| Quantity | Method / Formula |
|---|---|
| Hamiltonian | H = 4EC(n̂ − ng)² − EJcos(φ̂) |
| f₀₁, α, Δ | Exact numerical diagonalization (charge basis, 31 states) |
| Charge dispersion Δ | |f₀₁(ng=0) − f₀₁(ng=½)| |
| f₀₁ (transmon approx.) | √(8EJEC) − EC (valid for EJ/EC ≫ 1) |
| Critical current Ic | EJ = ℏIc / (2e) |
| EC from CΣ | EC = e² / (2CΣ) |
| CΣ from EC | CΣ = e² / (2EC) |
| Josephson inductance LJ | Φ0 / (2πIc) = ℏ / (2eIc) |
| Lamb shift | δL ≈ g² / (f₀₁ − fr) |
| Dispersive shift χ | g²α / ((f₀₁−fr)(α+f₀₁−fr)) |
| Purcell T1 | 1/T1Purcell = κr(g/Δ)², Δ = ω₀₁ − ωr (all angular frequencies) |
| g from Cc | g = (Cc/CΣ) · (e/ℏ) · Vzpf · (EJ/8EC)1/4 (transmon limit) |
| Resonator Vzpf | Vzpf = ωr√(ℏZr/2), Zr = 50 Ω |
| Charge-line T1 | 1/T1gate = ω01² Cg² Z0 / CΣ, CΣ = e²/(2EC), Z0 = 50 Ω |