Transport Spectroscopy of a Spin-Coherent Dot-Cavity System

Quantum engineering requires controllable artificial systems with quantum coherence exceeding the device size and operation time. This can be achieved with geometrically confined low-dimensional electronic structures embedded within ultraclean materials, with prominent examples being artificial atoms (quantum dots) and quantum corrals (electronic cavities). Combining the two structures, we implement a mesoscopic coupled dot-cavity system in a high-mobility two-dimensional electron gas, and obtain an extended spin-singlet state in the regime of strong dot-cavity coupling. Engineering such extended quantum states presents a viable route for nonlocal spin coupling that is applicable for quantum information processing.

Figure 1

(a) Scanning electron micrograph of the dot-cavity device. Schottky electrodes (bright) define an electronic cavity (red overlay) focused onto the tunnel barrier of a dot (yellow circle overlay). In a typical cavity quantum optics experiment (inset), a confocal cavity is coupled to an atom. Our setup resembles a hemispherical cavity where the cavity is focused onto an atom embedded in a plane mirror. Gate voltages $V_{\text{dot}}$ and $V_{\text{cav}}$ change the energies and occupancies of the dot and cavity; $V_{\mathrm{TBS}}$ and $V_{\mathrm{TBC}}$ tune the coupling ${\mathrm{\Gamma }}_S$ of the dot to source lead, whereas $V_{\mathrm{TBD}}$ and $V_{\mathrm{TBS}}$ tune the couplings ${\mathrm{\Gamma }}_D$ and $\mathrm{\Omega }$ of the dot to both drain and cavity modes simultaneously. We report measured differential conductance $g=dI/d{V}_{SD}$ of the dot-cavity system obtained at small bias $V_{SD}=-10\mu \mathrm{V}$ in two coupling configurations ${\mathrm{\Gamma }}_D\ll {\mathrm{\Gamma }}_S$ (b) and ${\mathrm{\Gamma }}_D\gtrsim {\mathrm{\Gamma }}_S$ (c). The dot is tuned to the Coulomb valley of the third charge state. As long as the cavity is not defined ($V_{\text{cav}}>50\mathrm{mV}$), constant transport is due to the Kondo effect. As the cavity voltage $V_{\text{cav}}$ depletes the 2DEG, a mirror forms that focuses quantized modes onto the tunnel barrier of the dot. (b) Tuning $V_{\text{cav}}$, the cavity modes are pushed through the chemical potential, causing peaks of increased Kondo conductance. These peaks are separated by a cavity-mode spacing ${\delta }_{\text{cav}}\approx 220\mu \mathrm{eV}$ and have a width ${\mathrm{\Gamma }}_{\text{cav}}\approx 40\mu \mathrm{eV}$. (c) Increasing ${\mathrm{\Gamma }}_D$, signatures of strong coupling develop when the coupling $\mathrm{\Omega }$ increases beyond the decay rate ${\mathrm{\Gamma }}_{\text{cav}}$. This strong coupling manifests in the appearance of split peaks with a gap $\mathrm{\Delta }\approx 80\mu \mathrm{eV}$, indicating the formation of a spin singlet on the dot-cavity hybrid that competes with and suppresses the Kondo effect. Different gray shades are applied to the background to highlight the effect of each cavity mode crossing the chemical potential. Note that the two measurements (b) and (c) are slightly shifted and stretched in $V_{\text{cav}}$ to correct for the different settings of $V_{\mathrm{TBS}}$ and $V_{\mathrm{TBD}}$.

Figure 2

(a) Differential conductance of the dot plotted as a function of $V_{SD}$ and $V_{\text{dot}}$ with the cavity switched off ($V_{\text{cav}}=+200\mathrm{mV}$) and symmetric tunnel coupling ${\mathrm{\Gamma }}_D\approx {\mathrm{\Gamma }}_S$. Characteristic (dark) regions of the Coulomb blockade are intersected by a pronounced zero-bias Kondo resonance at odd dot occupation ($N=3e^{-}$). (b) Same experimental parameters, but with cavity modes present ($V_{\text{cav}}=-200\mathrm{mV}$) in the drain of the dot. Additional resonance lines spaced by ${\delta }_{\text{cav}}\approx 220\mu \mathrm{eV}$ arise due to the cavity modes modifying the dot transport. Horizontal dashed lines at fixed $V_{\text{dot}}$ indicate the parameter settings used in Fig. 4 (blue line) and Fig. 4 (white line). Note that increasing $V_{\text{dot}}$ also increases the tunnel couplings, resulting in more pronounced transport signatures in the bottom of the figures.

Figure 3

(a) A sketch of an Anderson model depicting a dot with energies ${\epsilon }_d^{(i)}$, on-site interaction $U$, constant coupling ${\mathrm{\Gamma }}_S$ to the source, and energy-dependent coupling ${\mathrm{\Gamma }}_D(\epsilon )$ to the drain due to focused cavity modes [19]. In this model the effect of the cavity mirror on the dot is encoded as a structured tunneling amplitude, ${\mathrm{\Gamma }}_D(\epsilon )$, into the drain lead. (b) A sketch of a coherent dot-cavity model where the ${\mathrm{\Gamma }}_D(\epsilon )$ of the previous model is replaced by a noninteracting cavity with a discrete spectrum ${\epsilon }_c^{(j)}$, effectively coupled to the dot with $\mathrm{\Omega }$ and to the drain lead by ${\mathrm{\Gamma }}_{\mathrm{cav}}$. (c) Calculated ground state map $|N_{\text{cav}},N_{\text{dot}}⟩$ of the dot-cavity hybrid as a function of $V_{\text{cav}}$ and $V_{\text{dot}}$ with (relative) occupation numbers $N_{\text{dot}}$ ($N_{\text{cav}}$) determining the ground state of the dot (cavity) for ${\delta }_{\text{cav}}/U=0.5$ and $\mathrm{\Omega }/U=0.1$. Dark red shading marks the regimes of spin-singlet formation on the dot-cavity hybrid, where the Kondo resonance (orange) is suppressed. (d) Measurement of the linear conductance (with $V_{SD}=-10\mu \mathrm{V}$) in the strong-coupling regime. Lines of resonant transport match the boundaries between ground states in (c).

Figure 4

Finite bias differential conductance of the dot-cavity system (a)–(c) and control experiment with even dot occupancy (d)–(f). (a) In the weak dot-cavity coupling regime, the odd occupancy of the dot ($N=3e^{-}$) gives rise to a zero-bias Kondo resonance (faint horizontal line) and finite-bias cotunneling signatures (diagonal lines) due the presence of cavity modes. For increasing coupling (b), the hybridization of the dot and cavity leads to a splitting of Kondo and cotunneling features, increasing upon stronger coupling (c). (d)–(f): Control experiment with $N=2e^{-}$ in the dot, where the Kondo effect is absent and the cavity resonances do not exhibit an energy gap when the dot-cavity coupling is increased. Notably, replicas of the cavity-enhanced cotunneling lines appear at $V_{SD}$ larger than (or equal to) the dot’s singlet-triplet energy gap ${\delta }_{\text{dot}}^{\text{st}}$, signaling that the strong dot-cavity hybridization alters also the inelastic cotunneling through the triplet states of the dot. Note the adjusted color scales in (d),(e), and (f) to improve the visibility of the cotunneling currents. Dashed lines in (b) and (e) indicate the correspondence to Figs. 2 and 2.