Electromagnetic Coupling

Coupling outputs connect Q3D extraction to qubit-qubit, qubit-resonator, and package-mode interactions.

#	Parameter	Symbol / Unit	Extraction Method	Typical Q3D Value	Ideal / Optimal	Good Range	Worst Case	Why It Matters	Key Design Note
34	External Quality Factor (Q_ext)	Q_ext	HFSS eigenmode + Q3D coupling capacitance extraction	10³ – 10⁵	5×10³ – 2×10⁴ (dispersive readout)	10³ – 10⁵	< 500 or > 10⁶	Sets readout bandwidth κ = ω_r/Q_ext and Purcell loss rate; undercoupled → slow; overcoupled → Purcell	Purcell limit: T₁_Purcell = Q_ext/ω_r × (Δ/g)²; Purcell filter relaxes this trade-off
35	Internal Quality Factor (Q_int)	Q_int	VNA transmission measurement; HFSS loss tangent input	> 10⁶ (planar Al at 4K)	> 10⁶	10⁵ – 10⁶	< 10⁴	Intrinsic resonator loss from TLS, dielectric, radiation; directly sets T₁ floor via Purcell	Q_int > 10⁶ requires: HR-Si or sapphire substrate, clean metal deposition, minimal surface TLS
36	Loaded Quality Factor (Q_L)	Q_L	1/Q_L = 1/Q_int + 1/Q_ext; VNA S21 Lorentzian fit	10³ – 10⁴	10³ – 10⁴ (balanced readout)	500 – 2×10⁴	< 200 or > 10⁵	Determines resonator 3 dB bandwidth; BW = f_r/Q_L sets speed vs SNR tradeoff for readout	In practice Q_L ≈ Q_ext when Q_int > > Q_ext (under-coupled limit is common design choice)
37	CPW Characteristic Impedance (Z_0)	Z_0 / Ω	Q3D RLGC → Z_0 = √(L’/C’); verified by HFSS S11 calibration	50 Ω ± 1 Ω	50 Ω ± 1 Ω	45 – 55 Ω	< 30 or > 80 Ω	Impedance mismatch causes reflections degrading signal integrity; Z_0 controlled by trace/gap ratio	On 500 μm Si: 10 μm trace / 6 μm gap → Z_0 ≈ 50 Ω; wider trace → lower Z_0
38	Effective Permittivity (varepsilon_eff)	varepsilon_eff	Q3D electrostatic fill factor calculation; HFSS eigenmode	6.0 – 6.5 (CPW on Si)	6.0 – 6.5	5.5 – 7.0	< 4 or > 9	Sets propagation velocity v_ph = c/√varepsilon_eff and resonator physical length for target frequency	varepsilon_eff depends on substrate filling fraction; varepsilon_eff ≈ (1 + εr)/2 for CPW in air on substrate
39	Coupling Coefficient k^2	k^2 / ×10⁻³	Q3D capacitance ratio k^2 = C_g² / (C_1 × C_2)	1 – 10 ×10⁻³	1 – 10 ×10⁻³	0.5 – 20 ×10⁻³	< 0.1 or > 50 ×10⁻³	Power transfer efficiency between resonator and feedline; determines Q_ext directly	k^2 ∝ gap width at coupling capacitor; etch depth variation of 0.1 μm → δk²/k^2 ~ 5%