Quantum Otto engine - 2D cycle diagram

Equations

With k_B = ℏ = 1 a qubit with level gap ω in contact with a bath at temperature T relaxes to the excited population p_eq = 1/(1 + e^ω/T). During an isochore (gap fixed) p relaxes exponentially, p → p_eq + (p - p_eq)e^-γt, and the heat taken in is Q = ωΔp. During an adiabatic stroke the gap is swept with the population frozen, so the work on the qubit is ∫p dω. Over a closed cycle the first law gives W = Q_h + Q_c = (ω₂ - ω₁)(p_h - p_c) and the efficiency is exactly η = 1 - ω₁/ω₂, which stays below the Carnot value 1 - T_c/T_h whenever the cycle is an engine (ω₁/ω₂ ≥ T_c/T_h). The yellow/blue/purple/red corners come from solving the periodic steady state of the two isochores.

Honest simplifications: populations only (no coherence, so the 3D page's 'quantum friction' penalty is not modelled here), ideal instantaneous baths and a linear gap sweep, so finite-time power is only as accurate as the simple exponential-relaxation model.

• Load 'Heat pump' and see the area turn red: the cycle consumes work.
• Drag w1 toward w2 and watch work per cycle peak and then fall while efficiency keeps dropping.
• Shorten tau and see the cycle corners shrink as the isochores no longer thermalise.