L02 · Laser foundations

Three Light–Matter Transitions

Does stimulated emission always mean amplification? No. The light that triggers stimulated emission is also absorbed by atoms in the lower level. Net gain needs more atoms in the upper level than in the lower one. Even if you start with such an inversion, without a pump the emission soon uses it up, and the medium goes back to absorbing more than it emits.

Prepare 80 emitters, shine a light field on them, and count absorption, spontaneous emission and stimulated emission one event at a time. Compare one random run with the exact prediction for the ensemble.

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 $\text{—}$

The emitters are two-level systems with equal degeneracy and fast dephasing. The spontaneous lifetime is 1 ms. The external probe stays constant and is never depleted; the photon exchange count records what the emitters give to or take from that probe. The model does not solve for a cavity, coherent Rabi dynamics or the phase of the field. Each count is a physical event in the model; the animated 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 happen while the incident light is losing photons overall?
One-sentence intuition: Whether the light gains or loses depends on the difference between the upper and lower populations. Without a pump, spontaneous and stimulated emission use up any inversion you prepare.

Core mathematical model

Three event rates

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

Each rate counts events in the whole ensemble per millisecond. The spontaneous coefficient is one per millisecond. The driving rate is per emitter and includes the Lorentzian detuning factor.

Detuning weakens the driven processes

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

Detuning is measured in units of the frequency half-width at half maximum. At one half-width, the absorption and stimulated rates fall to half for the same populations. The intrinsic spontaneous lifetime does not change.

Exact mean for a fixed number of emitters

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

There are always 80 emitters in total. The dashed curve is the exact ensemble mean; the solid curve is one direct Gillespie random run. With a finite sample, the solid curve can stray from the dashed one.

Event counts conserve the emitters

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

Each event is counted once and moves one emitter. The net photon exchange with the driving mode is the stimulated count minus the absorption count; spontaneous photons are counted separately. The external probe is held fixed, so its strength does not respond to the photons it gains or loses.

Common difficulties

Stimulated emission is not a collision

Typical misconceptionA photon hits the atom like a ball and knocks out a copy of itself.

Better mental modelThe animation only marks an event in a rate model. Stimulated emission adds a photon to the driving optical mode. Absolute optical phase and coherent atomic amplitudes are not part of this model.

An average is not a single run

Typical misconceptionA random trace is wrong whenever it strays from the exact curve.

Better mental modelThe mean describes the average over many runs. Every single run must conserve the emitters, but agreement with the mean can only be tested statistically, across many seeds.

Run the experiment

01
Turn off the light field
Choose Field off. Step or advance, and watch the upper population and the spontaneous-event count.
What to observe: Excited emitters still decay with no incident field. The absorption and stimulated counts stay at zero.
02
Start at transparency
Choose Equal populations before running. Compare the absorption and stimulated rates at that instant, then step.
What to observe: The two driven rates start out equal. Spontaneous decay then breaks the balance, and the ensemble absorbs more than it emits, yet stimulated events keep happening.
03
Use up a prepared inversion
Choose Inverted sample and run. Watch the upper population, the signed photon exchange with the mode, and how the two driven rates compete later on.
What to observe: Without a pump, the initial gain does not last. The population settles toward a steady mean with no inversion.
04
Look at many runs, not one lucky trace
Repeat with the same settings and different random seeds. Also compare running on resonance with a detuning of one half-width.
What to observe: The event counts change from run to run, but the exact mean comes out the same every time. Detuning changes the driven rates, and spontaneous decay still happens.

Three Light–Matter Transitions

Count both sides of the competition

Core mathematical model

Three event rates

Detuning weakens the driven processes

Exact mean for a fixed number of emitters

Event counts conserve the emitters

Common difficulties

Stimulated emission is not a collision

An average is not a single run

Run the experiment

Turn off the light field

Start at transparency

Use up a prepared inversion

Look at many runs, not one lucky trace

Related experiments and articles

Events and their average