Transmon anharmonicity: α = ω_12 − ω_01 ≈ −E_C (leading order in (E_C/E_J)^(1/2))

Transmon (E_J/E_C ≫ 1) is effectively a weakly-anharmonic oscillator. Duffing-oscillator expansion of the H_trans = 4E_C(n̂−n_g)² − E_J cos φ̂ Hamiltonian around φ = 0 gives, to leading order in (E_C/E_J)^(1/2): ω_01 ≈ √(8 E_J E_C)/ℏ −…