Dynamical Instability of Helical Vortex Filaments and Impurity Trapping Dynamics in Periodically Modulated Bose-Einstein Condensates

This work investigates the dynamical stability, parametric excitation, and impurity (tracer) trapping kinetics of quantized vortex filaments in periodically modulated superfluid systems (Bose-Einstein Condensates and Helium-II). Within the framework of the Local Induction Approximation (LIA) and linear perturbation theory, we show that external quadrupole pressure/density modulation couples counter-propagating Kelvin waves, which is modeled via the Rotating Wave Approximation (RWA) as a damped Mathieu-type autonomous system. By employing the microscopic theory of Bogoliubov phonon scattering off a quantized vortex core (Pitaevskii-Fetter theory), we integrate a T⁵ temperature dependence of the mutual friction drag force. Accounting for this thermodynamic damping and the mechanical cavitation limit ε_max ≈ c_s/ξ, we analytically derive the minimum parametric excitation threshold ε_min(T) = C₁ T⁵. For a ⁸⁷Rb condensate, we establish that the stable helical phase is strictly bounded by a critical crossover temperature T_cross ≈ 34 pK, above which the thermal phonon damping demands a driving amplitude that inevitably causes structural cavitation. Furthermore, the kinetics of neutral impurity trapping into the vortex core are evaluated in two distinct regimes. In the viscous (overdamped) Helium-II regime (T ~ 1-1.5 K), we obtain the characteristic trapping time scaling τ_trap ∝ T⁴ R_part⁻¹. In the ballistic (underdamped) BEC regime (34 pK < T ~ 10-50 nK for straight vortices, and T < 34 pK for helical vortices), the trapping is driven by acoustic bremsstrahlung. By evaluating the inertial acceleration of the tracer via orbital force balance, we show that the trapping time scales quadratically with the density ratio, yielding τ_trap(BEC) ∝ (ρ_part / ρ_s)², which is numerically evaluated as ≈ 6.5 s for a density-matched macroscopic tracer. Finally, we estimate the critical trapping velocity to be v_c ≈ 0.1–1 mm/s. Thermodynamic Crossover (T_cross): Establishes that the stable helical vortex phase in a ⁸⁷Rb BEC is strictly bounded by T_cross ≈ 34 pK due to the competition between T⁵ phonon damping and the mechanical cavitation limit. Dual-Regime Impurity Trapping Kinetics: Derives the trapping time scaling laws for both the viscous Helium-II regime (τ_trap ∝ T⁴) and the ballistic BEC regime where capturing is mediated by acoustic bremsstrahlung (τ_trap(BEC) ∝ (ρ_part / ρ_s)²). Experimental Verification: Connects the theoretical boundaries to state-of-the-art microgravity BEC platforms (Cold Atom Lab and QUANTUS) and Feshbach tuning methods.