Exact Analytical Treatment of Collectively Pumped Dicke Superradiance

Abstract

We study the dynamics of an ensemble of $N$ two-level emitters undergoing collective decay at rate $Γ$ and collective incoherent pumping at rate $R$, confined to the fully symmetric Dicke ladder. Although the population dynamics are confined to a finite set of rate equations, their quantum-trajectory representation involves infinitely many pump-decay histories. We derive an exact finite continued fraction for the Laplace-transformed Dicke-state populations by resumming the infinitely many quantum trajectories generated by competing pump and decay processes. Our construction provides a physically transparent interpretation of the tridiagonal resolvent and extends to arbitrary permutationally symmetric initial states. In the strong- and weak-pumping regimes, we obtain perturbative transient solutions. We also identify exact relaxation eigenvalues and emission observables at balanced pump and decay strengths. Our solution provides a starting point for extending trajectory-based analytical methods to more general driven-dissipative collective systems.

The paper studies transient populations of $N$ two-level emitters subject to collective decay at rate $\Gamma$ and collective incoherent pumping at rate $R$, restricted to the fully symmetric $J=N/2$ Dicke ladder. Although populations obey $N+1$ classical birth–death equations, their quantum-trajectory expansion contains infinitely many pump–decay histories. The central construction organizes these histories into nested closed excursions and resums them into finite continued fractions for the Laplace-transformed populations. Equivalent tridiagonal-determinant expressions connect the trajectory construction to the rate-matrix resolvent. An appendix extends the construction from the ground state to arbitrary Dicke-diagonal initial populations. For $R,\Gamma>0$, similarity to a symmetric Jacobi matrix establishes real, nondegenerate eigenvalues, giving at most $N+1$ relaxation modes. At balanced pumping, $R=\Gamma$, Hahn polynomials yield the population eigenvalues $\lambda_j=-\Gamma j(j+1)$, $j=0,\ldots,N$. Detailed balance gives geometric steady-state populations, uniform at balance. There, the emission rate is $\Gamma N(N+2)/6$, $g^{(2)}(0)\to6/5$ for large $N$, and quantum regression gives a Lorentzian spectrum with full width at half maximum $2\Gamma$. Weak pumping from the ground state gives $I(t)\simeq RN[1-e^{-\Gamma Nt}]$; strong pumping reduces at leading order to pure upward Dicke dynamics. For generic unequal rates, the relaxation eigenvalues remain an implicit finite spectral problem rather than a simple closed expression. General-rate spectra require coherence-sector propagation. Coherent driving, local dissipative channels, and more complicated collective systems are proposed extensions, not treated here.