Mesoscopic Resonating Valence Bond System on a Triple Dot

We theoretically introduce a mesoscopic pendulum from a triple dot. The pendulum is fastened through a singly occupied dot (spin qubit). Two other strongly capacitively coupled islands form a double-dot charge qubit with one electron in excess oscillating between the two low-energy charge states $(1,0)$ and $(0,1)$. The triple dot is placed between two superconducting leads. Under realistic conditions, the main proximity effect stems from the injection of resonating singlet (valence) bonds on the triple dot. This gives rise to a Josephson current that is charge- and spin-dependent and, as a consequence, exhibits a distinct resonance as a function of the superconducting phase difference.

Figure 1

The pendulum with resonating singlet bonds under consideration: Dots 1 and 3 are strongly capacitively coupled and form a two-level system characterized by the degenerate orbital states $(1,0)$ or $(0,1)$. Dot 2 is singly occupied and the direct tunnel coupling between dots 1 and 2 (or 2 and 3) is negligible. The spin entanglement induced by the SC leads between dot 2 and, say, dot 1 (or dot 3) embodies the rod of the pendulum. The singlets can also resonate through the direct tunnel coupling ${\mathcal{T}}^{\prime }$ between dots 1 and 3. Consequences in a SQUID geometry are properly analyzed.

Figure 2

Main nonlocal Andreev mechanisms when the electron on the double-dot charge qubit is initially on dot 1 and $E_{10}\sim E_{01}=E_Q$. The numbered arrows indicate the direction and the order of the electron transfers. Since $ϵ\ll \Delta$ and $E_{00}-E_Q\ll \Delta$, we ignore individual contributions [10] from dot 2 giving a $\pi$ junction [22] and from the charge qubit [20].

Figure 3

$I_1/I_J$ versus $\theta -2\pi f$ for different values of $x=|{\mathcal{T}}^{\prime }|/J$. When ${\mathcal{T}}^{\prime }\sim 0$, $E^{\mathrm{eff}}$ does not depend on ${\varphi }_R-{\varphi }_L$ resulting in $I_1=0$. When $|{\mathcal{T}}^{\prime }|\gg J$, the sinuslike profile is reminiscent of the double-dot case [10]. For $|{\mathcal{T}}^{\prime }|=J$, when the interference $|⟨{\Psi }_J|{\Psi }_Q⟩|$ is fully destructive ($|⟨{\Psi }_J|{\Psi }_Q⟩|\to 0$) and $\sin (\theta -2\pi f)$ changes sign, an abrupt jump in $I_1$ occurs.