Turing mechanisms in a multimode open quantum system

Abstract: We investigate pattern formation in a finite chain of bosonic modes whose dynamics is governed by a Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) master equation. The model combines local parametric driving and nonlinear damping with nonlocal dissipative couplings between modes that work on different discrete spatial scales. In the classical limit, these mechanisms generate a reaction-diffusion-like dynamics, allowing the emergence of Turing-type instabilities. The key aspect of the analysis is the coexistence and competition of different unstable spatial modes. Depending on the range of parameters, the system may select different stationary nonuniform configurations, oscillatory wave-like states, or regimes in which multiple modes interact before a dominant pattern is established, thus providing a mechanism for pattern selection. We compare the deterministic bifurcation scenario, generated by a reaction-diffusion-like system derived from semiclassical drift dynamics, with the quantum dynamics, derived via the GKSL master equation, using phase-space methods and reduced Wigner functions. The results show how Turing instabilities, mode competition, and pattern selection can be extended to multimode open quantum systems, providing a bridge between nonlinear dynamical systems, dissipative quantum mechanics, and spatial self-organization.