Unitary quantum phase operators for bosons and fermions: A model study   on quantum phases of interacting particles in a symmetric double-well   potential
  We introduce unitary quantum phase operators for material particles.   We carry out a model study on quantum phases of interacting bosons in a symmetric double-well potential in terms of unitary and commonly-used non-unitary phase operators and compare the results for different number of bosons. We find that the results for unitary quantum phase operators are significantly different from those for non-unitary ones especially in the case of low number of bosons.   We introduce unitary operators corresponding to the quantum phase-difference between two single-particle states of fermions. As an application of fermionic phase operators, we study a simple model of a pair of interacting two-component fermions in a symmetric double-well potential.   We also investigate quantum phase and number fluctuations to ascertain number-phase uncertainty in terms of unitary phase operators. 
Biswajit Das, Bitan Ghosal, Subhasish Dutta Gupta and Bimalendu Deb
quant-ph cond-mat.quant-gas