Exact analytical eigenfunctions for the quantum theory of the laser
Phys. Rev. A 112, 053712 – Published 12 November, 2025
DOI: https://doi.org/10.1103/z2gh-xn45
Abstract
We present an exact solution of the quantum laser master equation using generating functions and confluent Heun equations. By reformulating the evolution of both diagonal and off-diagonal elements of the reduced density matrix, we derive closed-form expressions for the generating functions and identify the full spectrum of decay rates from a two-point connection problem for the confluent Heun equation. This method captures both transient dynamics and steady-state properties without truncation. In the thermodynamic limit of vanishing saturation, the spectrum separates into two analytically tractable eigenvalue families; their intersections mark exceptional points, reflecting nontrivial structure in the system's Liouvillian dynamics. At the lasing threshold, we observe a Liouvillian spectral collapse, signaling a nonequilibrium phase transition. Our results provide a unified, nonperturbative description of laser dynamics across different regimes.
