We study crossed Andreev reflection occurring in quantum dots connected to one superconducting lead and two normal leads at low temperatures $T$. Specifically, we derive an exact formula for the conductance up to order $T^2$ in the large-superconducting-gap limit, which is expressed in terms of the transmission probabilities of Cooper pairs and interacting Bogoliubov quasiparticles. Our formulation is based on the latest version of Fermi-liquid theory for the Anderson impurity model, which has clarified that the quasiparticle energy shifts of order ${\omega }^2$ and $T{}^2$—namely, corrections of the same order as those arising from the finite lifetime of quasiparticles—can be expressed exactly in terms of three-body correlations of impurity electrons. We also demonstrate how the three-body contributions evolve and affect the Cooper-pair tunneling as the Andreev level moves away from the Fermi level, using the numerical renormalization group approach. The results show that the Cooper-pair contribution to the $T{}^2$ terms of the local and nonlocal conductances becomes comparable to the Bogoliubov-quasiparticle contribution in the parameter region in which superconducting proximity effects dominate over the Kondo effect.

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• Andreev reflection
• Kondo effect
• Numerical Renormalization Group
• 0-dimensional systems
• Quantum dots
• Anderson impurity model
• Fermi liquid theory
• Nonequilibrium Green's function