  Interferometry with ultracold atoms promises the possibility of ultraprecise
and ultrasensitive measurements in many fields of physics, and is the basis of
our most precise atomic clocks. Key to a high sensitivity is the possibility to
achieve long measurement times and precise readout. Ultra cold atoms can be
precisely manipulated at the quantum level, held for very long times in traps,
and would therefore be an ideal setting for interferometry. In this paper we
discuss how the non-linearities from atom-atom interactions on one hand allow
to efficiently produce squeezed states for enhanced readout, but on the other
hand result in phase diffusion which limits the phase accumulation time. We
find that low dimensional geometries are favorable, with two-dimensional (2D)
settings giving the smallest contribution of phase diffusion caused by
atom-atom interactions. Even for time sequences generated by optimal control
the achievable minimal detectable interaction energy $\Delta E^{\rm min}$ is on
the order of 0.001 times the chemical potential of the BEC in the trap. From
there we have to conclude that for more precise measurements with atom
interferometers more sophisticated strategies, or turning off the interaction
induced dephasing during the phase accumulation stage, will be necessary.
