Spontaneous Emission

Einstein’s Coefficients

For the following

• system of particles in a lower state \(\psi_a\) state is \(N_a\)

• a system of particles in an upper state \(\psi_b\) state is \(N_b\)

with the coefficients,

• \(A\) : spontaneous emission rate out of \(\psi_b\)

• \(B\rho(\omega_0)\) : stimulated emission rate

• \(B_{ab}\rho(\omega_0)\) : rate of particles entering \(b\) from \(a\)

• \(B_{ba}\rho(\omega_0)\) : rate of particles leaving \(b\) to \(a\)

Then the number of particles entering the state \(\psi_b\) is,

\[ \dot N_b = -N_bA - N_bB_{ba}\rho(\omega_0) + N_aB_{ab}\rho(\rho_0) \]

If the system is in thermal equilibrium then \(dN_b/dt = 0\) such that,

\[ B_{ab} = B_{ba} \]

and we can solve for \(\rho(\omega_0)\) to find Plancks’ law,

\[ \rho(\omega) = \frac{A}{e^{\hbar\omega_0/\tau}B_{ab} - B_{ba}}\]

We see that the spontaneous emission rate can be written in terms of the stimulated emission rate,

\[\begin{split}\begin{align} \boxed{A = \frac{\omega_0^3 \hbar}{\pi^2c^3}B_{ba}}\\ \boxed{A = \frac{\omega_0^3 q\abs{r_{ba}}^2}{3\pi\epsilon_0\hbar c^3}} \end{align}\end{split}\]

This is as far as we can go without going into quantum field theory and electrodynamics. Einstein’s coefficient \(A\) is a useful equation such that it says spontaneous emission is similar to a system of stimulated emission off by a factor.

Lifetime and Decay Rate

Consider a system that is out of equilibrium but has a spontaneous emission rate \(A\) out of the state \(\psi_b\). You may treat \(A\) in this case as the decay rate where,

\[\begin{split}\begin{gather} \frac{dN_b}{dt} = -AN_b\\ N_b(t) = N_b(0)e^{-At} \end{gather}\end{split}\]

The period of decay or lifetime is then,

\[\begin{equation} \boxed{\tau = \frac{1}{A}} \end{equation}\]

Given that there is \(n\) states and not just one (\(b\)) then the decay rate simply adds up,

\[\begin{split} \begin{gather} A = \sum_n{A_n}\\ \end{gather} \end{split}\]