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After having set up a concise frame for pure and mixed all-order coherent states, we let them undergo a rather general squeezing transformation. These squeezed states are always non-classical, but some—apparently experimentally preparable—of them are still second-order coherent. They combine peculiar properties of their quantum fluctuations with the ordering features of coherence, and deserve special attention. Quite generally, several criteria for non-classicality are discussed and applied to the considered class of squeezed states. The connection between non-classicality and anti-bunching resp. sub-Poissonian photon statistics is shown to be more subtle than quantum optical usage suggests.

We compare the weakly inhomogeneous BCS-model of a previous investigation with a weak perturbation of a symmetrized Hubbard model in the pair formalism on the homogeneous lattice. In terms of our operator algebraic discussion we prove the existence of unique limiting Gibbs states at fixed particle density (and temperature) for both models (which guarantees the physical equivalence of the large finite models with their thermodynamic limit). A close similarity of the condensate structure between the two models is made explicit, especially the appearance of two critical densities. This likeness manifests itself also in the derived self-consistency equations for the momentum resp. position dependent, complex gap parameters. The results on a stable gap and macroscopic phase are thus transferable from the BCS- to this Hubbard model. Only the localization of the condensate of the Hubbard model at zero momentum bears some resemblance to the Bose-Einstein condensation.

Combining the signal and the idler modes we show that each nondegenerate squeezing quadratic Hamiltonian may be transformed into the form of a degenerate squeezing Hamiltonian, if one uses the smeared field formalism. For the case of infinitely many photon modes we discuss the existence of squeezing quadratic Hamiltonians in Fock space. This gives a limitation on the squeezing parameters, which guarantees that all squeezed vacua are representable as vectors in Fock space. If this condition is not satisfied (the case of strong squeezing) then all squeezed vacua are outside the Fock space and mutually inequivalent.

It is shown that the local existence of certain thermodynamic potentials allows for the complete diagonalization of thermodynamic stability conditions, so that only one-dimensional extrema have to be investigated.