The same real quantum Otto cycle as the 2D version, rendered as a 3D energy ladder. Each rung n sits at height Eₙ(ω)=ω(n+½); during the two adiabatic strokes every rung slides smoothly up or down together as ω is swept, with populations pₙ frozen (no particle exchange — pure work). During the two isochoric strokes the ladder's height stays fixed while it couples to a bath sphere (orange = hot, cyan = cold) at the side; particles stream between the bath and whichever rungs are gaining or losing occupation, visualizing the real heat flow ΔQ=ΣΔpₙEₙ.
Eₙ(ω) = ω(n+½)
pₙ(T,ω) = e^(−Eₙ/T) / Σₘ e^(−Eₘ/T)
η = W_net / Q_hot ≤ 1 − ω_cold/ω_hot
- ω_hot / ω_cold — how far apart the compressed vs. expanded rung spacing is; sets the Otto efficiency ceiling.
- T_hot / T_cold — bath temperatures driving the hot/cold equilibrium occupations shown by rung brightness.
- Bath coupling κ — relaxation rate toward equilibrium during each isochore; low κ leaves rungs visibly short of full brightness at stroke's end — a finite-time cycle with reduced net work.
Real-world relevance: this ladder is exactly what a trapped-ion or optomechanical quantum-Otto-engine experiment measures — the population of discrete motional Fock states as the trap's spring constant (ω) is modulated between two baths.