From Few to Many Emitters Cavity QED: Energy Levels and Emission Spectra From Weak to Deep-Strong Coupling

TL;DR

The paper addresses how a collection of $N$ identical two-level emitters coupled to a single cavity mode behaves across coupling regimes from weak to deep-strong. It develops gauge-invariant Hamiltonians in Coulomb and multipolar gauges, includes the self-polarization term, and analyzes both finite-$N$ and thermodynamic-limit behavior, using a gauge-invariant master equation to study incoherent pumping and emission. A key finding is that with local reservoirs there is a pronounced emission peak at the cavity frequency for even $N$, while increasing $N$ smoothly transitions the system toward the Hopfield bosonic model, erasing higher-order nonlinearities. The results bridge few-emitter quantum optics and many-body cavity QED, with implications for USC/DSC experiments and quantum technologies.

Abstract

We present a systematic study of the properties of systems composed of $N$ two-level quantum emitters coupled to a single cavity mode, for light-matter interaction strengths ranging from the weak to the ultrastrong and deep-strong coupling regimes. Beginning with an analysis of the energy spectrum as a function of the light-matter coupling strength, we examine systems with varying numbers of emitters, from a pair to large collections, approaching the thermodynamic limit ($N \to \infty$). Additionally, we explore the emission properties of these systems under incoherent excitation of the emitters, employing a general theoretical framework for open cavity-QED systems, which is valid across all light-matter interaction regimes and preserves gauge invariance within truncated Hilbert spaces. Furthermore, we study the influence of the emitter-environment interaction on the spectral properties of the system. Specifically, when each emitter interacts independently with its own reservoir, we observe the emergence of an emission peak at the cavity's resonant frequency for even values of $N$. Our analysis also clarify the evolution of the system as the number of emitters increases, ultimately converging towards an equivalent system composed of two interacting single-mode bosonic fields.

Figures (14)

• Figure 1: (a) Comparison between the lowest energy levels of the QRM (blue continuous line) and those of the JC model, as a function of the normalized coupling strength. (b,c) Comparison between the lowest energy levels of the generalized Dicke Hamiltonian in Eq. (\ref{['eq:H_D_Dicke']}) (blue continuous line) and those of the TC model (red dashed lines), as a function of the normalized coupling strength for $N=2$ (b) and N=3 (c) TLSs.
• Figure 2: Energy states, and their corresponding labeling, of the generalized Dicke Hamiltonian in Eq. (\ref{['eq:H_D_Dicke']}) for $N=2$ TLSs, as a function of the normalized coupling strength $\lambda \in \{0,1.5\}$. The energy levels are labeled using the notation $|\tilde{c}, \tilde{k} \rangle$, where $\tilde{c}$ is the excitation number of the multiplet at $\lambda = 0$ and $\tilde{k}$ indicates the levels in ascending order of energy within each multiplet. Two levels, $|{\tilde{1},\tilde{3}}\rangle$ and $|{\tilde{3},\tilde{1}}\rangle$, are highlighted in red to evidence an avoided level crossing occurring at approximately $\lambda \sim 0.65$. This anti-crossing is characterized by the formation of hybridized superposition states, denoted as $|{\Psi_{\tilde{c}\tilde{k};\tilde{c'}\tilde{k'}}^{\pm}}\rangle$. At the anti-crossing, the energy levels exchange character, and thus, we effectively swap the labels of the two states.
• Figure 3: Comparison between the lowest energy eigenvalues of the generalized Dicke Hamiltonian in Eq. (\ref{['eq:H_D_Dicke']}) (blue solid lines) and those of the standard Dicke Hamiltonian, without the self-energy polarization term (red dashed lines). The energy levels are shown as a function of the normalized coupling strength on a logarithmic scale for $N=1,2,3$. The energy levels are displayed considering the ground state energy as reference.
• Figure 4: (a-f) Lowest energy eigenvalues of the generalized Dicke Hamiltonian in Eq. (\ref{['eq:H_D_Dicke']}) for systems composed by $N=1$ up to $N=6$ TLSs, varying the normalized coupling strength $\lambda \in \{0,3\}$. All the energy levels are taken with the ground state energy as reference. Increasing the number of quantum emitters, the number of energy levels rises exponentially as $2^N$. Multi-dipole systems present avoided level crossings between energy levels corresponding to states with the same parity, highlighted in green for N=2 (b), and in orange and red for $N=3$ (c). These avoided crossings become narrower when the number of TLSs increases. (f) The red-highlighted energy levels correspond to the Dicke states belonging to the maximum total angular momentum manifold, $j=N/2$.
• Figure 5: Lowest energy eigenvalues of the generalized Dicke Hamiltonian belonging to the maximum angular momentum manifold (blue solid lines) for $N=10$ (a), $N=50$ (b), and $N=100$ (c), as a function of the normalized coupling strength. For comparison, the corresponding energy levels of the Hopfield model are also displayed (orange solid lines).