Machine Learning Catalysis of quantum tunneling

Keywords: Josephson effect, Quantum tunnelling, Josephson tunnelling, Tunnelling between two Bose-Einstein condensates, Two-sit Bose-Hubbard Hamiltonian, Bose-Hubbard Hamiltonian, s-wave scattering length, Reduced Dynamics, Jordan-Schwinger representation. More keywords to be added.

Questions

I have yet to answer the questions below. Help me ❓

1. On Page 3, why can $J_z^2=\frac{1}{4}{\sigma }_z^2$ be neglected?