Non-classical light state transfer in $su(2)$ resonator networks

We use a normal mode approach to show full and partial state transfer in a class of coupled resonator networks with underlying $su(2)$ symmetry that includes the so-called $J_{x}$ photonic lattice. Our approach defines an auxiliary Hermitian coupling matrix describing the network that yields the normal modes of the system and its time evolution in terms of orthogonal polynomials. These results provide insight on the full quantum state reconstruction time in a general $su(2)$ network of any size and the full quantum transfer time in the $J_{x}$ network of size $4 n + 1$ with $n=1,2,3,\ldots$ In the latter, our approach shows that the Fock state probability distribution of the initial state is conserved but the amplitudes suffer a phase shift proportional to $\pi/2$ that results in partial quantum state transfer for any other network size.