Transition rate for electrons between two energy level

Question

I am considering spontaneous emission between two energy eigenstates, $e \to g$, at finite temperature. I would imagine that the rate of transition depends on the occupancy of the initial and final states. For transition to happen, $e$ should be filled and $g$ should be empty? Since electrons are fermions, the maximum occupancy is 1. I would like to know if the transition rate is proportional to $$ n_e (1- n_g) $$

Answer 1

In short, the probabilities are linear. Check out the Einstein Coefficients , which define the probabilities of 'up' and 'down' transitions as well as stimulated absorption & emission. Quoting from that,

Spontaneous emission is the process by which an electron "spontaneously" (i.e. without any outside influence) decays from a higher energy level to a lower one. The process is described by the Einstein coefficient $A_{21}$ ($s^{−1}$) , which gives the probability per unit time that an electron in state 2 with energy $E_{2}$ will decay spontaneously to state 1 with energy $E_{1}$, emitting a photon with an energy E2 − E1 = hν. Due to the energy-time uncertainty principle, the transition actually produces photons within a narrow range of frequencies called the spectral linewidth. If $n_{i}$ is the number density of atoms in state i , then the change in the number density of atoms in state 2 per unit time due to spontaneous emission will be

As a side note, for some materials there's an event called "superfluorescence," where the upper level is highly loaded, leading to a stimulated but not fully coherent emission pulse.