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Mode entanglement exists naturally between regions of space in ultra-coldatomic gases. It has, however, been debated whether this type of entanglementis useful for quantum protocols. This is due to a particle numbersuperselection rule that restricts the operations that can be performed on themodes. In this paper, we show how to exploit the mode entanglement of just asingle particle for the teleportation of an unknown quantum state of a spatialmode. We detail how to overcome the superselection rule to create any initialquantum state and how to perform Bell state analysis on two of the modes. Weshow 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 phasematching condition. The teleportation of an unknown state of a quantum modethus only succeeds half of the time.

Teleportation of a quantum state of a spatial mode with a single massive particle