  Turing instability is a fundamental mechanism of nonequilibrium
self-organization. However, despite the universality of its essential
mechanism, Turing instability has thus far been investigated mostly in
classical systems. In this study, we show that Turing instability can occur in
a quantum dissipative system and analyze its quantum features such as
entanglement and the effect of measurement. We propose a degenerate parametric
oscillator with nonlinear damping in quantum optics as a quantum
activator-inhibitor unit and demonstrate that a system of two
activator-inhibitor units can undergo Turing instability when diffusively
coupled with each other. The Turing instability induces nonuniformity and
entanglement between the two units and gives rise to a pair of nonuniform
states that are mixed due to quantum noise. Further performing continuous
measurement on the coupled system reveals the nonuniformity caused by the
Turing instability. Our results extend the universality of the Turing mechanism
to the quantum realm and may provide a novel perspective on the possibility of
quantum nonequilibrium self-organization and its application in quantum
technologies.
