NORMALLY ORDERED FORMS OF SHAPIRO-WAGNER PHASE OPERATORS AND THEIR APPLICATION

Based on the quantum mechanical representation |\xi>=exp-\frac12|\xi|+\xi a_1^+ + \xi^*a_2^+ - a_1^+ a_2^+ |00> constructed by Fan Hong-yi and with the help of the technique of integration within an ordered product of operators, we derive the normally ordered expressions of the Shapiro-Wagner (SW) phase operators. They are in terms of the Bessel functions. As their application we discuss the minimal uncertain relation regarding the two-mode phase and number-difference operators.