all energy levels. The lowest three-level crossings are labeled $\gamma$, $\tau$, and $\pi$ in each spectrum.Reuse & Permissions

Figure 3

(Color online) Energy spectrum of dot 2 from device I [(a) and (b)], and energy spectrum of dot 2 from device II [(c) and (d)] both of which reveal widespread level mixing. The reason why there are upper and lower parts to each spectrum is explained in Sec. 4 of the main text. The approximately picoampere resonant current (nonresonant background current not removed) is shown. Dotted black lines identify barely resolvable portions of some weak spectral features. The energy scales are estimated by comparison to ideal calculated spectra (see Sec. 3 of the main text). The $2p^{-}$-like state and the related $\left(n_x,n_y\right)=\left(1,0\right)$-like state are marked in (b) and (d), respectively. The thick black lines at the bottom right of each panel are explained in Sec. 4 of the main text.Reuse & Permissions

Figure 4

(Color online) (a) Differential conductance, $dI/d{V}_{\text{SD}}$, in the $V_{\text{SD}}\text{−}{V}_{\text{G}}$ plane for device II at 0 T. Black, gray, and white, respectively, represents positive, zero, and negative conductance. The current is less than 200 pA (except in the gray region in the lower left of the figure). The $N=1$ and 2 Coulomb diamonds are marked, and the two arc-shaped regions of primary interest are highlighted and marked SET. The dot 1 (dot 2) spectrum can be measured in forward (reverse) bias. The first few resonance lines involving the lower energy levels of the probed dot are marked by triangles (and labeled with the atomic-orbital-like notation for simplicity although the dots’ spectra are elliptical in nature). (b) Cartoon of extended reverse bias SET region (in gray) showing three possible two-part vector-voltage line scans numbered i, ii, and iii [see Sec. 4 in the main text for discussion]. The situation depicted, with the spectrum of dot 2 from device II in mind [see Figs. 3c, 3d], is of the nearly vertical resonance lines cut at $\sim 2\text{T}$ if the dot 2 spectrum were that shown in Fig. 2b. For this condition, the resonance lines relevant to the $\tau$ and $\gamma$ crossings are minimally separated. When the resonance lines pass below the SET region into the region where other tunneling processes can additionally occur, they are shown as dashed lines, and not all resonance lines that may appear in this region are shown (Ref. 39).Reuse & Permissions

Figure 5

(Color online) (a) Energy-level [(differential conductance resonance (Ref. 38)] position versus $B$ field for the three-level crossing $\gamma$ from the spectrum of dot 2 from device II [see boxed region in Fig. 3c]. Black, gray, and white, respectively, represents positive, zero, and negative conductance. The energy scale bar corresponds to $\sim 0.8\text{meV}$, and the upper, center, and lower branches, respectively, are labeled by the symbols △, ○, and ▽. (b) Selected (smoothed) current traces which when numerically differentiated form vertical sections of the plot in (a) (nonresonant background current not removed). The peaks of the three resonances are also marked by △, ○, and ▽ for each trace, except where the peaks are too weak to identify. (c) Current values (resonant current with nonresonant background component subtracted) versus $B$ field for each branch extracted by strategy A from the current traces used to build up the plot in (a). The black lines are a guide to the eyes (and are generated by simple Gaussian fitting although no meaning is attached to the fitting procedure).Reuse & Permissions

Figure 6

Panels showing gray scale plots of the differential conductance, $dI/d{V}_{\text{SD}}$, in the relevant region of the $V_{\text{SD}}\text{−}{V}_{\text{G}}$ plane in the vicinity of the $\gamma$ crossing in the spectrum of dot 2 from device II at eight different $B$ fields. The gray scale is in the same sense as that in Fig. 4a. Dashed lines in the 1.9 T panel highlight the lower and upper edges of the region of interest in which SET dominates (Ref. 45). In this region, the upper, center, and lower branch resonance lines, respectively, are labeled by the symbols △, ○, and ▽. Due to the extent and shape of the SET region, the time required for the measurements is minimized by capturing data from a parallelogram-shaped region in the $V_{\text{SD}}\text{−}{V}_{\text{G}}$ plane rather than a (more conventional) rectangular-shaped region [as was done in Fig. 4a]. However, the data is displayed here in rectangular-shaped panels for which the start and end points of the $V_{\text{SD}}$ sweep are systematically shifted together for each value of $V_{\text{G}}$ [stepped from $-0.55$ to $-0.9\text{V}$ from bottom to top] such that at the bottom (top) of each panel, $V_{\text{SD}}=-47\text{mV}$ $\left(-100\text{mV}\right)$ on the left side and $V_{\text{SD}}=-31\text{mV}$ $\left(-84\text{mV}\right)$ on the right side of the panel. The crosses in the 2.4 T panel are explained in Sec. 5 of the main text in connection with Fig. 7.Reuse & Permissions

Figure 7

Current values (resonant current with nonresonant background component subtracted) versus $B$ field for each branch of the $\gamma$ crossing in the spectrum of dot 2 from device II extracted by strategy B at points (a) a quarter, (b) half, and (c) three quarters of the way along each of the three resonance lines inside the region of interest [see 2.4 T panel of Fig. 6 for definition of these points (Ref. 45)]. The average values of the currents at these three points are shown in (d).Reuse & Permissions

Figure 8

(Color online) (a) Energy-level [differential conductance resonance (Ref. 38)] position versus $B$ field for the three-level crossing $\tau$ from the spectrum of dot 1 from device I [not shown (Ref. 40)]. The grayscale is in the same sense as that in Fig. 5a. The energy scale bar corresponds to (a surprisingly large) $\sim 2.5\text{meV}$, and the upper, center, and lower branches, respectively, are labeled by the symbols △, ○, and ▽. (b) Current values (resonant current with nonresonant background component subtracted) versus $B$ field for each branch extracted by strategy A from the current traces used to build up the plot in (a). The upper branch current to the left of $\sim 2.2\text{T}$ is too small to determine. (c) Fit of energy-level positions. The colored lines estimating the position of the uncoupled basis levels provide a guide to the eyes. The energy of the point where the three uncoupled basis levels cross is set to zero energy. The energy-level position data extracted from (a) is also included (gray squares). (d) Fit of branch currents. The data points from (b) are also included (gray squares). (e) Reconstructed eigenvectors of the three branches showing the components of the uncoupled basis states on passing through the crossing region. $\Delta B=0\text{T}$ is at $\sim 2.68\text{T}$.Reuse & Permissions