Quantum Hall effect with small numbers of vortices in Bose-Einstein condensates

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.