Circuit QED with a quantum-dot charge qubit dressed by Cooper pairs

Coupling double-quantum-dot circuits to microwave cavities provides a powerful means to control, couple, and manipulate qubits based on the charge or spin of individual electrons. Here, we revisit this standard configuration by adding superconductivity to the circuit. We combine theory and experiment to study a superconductor–double-quantum-dot circuit coupled to microwave cavity photons. First, we use the cavity as a spectroscopic probe. This allows us to determine the low-energy spectrum of the device and to reveal directly Cooper-pair-assisted tunneling between the two dots. Second, we observe a vacuum Rabi splitting which is a signature of strong charge photon coupling and a premiere with carbon-nanotube-based quantum-dot circuits. We show that our circuit design intrinsically combines a set of key features to achieve the strong coupling regime to the cavity. A low charging energy reduces the device sensitivity to charge noise, while sufficient coupling is provided by the shaping of the spectrum of the double quantum dot by the superconducting reservoir. Our findings could be adapted to many other circuit designs and shed light on the coupling of superconducting nanoscale devices to microwave fields.

Figure 1

(a) Optical photograph of the layout of our cavity QED architecture on a large scale. (b), (c) SEM micrographs of our devices on two different scales in false colors. The “fork” coupling gate is colored in red. The superconducting electrode is colored in orange. The normal (nonsuperconducting) electrodes are colored blue. The gates are colored in green. (d) Circuit diagram of our hybrid double quantum dot highlighting the symmetric coupling scheme between the two dots and the resonator in red.

Figure 2

(a) Color scale map of the current $I_L$ flowing through the left $\left(L\right)$ normal metal contact as a function of bias voltage ${\mathrm{V}}_S$ and the gate voltage ${\mathrm{V}}_{\mathrm{\Sigma }}$ for sample A. From this map, we read off a superconducting gap $\mathrm{\Delta }\sim 150\mu \mathrm{eV}$. (b) Cotunneling scheme accounting for the renormalization of the hopping constant between the left and the right quantum dots.

Figure 3

(a), (b) [resp. (c), (d)] Microwave phase contrasts (resp. amplitude) maps at the bare resonator frequency $f_c$ for sample A with two different tunings Aa and Ab as a function of the gate voltages. The charge occupation (i,j) of the double dot is indicated in (a). The sign changes demonstrate resonant interaction between the hybrid double quantum dot and the cavity photons. The elongated 0-phase line demonstrates the dependence of hopping with ${\epsilon }_{\mathrm{\Sigma }}$. The black dashed lines were obtained from theoretical expressions for the conditions ${\omega }_{\mathrm{cav}}={\omega }_{S-\text{DQD}}$ and $E_N=E_{N+1}$ with parameters given in the main text. In (d), the blue dot indicated by a green arrow is the sign of photon gain.

Figure 4

(a) Diagram of the transition map of the hybrid double quantum dot intersecting with the cavity resonance frequency. This results in the phase contrast maps of Figs. 3 and 3. The axis are the orbital detuning ${\epsilon }_{\delta }$ and the average orbital energy ${\epsilon }_{\mathrm{\Sigma }}$ of the double dot. (b) Bloch sphere diagram depicting the active states of our hybrid double quantum dot and the tunable hopping strength. This symmetric coupling scheme is crucial for the strong electron-photon coupling.

Figure 5

(a) Top panel: vacuum Rabi splitting for sample B with $\mathrm{n}\sim 1$ photon. Bottom panel: saturation of the mode splitting for a large number of photons. The open blue circles are the data points and the black solid line is the theory. (b) Level structures explaining the strong coupling and its power dependence. The $K,K^{\prime }$ labels indicate the valley degree of freedom arising from the band structure of carbon nanotubes. (c) Power dependence of the mode splitting showing the gradual saturation of the coherent transition. Each cut can be fitted using the fully quantum light-matter interaction theory (qutip).

Figure 6

Large-scale characterization of sample A in the $V{}_L\text{−}V{}_R$ plane. Current (a) and phase contrast (b) color maps as a function of the two side gate voltages $V{}_L$ and $V{}_R$ for sample A. The black rectangle indicates the area Aa under study in the main text. The blue axis show the relative orientation of the $V_{\mathrm{\Sigma }}-V_{\delta }$ frame, which was used to measure all data presented in this paper for sample A. Note that for clarity the frame origin chosen at ${(V}_{\mathrm{L}}=20\mathrm{V};V_R=20\text{V})$ is shifted on the figure.

Figure 7

Double-dot stability diagram in transport measurement for sample A. Right (a) and left (b) differential conductance maps in the rotated gate-gate frame $V_{\mathrm{\Sigma }}\text{−}{V}_{\delta }$ with a bias voltage $V_S=-0.16$ mV applied to the superconducting electrode. This measurement is simultaneous with the cavity transmission measurement presented in Fig. 3 for the phase and 3 for the amplitude.

Figure 8

Phase map as a function of $V_{\delta }$ and $V_{\mathrm{\Sigma }}$ for sample A. The black dotted line indicates for which parameters the cavity spectra are shown on (b). (b) Cavity spectrum as a function $V_{\delta }$ for sample A. (c) Phase map as a function of $V_L$ and $V_{\mathrm{\Sigma }}$ for sample B. The black dotted line indicates for which parameters the cavity spectra are shown on (d). (d) Cavity spectrum as a function $V_L$ for sample B. The orange vertical dotted line shows the cut along which the spectrum showing the Rabi splitting on the left has been taken.