Microwave Detection of Electron-Phonon Interactions in a Cavity-Coupled Double Quantum Dot

Quantum confinement leads to the formation of discrete electronic states in quantum dots. Here we probe electron-phonon interactions in a suspended InAs nanowire double quantum dot (DQD) that is electric-dipole coupled to a microwave cavity. We apply a finite bias across the wire to drive a steady state population in the DQD excited state, enabling a direct measurement of the electron-phonon coupling strength at the DQD transition energy. The amplitude and phase response of the cavity field exhibit oscillations that are periodic in the DQD energy level detuning due to the phonon modes of the nanowire. The observed cavity phase shift is consistent with theory that predicts a renormalization of the cavity center frequency by coupling to phonons.

Figure 1

(a) Schematic representation of the suspended InAs nanowire DQD. (b) Tilted angle SEM image of the device. (c) Energy level configuration of the DQD, which is placed inside a microwave cavity and probed by a weak microwave field. (d) Optical image of the microwave cavity in (c) (upper panel), and tilted angle SEM image showing the electrical connection between the drain contact of the DQD and the cavity centerpin (lower panel). The red box indicates the region shown in (b).

Figure 2

(a) Current $I$ through the DQD as a function of $V_L$ and $V_R$ with $V_{\mathit{SD}}=2.5\mathrm{mV}$. The interdot level detuning $\epsilon$ is adjusted along the white arrow. (b) Second derivative ${\partial }^{2}I/(\partial V_L\partial V_R)$ of the data in (a) show periodic oscillations as a function of $\epsilon$. (c) $I$ as a function of $\epsilon$ (red) accompanied by a simple theory prediction (black) that accounts for phonon emission. In the theory curve, we use the parameters: $t_c=22\mu \mathrm{eV}$, $J(2t_c/\hslash )/2\pi =0.3\mathrm{GHz}$, $d=120\mathrm{nm}$, ${\omega }_0/2\pi =130\mathrm{GHz}$, $c_n=2100\mathrm{m}/\mathrm{s}$, $r=4\times {10}^{-3}$, ${\mathrm{\Gamma }}_L/2\pi ={\mathrm{\Gamma }}_R/2\pi =90\mathrm{MHz}$ and a nanowire temperature of 200 mK. Inset: $dI/d\epsilon$ as a function of $\epsilon$ showing oscillations with approximately $60\mu \mathrm{eV}$ periodicity in $\epsilon$.

Figure 3

(a) Cavity phase response $\mathrm{\Delta }\varphi$ as a function of $V_L$ and $V_R$ near the lower finite bias triangle. (b) Second derivative ${\partial }^2\mathrm{\Delta }\varphi /(\partial V_L\partial V_R)$ of the data in (a). (c) $\mathrm{\Delta }\varphi$ and $A/A_0$ as a function of detuning $\epsilon$. Inset: Full range of data. The subset of data shown in the main panel is outlined by the dashed grey box. $A/A_0$ is normalized to the value in Coulomb blockade.

Figure 4

(a) A comparison of $I(\epsilon )$ and $A(\epsilon +h{f}_c)/A_0$. Inset: DQD energy level diagram showing the first order emission of a phonon (blue) or the second order emission of the same phonon and a photon (green) of energy $h{f}_c$, which occurs when $\epsilon$ increases by approximately $h{f}_c$. (b) $\mathrm{\Delta }\varphi$ as a function of $\epsilon$ (red) and theory fit (black). The theory curve takes into account the dispersive cavity shift (inset) that is caused by the coupling of cavity photons to nanowire phonon modes. These data, and the Fig. 2 data, are simultaneously fit.