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Geometry in the entanglement dynamics of the double Jaynes-Cummings model

We report on the geometric character of the entanglement dynamics of to pairs of qubits evolving according to the double Jaynes-Cummings model. We show that the entanglement dynamics for the initial states |ψ_0> = Cosα |1 0> + Sinα |0 1> and |ϕ_0> = Cosα |1 1> + Sinα |0 0> cover 3-dimensional surfaces in the diagram C_ij\timesC_ik\timesC_il, where C_mn stands for the concurrence between the qubits m and n, varying 0\leqα\leqπ/2. In the first case projections of the surfaces on a diagram C_ij\timesC_kl are conics. In the second case the curves can be more complex. We relate those conics with a measurable quantity, the {\it predictability}.We also derive inequalities limiting the sum of the squares of the concurrence of every bipartition and show that sudden death of entanglement is intimately connected to the size of the radius of a hyper-sphere.