Number-phase uncertainty and quantum dynamics of bosons and fermions interacting with a finite range and large scattering length in a double-well potential

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.