The Dual Pairing Principle of Quantum State Engineering: Cooper Pairs and Bose-Einstein Condensates

The non-invasive diagnostic protocols of magnetic resonance tomography and spectroscopy have been made effective for the clinical routine by means of the chirality reversing concepts of spin echo and gradient echo. Because Cooper pairs in a spin singlet are formed from time-reversed quantum states, the coadjoint orbit picture of the unitary dual N of the real Heisenberg step 2 nilpotent Lie group N gives rise to the symmetry group SU(2, C) ~ Spin(R³ ). Due to the Hopf fibration of the unit sphere S₃ → C C over the Bloch sphere S₂ → R C with fiber S₁ → C, the compact Lie group SU(2,C) acts via U(l,C) gauge transformations on the complex line bundle associated with N. The metaplectic symmetries of the symplectic spinor bundle configuration, conjugated by indistinguishable pairs of contragredient flat double-layers occurring in the foliation of N, permit a natural approach to the quantum information transfer within pairs of Bose-Einstein condensates.