  We study the dissipative Bose-Hubbard model on a small ring of sites in the
presence of a chiral drive and explore its long-time dynamical structure using
the mean field equations and by simulating the quantum master equation.
Remarkably, for large enough drivings, we find that the system admits, in a
wide range of parameters, a chaotic attractor at the mean-field level, which
manifests as a complex Wigner function on the quantum level. The latter is
shown to have the largest weight around the approximate region of phase space
occupied by the chaotic attractor. We demonstrate that this behavior could be
revealed via measurement of various bosonic correlation functions. In
particular, we employ open system methods to calculate the out-of-time-ordered
correlator, whose exponential growth signifies a positive quantum Lyapunov
exponent in our system. This can open a pathway to the study of chaotic
dynamics in interacting systems of photons.
