Quantum Otto Engine — Indicator Diagram (2D)

This is an independent 2D companion to the 3D quantum Otto engine, built around a different piece of physics machinery entirely: instead of rendering an ensemble of particles hopping between energy levels, it plots the qubit's own trajectory through its (Δ, p_e) state space — the quantum equivalent of a P–V indicator diagram from classical thermodynamics. Two adiabatic strokes sweep the level spacing Δ with the excited-state population p_e frozen (horizontal legs of the loop); two isochoric strokes hold Δ fixed while p_e relaxes toward the Gibbs equilibrium of whichever bath is coupled (near-vertical legs). By Green's theorem the net work extracted per cycle equals the signed area enclosed by this loop, and the simulator proves that live: it integrates the loop with both a trapezoidal path integral and an independent shoelace-polygon formula, then checks both against the closed-form stroke bookkeeping W = (Δh−Δc)(p₂−p₁) already used elsewhere on the site — every run, to floating-point precision. A companion strip chart plots Δ(t) and p_e(t) directly, so each segment of the loop can be traced back to the exact stroke that drew it, and raising the cycle-speed slider shows the loop collapsing toward zero area while its aspect ratio — and therefore the efficiency η = 1−Δc/Δh — stays exactly fixed, a subtlety this indicator-diagram view makes precise that a particle-cloud rendering cannot.