Even-k Suppression in Parametrically Driven Spherical Cavities

This work provides the full rigorous derivation of the suppression of even-order (even-k) parametric resonances in sinusoidally driven spherical electromagnetic cavities, completing the analysis initiated in the companion paper: C. R. Singleton, “Parametric Instabilities and Mode Coupling in Oscillating Spherical Cavities: Theoretical Framework, ” Phys. Rev. A 112, 063533 (2025), https: //doi. org/10. 1103/4dc7-hx5b. That earlier work derived exact coupling coefficients λₙₙ’⁽ˡ⁾ (t) and identified instability tongues at Ω = 2ωₙₗ/k (k = 1, 2, 3, …), with numerical Floquet evidence suggesting even-k resonances are suppressed by a factor ~ε^k-1 relative to odd-k. The analytical proof was deferred. Here we deliver it explicitly: • Closed-form, phase-independent evaluation of the overlap integral Iₖ = ∫ sin (Ωt) cos (kΩt/2) dt over one period, vanishing for even k (Theorem 1). • Complete trigonometric expansion showing the ˙λ term vanishes for all k ≥ 2 (stronger than even-k alone). • First-principles Floquet recurrence with exact integer weights (m±2) ; explicit Schur complement for k=2 (effective coupling O (ε²) with prefactor 4) ; structural zero for k=4 Mechanism 2 chain. • All-orders, all-modes suppression via Fourier parity of interaction vertices and temporal ℤ₂ symmetry under half-period shift (Appendix A). • Universality: the suppression holds for any resonator with λ (t) ∝ sin (Ωt) time dependence, independent of geometry or Bessel-form factors. The results predict an experimentally decisive growth-rate hierarchy in high-Q cavities (e. g. , 4. 77 GHz superconducting with Q=10⁹, ε=2×10⁻³), enabling mode-selective control and suppression of instabilities in cavity optomechanics, quantum state engineering, and dynamical Casimir setups. The temporal symmetry identified here suggests broader analogies in Floquet-driven systems.