Suppression of the two-stream instability by quantum effects: Numerical analysis with Wigner-Poisson and Nyquist method

The quantum two-stream instability in electrostatic plasmas is investigated through high-resolution numerical simulations of the Wigner-Poisson system. A fourth-order operator splitting scheme, which combines exact Fourier treatment of the free-streaming term with a semiclassical approximation of the quantum force, is implemented to achieve high temporal accuracy and minimal numerical dissipation. This approach enables precise extraction of linear growth rates and detailed characterization of the long-time nonlinear evolution. Systematic simulations demonstrate that quantum diffraction effects, parameterized by the dimensionless quantum parameter ħ, significantly reduce the growth rate of the two-stream instability and lead to its complete suppression above a critical value ħc ≈ 0.78. The stabilization mechanism is independently confirmed by high-resolution Nyquist diagrams constructed directly from the spatially averaged simulated Wigner function, showing excellent quantitative agreement (typically within 3.5%) with growth rates extracted from the time evolution of the electrostatic field energy E field (t). While the semiclassical approximation remains accurate to within 2% for ħ ≤ 0.6, its relative error increases to 10-25% near the stabilization threshold. Simulations employing the full non-local quantum operator yield a slightly lower threshold, ħc ≈ 0.74 ± 0.02. The combined effect of quantum diffraction and Bhatnagar-Gross-Krook (BGK) collisions is also examined, revealing a synergistic stabilization that further lowers the critical quantum parameter. In the nonlinear regime, the Wigner function displays distinct quantum signatures, including smoother filamentation and reduced vortex formation compared to the classical case. These findings provide new benchmark values for quantum stabilization thresholds and underscore the importance of full kinetic treatments over fluid models based on the Bohm potential. Potential implications for inertial confinement fusion, high-energy-density plasmas, and nanoscale semiconductor devices are discussed.