  Mode entanglement exists naturally between regions of space in ultra-cold
atomic gases. It has, however, been debated whether this type of entanglement
is useful for quantum protocols. This is due to a particle number
superselection rule that restricts the operations that can be performed on the
modes. In this paper, we show how to exploit the mode entanglement of just a
single particle for the teleportation of an unknown quantum state of a spatial
mode. We detail how to overcome the superselection rule to create any initial
quantum state and how to perform Bell state analysis on two of the modes. We
show that two of the four Bell states can always be reliably distinguished,
while the other two have to be grouped together due to an unsatisfied phase
matching condition. The teleportation of an unknown state of a quantum mode
thus only succeeds half of the time.
