  The interaction between a quantum charge and a quantized source of a magnetic
field is considered in the Aharonov-Bohm scenario. It is shown that, if the
source has a relatively small uncertainty while the particle encircles it, an
effective magnetic vector potential arises and the final state of the joint
system is approximately a tensor product. Furthermore, if a post-selection of
the source is considered, the effective vector potential is, in general,
complex-valued. This leads to a new prediction in the Aharonov-Bohm scenario
before the magnetic field is fully enclosed that has a parallel with Berry
phases in open quantum systems. Also, new insights into the correspondence
principle, which makes complex vector potentials relevant in the study of
classical systems, are discussed.
