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arXiv:2511.02390 · Physical Review Letters

Symbolic Quantum-Trajectory Method for Multichannel Dicke Superradiance

Raphael Holzinger, Nico S. Bassler, Julian Lyne, Susanne F. Yelin, Claudiu Genes

quant-phSubmitted Nov 4, 2025• Announced Nov 5, 2025• 0 citations• Revised Apr 15, 2026 (v3)
v1 · Nov 4, 2025v2 · Feb 19, 2026v3 · Apr 15, 2026v4 · Aug 27, 2026
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Abstract

We solve Dicke superradiance with two or more competing collective decay channels of tunable rates using a symbolic quantum-trajectory construction. The method yields closed time-domain populations and observables as finite sums of exponentials for arbitrary numbers of emitters and arbitrary decay rates. For two channels, the behavior of the stationary ground-state distribution resembles a first-order phase transition at the point where the channel-rate ratio is equal to unity. For balanced $d$-channel decay, we obtain scaling laws for the superradiant peak time and intensity. These results unify and extend single-channel Dicke dynamics to multilevel emitters and provide a compact tool for cavity and waveguide experiments, where permutation-symmetric reservoirs engineer multiple collective decay paths.

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We solve Dicke superradiance with two or more competing collective decay channels of tunable rates using a symbolic quantum-trajectory construction. The method yields closed time-domain populations and observables as finite sums of exponentials for arbitrary numbers of emitters and arbitrary decay rates. For two channels, the behavior of the stationary ground-state distribution resembles a first-order phase transition at the point where the channel-rate ratio is equal to unity. For balanced ddd-channel decay, we obtain scaling laws for the superradiant peak time and intensity. These results unify and extend single-channel Dicke dynamics to multilevel emitters and provide a compact tool for cavity and waveguide experiments, where permutation-symmetric reservoirs engineer multiple collective decay paths.
We study the dynamics of an ensemble of NNN two-level emitters undergoing collective decay at rate ΓΓΓ and collective incoherent pumping at rate RRR, 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.
Determining the peak photon emission time and rate for an ensemble of NNN quantum systems undergoing collective superradiant decay typically requires tracking the time evolution of the density operator. Generally, the dimension of the density operator grows exponentially (∼ ⁣2N\sim \! 2^N∼2N) with the number of emitters, in the absence of any symmetries such as in Dicke superradiance with full or partial permutational symmetry. We present a detailed study of the superradiant peak emission rate and time for initially fully excited quantum emitter ensembles, for one-, two- and three-dimensional arrays in free-space and emitter chains coupled to waveguide reservoirs. For few emitters (N≲14N\lesssim 14N≲14) we utilize the full quantum master equation, and for mesoscopic emitter numbers (N≲400N\lesssim 400N≲400) we use a second- and third-order cumulant expansion of the operator averages to track the time evolution of the system.
We study a bosonic analog of the paradigmatic Dicke model of superradiance, comprising interacting bosonic modes subject to fully symmetric collective decay. Depending on the interaction strength, we uncover qualitatively distinct regimes of emission. For strong interactions, the emission closely resembles Dicke superradiance, with perturbative corrections arising from the presence of additional levels. For weaker interactions, the bosonic statistics qualitatively changes the dynamics, leading to a crossover to subradiant emission. Remarkably, we show that the dynamics in this regime can be described by rate equations analogous to those of the Dicke model despite the large accessible bosonic Hilbert space. Our findings are based on a combination of analytical arguments and large-scale numerics enabled by the permutational symmetry of the problem and may be probed in circuit QED experiments.
Abstract We revisit the Dicke superradiance problem, where an ensemble of N identical two-level systems undergoes collective spontaneous decay. While an exact analytical solution has been known since 1977, its algebraic complexity has hindered practical use. Here we present a compact, closed-form solution that expresses the dynamics as a finite sum over residues or, equivalently, a complex contour integral. The method yields explicit populations of all Dicke states at arbitrary times and system sizes, and generalizes naturally to arbitrary initial conditions. Our formulation is computationally efficient and offers structural insights into the role of spectral degeneracies and Lindbladian eigenmodes in collective decay.
When multiple quantum emitters couple to a common electromagnetic environment, interference in their collective radiative dynamics gives rise to superradiance and subradiance. In regimes where coherent interactions and collective dissipation compete, the microscopic many-body dynamics and quantum correlations among the emitters that underlie superradiance and subradiance are theoretically challenging and remain experimentally elusive, even though collective emission has been observed in many physical systems. Here, we realize a superconducting qubit array coupled to a common microwave waveguide that mediates collective dissipation, with simultaneous access to coherent interactions and microscopic measurements of many-body dynamics. Engineered qubit-waveguide couplings with tunable amplitude and phase enable control of collective interference and the resulting super- and subradiant states. Leveraging site-resolved control and readout, we directly observe the microscopic decay dynamics of multi-qubit states across different excitation manifolds and track the evolution of populations and tunable quantum correlations. We reveal collective decay in regimes beyond the ideal Dicke model, where strong qubit-qubit interactions stabilize superradiance and subradiance against local dephasing and reshape decay pathways through spatially and spectrally structured many-body eigenstates. Our results establish a flexible platform for exploring collective phenomena in many-body quantum optics and driven-dissipative approaches to robust quantum information processing.
Collective superradiant decay of a tightly packed inverted quantum emitter ensemble is among the most intensely studied phenomena in quantum optics. Since the seminal work of Dicke more than half a century ago, a plethora of theoretical calculations in quantum many-body physics have followed. Widespread experimental efforts range from the microwave to the X-ray regime. Nevertheless, accurate calculations of the time dynamics of the superradiant emission pulse still remain a challenging task requiring approximate methods for large ensembles. Here, we benchmark the cumulant expansion method for describing collective superradiant decay against a newly found exact solution. The application of two variants of the cumulant expansion exhibits reliable convergence of time and magnitude of the maximum emission power with increasing truncation order. The long-term population evolution is only correctly captured for low emitter numbers, where an individual spin-based cumulant expansion proves more reliable than the collective spin-based variant. Surprisingly, odd orders show unphysical behavior. At sufficiently high spin numbers, both chosen cumulant methods agree, but still fail to reliably converge to the numerically exact result. Generally, on longer time scales the expansions substantially overestimate the remaining population. While numerically fast and efficient, cumulant expansion methods need to be treated with sufficient caution when used for long-time evolution or reliably finding steady states.