Dipole Coupling of a Double Quantum Dot to a Microwave Resonator

We demonstrate the realization of a hybrid solid-state quantum device, in which a semiconductor double quantum dot is dipole coupled to the microwave field of a superconducting coplanar waveguide resonator. The double dot charge stability diagram extracted from measurements of the amplitude and phase of a microwave tone transmitted through the resonator is in good agreement with that obtained from transport measurements. Both the observed frequency shift and linewidth broadening of the resonator are explained considering the double dot as a charge qubit coupled with a strength of several tens of MHz to the resonator.

Figure 1

(a) Optical micrograph of the microwave resonator ($R$), with integrated double quantum dot, Ohmic contacts ($M$), top gates ($C$), ground plane (GND), and on-chip inductor ($I$). Inset: Magnified view of inductor ($I$). (b) Enlarged view of the device near the double quantum dot. The mesa edge is highlighted with a dashed line. (c) Scanning electron micrograph of the gate structure defining the double quantum dot (LD, RD). RG marks the gate connected to the resonator, $V_L$, $V_{\mathrm{LP}}$, $V_C$, $V_{\mathrm{RP}}$, $V_R$ label top gate voltages, and $S$, $D$ the 2DEG source and drain. (d) Electric circuit representation of the double quantum dot coupled to the resonator. The double quantum dot is tuned with voltages $V_L$, $V_{\mathrm{LP}}$, $V_C$, $V_{\mathrm{RP}}$, $V_R$ and connected to the resonator via the capacitance $C_{\mathrm{RG}}$. The resonator is driven with a microwave signal at frequency ${\nu }_R$. The transmitted signal passes through a circulator, is amplified and mixed with the local oscillator ${\nu }_{\mathrm{LO}}$ to obtain the field quadratures $I$ and $Q$.

Figure 2

(a) Measurement of the direct current through the double quantum dot versus $V_L$, $V_R$ (no microwave signal applied). A small background current is subtracted from each horizontal line of the data set. The dashed line outlines a region with fixed electron number ($N,M$). The $ϵ$ and $\delta$ axes indicate the direction of the mean and the detuning energy of the two quantum dots. (b) Resonator transmission amplitude at fixed measurement frequency ${\nu }_M$ in the same gate voltage range as in (a). The red dashed lines highlight the same hexagon as in (a). (c) Transmission phase change with respect to ${\nu }_M$, gate voltages as in (a). Red dashed lines as in (a). (d),(e) Measured transmission spectra of the microwave resonator at positions (1) (blue rectangles) and (2) (red open points), indicated in (a). (d) Transmission amplitude and (e) phase.

Figure 3

(a) Measured transmission phase as a function of ${\mathrm{V}}_{\mathrm{LP}}$ and ${\mathrm{V}}_{\mathrm{RP}}$ at ${\mathrm{V}}_C=-120\mathrm{mV}$ (with $2t<h{\nu }_0$, as extracted from simulations). Black dashed line indicates axis swept in (c),(e). (b) Transmission phase measured at ${\mathrm{V}}_C=-119\mathrm{mV}$ (with $2t>h{\nu }_0$ from simulation). Black dashed line indicates axis swept in (d),(f). (c),(d) Measured resonator frequency shift $\Delta {\nu }_0$ (blue points) along the axes shown in (a) and (b), respectively. (e),(f) Broadened resonator linewidth ${\kappa }^{*}$ (blue points) measured along the axes shown in (a) and (b), respectively. Solid red lines in (c)–(f) are results of numerical simulations.

Figure 4

(a) Schematic of the two lowest eigenenergies of the coupled system for $2t<h{\nu }_0$. Horizontal green dashed line indicates the bare resonator frequency ${\nu }_0$. Red dotted line shows the bare transition frequency ${\nu }_q$ between the double dot charge states. The two sloped blue dash-dotted lines indicate the detuning $\delta$. Insets I and III: Schematic of the two double dot charge states detuned by energy $\delta$, in the limit $\delta \gg t$. Inset II: Schematic of the hybridized charge states in the double quantum dot, split by $2t$ at $\delta =0$. (b) Transition energy of the charge qubit for $2t>h{\nu }_0$ in the presence of the nearby resonator. Dashed and dotted lines as in (a). (c) Resonator frequency in the presence of the nearby charge qubit transition for the same case as in (b).