Source-verifiedLevel 4

Anharmonicity

Anharmonicity is the deviation of an oscillator's energy-level spacings from equality; in a transmon, α = ω12 − ω01 < 0 is the only spectral handle that separates the qubit transition from higher levels.

What it means

Anharmonicity is the deviation of an oscillator's energy-level spacings from perfect equality.A harmonic oscillator, with identical spacing between all levels, cannot serve as a qubit: any resonant drive that excites 0→1 also drives 1→2, 2→3 and beyond, marching population up the entire ladder with no way to isolate two states.In a transmon, the Josephson junction's nonlinearity — a quartic correction to the potential — breaks this degeneracy, so the 1→2 transition frequency ω12 sits below the 0→1 frequency ω01 by the anharmonicity α: ω12 = ω01 + α with α < 0.Typical magnitudes are |α|/2π ≈ 200–300 MHz, only about 5% of the ~5 GHz qubit frequency — hence 'weakly' anharmonic.That small 5% offset is the sole spectral handle that lets control pulses address the computational subspace {|0⟩, |1⟩} while avoiding |2⟩.Because the offset is so small, short pulses with wide spectra inevitably brush the 1→2 transition, which is why the fight against leakage never truly ends.

Everyday analogy

Imagine a ladder whose rungs are almost — but not exactly — evenly spaced: the step from floor 1 to floor 2 differs from the step from floor 2 to floor 3 by just 5%. That tiny difference is our only hope. It lets us knock in a way that opens only the 1→2 door and leaves the 2→3 door shut. If every step were identical, no knock could ever tell them apart.
'Anharmonicity' comes from Greek an- 'without' + harmonia 'joint, harmony' — a word that originally meant a carpenter's joint before it meant musical concord. An anharmonic ladder is one whose rungs no longer join at equal intervals — and for a qubit, that broken joinery is a feature, not a flaw.

Common misconceptions

  • Anharmonicity is NOT an imperfection to be engineered away — it is the very feature that makes a multi-level circuit usable as a qubit; a perfectly harmonic oscillator could never be one, since a resonant drive would climb its entire ladder.
  • The transmon's anharmonicity is NOT large — at roughly 5% of the qubit frequency it is weak by deliberate design (anharmonicity was traded for charge-noise insensitivity), which is why leakage must be actively managed rather than assumed away.

Key takeaways

  • ω12 = ω01 + α with α < 0: the 1→2 step of the transmon ladder is lower than the 0→1 step by the anharmonicity α.
  • Typical magnitude |α|/2π ≈ 200–300 MHz — only about 5% of the ~5 GHz qubit frequency, which is what 'weakly anharmonic' means.
  • Anharmonicity is the sole reason a computational subspace {|0⟩, |1⟩} can be selected at all — and its smallness is the reason pulse shaping matters.

Check your understanding

Why can a harmonic oscillator not be used as a qubit?

  1. A.Its energy levels are too far apart to drive
  2. B.Its equal level spacing means any resonant drive also excites every higher transition, climbing the whole ladder
  3. C.It has only two energy levels
  4. D.It cannot be cooled to millikelvin temperatures
Show the answer

Answer: B. Its equal level spacing means any resonant drive also excites every higher transition, climbing the whole ladder

Why: With identical spacing, the frequency that drives 0→1 also drives 1→2, 2→3 and so on — resonant driving pushes population up the entire ladder, so there is no way to isolate {|0⟩, |1⟩}. The transmon's anharmonicity α provides exactly that distinguishability.

A transmon has qubit frequency ω01/2π ≈ 5 GHz and anharmonicity |α|/2π ≈ 250 MHz. Roughly what fraction of the qubit frequency is the anharmonicity?

  1. A.About 0.5%
  2. B.About 5%
  3. C.About 20%
  4. D.About 50%
Show the answer

Answer: B. About 5%

Why: 250 MHz / 5 GHz = 0.05, i.e. about 5%. This small ratio is what the word 'weakly' in 'weakly anharmonic' means — and why leakage into |2⟩ remains a constant concern.

Builds on

Transmon

Primary source: Koch et al., Phys. Rev. A 76, 042319 (2007), doi:10.1103/PhysRevA.76.042319

omega_12 = omega_01 + alpha, |alpha|/2pi ~ 200-300 MHz per Krantz 2019 SII and Patra & Raina arXiv:2604.21565 (verified on arXiv 2026-07-03).

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