Derive Rate Equations for a Two-Level Atom Including Both Emissions

 

Rate equations describe how the populations of energy levels change with time. Consider a simple two-level system with lower level (1) and upper level (2), whose populations are N1 and N2 respectively. The total number of atoms is constant: N= N1+ N2 .

Populations change through three processes—absorption, spontaneous emission, and stimulated emission— in the presence of a light field of spectral energy density ρ(ν).

The rates for each process are:

• 1. Absorption (1 → 2): increases N2, decreases N1; rate = B12 ρ(ν) N1 .
• 2. Spontaneous Emission (2 → 1): decreases N2, increases N1; rate = A21 N2 .
• 3. Stimulated Emission (2 → 1): decreases N2, increases N1; rate = B21 ρ(ν) N2 .

Rate Equation for the Upper State (N₂)

The rate of change of N2 is the population gain (absorption) minus the losses (spontaneous + stimulated emission):

dN2 dt = B12 ρ(ν) N1 − A21 N2 − B21 ρ(ν) N2

Rate Equation for the Lower State (N₁)

Likewise, the rate of change of N1 equals population gains (spontaneous + stimulated emission) minus loss (absorption):

dN1 dt = A21 N2 + B21 ρ(ν) N2 − B12 ρ(ν) N1

Note: The total population N=N1+N2 is constant, so d(N1+N2) dt =0 , implying dN1 dt =− dN2 dt , confirming consistency.