Coherent Control and Spectroscopy of a Semiconductor Quantum Dot Wigner Molecule

Semiconductor quantum dots containing more than one electron have found wide application in qubits, where they enable readout and enhance polarizability. However, coherent control in such dots has typically been restricted to only the lowest two levels, and such control in the strongly interacting regime has not been realized. Here we report quantum control of eight different transitions in a silicon-based quantum dot. We use qubit readout to perform spectroscopy, revealing a dense set of energy levels with characteristic spacing far smaller than the single-particle energy. By comparing with full configuration interaction calculations, we argue that the dense set of levels arises from Wigner-molecule physics.

Figure 1

(a) False-color micrograph of a device lithographically identical to that measured here. Quantum dots are formed under $P1$ and $P2$. A current $I_{\mathrm{CS}}$ flows through the dot controlled by gate CS and is used to detect the electron occupation of the $P1$ and $P2$ dots. (b) The ground state for the five-electron system at negative $ϵ$ is the charge configuration (4,1), used for initialization ($I$). (c) The ground state at positive $ϵ$ is (3,2), used for manipulation ($M$). (d) Readout of the ground state, $R_0$, maps onto (4,1), while the excited states, $R_1$, maintain the (3,2) configuration. The tunnel rates to both reservoirs are tuned for a long decay time from (3,1), enabling latched readout. (e) Rabi pulse sequence used in this work. (f) Ramsey pulse sequence used in this work, composed of two $\pi /2$ pulses and a detuning pulse with amplitude $P$. (g)–(j) Rabi and detuned-Ramsey measurements of two coexisting states. For (g),(i), $\delta V_{P2}$ corresponds to a relative shift of the entire pulse sequence; for (h),(j) $\delta V_{P2}$ corresponds to relative changes in $P$. Dashed lines in (h),(j) denote the value of $\delta V_{P2}$ for which $P=0$. (g),(h) Rabi and Ramsey oscillations using $f_R=8.33\mathrm{GHz}$. (i),(j) Rabi and Ramsey oscillations using $f_R=7.30\mathrm{GHz}$, taken at the same device tuning but different $ϵ$ from (g),(h).

Figure 2

(a) Rabi oscillations with $f_R=6.15\mathrm{GHz}$, with the centers of two on-resonance oscillations marked by dashed lines. (b) Rabi oscillations with $f_R=6.10\mathrm{GHz}$, where the two resonances move closer together in $ϵ$ as compared with (a). (c) Rabi oscillations with $f_R=6.00\mathrm{GHz}$, where the on-resonance locations have completely merged. (d)–(f) Simulated Rabi oscillations corresponding to (a)–(c) using a simplified, four-level model.

Figure 3

FCI calculations performed using parabolic confinement potentials with $E_{\mathrm{orb}}/h={\omega }_x/2\pi =59.2\mathrm{GHz}$ and ${\omega }_y=1.07{\omega }_x$. (a) Two-electron excited state energies $(E_1-E_0)$ plotted as a function of the relative interaction strength $(e^{*}/e{)}^2$ (bottom) and the Wigner parameter $R_W$ (top). (b) Excited state energies with ($e^{*}=e$, blue) and without ($e^{*}=0$, yellow) interactions. The spectrum is grouped into manifolds, indicated by boxes, with the lowest manifold in the inset. (c) The single-electron energy levels used to construct the interacting two-electron wave functions in (b), with the orbitals labeled as $n_x$, $n_y$. Each grouping of orbital excitations forms a valley-split doublet. (d) Electron density distributions of the lowest-energy singlet wave functions in each of the four manifolds in (b), identified by the corresponding border style. The bar plots below indicate the contributions from each of the single-electron wave functions in (c). (e) Two-electron singlet-triplet splitting with (blue) and without (yellow) electron-electron interactions as a function of orbital confinement, demonstrating large variation as a function of $E_{\mathrm{orb}}$ when interactions are strong.

Figure 4

(a) Energy eigenvalues versus $ϵ$ of a Hamiltonian motivated by the electron interaction effects reported here. Measurement locations from Fig. 1 are indicated. Inset: excited state spectrum with example transitions shown. (b) Frequencies versus $ϵ$ of the eight transitions reported here, $E_{01}$–$E_{05}$, and $E_{12}$–$E_{15}$, and $E_{01}$ which is inferred but not directly observed. The dark teal lines are the energy differences between the ground state and energy level $n$, defined in (a); the light teal lines are differences between the first excited level and the same energy levels in (a). Symbols plotted correspond to frequencies extracted from the experiment, as described in the legend.