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

Hybrid systems where a quantum material strongly couples to a nonlocal cavity photon mode have emerged as a new frontier for controlling and probing quantum correlations, yet the structure and scaling of light-matter entanglement produced by the nonlocal coupling remains poorly understood. We address this problem through an exactly solvable framework based on reinterpreting the Power–Zienau–Woolley (PZW) transformation as a $\mathit{\text{light-matter}}\mathit{\text{quantum}}\mathit{\text{circuit}}$ that couples the photonic position quadrature $X\sim a+a^{†}$ to the many-body dipole $\mathcal{P}$ of a one-dimensional quantum chain. We derive a closed-form expression for the reduced density matrix valid at all coupling strengths, in which off-diagonal elements between matter states of unequal dipole are suppressed by a Gaussian factor encoding the full weak-to-ultrastrong coupling crossover. At weak coupling, the reduced density matrix takes a Lindbladian form with $\mathcal{P}$ as the jump operator, and the entanglement entropy is controlled by the dipole variance. At ultrastrong coupling, the density matrix becomes exactly block-diagonal in dipole sectors, reflecting an $\mathit{\text{emergent}}\mathit{\text{dipole}}\mathit{\text{symmetry}}$ dynamically imposed by the photon field, with entanglement entropy given exactly by the Shannon entropy of the dipole-sector weight distribution. Applying this framework to a half-filled Su–Schrieffer–Heeger chain, we show that, at strong coupling, both the light-matter entanglement and the spatial entanglement of the photon-dressed matter state scale logarithmically with system size, $S_{\mathrm{\infty }}\sim \frac{\alpha }{2}\log L$, robust across the SSH phase diagram. The logarithm originates from the photon resolving a single collective coordinate $\mathcal{P}$ whose fluctuations grow as $L^{\alpha \mathrm{/}2}$, a distinct mechanism from the logarithmic entanglement of critical one-dimensional systems.