All-electrical generation and control of odd-frequency $s$-wave Cooper pairs in double quantum dots

We propose an all-electrical experimental setup to detect and manipulate the amplitude of odd-frequency pairing in a double quantum dot. The odd-frequency pair amplitude is induced from the breakdown of orbital symmetry when Cooper pairs are injected in the double dot with electrons in different dots. When the dot levels are aligned with the Fermi energy, i.e., on resonance, nonlocal Andreev processes are directly connected to the presence of odd-frequency pairing. Therefore, their amplitude can be manipulated by tuning the level positions. The detection of nonlocal Andreev processes by conductance measurements contributes a direct proof of the existence of the odd-frequency pair amplitude and is available using current experimental techniques.

Figure 1

Proximity-induced superconductivity in a DQD three-terminal device. (a) Schematics of a DQD with level positions ${\epsilon }_L$ and ${\epsilon }_R$ contacted by a superconducting lead S and two normal leads L and R. (b) In a local Andreev process, the electrons of a Cooper pair tunnel through one dot into the normal lead. The amplitude for these processes, $F_{LL,RR}$ (blue dashed lines), is enhanced for symmetric devices, i.e., those where each dot is similarly coupled to the leads. (c) In a nonlocal Andreev process, each electron of the Cooper pair tunnels to a different lead. The nonlocal amplitude, $F_{LR}$ (red solid line), is enhanced in asymmetric devices. On resonance (${\epsilon }_L={\epsilon }_R=0$), $F_{LR}$ is odd in frequency if S is a BCS superconductor.

Figure 2

Cooper pair splitter configuration. (a) A voltage $V$ is applied symmetrically to both normal electrodes which allows us to measure the conductances $G_L\left(V\right)$ and $G_R\left(V\right)$ (left). Right: In an asymmetric DQD system, odd-frequency nonlocal $F_{LR}$ (red line) can be enhanced on dot L and $G_L$ is mainly due to nonlocal Andreev processes (red arrow). (b), (c) Map of the (b) ratio $R_L$ and (c) conductance $G_L$ as a function of applied voltage and level position $\delta \epsilon ={\epsilon }_L+3{\epsilon }_R$. $R_L>1$ inside the white dashed line. (d) Conductance (black solid lines), nonlocal Andreev (red dashed lines), and local processes (blue dotted-dashed lines) for $eV=0$ (left) and $0.75\mathrm{\Delta }$ (right). For all plots, $T=0, {\mathrm{\Gamma }}_L/\mathrm{\Delta }=5, {\mathrm{\Gamma }}_R=\mathrm{\Delta }=1, {\mathrm{\Gamma }}_{SL}/\mathrm{\Delta }=0.1, {\mathrm{\Gamma }}_{SR}/\mathrm{\Delta }=0.9$, and ${\mathrm{\Gamma }}_{LR}/\mathrm{\Delta }=0.5$.

Figure 3

Nonlocal conductance measurement. (a) A voltage is applied to lead R, allowing one to measure the conductance $G_{LR}$ on lead L (left). Right: On resonance, odd-frequency nonlocal Cooper pairs provide a dominant contribution to $G_{LR}$ (red arrow), while electron tunneling dominates out of resonance (blue arrow). (b) Map of $G_{LR}$ as a function of the applied voltage and the position of dot R, ${\epsilon }_R$, for ${\epsilon }_L=0$. $G_{LR}>0$ inside the black dashed line. (c) Ratio on dot L (top) and nonlocal conductance (bottom) for ${\epsilon }_L={\epsilon }_R$ (red solid line) and ${\epsilon }_L\ne {\epsilon }_R$ (blue dashed line). For all plots, ${\mathrm{\Gamma }}_L/\mathrm{\Delta }=5, {\mathrm{\Gamma }}_R/\mathrm{\Delta }=0.2, {\mathrm{\Gamma }}_{SL}={\mathrm{\Gamma }}_{SR}={\mathrm{\Gamma }}_{LR}=\mathrm{\Delta }=1$, and $T=0$.