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A general analysis of squeezing transformations for two mode systems is givenbased on the four dimensional real symplectic group $Sp(4,\Re)\/$. Within theframework of the unitary metaplectic representation of this group, adistinction between compact photon number conserving and noncompact photonnumber nonconserving squeezing transformations is made. We exploit the$Sp(4,\Re)-SO(3,2)\/$ local isomorphism and the $U(2)\/$ invariant squeezingcriterion to divide the set of all squeezing transformations into a twoparameter family of distinct equivalence classes with representative elementschosen for each class. Familiar two mode squeezing transformations in theliterature are recognized in our framework and seen to form a set of measurezero. Examples of squeezed coherent and thermal states are worked out. The needto extend the heterodyne detection scheme to encompass all of $U(2)\/$ isemphasized, and known experimental situations where all $U(2)\/$ elements canbe reproduced are briefly described.Comment: Revtex 37 pages, Latex figures include

Two-mode quantum systems: Invariant classification of squeezing transformations and squeezed states