Exact solution to laser rate equations: three-level laser as a Morse-like oscillator

It is shown how the rate equations that model a three-level laser can be cast into a single second-order differential equation, whose form describes a time-dependent harmonic oscillator. Using this result, we demonstrate that the resulting equation can be identified as a Schrodinger equation for a Morse-like potential, thus allowing us to derive exact closed-form expressions for the dynamics of the number of photons inside the laser cavity, as well as the atomic population inversion.