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arXiv:2312.03833 · Physical Review A

Integrability, multifractality, and two-photon dynamics in disordered Tavis-Cummings models

Agnieszka Wierzchucka, Francesco Piazza, Pieter W. Claeys

Published Mar 19, 2024• 8 citations
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Abstract

The Tavis-Cummings model is a paradigmatic central-mode model in which a set of two-level quantum emitters (spins) are coupled to a collective cavity mode. Here we study the eigenstate spectrum, its localization properties, and the effect on dynamics, focusing on the two-excitation sector relevant for nonlinear photonics. These models admit two sources of disorder: in the coupling between the spins and the cavity, and in the energy shifts of the individual spins. While this model was known to be exactly solvable in the limit of a homogeneous coupling and inhomogeneous energy shifts, we establish here the solvability in the opposite limit of a homogeneous energy shift and inhomogeneous coupling, presenting the exact solution and corresponding conserved quantities. We identify three different classes of eigenstates, exhibiting different degrees of multifractality and semilocalization closely tied to the integrable points, and we study their stability to perturbations away from these solvable points. The dynamics of the cavity occupation number away from equilibrium, exhibiting boson bunching and a two-photon blockade, is explicitly related to the localization properties of the eigenstates, and it illustrates how these models support a collective spin description despite the presence of disorder. Published by the American Physical Society 2024

