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In the differential approach elaborated, we study the evolution of the parameters of Gaussian, mixed, continuous variable density matrices, whose dynamics are given by Hermitian Hamiltonians expressed as quadratic forms of the position and momentum operators or quadrature components. Specifically, we obtain in generic form the differential equations for the covariance matrix, the mean values, and the density matrix parameters of a multipartite Gaussian state, unitarily evolving according to a Hamiltonian H ^ . We also present the corresponding differential equations, which describe the nonunitary evolution of the subsystems. The resulting nonlinear equations are used to solve the dynamics of the system instead of the Schrödinger equation. The formalism elaborated allows us to define new specific invariant and quasi-invariant states, as well as states with invariant covariance matrices, i.e., states were only the mean values evolve according to the classical Hamilton equations. By using density matrices in the position and in the tomographic-probability representations, we study examples of these properties. As examples, we present novel invariant states for the two-mode frequency converter and quasi-invariant states for the bipartite parametric amplifier.

We characterize spontaneous parametric downconversion in a domain-engineered, type-II periodically poled lithium niobate (PPLN) crystal using seeded emission and single-photon techniques. Using continuous-wave (CW) pumping at 775 nm wavelength, the signal and idler are at 1532.5 nm and 1567.5 nm, respectively. The domain-engineered crystal simultaneously phasematches signal and idler pairs: [H(1532.5 nm), V(1567.5 nm)] and [V(1532.5 nm), H(1567.5 nm)]. We observe the tuning curves of these processes through difference-frequency generation and through CW fiber-assisted, single-photon spectroscopy. These measurements indicate good matching in amplitude and bandwidth of the two processes and that the crystal can in principle be used effectively to generate polarization-entangled photon pairs.