Einstein Coefficients 2D: Spontaneous vs Stimulated Emission

Photon Bath

Temperature T 3000 K

Transition energy ΔE 1.20 eV

External Pump

Pump rate (units of A₂₁) 0.00

Ensemble size 220 atoms

Level Populations

N₁ (ground) 100%N₂ (excited) 0%

Mean photon number n̄

0.00

N₂ / N₁ ratio

0.00

Spontaneous /s

0.0

Stimulated /s

0.0

Population inversion (N₂>N₁)

Drag the atom cloud to pan, scroll/pinch to zoom. The level diagram (bottom-left) shows absorption, pump, stimulated and spontaneous transition arrows scaled to their real-time rates; the strip chart (bottom-right) traces N₂ population over the last 20 s.

How it works

Each atom has a ground state |1⟩ and an excited state |2⟩ separated by ΔE = hf, immersed in blackbody radiation at temperature T. Einstein (1917) showed that consistency with Planck's law fixes three rates in terms of a single mean photon occupation number of the resonant mode:

n̄(f,T) = 1 / (exp(hf/k_BT) − 1)      [Bose-Einstein occupation]

Absorption rate      (1→2):  W_abs  = B·ρ(f)      = A₂₁ · n̄
Stimulated emission   (2→1):  W_stim = B·ρ(f)      = A₂₁ · n̄     (B₁₂ = B₂₁)
Spontaneous emission (2→1):  W_spon = A₂₁                        (field-independent)

Rates are shown in units of A₂₁ (one "spontaneous lifetime" = 1 time unit). Each atom is simulated individually: at every step a ground atom jumps up with probability 1−exp(−(n̄+pump)·dt), and an excited atom decays with probability 1−exp(−(n̄+1)·dt), split stochastically between the stimulated channel (probability n̄/(n̄+1)) and the spontaneous channel (probability 1/(n̄+1)).

Detailed balance at thermal equilibrium gives the Boltzmann ratio:

N₂/N₁ = n̄/(n̄+1) = exp(−ΔE / k_BT)   →   always < 1

No amount of heating a passive thermal bath can push N₂ above N₁ — the exponential ceiling always favours the ground state. Real lasers escape this with an external pump that excites atoms independently of the thermal field (the Pump slider here); once the pump rate exceeds the thermal + spontaneous decay, N₂ crosses N₁ and population inversion is reached — the precondition for net stimulated (laser) emission.

Ground atom |1⟩ Excited atom |2⟩ Emitted photon

drag to pan · scroll to zoom

Energy levels & transition rates

N₂ population, last 20 s