Resonator & Cavity Parameters

Resonator and cavity checks that control readout speed, coupling, Q factors,

impedance, and frequency placement.

#	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.