Resonator & Cavity Parameters
=============================

Resonator and cavity checks that control readout speed, coupling, Q factors,
                impedance, and frequency placement.

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

   * - #
     - VER ID
     - Parameter
     - Severity
     - Design Rule / Constraint
     - Ideal / Optimal Value
     - Acceptable Range
     - Good
     - Bad
     - Why It Matters
   * - 9
     - HFSS-R-001
     - Resonant Frequency f₀
     - Critical
     - 4.0 GHz ≤ f₀ ≤ 8.0 GHz; detuned ≥ 300 MHz from qubit
     - 5 – 7 GHz
     - 4 – 8 GHz
     - 5–7 GHz: low thermal photon occupancy; standard coax hardware
     - < 1 GHz: thermal excitation; > 15 GHz: lossy substrate
     - Resonator frequency sets readout photon energy, hardware requirements, and
                        Purcell rate via qubit–resonator detuning.
   * - 10
     - HFSS-R-002
     - Loaded Q (Q_L)
     - Critical
     - Q_L ~ 5,000–20,000 (readout); > 10⁶ (memory)
     - 5,000 – 20,000
     - 1,000 – 50,000
     - 5k–20k: readout BW 250–1000 kHz; fast measurement with acceptable Purcell
     - < 500: too leaky; rapid Purcell decay; > 10⁶ readout: extremely slow
     - Loaded Q determines readout bandwidth κ = ω₀/Q_L. Governs measurement time
                        and Purcell-limited qubit T₁.
   * - 11
     - HFSS-R-003
     - Internal Q (Q_i)
     - Critical
     - Q_i ≥ 10⁵ (2D planar); ≥ 10⁷ (3D cavity)
     - > 10⁶
     - 10⁵ – 10⁷
     - > 10⁶: resonator loss << Purcell loss; qubit T₁ not
                        resonator-limited
     - < 10⁴: resonator dominates T₁ budget; unacceptable in planar SC circuits
     - Internal Q reflects intrinsic material, TLS, and vortex losses in resonator
                        walls. Sets upper limit on qubit T₁ via Purcell.
   * - 12
     - HFSS-R-004
     - External Q (Q_e)
     - High
     - Q_e ~ 2,000 – 50,000 (readout)
     - 2,000 – 20,000
     - 500 – 100,000
     - 2k–20k: controllable readout rate; Purcell rate < qubit decay rate
     - < 100: over-coupled; Purcell T₁ < 1 µs; > 10⁶: under-coupled
     - External Q sets coupling to transmission line. With Q_i >> Q_e
                        (over-coupled), resonator is readout-limited not loss-limited.
   * - 13
     - HFSS-R-005
     - Coupling Strength g
     - Critical
     - 50 MHz ≤ g / 2pi ≤ 200 MHz (strong coupling)
     - 50 – 150 MHz
     - 10 – 300 MHz
     - 50–150 MHz: well in strong coupling; g/κ > 10 and g/γ > 10 confirmed
     - < 1 MHz: weak coupling; cQED regime not achieved; readout fidelity <
                        90%
     - Qubit–resonator coupling. Strong coupling (g >> κ, γ) is fundamental
                        requirement for circuit QED dispersive readout.
   * - 14
     - HFSS-R-006
     - Dispersive Shift chi
     - Critical
     - 0.5 MHz ≤ \|chi\|/2π ≤ 10 MHz
     - 1 – 5 MHz
     - 0.1 – 20 MHz
     - 1–5 MHz: large IQ-plane separation; high-fidelity single-shot readout
     - < 0.01 MHz: states indistinguishable; > 50 MHz: photon-induced
                        dephasing
     - State-dependent resonator frequency shift enables QND readout. chi = g²/Δ
                        sets IQ-plane angle; drives single-shot fidelity.
   * - 15
     - HFSS-R-007
     - Photon Decay Rate κ
     - High
     - κ/2π = 1 – 5 MHz (readout resonator)
     - 1 – 5 MHz
     - 0.1 – 20 MHz
     - 1–5 MHz: readout ring-up/ring-down time ~100–500 ns; compatible with 1 µs
                        cycles
     - < 10 kHz: readout too slow; > 100 MHz: broad resonator; Purcell
                        collapse
     - Resonator energy decay rate sets readout speed. Too small → slow readout;
                        too large → Purcell-limited T₁.
   * - 16
     - HFSS-R-008
     - Impedance Z_0
     - Medium
     - Z_0 = 50 Ω ± 2 Ω (matched to coax)
     - 50 Ω
     - 45 – 55 Ω
     - 50 Ω ± 1 Ω: VSWR < 1.05; full power coupling, no reflections in cryo
                        lines
     - < 25 Ω or > 100 Ω: VSWR > 2; large reflections; effective κ shifts
                        from design
     - Characteristic impedance matching to 50 Ω coaxial environment. Mismatch
                        reduces coupling efficiency and shifts κ from design.
   * - 17
     - HFSS-R-009
     - Frequency Pulling Δf
     - Medium
     - Δf < 1 MHz from design target
     - < 0.5 MHz
     - < 2 MHz
     - < 0.5 MHz: resonator on-frequency; readout pulse pre-calibrated
     - > 5 MHz: readout tone off-resonance; SNR degraded; tone calibration
                        required
     - Frequency shift due to coupling, fabrication tolerances, or dielectric
                        loading. Excess pulling requires per-device calibration.
   * - 18
     - HFSS-R-010
     - Kinetic Inductance α
     - Low
     - 0.01 ≤ α ≤ 0.3 for standard Al/Nb resonators
     - 0.05 – 0.2
     - 0.001 – 0.5
     - 0.05–0.2: moderate nonlinearity; resonator frequency stable vs power
     - > 0.8: strong nonlinearity; resonator bifurcates at readout photon
                        numbers
     - Kinetic inductance fraction α = L_k/(L_k+L_geo). Controls resonator
                        nonlinearity, power handling, and anharmonicity contribution.