Electron dynamics induced by quantum cat-state light

We present an effective theory for describing electron dynamics driven by an optical external field in a Schrödinger's cat state. We show that the reduced electron density matrix evolves as an average over trajectories $\{ρ_α\}$ weighted by the Sudarshan--Glauber $P$ distribution $P(α)$ in the weak light--matter coupling regime. Each trajectory obeys an equation of motion, $\mathrm{i} \partial_tρ_α=\mathcal{H}_α ρ_α-ρ_α\mathcal{H}_α$, where an effective Hamiltonian $\mathcal{H}_α$ becomes non-Hermitian due to quantum interference of light. The optical quantum interference is transferred to electrons through the asymmetric action between the ket and bra state vectors in $ρ_α$. This non-Hermitian dynamics differs from the conventional one observed in open quantum systems, described by $\mathrm{i} \partial_tρ=\mathcal{H}ρ-ρ\mathcal{H}^\dagger$, which has complex conjugation in the second term. We confirm that the reduced, trajectory-resolved effective theory agrees with full electron-photon simulations for the few-electron Dicke model, thereby validating the interferential non-Hermitian description in the weak-coupling regime.