Non-Hermitian dynamics of a Cooper pair splitter

We propose a non-Hermitian model for Cooper pair splitters in which the process of electron tunneling into electrodes is characterized by non-Hermitian terms. We find that across a broad range of parameters, the energy levels consistently remain real, and coalescing states are always present. The Coulomb repulsion between electrons in a quantum dot affects the order of the coalescing states. This gives rise to two distinct dynamic behaviors: (i) when the initial state is an empty state, the final state supports a nonzero electron-escaping rate; (ii) the electron-escaping rate is zero for a single-electron initial state. In the former case, our exact solutions reveal that the average electron-escaping rate vanishes along a set of hyperbolic curves in the plane of the chemical potentials of the two quantum dots. The stability of the results in the presence of disordered perturbation is also investigated. Our findings pave the way for investigating Cooper pair splitters within the framework of non-Hermitian quantum mechanics.