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
(3N+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.
We develop a nonperturbative framework for generic nonintegrable many-body systems that reorganizes the expansion of diagonal Green's functions. Starting from exact projection identities and the spectral representation of the resolvent, we derive a recursive hierarchy for the self-energy in which cross-correlated propagation processes are systematically rewritten in terms of diagonal resolvents. Under a diagonal closure approximation, the hierarchy becomes formally closed yet remains systematically improvable.
The framework combines two nonperturbative mechanisms. First, a Lanczos continued-fraction representation provides a recursive single-resolvent structure that naturally produces non-Lorentzian spectral features beyond self-consistent Born approximations. Second, an exact projected multi-resolvent hierarchy introduces nonlocal frequency couplings through products of resolvents and their Hilbert transforms. These contributions mix parity sectors under energy reflection and generate spectral skewness, which is absent in single-resolvent closures.
To solve the resulting equations, we employ a hierarchy of Lorentzian, Gaussian, and Voigt-type ansätze together with an effective Faddeeva self-energy representation ensuring analyticity and causality. Spectral broadening, distribution tails, and higher-order fluctuations emerge from the interplay between continued-fraction recursion and multi-resolvent correlations.
The framework requires no small expansion parameters or diagrammatic truncations, relying instead on ETH-type statistical assumptions appropriate for dense chaotic spectra. It provides a unified route from microscopic interactions to emergent spectral structure, revealing a progression from single-pole self-consistent dynamics to continued-fraction renormalization and finally to multi-resolvent interference effects.
A generalized model of the continuously generating superradiant Raman laser is considered. The model includes competing individual and cooperative atomic processes arising from the interaction of atoms with external light fields. We show that proper selection of the model parameters leads to cooperative effects in the atomic active laser medium. We calculate the steady state values of the atomic polarization and two-atom correlations. Using the quantum regression theorem, we determine the spectral characteristics of the collectively emitted light. We use cumulant expansion to estimate the higher-order correlations between atoms. Keywords: superradiant laser, correlated states of atoms, cooperative effects, cumulant expansion.