Cavity Spectroscopy for Strongly Correlated Systems

TL;DR

This work develops an all-optical framework to probe strongly correlated materials embedded in optical cavities by connecting emitted photons to the embedded matter's static and dynamic properties. By combining the Input-Output formalism with a Schrieffer-Wolff treatment of a cavity-coupled Hubbard system, the authors derive explicit relations between bath photons and matter observables, and construct effective spin-cavity Hamiltonians that capture photon dressing of exchange and anisotropic interactions. They demonstrate how the cavity occupation and dynamical photon correlators can diagnose entanglement transitions in H$_2$-like dimers and reveal spin dynamics via nonuniform light-matter coupling and LS interactions. The results provide a practical all-optical protocol to access static and dynamic properties of cavity-embedded materials, including selection-rule filtered excitations and doublon-polariton processes, with broad implications for quantum materials and cavity QED spectroscopy.

Abstract

Embedding materials in optical cavities has emerged as an intriguing perspective for controlling quantum materials, but a key challenge lies in measuring properties of the embedded matter. Here, we propose a framework for probing strongly correlated cavity-embedded materials through direct measurements of cavity photons. We derive general relations between photon and matter observables inside the cavity, and show how these can be measured via the emitted photons. As an example, we demonstrate how the entanglement phase transition of an embedded H$_2$ molecule can be accessed by measuring the cavity photon occupation, and showcase how dynamical spin correlation functions can be accessed by measuring dynamical photon correlation functions. Our framework provides an all-optical method to measure static and dynamic properties of cavity-embedded materials.

Figures (6)

• Figure 1: Theoretical setup. An incoming photonic wave-packet $b\textunderscore{in}(t)$ (can also be the vacuum) interacts with the system $H\textunderscore{cav}$ consisting of a material embedded in a cavity, and is scattered into a new state $b\textunderscore{out}(t)$. The input-output formalism provides a relation between $b\textunderscore{in}(t)$ and $b\textunderscore{out}(t)$ in terms of cavity photon correlation functions, which are in turn related to matter correlations.
• Figure 2: Classical analogy for Input-Output formalism. A harmonic oscillator at $x = 0$ interacts with a string at $x > 0$. The string acts as a thermal bath and a driving force for the harmonic oscillator and resembles the vacuum field outside the cavity in the setup discussed in the main text.
• Figure 3: Dimer entanglement transition with finite $\boldsymbol{L \cdot S}$ interactions for $U = 10, \Omega = 0.5, g = 0.5$ and various uniform spin orbit interactions $\boldsymbol{\alpha} = \alpha e^{i \pi / 4} \mathbb{I}_3$ [see \ref{['eq:appdx_sw_hubbard']} for full Hamiltoninan]. (a) Lattice entanglement entropy across the phase transition, with colors denoting different strengths of spin-orbit interaction in comparison with (b) the color matched cavity occupation. We observe excellent agreement and preservation of the sharp transition for weak spin-orbit strength.
• Figure 4: Ground state photon occupation distribution for $U/t = 10$ and $\Omega/t = 0.5$ without spin-orbit coupling at various $g$ with $n\textunderscore c = 120$ the bosonic cutoff. The majority of weight resides in the $\ket{n = 0}$ state, while the occupation in all higher order number states is at least $10^2$ times smaller. For larger $g$, more $n > 0$ states become important, but the separation of scales for $n = 0$ and $n > 0$ is universal.
• Figure 5: Photonic $\boldsymbol{N_2}(\omega)$ correlation function for the Hubbard Plaque at $U = 25, \Omega = 1.1, \boldsymbol{\alpha}_{ij} = 0, g = 0.1$ and $\eta = 10^{-3}$. Vertical gray lines indicate cavity resonances at $n \cdot \Omega$, while the red dashes indicate the selection rule allowed low energy excitation of the matter, which appear as a bare resonance at $\Delta E$ and a polariton-spin excitation at $\Delta E + 1 \cdot \Omega$. (a) Exact electronic frequency space representation of $N_2(\omega)$ (black) and approximate low-energy expression $N\textunderscore{2,eff}(\omega)$ [\ref{['eq:corr_N2_w']}] for the spin model (blue). Panels (b) - (e) show the contribution resolved $N\textunderscore{2,eff}(\omega)$, illustrating the importance of the different terms. Expectation values evaluated in full spin ground state, since SW$_0$ leads to no significant changes.