Electromagnetic Field Outputs

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

#	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.