Quantum entanglement in nuclear Cooper-pair tunneling with $\gamma$ rays

While Josephson-like junctions, transiently established in heavy-ion collisions (${\tau }_{\mathrm{coll}}\approx {10}^{-21}\mathrm{s}$) between superfluid nuclei—and through which Cooper-pair tunneling ($Q$-value $Q_{2n}$) proceeds mainly in terms of successive transfer of entangled nucleons—are deprived of the macroscopic aspects of a supercurrent, they display many of the special effects associated with spontaneous symmetry breaking in gauge space (BCS condensation), which can be studied in terms of individual quantum states and of tunneling of single Cooper pairs. From the results of studies of one- and two-neutron transfer reactions carried out at energies below the Coulomb barrier we estimate the value of the mean-square radius (correlation length) of the nuclear Cooper pair. A quantity related to the largest distance of closest approach for which the absolute two-nucleon tunneling cross section is of the order of the single-particle one. Furthermore, emission of $\gamma$ rays of (Josephson) frequency ${\nu }_J=Q_{2n}/h$ distributed over an energy range $\hslash /{\tau }_{\mathrm{coll}}$ is predicted.

Figure 1

(a) Double differential cross section for $\gamma$ emission at ${\theta }_{\mathrm{c}.\mathrm{m}.}={140}^{\circ }$ as a function of the energy of the emitted $\gamma$ ray, calculated with Eqs. (11) and (13) (dashed curve). The reduced strength (continuous curve) has been obtained by dividing out from $d\sigma /d\mathrm{\Omega }d{E}_{\gamma }$ the phase-space factor $\sim E_{\gamma }^2$, and multiplying it by the corresponding quantity with $E_{\gamma }=1.307\left(\mathrm{MeV}\right)$ (see the text for details). The reduced $\gamma$-strength function is shown in (b) with a different scale where the width and the position of the centroid are more apparent.