Microwave-Resonator-Detected Excited-State Spectroscopy of a Double Quantum Dot

As an application in circuit quantum electrodynamics coupled systems, superconducting resonators play an important role in high-sensitivity measurements in a superconducting-semiconductor hybrid architecture. Taking advantage of a high-impedance $\mathrm{Nb}\text{−}\mathrm{Ti}\text{−}\mathrm{N}$ resonator, we perform excited-state spectroscopy on a $\mathrm{Ga}\mathrm{As}$ double quantum dot (DQD) by applying voltage pulses to one gate electrode. The pulse train modulates the DQD energy detuning and gives rise to charge state transitions at zero detuning. Benefiting from the outstanding sensitivity of the resonator, we distinguish different spin-state transitions in the energy spectrum according to the Pauli exclusion principle. Furthermore, we experimentally study how the interdot tunneling rate modifies the resonator response. The experimental results are consistent with the simulated spectra based on our model.

Figure 1

Hybrid device and stability diagram. (a) False-color scanning electron micrograph of the device, showing the resonator (azure) and gate electrodes (yellow) of the DQD. The resonator capacitively couples to the drive line at one end and is grounded at the other end. The DQD region is indicated by a red rectangle. (b) DQD defined by metallic top gates. The left plunger gate highlighted in blue is connected to the $\mathrm{Nb}\text{−}\mathrm{Ti}\text{−}\mathrm{N}$ superconducting resonator. Pulses are applied to gate R to perform excited-state spectroscopy. (c) $\mathrm{Nb}\text{−}\mathrm{Ti}\text{−}\mathrm{N}$ resonator with a narrow width of 260 nm. (d) Transition line between charge configurations (2,2) and (1,3) measured by the resonator. The detuning $\epsilon$ indicated by an arrow can be controlled by the gate voltage $V_R$. Dashed lines indicate the electron tunneling between the DQD and the leads.

Figure 2

(a) Energy levels of two-electron spin states as a function of detuning $\epsilon$. A pulse train is applied to gate R, which is generated from an arbitrary wave-function generator with width ${\tau }_0$ and peak-to-peak voltage $V_{\mathrm{p.p.}}$. (b) Reflection signal $\mathrm{\Delta }|S_{11}|$ measured as a function of detuning $\epsilon$ and pulse voltage $V_{\mathrm{p.p.}}$ after subtracting the background in the Coulomb blockade regime. (c) Horizontal line cut of the spectrum at $V_{\mathrm{p.p.}}=0.1\mathrm{V}$. (d)–(g) Schematic electrochemical potential diagrams for different transition lines in (b) labeled $A$, $B$, $C$, and $D$.

Figure 3

(a),(b) Excited-state spectroscopy for $2t/h=7.5\mathrm{GHz}$ and $5.0\mathrm{GHz}$, respectively. (c),(d) Simulated results for the parameters used in panels (a) and (b).

Figure 4

(a) Frequency shift $\mathrm{\Delta }{f}_r$ as a function of $2t$ and $\epsilon$. The white and red dashed lines correspond to the values of $2t$ in Figs. 3 and 3, respectively. Other parameters are the same as those extracted from the experiment. (b) Frequency shift $\mathrm{\Delta }f$ as a function of $2t$ for coupling strengths $g_0/2\pi =50\mathrm{MHz}$ (blue line), $g_0/2\pi =104.5\mathrm{MHz}$ (black line), and $g_0/2\pi =150\mathrm{MHz}$ (red line), along the zero detuning in panel (a).