  The interaction between a photon and a qubit in the Janeys-Cummings (JC)
model generates a kind of quasiparticle called polariton. While they are widely
used in quantum optics, difficulties in engineering controllable coupling of
them severely limit their applications to simulate spinful quantum systems.
Here we show that, in the superconducting quantum circuit context, polariton
states in the single-excitation manifold of a JC lattice can be used to
simulate a spin-1/2 system, based on which tunable synthetic spin-orbit
coupling and novel topological polaritons can be generated and explored. The
lattice is formed by a sequence of coupled transmission line resonators, each
of which is connected to a transmon qubit. Synthetic spin-orbit coupling and
effective Zeeman field of the polariton can both be tuned by modulating the
coupling strength between neighbouring resonators, allowing for the realization
of a large variety of polaritonic topological semimetal bands. Methods for
detecting the polaritonic topological edge states and topological invariants
are also proposed. Therefore, our work suggests that the JC lattice is a
versatile platform for exploring spinful topological states of matter, which
may inspire developments of topologically protected quantum optical and
information processing devices.
