Nuclear Josephson-like $\gamma$-emission

Josephson-like junctions, transiently established in heavy ion collisions between superfluid nuclei, few MeV below the Coulomb barrier, allow for the back and forth transfer of a nuclear Cooper pair of effectively charged nucleons and thus the emission of $\gamma$-rays. The second order DWBA $T$-matrix formulation of single Cooper pair alternating current is shown to contain the gauge phases and gauge rotational frequencies as required by the Josephson (ac) effect, in keeping with the derivation of the transfer (tunneling) Hamiltonian in a gauge invariant representation. We describe the emergence of two strongly convergent parameters (conserved quantities) within the time the abnormal densities of the two superfluid nuclei overlap: a) the correlation length (dc); b) the number of emitted $\gamma$-rays per cycle (ac), and thus the dipole moment of the successively transferred nucleons. Result which leads to a nuclear parallel with the direct current (dc) and alternating current (ac) Josephson effects, and which testifies to the validity of BCS theory of superconductivity down to few Cooper pair condensates, and single Cooper pair alternating currents. The physics at the basis of a quantitative description of Cooper pair tunneling between weakly coupled superconductors or superfluid nuclei at energies below the Coulomb barrier, is that the process is dominated by the successive transfer of the two partner fermions entangled over distances of the order of the coherence length, $\approx10^4$\AA ${}$ in the case of lead, and 13.5 fm in the case of the reaction $^{116}\text{Sn}+^{60}\text{Ni}\to^{114}\text{Sn(gs)}+^{62}\text{Ni(gs)}$ at few MeV below the Coulomb barrier.