Magnetic Field Tuning and Quantum Interference in a Cooper Pair Splitter

Cooper pair splitting (CPS) is a process in which the electrons of the naturally occurring spin-singlet pairs in a superconductor are spatially separated using two quantum dots. Here, we investigate the evolution of the conductance correlations in an InAs CPS device in the presence of an external magnetic field. In our experiments the gate dependence of the signal that depends on both quantum dots continuously evolves from a slightly asymmetric Lorentzian to a strongly asymmetric Fano-type resonance with increasing field. These experiments can be understood in a simple three-site model, which shows that the nonlocal CPS leads to symmetric line shapes, while the local transport processes can exhibit an asymmetric shape due to quantum interference. These findings demonstrate that the electrons from a Cooper pair splitter can propagate coherently after their emission from the superconductor and how a magnetic field can be used to optimize the performance of a CPS device. In addition, the model calculations suggest that the estimate of the CPS efficiency in the experiments is a lower bound for the actual efficiency.

Figure 1

(a),(b) Schematics of a Cooper pair splitter and the corresponding three-site model in which $S$ is coupled to the QDs by a central NW region. The dashed and solid lines represent two different interfering single-electron paths from $S$ to $N1$, which can result in Fano-type conductance resonances. ${\epsilon }_{1,2}$ are the QD energy levels, ${\epsilon }_m$ is the level position of the middle region, ${\gamma }_{1,m,2}$ are the tunnel coupling strengths to the respective reservoirs, and $t_{1m,2m}$ are the couplings between the middle region and the QDs.

Figure 2

(a),(b) $G_1$ and $G_2$ as a function of $V_{g1}$ and $V_{g2}$ for $B=0$ and $B=1\mathrm{T}$, respectively. (c),(d) $G_1$ and $G_2$ as a function of the respective far gate $V_{g2}$ and $V_{g1}$ for a series of magnetic fields $B$ with the local gates set to a QD resonance, as indicated by black arrows in (a) and (b). The dashed lines indicate the resonance crossings. All curves are shifted vertically for clarity.

Figure 3

Conductances through QD1 and QD2 in the three-site model with parameters chosen to qualitatively reproduce the experiments in Fig. 2. (a),(b) $G_1$ and $G_2$ as a function of both QD level positions. (c),(d) On-resonance conductance variations of QD1 and QD2, respectively, as a function of the level position of the other QD, for increasing values of the magnetic field. The parameters used in the calculations are ${\epsilon }_m=-0.03\mathrm{\Delta }$, $2t_{1m}=t_{2m}=0.1\mathrm{\Delta }$, $U=3\mathrm{\Delta }$, and the coupling to the leads ${\gamma }_1=0.15\mathrm{\Delta }$, ${\gamma }_2=0.2\mathrm{\Delta }$, and ${\gamma }_m=0.11\mathrm{\Delta }$.

Figure 4

(a) Total conductance through the QD2, $G_2$ (solid line), and the decomposition into the local (dashed line) and CPS (dotted line) contributions, as a function of $-{\epsilon }_1$ for $B=0$ (dark) and $B=1.2\mathrm{T}$ (green). (b) $G_2$ vs $V_{g1}$ for different coupling strengths ${\gamma }_m$ to the superconductor. The curves are offset vertically for clarity. The value at which ${\gamma }_m=t_{im}$ is pointed out by an arrow. (c),(d) Correct and estimated CPS efficiencies $\chi$ and ${\chi }_e$ for the numerical results in (b).