  We study why it is quite so hard to make a superposition of superfluid flows
in a Bose-Einstein condensate. To do this we initially investigate the quantum
states of $N$ atoms trapped in a 1D ring with a barrier at one position and a
phase applied around it. We show how macroscopic superpositions can in
principle be produced and investigate factors which affect the superposition.
We then use the Bose-Hubbard model to study an array of Bose-Einstein
condensates trapped in optical potentials and coupled to one another to form a
ring. We derive analytic expressions for the quality of the superposition for
this system, which agrees well with direct diagonalisation of the Hamiltonian
for relatively small numbers of atoms. We show that for macroscopic
superpositions to be realised there are essentially three straightforward
requirements, other than an absence of decoherence, which become harder to
achieve as the system size increases. Firstly, the energies of the two distinct
superfluid states must be sufficiently close. Secondly, coupling between the
two states must be sufficiently strong, and thirdly, other states must be well
separated from those participating in the superposition.
