Phase-difference between two quantized harmonic oscillators

We consider two quantized harmonic oscillators and their phase-difference. In our treatment, we use the phase-difference operator defined by Luis and Sanchez-Soto [Phys. Rev. A, 1993, vol. 48, p. 4702]. A setup for direct measurement of the phase-difference is suggested, and its limitation discussed. We also derive the phase-difference distributions for some states and compare the results to the single mode case. Finally, we consider the general uncertainty relation for phase-difference and photon number-difference and derive the two classes of minimum uncertainty states in the first manifold.