Exact damping-basis continued fractions for few-atom lasers: collective correlations beyond second-order cumulant closure

Abstract

We combine the Briegel--Englert damping basis of a lossy cavity mode with the permutation-invariant Liouville basis of \(N\) identical, incoherently pumped two-level emitters. The resulting Liouvillian is exactly block tridiagonal in a single radial field index, with atomic blocks of dimension \(\binom{N+3}{3}\). The stationary state follows from a matrix continued fraction whose asymptotic closure is a linear matrix equation that we prove to be uniquely solvable throughout the dissipative regime \(A>0\), and the same construction in the coherence-one sector yields the emission spectrum. The method is validated against direct diagonalization, weak-pump perturbation theory, and closed-system spectra, reaches \(N=16\) in tens of seconds in the few-quanta regime, and has a quantified precision limit at large photon number. We use it to derive an exact two-atom photon-balance identity that isolates the cavity-induced pair coherence and relates it to a singlet--triplet population difference; to establish an exact pump-order hierarchy of connected correlations for a zero-temperature cavity and a parity property of the minimal collective-spin sector; and to benchmark the second-order cumulant closure used in few-emitter laser theory, which we find to be semi-quantitative for the photon number but to predict the wrong sign of the inter-emitter coherence above inversion. In a good-cavity regime the exact solution shows near-Poissonian statistics and strong line narrowing for two and three emitters.