  We analyze a Josephson junction between two superconductors interconnected
through a normal-state nanostructure made of two parallel nanowires with
embedded quantum dots. We study the influence of interference effects due to
the Aharonov-Bohm (AB) and Aharonov-Casher (AC) phases for local and nonlocal
(split) Cooper pairs. In the AB effect the phase of electron is affected by
magnetic flux, while in the AC effect the phase of the electron in solid state
can be modified due to the Rashba spin-orbit coupling. In the low-transmission
regime the AB and AC effects can be related to only local or nonlocal Cooper
pair transport, respectively. We demonstrate that by the addition of the
quantum dots the Cooper pair splitting can be made perfectly efficient, and
that the AC phase is different for non-spin-flip and spin-flip transport
processes.
