A single quantum harmonic oscillator (levels Eₙ=ℏω(n+½)) is carried around a real 4-stroke Otto cycle: (1) adiabatic compression ω_cold→ω_hot with level occupations pₙ frozen — no heat, only work as the level spacing widens; (2) isochoric heating at fixed ω_hot while the oscillator relaxes toward the hot bath's Gibbs distribution pₙ∝e^(−Eₙ/T_hot) — pure heat absorption Q_hot; (3) adiabatic expansion ω_hot→ω_cold, pₙ frozen again — work extracted; (4) isochoric cooling at fixed ω_cold, relaxing toward the cold Gibbs distribution — heat Q_cold rejected.
Eₙ(ω) = ω(n+½)
pₙ(T,ω) = e^(−Eₙ/T) / Σₘ e^(−Eₘ/T)
ΔW (adiabatic) = Σₙ pₙ·ΔEₙ
ΔQ (isochoric) = Σₙ Δpₙ·Eₙ
η = W_net/Q_hot ≤ 1 − ω_cold/ω_hot
- ω_hot / ω_cold — the compressed and expanded trap frequencies; a bigger ratio raises the theoretical efficiency ceiling.
- T_hot / T_cold — bath temperatures; the engine only produces net positive work when T_hot/T_cold exceeds ω_hot/ω_cold.
- Bath coupling κ — how fast populations relax during the isochores; low κ (or short strokes) leaves the working substance short of equilibrium each stroke — real finite-time thermodynamics, and it measurably lowers both power and efficiency.
- Stroke duration — sets how much wall-clock time each of the 4 strokes gets, which — together with κ — decides how close each isochore gets to full thermalisation.
Real-world relevance: this exact model (a trapped ion or optomechanical oscillator whose trap stiffness ω is modulated) is the leading experimental platform for demonstrated quantum heat engines, used to probe how far quantum coherence and finite-time driving can push thermodynamic efficiency beyond — or below — the classical Otto bound.