Investigation of entanglement in pure final polarization states from neutron-deuteron elastic ccattering and exclusive deuteron break

TL;DR

This paper investigates entanglement in final spin states arising from elastic $nd$ scattering and exclusive deuteron breakup, using the three-nucleon Faddeev formalism with the CD-Bonn $NN$ potential to compute final-state spin density matrices from six pure axially polarized incoming $nd$ states along the $y$-axis. While elastic $nd$ scattering does not yield robust Bell-like entangled final states, the exclusive breakup process in carefully chosen kinematics (notably QFS($nn$) and FSI($np$)) produces strongly entangled Bell-like states, driven by the dominance of specific $NN$ partial waves, especially the singlet $^1S_0$ channel. The entanglement is quantified through the entanglement power $oldsymbol{ ilde ho}$ and concurrence $ar C$, and corroborated by norm decompositions of the final-state amplitudes, revealing regions in angle and energy where entanglement is maximized with minimal admixture from entanglement-breaking components. The findings suggest experimental pathways to realize and study entangled multi-nucleon states with highly polarized beams and motivate extending such analyses to four-nucleon systems and related reaction channels.

Abstract

We investigate the pure polarization states of the outgoing neutron deuteron pair in elastic polarized neutron polarized deuteron scattering, as well as the pure polarization states of the three free nucleons produced in the corresponding deuteron breakup reaction. Our aim is to provide clear evidence of entanglement in their spin degrees of freedom. These final states can be generated from pure spin states of the incoming nd system, where both the neutron and the deuteron are strongly polarized by maximizing the occupation of a specific magnetic substate. To fully characterize the final configurations, we compute the corresponding spin density matrices using the high precision CD Bonn nucleon-nucleon potential. Among these pure final spin states, we identify strongly entangled, Bell-like states that include at most small admixtures of components diminishing entanglement.

Figures (9)

• Figure 1: (color online) Angular distributions of the contributions $|\bar{\alpha}_{m_n m_d}|^2$ to the norm for the pure final $nd$ states of Eq. (\ref{['new_a1']}). These states are formed in the $\vec{d} (\vec{n}, n)d$ scattering at an incoming neutron laboratory energy of $E = 37.5$ MeV, from pure cross-product initial states with the neutron and deuteron polarizations $(p_y^n,p_y^d,p_{yy})$ shown in a)-d). The solid, dashed, and dotted lines in a)–d) show the contributions of $|\bar{\alpha}_{+\frac{1}{2} +1}|^2 = |\bar{\alpha}_{-\frac{1}{2} -1}|^2$, $|\bar{\alpha}_{+\frac{1}{2} 0}|^2 = |\bar{\alpha}_{-\frac{1}{2} 0}|^2$, and $|\bar{\alpha}_{+\frac{1}{2} -1}|^2 = |\bar{\alpha}_{-\frac{1}{2} +1}|^2$, respectively. The angular distributions of the entanglement power $\epsilon(\rho_{nd})$ and the concurrence ${\bar{C}}(\rho_{nd})$ for a)–d) are presented in e) and f), by the solid, short-dashed, long-dashed, and dotted lines of the same colors as in a), b), c), and d), respectively. The predictions were obtained using the CD-Bonn potential.
• Figure 2: (color online) Angular distributions of the entanglement power $\epsilon(\rho_{nd})$ for the elastic scattering pure final spin states of Eq. (\ref{['new_a1']}). These states are formed in the $\vec{d} (\vec{n}, n)d$ scattering at incoming neutron laboratory energies of $E = 8.5$ MeV (a), $E = 37.5$ MeV (b), $E = 65$ MeV (c), and $E = 135$ MeV (d), from pure cross-product initial states with the neutron and deuteron polarization combinations given in Table \ref{['tab1']}. Their entanglement power $\epsilon(\rho_{nd})$ is shown by different lines: (a) red solid, (b) blue long-dashed, (c) green short-dashed, (d) black dash-dotted, (e) orange double-dash-dotted, and (f) magenta double-dash-dotted. The predictions were obtained using the CD-Bonn potential.
• Figure 3: (color online) Contributions $|\bar{\alpha}_{m_1 m_2 m_3}|^2$ to the norm of the breakup pure final spin states of Eq. (\ref{['new_a4']}) in QFS(nn) kinematically complete configurations, shown as functions of the laboratory angle of the first neutron $\theta_1^{lab}$. These states are formed in the $\vec{d} (\vec{n}, n_1 n_2)p$ breakup reaction at an incoming neutron laboratory energy of $E = 65$ MeV, from pure cross-product initial states with the neutron and deuteron maximal polarizations given in Table \ref{['tab1']} and shown in a)-f). The blue solid, green long-dashed, magenta short-dashed, and orange dash-dotted lines in a)–f) show the contributions of $|\bar{\alpha}_{---}|^2 = |\bar{\alpha}_{+++}|^2$, $|\bar{\alpha}_{--+}|^2 = |\bar{\alpha}_{++-}|^2$, $|\bar{\alpha}_{-+-}|^2 = |\bar{\alpha}_{+-+}|^2$, and $|\bar{\alpha}_{-++}|^2 = |\bar{\alpha}_{+--}|^2$, respectively. The predictions were obtained using the CD-Bonn potential.
• Figure 4: (color online) The same as in Fig. \ref{['fig2']}, but for the QFS(np) case.
• Figure 5: (color online) The same as in Fig. \ref{['fig2']}, but for the FSI(np) case.