Quantum Otto Engine: A Qubit Heat Engine

Cycle

Stage: —

Baths & drive

Hot bath T_h 2.20

Cold bath T_c 0.60

Max gap Δ_h 2.40

Min gap Δ_c 1.00

Cycle speed 1.0×

Live readouts

Excited pop. p

0.00

Gap Δ(t)

0.00

Cycles done

Net work / cycle

0.00

Efficiency η

—

η bound (1−Δc/Δh)

—

How it works

A single qubit (two-level system, energies 0 and Δ) is carried around a quantum Otto cycle: two isochoric strokes (gap held fixed while the qubit thermalizes with a bath) alternate with two quantum-adiabatic strokes (gap swept while no transitions occur, by the quantum adiabatic theorem).

Excited-state population target:
  p_eq(Δ,T) = 1 / (1 + exp(Δ / T))     (k_B = 1)

Isochore (Δ fixed): relax toward p_eq
  p(t) = p_eq + (p0 − p_eq) e^(−γt)

Adiabat (p fixed, Δ swept linearly):
  dU = p · dΔ            (no heat exchanged, dQ = 0)

Mean energy: U = p·Δ
  Heat strokes:     dU = Δ·dp   → accumulates as Qh or Qc
  Adiabatic strokes: dU = p·dΔ  → work done ON the qubit

Net work OUT per cycle:  W = Qh + Qc
Efficiency:               η = W / Qh   (when Qh > 0)
Engine bound:              η ≤ 1 − Δc/Δh

1. Hot isochore — gap held at Δh, qubit heats up toward the hot bath's thermal population.
2. Adiabatic expansion — gap is swept Δh → Δc; population frozen, the qubit does work on the outside world.
3. Cold isochore — gap held at Δc, qubit cools toward the cold bath's (lower) thermal population.
4. Adiabatic compression — gap swept Δc → Δh; population frozen, work is done back on the qubit.

The spheres are an ensemble of identical qubits: their split between the lower (blue) and upper (orange) plate visualizes the population p. This is the same cycle realized experimentally with trapped ions and superconducting qubits as nanoscale heat engines.

Ground state |0⟩ Excited state |1⟩

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