Projection operators for asymmetric two-mode coherent states and two-mode Fock states

Abstract

Operators play an important role in the formalism of quantum mechanics. By means of the technique of integration within an ordered product of operators and Dirac notation, we analytically prove that a new kind of asymmetric two-mode projection integral operator and another asymmetric two-mode Fock state projection summation operator are equal to the same Hermitian operator. For such Hermitian operator, we also rigorously demonstrate it corresponds to a parity measurement of a two-mode quantum state after passing through a beam splitter. The method used in this work provides a good exercise in quantum mechanics at the graduate level.