Electromagnetic Coupling
========================

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

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

   * - #
     - 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%