  We define the standard quantum limit (SQL) for phase and number fluctuations,
and describe two-mode squeezing for number and phase variables. When phase is
treated as a unitary quantum-mechanical operator, number and phase operators
satisfy an uncertainty relation. As a result, the usual definition of number
squeezing parameter becomes modified. Two-mode number squeezing occurs when the
number fluctuation goes below the SQL at the cost of enhanced phase
fluctuation. As an application of number-phase uncertainty, we consider bosons
or fermions trapped in a quasi-one dimensional double-well (DW) potential
interacting via a 3D finite-range two-body interaction potential with large
scattering length $a_s$. Under tight-binding or two-mode approximation, we
describe in detail the effects of the range of interaction on the quantum
dynamics and number-phase uncertainty in the strongly interacting or unitarity
regime $a_s \rightarrow \pm \infty$. Our results show intriguing coherent
dynamics of number-phase uncertainty with number-squeezing for bosons and phase
squeezing for fermions. Our results may be important for exploring new quantum
interferometry, Josephson oscillations, Bose-Hubbard and Fermi-Hubbard physics
with ultracold atoms in DW potentials or DW optical lattices. Particularly
interesting will be the question of the importance of quantum phase operators
in two-atom interferometry and entanglement.
