Quantum Transport and Entanglement in Cooper Pair Splitters*

Recently, there has been a significant interest in generating entanglement in the solid state. Such proposals rely on superconducting hybrid systems, where crossed Andreev processes are utilized. Recent experiments [1] have attempted to engineer Cooper pair splitting (CPS) by sandwiching quantum dots between the N-S junctions as resonant probes of Andreev processes. Theoretical works study CPS either with semi-classical rate equations [2] or fully quantum Lindblad master equations [3]. We point out that master equation based modelling is only applicable in certain regimes of the transport, and omits the physics of quantum broadening. We formulate a study of such an experimentally relevant setup via the Keldysh non-equilibrium Green’s function (NEGF) approach. We build upon previously established results [4] on N-QD-S systems and extend the analysis to N-QD-S-QD-N systems.  Establishing parallels between the NEGF method and the master equation approaches, we build a correlation matrix to calculate the steady-state entanglement entropy in the transport. The correlation matrix and entanglement measures inform the search for control parameters, thereby rendering our theoretical tools relevant for systematically guiding experiments for achieving high CPS efficiency. 

[1] Nature Comms. 12, 138 (2021).

[2] Beilstein J. Nanotechnol. 2019, 10, 363–378

[3] Physical Review Letters 127.23 (2021): 237701

[4] Physical Review B 59.5 (1999): 3831