Electromagnetic Field Outputs
=============================

Field-solution checks for peak fields, surface/interface participation, EPR
                participation, and radiation quality.

.. list-table::
   :header-rows: 1

   * - #
     - VER ID
     - Parameter
     - Severity
     - Design Rule / Constraint
     - Ideal / Optimal Value
     - Acceptable Range
     - Good
     - Bad
     - Why It Matters
   * - 19
     - HFSS-E-001
     - Peak E-Field \|E\|max
     - High
     - \|E\|max < 10⁶ V/m at junction (single photon)
     - < 10⁵ V/m
     - < 10⁷ V/m
     - < 10⁵ V/m: well below oxide breakdown; TLS excitation rate suppressed
     - > 10⁸ V/m: dielectric breakdown risk; TLS saturation
     - Peak E-field at Josephson junction. High fields excite TLS defects in AlOx
                        barrier and substrate, directly increasing T₁⁻¹.
   * - 20
     - HFSS-E-002
     - H-Field Distribution \|H\|
     - Medium
     - \|H\| < 10³ A/m at SC surface; spatially uniform
     - < 500 A/m
     - < 5,000 A/m
     - < 500 A/m uniform: no vortex nucleation; surface current below
                        pair-breaking
     - > 10⁵ A/m: vortex trapping in SC film; each vortex adds ~1 kHz to κ
     - H-field concentration at superconductor surface. Hotspots exceed H_c1
                        locally, nucleating Abrikosov vortices that add loss.
   * - 21
     - HFSS-E-003
     - Interface Participation pᵢ
     - Critical
     - p_SA < 10⁻³; p_MS < 10⁻³
     - < 10⁻³
     - 10⁻³ – 10⁻²
     - < 10⁻³: interface loss contribution < 1 kHz to T₁⁻¹; T₁ > 1 ms
                        achievable
     - > 10⁻¹: interface dominates all loss channels; T₁ < 10 µs
     - Fraction of electric field energy at lossy interface. pᵢ × tan δᵢ × ω = loss
                        rate. Dominant predictor of TLS-limited T₁.
   * - 22
     - HFSS-E-004
     - Surface Participation p_MA
     - Critical
     - p_MA (metal–air) < 10⁻⁴
     - < 10⁻⁴
     - 10⁻⁴ – 10⁻³
     - < 10⁻⁴: native oxide TLS loss negligible; consistent with T₁ > 500 µs
     - > 10⁻²: native oxide (AlOx) TLS dominates decay; T₁ < 50 µs
     - Energy stored in metal–air (native oxide) interface. Key TLS loss channel in
                        Al transmons; reduced by surface treatment.
   * - 23
     - HFSS-E-005
     - Bulk Participation p_bulk
     - High
     - p_bulk < 10⁻² for Si/sapphire substrates
     - < 5×10⁻³
     - 10⁻² – 5×10⁻²
     - < 5×10⁻³: bulk dielectric loss < 1 kHz; substrate does not limit T₁
     - > 0.1: bulk dominates; even ultra-pure Si limits T₁ < 100 µs
     - Energy fraction in bulk substrate dielectric. Multiplied by tan δ gives bulk
                        T₁ contribution. Minimised by geometry.
   * - 24
     - HFSS-E-006
     - EPR (Junction EPR)
     - Critical
     - Junction EPR ≥ 0.9 for dominant mode
     - 0.95 – 1.0
     - 0.8 – 1.0
     - 0.95–1.0: junction hosts >95% inductive energy; anharmonicity and χ
                        predicted accurately
     - < 0.5: junction energy shared with stray inductances; Hamiltonian
                        extraction unreliable
     - Fraction of inductive energy in Josephson junction vs total. Drives EPR
                        method for extracting dispersive shifts and decay rates.
   * - 25
     - HFSS-E-007
     - Radiation Q (Q_rad)
     - High
     - Q_rad ≥ 10⁶ for on-chip resonators (shielded)
     - > 10⁶
     - > 10⁵
     - > 10⁶: radiation loss < 1 kHz; package design validated; Q_i not
                        radiation-limited
     - < 10⁴: radiation loss dominates internal Q; chip must be redesigned
     - Quality factor limited by power radiated from chip. Poor shielding or
                        slot-line modes radiate energy, reducing Q_i and T₁.