L02 · Laser foundations

Three Light–Matter Transitions

Prepare 80 emitters, apply a light field, and count absorption, spontaneous emission and stimulated emission one event at a time. Compare one stochastic realization with the exact ensemble prediction.

Model time $\text{—}$

Observed upper fraction $\text{—}$

Ensemble-mean upper fraction $\text{—}$

Absorption rate $\text{—}$

Spontaneous rate $\text{—}$

Stimulated rate $\text{—}$

Net drive-mode photon exchange $\text{—}$

Total counted transitions $\text{—}$

Equal-degeneracy, rapidly dephased two-level emitters. Spontaneous lifetime: 1 ms. The external probe is held constant and undepleted; the mode ledger counts exchange with that reservoir. No cavity, coherent Rabi dynamics or field phase is solved. Counts are physical model events; animation pulses and emission directions are schematic. Changing a setting prepares a new ensemble with the selected seed.

Start with the essentials

Focus question: Can stimulated emission occur while the incident mode has a net loss?
One-sentence intuition: Net stimulated gain depends on the population difference. In an unpumped ensemble, spontaneous and stimulated emission deplete the prepared inversion.

Core mathematical model

Three event rates

R_a=WN_l,\quad R_s=WN_u,\quad R_{sp}=AN_u

The rates count ensemble events per millisecond. The spontaneous coefficient is one per millisecond. The driving rate is per emitter and includes the Lorentzian detuning factor.

Detuning changes the driven processes

W=\frac{W_0}{1+(\delta/\gamma)^2}

The detuning unit is the frequency half-width at half maximum. One half-width halves absorption and stimulated rates at fixed populations; it does not change the intrinsic spontaneous lifetime.

A closed ensemble and an exact mean

\dot n_u=W(1-2n_u)-An_u

The total number of emitters stays at 80. The dashed trace is the analytic ensemble mean; the solid trace is one direct Gillespie realization. Finite-sample fluctuations need not follow the dashed line exactly.

Conservation through the event ledger

N_u(t)-N_u(0)=C_a-C_s-C_{sp}

Every event is counted once and moves one emitter. The drive-mode exchange is stimulated count minus absorption count; spontaneous photons are counted separately. The external probe is held fixed, so this is not a self-consistent finite-field energy simulation.

Common difficulties

A stimulated photon is not a collision product

Typical misconceptionA photon mechanically hits an atom and produces a copied particle.

Better mental modelThe drawing visualizes a rate-model event. Stimulated emission increases occupation of the driving optical mode. Absolute optical phase and coherent atomic amplitudes are outside this model.

A mean is not a trajectory

Typical misconceptionA random trace is incorrect whenever it departs from the exact curve.

Better mental modelThe mean predicts an ensemble of realizations. Conservation must hold in every realization; agreement with the mean is a statistical test across many seeds.

Run the experiment

01
Remove the drive
Choose Field off. Step or advance, watching the upper population and spontaneous-event counter.
What to observe: Excited emitters decay without an incident field. Absorption and stimulated counts remain zero.
02
Prepare transparency
Choose Equal populations before running. Compare instantaneous absorption and stimulated rates, then step.
What to observe: The two driven rates begin equal. Spontaneous decay breaks the initial balance; stimulated events still occur in a net-absorbing ensemble.
03
Spend a finite inversion
Choose Inverted sample and run. Watch the upper population, signed mode exchange, and eventual competition between the two driven rates.
What to observe: Initial gain does not last without pumping. The population approaches a non-inverted stationary mean.
04
Compare a distribution, not one lucky trace
Repeat at the same settings with different random seeds; also compare resonance with a detuning of one half-width.
What to observe: Event counts vary between realizations. The exact mean remains reproducible, and detuning changes driven rates while leaving spontaneous decay available.