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arXiv:2607.07449

Turing mechanisms in a multimode open quantum system

Giorgia Comparato, Francesco Gargano, Rosario Lo Franco

quant-phmath-phSubmitted Jul 8, 2026• Announced Jul 9, 2026• 0 citations
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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.

Comments
14 + 4 pages, 13 figures

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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.
An open quantum system can be described by a master equation, of which one of the most popular is the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) equation. We revisit description of driven-dissipative quantum systems focusing on the appropriate choice of the system's basis and the respective transformations. We consider the GKSL equation in different bases and calculate the dynamics for a qubit and for a qudit. An appropriate choice of the basis is a fundamental problem for theoretical consideration of open quantum systems and provides an opportunity to obtain the desired evolution in practice.
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 such 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.
Dissipative quantum dynamics is usually described through loss, decoherence, relaxation, and the flow of information into an environment. This work proposes a conservative but conceptually important shift in emphasis: in suitably structured models, dissipation can also be read as a mechanism of selection. The paper focuses on effective generators of the form G_eff = -iD - Γ, where D = D* is the coherent self-adjoint part and Γ ≥ 0 is a dissipative term. The algebraic setting is not modified. These generators arise naturally inside the standard Gorini–Kossakowski–Lindblad–Sudarshan, or GKLS, formalism through the effective no-jump sector. The novelty is therefore not a new quantum master equation, but a different architectural reading of a familiar structure. Instead of interpreting Γ only as a loss rate, the paper treats a structured dissipator as an auditor of admissible modes. Modes that are strongly penalized decay, while compatible sectors survive the dissipative test. In finite dimensions, the work proves an exact selection mechanism under an invariant-sector hypothesis and identifies the persistent sector selected by the effective dynamics. The paper also clarifies what remains open in the infinite-dimensional setting. Extending this viewpoint to Dirac–Roco type architectures requires analytic control of domains, semigroup generation, spectral separation, and dissipative estimates. For readers in mathematical physics, open quantum systems, non-Hermitian dynamics, and operator theory, the note provides a bridge between standard GKLS dynamics and the broader idea of selective dissipation: the environment is not only a sink for lost information, but may also enforce the structural survival of particular modes.
This paper starts with surveying the evolution of quantum-like models of cognition and decision making, transitioning from static kinematic representations to a robust dynamical framework based on open quantum systems. We provide a comprehensive analysis of the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) master equation's application in cognitive psychology and decision making, illustrating how it models mental state evolution as a dissipative process influenced by an informational environment. We categorize dynamical regimes into Passive and Active Hamiltonians, demonstrating how non-commutation with projections on decision basis serves as a mathematical signature of cognitive agency and Quantum Escape from classical equilibria. The utility of this framework is further explored through its ability to stabilize non-Nash outcomes in strategic games, such as the Prisoner's Dilemma. Building upon this dynamical foundation, we identify ``cognitive beats'' as a signature of the internal struggle between competing ``flows of mind'' deliberated at approximately equal frequencies. Distinct from the damped oscillations of simple interference, these beats emerge from a structural tension between Liouvillian channels that generates a secondary, slow-scale modulation of conviction. This beat envelope dictates the timing of peak readiness and hesitation, providing a mathematical map of the transition between conflicting cognitive states. By resolving these nested time scales, we provide a new spectral diagnostic for the depth of cognitive agency and the complexity of the underlying deliberation process. This paper develops a theoretical framework linking GKSL dynamics with quantum-like cognition and decision-making (QCDM), highlighting how dissipative quantum models can capture features of human thought and decision processes.
This paper investigates the properties of the Gorini–Kossakowski–Sudarshan–Lindblad (GKSL) equation within the camel-like framework, with a focus on quantum correlations in the context of open quantum systems. Here, we compute quantum correlations such as quantum discord and quantum steering, analyzing their behavior under decoherence and environmental interaction for three sets of quantum states. Our results indicate that the sign of the entanglement entropy's derivative serves as an indicator of the system's drift toward classical or quantum information exchange—an insight with important implications for quantum error correction and dissipation processes in quantum thermal machines. \\ Moreover, we parametrize quantum states using both single-parameter and Bloch-sphere representations. The Bloch-sphere analysis is particularly developed by examining the topology of the quantum states as elements on the S2\mathbb{S}^2S2 sphere, yielding a gradient map and a topological basin map which illustrate how quantum states evolve under the camel-like framework and how this can be partitioned into basins on the Bloch sphere, by stability/instability against decoherence. The decoherence-unstable regions form a Braiding ring around the Bloch sphere at the circle defined by θ=3π4\theta=\frac{3\pi}{4}θ=43π​. Notably, these unstable states on the Braiding ring form attraction points for the evolution of states by a constructed Lyapunov function, which gives an illustration of the complex interplay between geometry and dynamics and which shapes the topological geometry on the Bloch sphere. This deeper analysis underscores that quantum dynamics are governed not only by local energetics but also by the global structure of the state space. Moreover, we also derive a testable rudimentary experimental setup from the camel-like entropy, which describes the unitary evolution of pure states on the Bloch surface and add the analytical solutions of selected quantum states.
We develop a novel framework to engineer persistent oscillatory modes in Markovian open quantum systems governed by a time-independent Lindblad master equation. We show that oscillatory modes can be created when the Hamiltonian and jump operator can be expressed in the same block-diagonal form. A key feature of the framework is that the dissipator of the Lindblad master equation are generally non-zero. We identify the weak and strong conditions, where the onset of the oscillatory modes is dependent and independent of the parameters of the system, respectively. Our method extends beyond the typical decoherence-free subspace approach, in which the dissipator is zero. We demonstrate the applicability of this framework using various models, showing how carefully tailored system-environment interactions can produce sustained coherent oscillations.