# Logarithmic Entanglement and Emergent Dipole Symmetry from a Strongly Coupled Light-Matter Quantum Circuit

**Authors:** Luiz H. Santos  
**arXiv:** [2604.18670](https://arxiv.org/abs/2604.18670)  
**Date:** 2026-04-20  
**Summary page:** https://www.summarizepaper.com/en/paper/2604.18670v1/

## Key points

- Study focuses on the coupling of a matter system to a nonlocal cavity photon mode
- Emergence of new frontier in controlling and probing structure and scaling of the system
- Closed-form expression for reduced density matrix derived, valid across all coupling strengths
- Weak coupling results in Lindbladian form with dipole as jump operator; entanglement entropy influenced by dipole variance
- Ultrastrong coupling leads to block-diagonal density matrix in dipole sectors due to emergent dipole symmetry enforced by photon field; entanglement entropy determined by Shannon entropy of dipole-sector weight distribution
- Application to Su-Schrieffer-Heeger chain shows logarithmic scaling of entanglement measures with system size under strong coupling conditions
- Logarithmic scaling attributed to resolution of single collective coordinate by photons with fluctuations growing proportionally to system size raised to an exponent

## Summary

In this study, we focus on  where a  strongly couples to a nonlocal cavity photon mode. This leads to the emergence of a new frontier in controlling and probing 

The structure and scaling of , resulting from this nonlocal coupling, have been a subject of limited understanding until now. Our circuit couples the photonic position quadrature to the many-body dipole of a one-dimensional quantum chain. Through our analysis, we have derived a closed-form expression for the reduced density matrix that remains valid across all coupling strengths. We observed that off-diagonal elements between matter states with varying dipoles are suppressed by a Gaussian factor, encapsulating the entire weak-to-ultrastrong coupling crossover. At weak coupling, the reduced density matrix takes on a Lindbladian form with the dipole as the jump operator. The entanglement entropy is influenced by dipole variance. In contrast, at ultrastrong coupling, we found that the density matrix becomes precisely block-diagonal in dipole sectors due to an emergent dipole symmetry dynamically enforced by the photon field. The entanglement entropy in this regime is determined by the Shannon entropy of the dipole-sector weight distribution. Applying this framework to a half-filled Su-Schrieffer-Heeger chain, we demonstrated that under strong coupling conditions, both  of photon-dressed matter states scale logarithmically with system size. This logarithmic scaling behavior was attributed to the resolution of a single collective coordinate by photons whose fluctuations grow proportionally to system size raised to an exponent. This mechanism stands distinct from other instances of logarithmic entanglement observed in critical one-dimensional systems. Overall, these findings shed light on how  impact spatial entanglement scaling within matter sectors. They provide valuable insights into understanding and manipulating  in hybrid systems.