Quantum mechanics is needed before the ionization of the electron to properly describe a laser.
See "Introduction to Quantum Features of Laser Physics" (Phys. Scr. 1986) Stig Stenholm for a single page recap of the historical description of lasing, and a short list of reference articles.
For an explanation that is as classical as possible see: "From a quantum to a classical description of intense laser–atom physics with Bohmian trajectories" (20 Nov 2009), by Lai, Cai and Zhan:
In this paper, Bohmian mechanics is applied to intense laser–atom physics. The motion of an atomic electron in an intense laser field is obtained from the Bohm–Newton equation. We find that the quantum potential that dominates the quantum effect of a physical system becomes negligible as the electron is driven far from the parent ion by the intense laser field, i.e. the behavior of the electron smoothly tends towards classical soon after the electron is ionized. Our numerical calculations present direct positive evidence for semiclassical trajectory methods in intense laser–atom physics where the motion of the ionized electron is treated by classical mechanics, while quantum mechanics is needed before the ionization.
The only difference between the Bohm–Newton equation and the Newton equation is that there is an extra term in the Bohm–Newton equation called the quantum potential. When the quantum potential is negligible, the Bohm–Newton equation will reduce to the standard Newton equation and then the motion of the particles can be described by classical mechanics.
See also our question: Laser beam in terms of maxwell's equations