Single-electron double quantum dot dipole-coupled to a single photonic mode

We have realized a hybrid solid-state quantum device in which a single-electron semiconductor double quantum dot is dipole coupled to a superconducting microwave frequency transmission line resonator. The dipolar interaction between the two entities manifests itself via dispersive and dissipative effects observed as frequency shifts and linewidth broadenings of the photonic mode respectively. A Jaynes-Cummings Hamiltonian master equation calculation is used to model the combined system response and allows for determining both the coherence properties of the double quantum dot and its interdot tunnel coupling with high accuracy. The value and uncertainty of the tunnel coupling extracted from the microwave read-out technique are compared to a standard quantum point contact charge detection analysis. The two techniques are found to be consistent with a superior precision for the microwave experiment when tunneling rates approach the resonator eigenfrequency. Decoherence properties of the double dot are further investigated as a function of the number of electrons inside the dots. They are found to be similar in the single-electron and many-electron regimes suggesting that the density of the confinement energy spectrum plays a minor role in the decoherence rate of the system under investigation.

Figure 1

(a) Circuit diagram of a double dot (center panel) coupled to a resonator (top panel) and a quantum point contact (bottom panel). The double dot is tuned with gate voltages $V_{\mathrm{LPG}}$, $V_{\mathrm{RPG}}$, $V_{\mathrm{CG}}$, $V_{\mathrm{SDB}}$, $V_{\mathrm{LSG}}$, $V_{\mathrm{RSG}}$. It is coupled to the resonator via the capacitance $C_{\mathrm{LPG}}$. The resonator is driven with a microwave signal at frequency ${\nu }_{\mathrm{r}}$. The transmitted signal passes through a circulator, is amplified, and is mixed with the local oscillator ${\nu }_{\mathrm{LO}}$ to obtain the field quadratures $I$ and $Q$. The QPC is tuned with voltage $V_{\mathrm{QPC}}$. (b) Scanning electron microscope picture of a double dot gate design similar to the one used in the experiment. The gate extending from the resonator is shown in orange. The gate QPC used for charge detection is shown in blue.

Figure 2

(a) Transconductance $d{I}_{\mathrm{QPC}}/d{V}_{\mathrm{LPG}}$ vs $V_{\mathrm{LPG}}$ and $V_{\mathrm{RPG}}$ obtained with the QPC. (b) Normalized transmitted amplitude through the resonator vs $V_{\mathrm{LPG}}$ and $V_{\mathrm{RPG}}$ in the range indicated in (a). In both cases, black dashed lines highlight boundaries between different charge states extracted from the measurement shown in (a).

Figure 3

(a) QPC readout: transconductance $d{I}_{\mathrm{QPC}}/d{V}_{\mathrm{LPG}}$ of the QPC with voltage $\Delta V_{\mathrm{LPG}}-\Delta V_{\mathrm{RPG}}$ swept along the detuning line $\delta$ shown in Fig. 2b. The voltage $V_{\mathrm{CG}}$ allows to tune the tunnel coupling $t$ between the dots. (b) Microwave readout: Transmission spectra experimentally obtained for two values of $\delta$ chosen either far from the charge degeneracy point (triangles) or at the charge degeneracy point (triangles) and associated Lorentzian fits (lines). (c) QPC readout: occupation probabilities of the left dot experimentally obtained from a numerical integration of the signal shown in (a) (see text for details). (d) Microwave readout: frequency shifts extracted from fits to resonance transmission spectra of the resonator recorded along the detuning line $\delta$ for different tunneling rates $2t/h$. A vertical offset of 250 kHz has been added between each curves for clarity. The conversion of the applied plunger gate voltages to frequency $\alpha$ follows from the lever arms consistently extracted from finite-bias triangles[24] and electronic temperature broadening of the charge detection linewidth,[32] i.e. $\alpha \approx 0.11\text{e}/\text{h}$.

Figure 4

(a) Tunneling rates $2t/h$ between the dots measured vs $V_{\mathrm{CG}}$ for the two methods: charge detection with a QPC and microwave polarizability measurement. The black line corresponds to an exponential fit of $t$ vs $V_{\mathrm{CG}}$. (b) Dephasing rates ${\gamma }_{\phi }/2\pi$ vs tunneling $t$ in the single-electron and many-electron ($\approx$50 in each dot) regimes.

Figure 5

Comparison of tunnel coupling extracted from QPC and microwave measurements. Colored regions highlight the $2t/h{\nu }_0$ ranges over which $\Delta \nu$ and $\Delta {\kappa }^{\prime }$ signals were observable. An observable signal corresponds to a signal-to-noise ratio SNR $>1$ when each point of the spectrum is acquired with an integration time $\tau =168\text{ms}$.