A superconducting transmon qubit is a single Josephson junction shunted by a large capacitor. Its Hamiltonian in the phase basis is
H = 4 E_C n² − E_J(Φ) cos φ
n = Cooper-pair number operator (charge)
φ = superconducting phase difference
E_C = e²/2C (charging energy)
E_J(Φ) = E_J,max·|cos(πΦ_ext/Φ_0)| (SQUID loop, flux-tunable)
The −E_J cos φ term is the "washboard potential" plotted below. Near its bottom it is nearly parabolic (a harmonic oscillator), but the quartic correction from cos φ makes the levels unevenly spaced — that anharmonicity is what lets f₀₁ be addressed without also driving f₁₂, which is what makes a two-level qubit out of what would otherwise be a linear oscillator.
In the transmon regime (E_J/E_C ≫ 1) the levels have a closed-form approximation:
E_m ≈ √(8 E_J E_C)·(m + ½) − (E_C/12)·(6m² + 6m + 3)
f_01 = (E_1 − E_0)/h ≈ [√(8 E_J E_C) − E_C]/h
α = f_12 − f_01 ≈ −E_C/h
The turning points of each level (where the classical energy equals the potential) are drawn as horizontal rungs across the trough; the bright sphere integrates Newton's law φ̈ = −8(E_J/E_C)·sinφ in these normalized units — the classical trajectory whose small-oscillation frequency reproduces the same √(8 E_J E_C) that sets f₀₁.
- E_J,max/E_C — the central knob. Low ratio (~5–10) is the charge-sensitive Cooper-pair-box regime; high ratio (~50–100) is the transmon sweet spot real hardware uses, trading some anharmonicity for near-immunity to charge noise.
- Φ_ext/Φ_0 — for a SQUID-loop (split) transmon, threading half a flux quantum collapses E_J to zero and the qubit frequency dives with it — exactly how tunable transmons are frequency-tuned in real chips.
- E_C/h — sets the absolute frequency scale (typical hardware: 200–350 MHz).
- n=0…3 buttons — pick which level's turning points and classical trajectory are drawn; a level whose turning points exceed the barrier top is flagged "unbound" (phase runs away — the real-junction analogue of switching to the voltage state).