Microwave quantum optics as a direct probe of the Overhauser field in a quantum dot circuit quantum electrodynamics device

We show theoretically that a quantum dot circuit quantum electrodynamics device can be used as a probe of the Overhauser field in quantum dots. By coupling a transmission line to the interdot tunneling gate, an electromagnetically induced transparency scheme can be established, whose Fano-type interference leads to a sharp curvature in the reflection spectrum around resonance. This sharp feature persists even in the presence of the fluctuating spin bath, rendering a high-resolution method to extract the bath's statistical information. For strong nuclear spin fields, the reflection spectrum exhibits an Autler-Townes splitting, where the peak locations indicate the strengths of the Overhauser field gradient.

Figure 1

Schematic of a quantum dot CQED device. A transmission line is coupled to the interdot tunneling gate of a double quantum dot system via a finger-shaped electrode extended from the transmission line. The quantum dots contain two electrons, both of which experience effective magnetic fields provided by the local nuclear spins.

Figure 2

(a) Relevant energy levels in a dot-state basis. The probe field with frequency ${\omega }_{\mathrm{p}}$ addresses the transition between the two singlets, where a frequency detuning $\mathrm{\Delta }$ is allowed. An OFG with magnitude $\mathrm{\Delta }h$ mixes the two qubit states. Charge noise renders a dominant decay channel from state ${|1,1\rangle }_{\mathrm{S}}$ to ${|0,2\rangle }_{\mathrm{S}}$ with rate $\mathrm{\Gamma }$. Dressed-state picture for (b) EIT and (c) ATS regimes. The OFG couples the single-triplet qubit states, leading to two nuclear-spin-mediated states, $\overline{|\pm \rangle }$. The nuclear-spin-mediated states have the same energy but distinctive widths in the EIT regime (weak $\mathrm{\Delta }h$), while they develop different frequencies separated by $2\mathrm{\Delta }h$ but the same width ${\mathrm{\Gamma }}_{\pm }=\mathrm{\Gamma }/2$ in the ATS regime (strong $\mathrm{\Delta }h$).

Figure 3

(a) Density plot of reflection $\mathcal{R}$ as a function of both detuning and OFG. The spectrum exhibits a double-peak structure. As $\mathrm{\Delta }h$ increases, the spectral peaks move far apart, and the system develops from an EIT regime to an ATS regime. (b) Reflection as a function of detuning. For weak OFG with $\mathrm{\Delta }h/\left(g{\mu }_{\mathrm{B}}\right)=10\mathrm{mT}$ (blue line), the spectrum exhibits a sharp dip around the resonance $\mathrm{\Delta }=0$ due to Fano-type interference. For strong OFG with $\mathrm{\Delta }h/\left(g{\mu }_{\mathrm{B}}\right)=80\mathrm{mT}$ (green line), the peaks are well separated, and the spectrum becomes rather flat at the resonance.

Figure 4

(a) Ensemble-averaged reflection ${\langle \mathcal{R}\rangle }_{\mathrm{nucl}}$ as a function of detuning with OFG variance $\sigma =75\mathrm{neV}$ (solid lines) and $\sigma =250\mathrm{neV}$ (dashed lines). The spectral peaks get broadened by increasing variance. For small mean value $h_0/\left(g{\mu }_{\mathrm{B}}\right)=10\mathrm{mT}$ the sharp feature around the resonance persists even in the presence of variance. As the mean value increases, the spectrum starts to develop a flat profile at the resonance. Throughout the paper we choose $\mathrm{\Gamma }/2\pi =400\mathrm{MHz}$ and consider GaAs quantum dots with $\left|g\right|=0.44$. (b) Reflection sensitivity to statistical variance $\sigma$ for different mean values $h_0$ at detuning $\mathrm{\Delta }/2\pi =2\mathrm{MHz}$. This sensitivity reduces with increasing mean value, where the spectrum becomes rather flat around the resonance.