Ground state of the double-well condensate for quantum metrology

We discuss theoretically the ground state of a Bose-Einstein condensate with attractive atom-atom interactions in a double-well trap as a starting point of Heisenberg-limited atom interferometry. The dimensionless parameter governing the quality of the ground state for this purpose is identified. The near degeneracy between the ground state and the first excited state severely curtails the prospects of the thermally prepared ground state in quantum metrology.