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We define the standard quantum limit (SQL) for phase and number fluctuations,and describe two-mode squeezing for number and phase variables. When phase istreated as a unitary quantum-mechanical operator, number and phase operatorssatisfy an uncertainty relation. As a result, the usual definition of numbersqueezing parameter becomes modified. Two-mode number squeezing occurs when thenumber fluctuation goes below the SQL at the cost of enhanced phasefluctuation. As an application of number-phase uncertainty, we consider bosonsor fermions trapped in a quasi-one dimensional double-well (DW) potentialinteracting via a 3D finite-range two-body interaction potential with largescattering length $a_s$. Under tight-binding or two-mode approximation, wedescribe in detail the effects of the range of interaction on the quantumdynamics and number-phase uncertainty in the strongly interacting or unitarityregime $a_s \rightarrow \pm \infty$. Our results show intriguing coherentdynamics of number-phase uncertainty with number-squeezing for bosons and phasesqueezing for fermions. Our results may be important for exploring new quantuminterferometry, Josephson oscillations, Bose-Hubbard and Fermi-Hubbard physicswith ultracold atoms in DW potentials or DW optical lattices. Particularlyinteresting will be the question of the importance of quantum phase operatorsin two-atom interferometry and entanglement.

journal of physics. b, atomic molecular and optical physics/journal of physics. b, atomic, molecular and optical physics

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