  We study the best attainable measurement precision when a double-well trap
with bosons inside acts as an interferometer to measure the energy difference
of the atoms on the two sides of the trap. We introduce time independent
perturbation theory as the main tool in both analytical arguments and numerical
computations. Nonlinearity from atom-atom interactions will not indirectly
allow the interferometer to beat the Heisenberg limit, but in many regimes of
the operation the Heisenberg limit scaling of measurement precision is
preserved in spite of added tunneling of the atoms and atom-atom interactions,
often even with the optimal prefactor.
