Compact cavity-dressed Hamiltonian framework at arbitrarily strong light-matter coupling

TL;DR

The paper introduces a nonperturbative cavity-dressed Hamiltonian (CDH) framework that uses a polaron-like unitary transformation to entangle light and matter, yielding compact, closed-form representations of strongly coupled quantum systems. By block-truncating each multimode cavity to $M$ levels and evaluating blocks in a momentum representation, CDH achieves rapid convergence across weak, ultrastrong, and deep-strong coupling, outperforming bare truncations. The authors demonstrate the approach on canonical models—the quantum Rabi model and the Dicke–Heisenberg lattice—showing accurate spectra and thermodynamic observables with significantly reduced computational cost, and they extend the method to multimode and dissipative (leaky) cavities. They further map and analyze various Dicke-family models (XX, XXX, and Ising variants), extracting phase diagrams via magnetization and entanglement metrics and highlighting how cavity-induced all-to-all interactions reshape spin correlations. Overall, CDH provides physical insight and computational efficiency for exploring strongly coupled light–matter systems relevant to chemistry, materials science, and quantum technologies.

Abstract

We present a non-perturbative Hamiltonian mapping method for quantum systems strongly coupled to a quantized field mode (cavity), yielding compact closed-form representations of hybrid light-matter systems. The mapping method builds on an entangling transformation of photonic and atomic degrees of freedom. By truncating the resulting cavity-dressed Hamiltonian (CDH) to successively larger excitation sectors, we construct a series of compact models that converge to the exact limit, outpacing conventional approaches even in the challenging resonant and ultrastrong light-matter regime. The mapping principle also applies to multimode cavities coupled to matter through noncommuting operators and to leaky cavities. We benchmark the CDH framework on the quantum Rabi model, demonstrating accurate spectral predictions in both weak and strong coupling regimes, together with converging ground-state and thermal observables. We study the Dicke-Heisenberg lattice model and determine its phase diagram under resonant and strong light-matter coupling, achieving significant computational savings over brute-force simulations and identifying cavity-mediated spin correlations both analytically and numerically. The closed-form and compactness of the CDH provide both physical insight and enhanced computational efficiency, facilitating studies of strongly coupled hybrid light-matter systems.

Figures (10)

• Figure 1: (a) The quantum Rabi model with a spin coupled to a single boson, and (b) the Dicke-Heisenberg model.
• Figure 2: (a)-(c) Eigenenergies of the quantum Rabi model against the coupling strength $\lambda$ for a resonant setting, $2\Delta=\Omega$ using the bare representation with (a) $N=4$ or (b) $N=10$ levels and (c) a CDH of degree $M=4$. We compare results in the bare or CDH representation (full) to exact results Braak2011 (dashed). (d) $\varepsilon_0$ defined as the difference between the zero eigenenergy predicted by the CDH with $M\in[1,4]$ or the bare model with $N\in[1,4]$, and the exact result. (e) Maximal error $\varepsilon_\nu$ over $\lambda\in[0,5]$ with respect to $M$ and $N$ for the first three eigenenergies, $\nu=0,1,2$. (f) Equilibrium magnetization $\langle \hat{\sigma}^z \rangle_{eq}$ at $T=1$ using the CDH with $M\in[1,4]$, compared to brute force converging solution (dashed). Parameters are $\Omega=2$, $\Delta=1$.
• Figure 3: Ground state phase diagrams of the Dicke-XX Heisenberg model (a) in the bare representation with $N=20$, (b) and using the $M=3$ CDH. The order parameter is the average magnetization, $M_z$. (c) Structure factors $S_\alpha$ for a resonance situation, $2\Delta=\Omega$. We compare numerically-converged simulations at $N=20$ (solid) to the CDH with $M=1$ (dashed) and $M=3$ (dotted), with $M_P=M$. Fixed parameters are $L=8$, $\gamma_x=\gamma_y=\Omega/8$, $\gamma_z=0$, $\Omega=2$, and we vary $\lambda$ and $\Delta$.
• Figure 4: Average magnetization, $\langle \hat{\sigma}^z \rangle$, as a function of the normalized coupling strength, $\lambda / \Omega$ (a) at equilibrium, $T=1$, and off resonance $\Omega=10\Delta$, (b) at zero temperature and off resonance $\Omega=10\Delta$, and (c) at zero temperature, in resonance, $\Omega=2\Delta$. Colors indicate CDH results for increasing $M$ values. Converged numerical results from the bare representation appear in dashed. We used $\Delta=1$.
• Figure 5: Average ground state magnetization, $\langle \hat{\sigma}^z \rangle_{GS}$, as a function of the normalized coupling strength, $\lambda / \Omega$, at resonance $\Omega = 2\Delta$. We vary $M_{P}$, the dimensionality of the cavity Hilbert space used to rotate operators with (a) $M_P=5$, (b) $10$, and (c) $20$. Colors indicate CDH results for different $M$. Converged numerical results appear in dashed. We used $\Delta=1$.