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Sandbox Physics

Q050 · Compress / heat / expand / cool

Two-Level Quantum Otto Engine

Open an energy gap, add heat, close the gap and cool the spin. Follow each stroke and keep the work, heat and energy change in one ledger.

Interactive modelTwo-Level Quantum Otto Engine
Model reference—\text{—}
Protocol reference—\text{—}

02 / FOLLOW THE FOUR STROKES

Change the gap. Then change the occupation.

Excited population versus gap · model ϵ/ϵ0\epsilon/\epsilon_0

Drag near a stroke to inspect it. This is not a pressure-volume plot. Loop area alone is not the work when driving also rotates the field. The ledger uses changes in energy at every boundary.

—\text{—}

Circles show probability weight, not multiple particles. The gap is drawn on a fixed scale across the cycle.

03 / BALANCE THE LEDGER

Per cycleRecords · mean ± one SEModel
Work delivered—\text{—}—\text{—}
Heat from hot bath—\text{—}—\text{—}
Heat from cold bath—\text{—}—\text{—}
Internal energy change—\text{—}—\text{—}

All energies use the reference gap. Heat into the spin and work out are positive. Each trial starts independently in the stationary boundary mixture; five measured energy-state bits determine its entire ledger.

—\text{—}

04 / MEASURE THE FLUCTUATIONS

A single cycle can run against the average.

Recorded work distribution Wout/ϵ0W_{\rm out}/\epsilon_0

Stationary model · not a fit

—\text{—}—\text{—}—\text{—}

Efficiency is shown only in engine mode; cooling performance only in refrigerator mode. Sampled heat close to zero makes an efficiency ratio unstable, so sampled heat and work remain separate.

05 / KEEP THE EVIDENCE

Save. Change one thing. Compare.

Eight saved samples maximum. Saving freezes a run; new acquisition starts another. Export includes all raw records and preparation settings. Reusing a seed reproduces draws, so repeated copies are not independent evidence.

Model, units & sources
ΔU=Qh+Qc−Wout,ϵc=ϵ0\Delta U=Q_h+Q_c-W_{\rm out},\qquad \epsilon_c=\epsilon_0

Energies use the low gap, temperatures use that energy divided by Boltzmann’s constant, and times use reduced Planck’s constant divided by that energy. The population relaxation rate in both baths is one inverse reference time. Gaps vary linearly; the field axis rotates linearly in one plane. Midpoint unitary steps preserve state length; numerical convergence is checked separately.

Projective energy readings at stroke boundaries remove inter-stroke coherence without changing mean energy. Baths are ideal Markov contacts with Gibbs detailed balance. Every sampled cycle starts independently in the stationary measured mixture. Bath generation, electronics, measurement memory and preparation costs are omitted; displayed power counts stroke durations only.

With both contacts closed and no drive-induced transitions, the stationary mixture is not unique; the cold Gibbs mixture is the declared convention. This model is inspired by spin heat engines, not a numerical reproduction of the cited NMR experiment.

Peterson et al. · Experimental characterization of a spin quantum heat engine

Physics tutorial

Two-Level Quantum Otto Engine

BackgroundOpen an energy gap, add heat, close the gap and cool the spin. Follow each stroke and keep the work, heat and energy change in one ledger.

Why it mattersCompare individual records with the declared ensemble model.

Start with the essentials

Focus question
Can one spin turn heat into work?
One-sentence intuition
Change one setting, preserve a comparison, and reconstruct from raw records.

Core mathematical model

Controlled Hamiltonian

H(t)=ϵ(t)2[I+n(t)⋅σ]H(t)=\tfrac{\epsilon(t)}2[ I+\mathbf n(t)\cdot\boldsymbol\sigma]

The energies are zero and the controlled gap. The field axis rotates in one plane during isolated strokes.

Thermal contact

pe(t)=pβ+[pe(0)−pβ]e−γt,pβ=11+eϵ/(kBT)p_e(t)=p_\beta+[p_e(0)-p_\beta]e^{-\gamma t},\quad p_\beta=\frac1{1+e^{\epsilon/(k_BT)}}

A Markov relaxation channel obeying Gibbs detailed balance. The reference relaxation rate is one in the declared time units.

First law

ΔU=Qh+Qc−Wout\Delta U=Q_h+Q_c-W_{\rm out}

Heat into the spin and work out of the spin are positive. Individual cycles need not return to the same measured state.

Transitionless Otto limit

ηOtto=1−ϵcϵh,1<ϵhϵc<ThTc\eta_{\rm Otto}=1-\frac{\epsilon_c}{\epsilon_h},\quad 1<\frac{\epsilon_h}{\epsilon_c}<\frac{T_h}{T_c}

The inequality is the work-producing regime for transitionless driving. Noncommuting fast ramps can reduce output.

Entropy ledger

Σ=−QhTh−QcTc≥0\Sigma=-\frac{Q_h}{T_h}-\frac{Q_c}{T_c}\geq0

For mean heat in a stationary measured cycle. This is not a constraint on the sign of every fluctuating single-cycle record.

Common difficulties

Free extra efficiency?

Typical misconceptionOne lucky cycle beats the thermodynamic bound.

Better mental modelEfficiency uses stationary ensemble heat and work. A sample ratio near zero heat is unstable; the UI reports sampled work and heat separately.

A complete machine?

Typical misconceptionThe output includes all engineering costs.

Better mental modelBath preparation, control electronics, measurement and resetting costs are outside this working-medium model.

Run the experiment

  1. 01

    Follow the cycle

    Scrub the four strokes and locate when energy changes as work or heat.

    What to observe: The horizontal diagram axis is energy gap, not volume.
  2. 02

    Count energy changes

    Record 500 independent cycles, export five energy-state readings per cycle, and rebuild the ledger.

    What to observe: Finite samples have nonzero mean energy drift; keep that term in the first law.
  3. 03

    Find a failure

    Rush the drive. Then fix its axis, or choose a gap ratio larger than the temperature ratio.

    What to observe: A fast commuting ramp has no transition penalty. A suitable large gap ratio changes the device into a refrigerator.