DOI
10.1103/physreva.109.033716

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The Tavis-Cummings model is a paradigmatic central-mode model in which a set of two-level quantum emitters (spins) are coupled to a collective cavity mode. Here we study the eigenstate spectrum, its localization properties, and the effect on dynamics, focusing on the two-excitation sector relevant for nonlinear photonics. These models admit two sources of disorder: in the coupling between the spins and the cavity, and in the energy shifts of the individual spins. While this model was known to be exactly solvable in the limit of a homogeneous coupling and inhomogeneous energy shifts, we establish here the solvability in the opposite limit of a homogeneous energy shift and inhomogeneous coupling, presenting the exact solution and corresponding conserved quantities. We identify three different classes of eigenstates, exhibiting different degrees of multifractality and semilocalization closely tied to the integrable points, and we study their stability to perturbations away from these solvable points. The dynamics of the cavity occupation number away from equilibrium, exhibiting boson bunching and a two-photon blockade, is explicitly related to the localization properties of the eigenstates, and it illustrates how these models support a collective spin description despite the presence of disorder. Published by the American Physical Society 2024
Confined electromagnetic modes strongly couple to collective excitations in ensembles of quantum emitters, producing light–matter hybrid states known as polaritons. Under such conditions, the discrete multilevel spectrum of molecular systems offers an appealing playground for exploring multiphoton processes. This work contrasts predictions from the Tavis–Cummings model in which the material is a collection of two-level systems, with the implications of considering additional energy levels with harmonic and anharmonic structures. We discuss the exact eigenspectrum, up to the second excitation manifold, of an arbitrary number N of oscillators collectively coupled to a single cavity mode in the rotating-wave approximation. Elaborating on our group-theoretic approach [New J. Phys. 23, 063081 (2021)], we simplify the brute-force diagonalization of N2 × N2 Hamiltonians to the eigendecomposition of, at most, 4 × 4 matrices for arbitrary N. We thoroughly discuss the eigenstates and the consequences of weak and strong anharmonicities. Furthermore, we find resonant conditions between bipolaritons and anharmonic transitions where two-photon absorption can be enhanced. Finally, we conclude that energy shifts in the polaritonic states induced by anharmonicities become negligible for large N. Thus, calculations with a single or few quantum emitters qualitatively fail to represent the nonlinear optical response of the collective strong coupling regime. Our work highlights the rich physics of multilevel anharmonic systems coupled to cavities absent in standard models of quantum optics. We also provide concise tabulated expressions for eigenfrequencies and transition amplitudes, which should serve as a reference for future spectroscopic studies of molecular polaritons.
Abstract The interaction between anharmonic quantum emitters (e.g. molecular vibrations) and confined electromagnetic fields gives rise to quantum states with optical and chemical properties that are different from those of their precursors. The exploration of these properties has been typically constrained to the first excitation manifold, the harmonic approximation, ensembles of two-level systems [Tavis–Cummings (TC) model], or the anharmonic single-molecule case. The present work studies, for the first time, a collective ensemble of identical multi-level anharmonic emitters and their dipolar interaction with a photonic cavity mode, which is an exactly solvable many-body problem. The permutational properties of the system allow identifying symmetry classified submanifolds in the energy spectrum. Notably, in this approach, the number of particles, typically in the order of several millions, becomes only a parameter from the operational standpoint, and the size of the dimension of the matrices to diagonalize is independent of it. The formalism capabilities are illustrated by showing the energy spectrum structure, up to the third excitation manifold, and the calculation of the photon contents as a permutationally invariant quantity. Emphasis is placed on (a) the collective (superradiant) scalings of light–matter couplings and the various submanifolds of dark (subradiant) states with no counterpart in the single-molecule case, as well as (b) the delocalized modes containing more than one excitation per molecule with no equivalent in the TC model. We expect these findings to be applicable in the study of non-linear spectroscopy and chemistry of polaritons.
We present a comprehensive theoretical and computational study of the Tavis–Cummings model in the collective-spin basis, addressing both its static spectral properties and dynamic behavior under unitary and open-system evolution. By exploiting the commutation of the total-excitation operator with the Hamiltonian, we construct block-diagonal representations for systems of two-level molecules coupled to a single quantized cavity mode. We derive analytic expressions for the degeneracies, block dimensions, and secular polynomials, introducing a family of orthogonal “Félix–Sanz-Vicario” new class of orthogonal polynomials (here baptized Félix-Sanz-Vicario polynomials) whose roots furnish the energy eigenvalues and whose finite-degree structure allows precise control over photon-number truncation in small manifolds. Numerical implementations in Python (QuTiP) confirm exact agreement with the original Tavis–Cummings solution in each excitation manifold, while also highlighting the role of lower-spin subspaces—absent in the bosonized highest-spin description—in mediating configurational molecular interactions via a rank-1 spherical tensor operator. Time-dependent simulations compare dynamics under the rotating wave aproximation (Tavis-Cummings model) and without it (Dicke model), demonstrating population transfer, Rabi oscillations, and collapse–revival phenomena. The dynamics of open quantum system, via a Lindblad master equation, illustrate cavity-loss effects on photon-number decay, von Neumann entanglement entropy, and coherent exchange under dissipation. Our results establish that the collective-spin framework, using a set of solutions in terms of orthogonal polynomials, provides both analytical insight and computational efficiency for spectroscopic modeling, ultrastrong coupling studies, and the design of low-photon-number quantum simulations.
We analyze the spectral and transport properties of the interacting disordered Tavis-Cummings model at half excitation filling. We demonstrate that a Poissonian-level statistics coexists with eigenfunctions that are multifractal (extended, but nonergodic) in the Hilbert space, for all strengths of light-matter interactions. This is associated with a lack of thermalization for a local perturbation. We find that the bipartite entanglement entropy grows logarithmically with time, similarly to many-body localized systems, while the spin imbalance tends to zero for strong coupling, in analogy to ergodic phases. We show that these effects are due to the combination of finite interactions and integrability of the model. When a small integrability-breaking perturbation (nearest-neighbor hopping) is introduced, typical eigenfunctions become ergodic, seemingly turning the system into a near-perfect conductor, contrary to the single-excitation noninteracting case. We propose a realization of this model with cold atoms.
The interaction of an ensemble of NNN two-level quantum emitters with a single-mode electromagnetic field is described by the Tavis-Cummings model. There, the collectively enhanced light-matter coupling strength is given by gN=Ng‾1{g}_{N}=\sqrt{N}{\overline{g}}_{1}gN​=N​g​1​, where g‾1{\overline{g}}_{1}g​1​ is the ensemble-averaged single-emitter coupling strength. This model has been employed to describe and to analyze numerous cavity-based experiments. Here, we show that this is only justified if the effective scattering rate into noncavity modes is negligible compared to the cavity's free-spectral range. In terms of experimental parameters, this requires that the optical depth of the ensemble is low, a condition that is violated in several state-of-the-art experiments. We give quantitative conditions for the validity of the Tavis-Cummings model and derive a more general Hamiltonian description that takes into account the cascaded interaction of the photons with all consecutive emitters. We show that the predictions of our cascaded model can differ quantitatively and even qualitatively from those obtained with the Tavis-Cummings model. Finally, we present experimental data, for which the deviation from the predictions of the Tavis-Cummings model is apparent. Our findings are relevant for all experiments in which optically dense ensembles of quantum emitters are coupled to an optical resonator.
N\mathcal{N}N emitters collectively coupled to a quantized cavity mode are described by the Tavis-Cummings model. We present a complete analytical solution of the model in the presence of inhomogeneous couplings and energetic disorder. We derive the exact expressions for the bright and the dark sectors that decouple the disordered model and find that, in the thermodynamic limit, the energetic disorder transforms the bright sector to Fano's model that can be easily solved. We thoroughly explore the effects of energetic disorder assuming a Gaussian distribution of emitter transition energies. We compare the Fano resonances in optical absorption and inelastic electron scattering in both the weak and the strong coupling regimes. We study the evolution of the optical absorption with an increase in the disorder strength and find that it changes the lower and upper polaritons to their broadened resonances that finally transform to a single resonance at the bare cavity photon energy, thus taking the system from the strong to the weak coupling regime. Interestingly, we learn that Rabi splitting can exist even in the weak coupling regime, while the polaritonic peaks in the strong coupling regime can represent almost excitonic states at intermediate disorder strengths. We also calculate the photon Green's function to see the effect of cavity leakage and nonradiative emitter losses and find that the polariton line width exhibits a minimum as a function of detuning when the cavity leakage is comparable to the Fano broadening.