New formalism for two-photon quantum optics. II. Mathematical foundation and compact notation

This paper provides the mathematical foundation for the two-mode formalism introduced in the preceding paper. A vector notation is introduced; it allows two-mode properties to be written as compactly as the comparable properties for a single mode. The fundamental unitary operators of the formalism are described and their properties are examined; particular attention is paid to the two-mode squeeze operator. Special quantum states associated with the formalism are considered, with emphasis on the two-mode squeezed states.