  When vortices are displaced in Bose-Einstein condensates (BEC), the Magnus
force gives the system a momentum transverse in the direction to the
displacement. We show that Bose-Einstein condensates (BEC) in long channels
with vortices exhibit a quantization of the current response with respect to
the spatial vortex distribution. The quantization originates from the
well-known topological property of the phase around a vortex --- it is an
integer multiple of $ 2 \pi $. In a similar way to the integer quantum Hall
effect, the current along the channel is related to this topological phase, and
can be extracted from two experimentally measurable quantities: the total
momentum of the BEC and the spatial distribution. The quantization is in units
of $ m/2h $, where $ m $ is the mass of the atoms and $ h $ is Planck's
constant. We derive an exact vortex momentum-displacement relation for BECs in
long channels under general circumstances. Our results presents the possibility
that the configuration described here can be used as a novel way of measuring
the mass of the atoms in the BEC using a topological invariant of the system.
If an accurate determination of the plateaus are experimentally possible, this
gives the possibility of a topological quantum mass standard and precise
determination of the fine structure constant.
