A universal gauge structure in the photon content of light-matter systems

After truncating the matter Hilbert space of a single-mode light-matter model, gauge transformations leave the spectrum invariant whereas the ground-state photon number is gauge dependent, hence, its meaning remains disputed. We show that this gauge dependence obeys an exact algebraic law whenever the gauge transformation acts as a displacement of the field coupled to a matter operator $G$. Namely, the ground-state photon number is a quadratic polynomial in the gauge parameter $λ$, $\langle a^{\dagger} a\rangle_λ$ $=$ $\langle a^{\dagger} a\rangle_{0}$ $+$ $Cλ$ $+$ $Qλ^2$. Here the coefficients are the correlator $C=\langle G\,p\rangle$ of $G$ and the field momentum $p$, and $Q=\langle G^2\rangle$. Its minimum over all gauges is naturally gauge-invariant, and for the quantum Rabi model it is determined by the ground-state energy alone. We show that it is strictly positive for the quantum Rabi and Dicke models with level splitting, so that a residual photon content survives in every gauge. At weak coupling the gauge of minimal photon number is the Jaynes-Cummings gauge, and for the Dicke model it remains so at all couplings in the thermodynamic limit.