The Einstein rate equations govern the absorption and emission of photons from a two-level system of energy levels.

It states that the rate of absorption of photons is given by $\textrm{N}_{1}\rho\left(\omega_{12}\right)\textrm{B}_{12}$, the rate of stimulated emission is $\textrm{N}_{2}\rho\left(\omega_{12}\right)\textrm{B}_{21}$ and the rate of spontaneous emission is $\textrm{N}_{2}\textrm{A}_{12}$. Here, $\textrm{N}_{1}, \textrm{N}_{2}$ are the number of atoms in the ground and excited state of the two-level system respectively, $\rho\left(\omega_{12}\right)$ is the spectral energy density (the energy per unit volume integrated over time passing through the system) and $\textrm{B}_{12}, \textrm{B}_{21}, \textrm{A}_{12}$ are the Einstein coefficients.

Certain literature (1) (2) state that the light is broadband, but why is this required? What happens if the light is only at the resonant frequency of the transition ($\omega_{12}$)?

  • 1
    $\begingroup$ Light cannot be a delta function at a single frequency. $\endgroup$
    – ProfRob
    Apr 11 '19 at 10:08
  • $\begingroup$ Oh noo then all the perfect plane waves that we studied do not exist :(( $\endgroup$
    – patta
    Apr 11 '19 at 12:32
  • $\begingroup$ @RobJeffries In these cases, does lifetime-broadened light count as broadband? I've always assumed that broadband light is of the order of ~THz. $\endgroup$ Apr 12 '19 at 8:00

The Einstein coefficients are dependent on the photon wavelength, so you send broadband light and look at the absorption at each frequency. You can well send a narrow single frequency and check how the system absorb; then you try for all other frequencies and you get an absorption spectrum. When you hit the resonant frequency, you'll see absorption.

In detail, each photon has its (approximate) frequency, thus its absorption/emission probability. Broadband light is a collection of many different photons, so you can check all different frequencies at once.

Another reason to use broadband is hystorical, as Einstein needed to explain why atoms absorb some light frequency and not others; and at its time main light sources were the sun and candles, which are broadbands. Almost-single-frequency lasers came much later


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