Coupling a Quantum Dot, Fermionic Leads, and a Microwave Cavity on a Chip

We demonstrate a hybrid architecture consisting of a quantum dot circuit coupled to a single mode of the electromagnetic field. We use single wall carbon nanotube based circuits inserted in superconducting microwave cavities. By probing the nanotube dot using a dispersive readout in the Coulomb blockade and the Kondo regime, we determine an electron-photon coupling strength which should enable circuit QED experiments with more complex quantum dot circuits.

Figure 1

(a) Schematics of the quantum dot embedded in the microwave cavity. (b) Scanning electron microscope (SEM) picture in false colors of the coplanar waveguide resonator. Both the typical coupling capacitance geometry of one port of the resonator and the 3-terminals geometry are visible. (c) False colors SEM picture of a SWNT dot inside an on-chip cavity embedded in a schematics of the measurement setup.

Figure 2

(a) Color scale plot of the differential conductance in units of $2e^2/h$ measured along three charge states exhibiting the conventional transport spectroscopy. A Kondo ridge is visible at zero bias around $V_g=-2.3\mathrm{V}$. (b) Color scale plot of the phase of the microwave signal at $f=4.976\mathrm{GHz}$, measured simultaneously with the conductance of Fig. 2a. (c) Differential conductance and phase of the transmitted microwave signal at $f=4.976\mathrm{GHz}$ as a function of source-drain bias $V_{sd}$ for $V_g=-2.32\mathrm{V}$.

Figure 3

(a) Capacitive coupling of the quantum dot to the cavity. Both the fermionic leads and the quantum dot are coupled to the resonator, resulting in an ac modulation of both $V_{sd}$ and $V_g$ (shadings). (b) Upper panel, calculated frequency dependence of the microwave signal phase for a standard resonance. Reference resonance (black dashed line), shifted by $\delta f_R$ (blue [dark gray] line) as a result of dispersion and broadened by $\delta f_D$ (red [medium gray] line) as a result of dissipation. Lower panel: the even part (blue [dark gray] curve) and odd part (red [medium gray] curve) as a function of frequency. (c) Even and odd parts of the phase contrast $\delta \varphi$ as a function of frequency on the coulomb peak at $V_g=-2.44\mathrm{V}$ on the spectroscopy of Fig. 2. The even part exhibits a resonance centered on $f=4.976\mathrm{GHz}$. The odd part shows residual modulation due to imperfection in the amplification lines.

Figure 4

(a) Color scale plot of the even part of the phase contrast $\delta \varphi$ of two Coulomb peaks as a function of the gate voltage $V_g$ and the frequency of the microwave signal in the vicinity of the cavity resonance. $\delta \varphi$ is taken with respect to a reference phase in the empty orbital at $V_g=2.4\mathrm{V}$. (b) Gate dependence of the frequency shift (blue [dark gray] dots) and the frequency broadening (red [medium gray] dots) of the cavity mode extracted, respectively, from the area under the even part [Fig. 4a] and the area under half of the corresponding odd part. Formulae of the main text for $\delta f_R$ (light blue [light gray] line) and $\delta f_D$ (orange [medium-light gray] line) give $C_0=18\mathrm{aF}$, $\alpha =0.003$. Comparison with EOM theory (dashed dark green line) and Bethe ansatz (horizontal dashed purple line). (c) Color scale plot of the even part of the phase contrast $\delta \varphi$ of the Kondo spectroscopy shown in Fig. 2 as a function of the gate voltage $V_g$ and the frequency $f$. The line cut corresponds to the curves of Fig. 3c. (d) Gate dependence of the frequency shift (blue [dark gray] dots) and the frequency broadening (red [medium gray] dots) of the cavity mode extracted, respectively, from the area under the even part [Fig. 4c] and the area under half of the corresponding odd part. Formulae of the main text for $\delta f_R$ (light blue [light gray] line) and $\delta f_D$ (orange [medium-light gray] line) give $C_0=22\mathrm{aF}$, $\alpha =0.004$.