  Recent experimental advances enable the manipulation of quantum matter by
exploiting the quantum nature of light. However, paradigmatic exactly solvable
models, such as the Dicke, Rabi or Jaynes-Cummings models for quantum-optical
systems, are scarce in the corresponding solid-state, quantum materials
context. Focusing on the long-wavelength limit for the light, here, we provide
such an exactly solvable model given by a tight-binding chain coupled to a
single cavity mode via a quantized version of the Peierls substitution. We show
that perturbative expansions in the light-matter coupling have to be taken with
care and can easily lead to a false superradiant phase. Furthermore, we provide
an analytical expression for the groundstate in the thermodynamic limit, in
which the cavity photons are squeezed by the light-matter coupling. In
addition, we derive analytical expressions for the electronic single-particle
spectral function and optical conductivity. We unveil quantum Floquet
engineering signatures in these dynamical response functions, such as analogs
to dynamical localization and replica side bands, complementing paradigmatic
classical Floquet engineering results. Strikingly, the Drude weight in the
optical conductivity of the electrons is partially suppressed by the presence
of a single cavity mode through an induced electron-electron interaction.
