  The coupling of a two-level system with an electromagnetic field, whose fully
quantized version is the quantum Rabi model, is among the central topics of
quantum physics. When the coupling strength becomes large enough that the field
mode frequency is reached, the deep strong coupling regime is approached, and
excitations can be created from the vacuum. Here we demonstrate a periodic
variant of the quantum Rabi model in which the two-level system is encoded in
the Bloch band structure of cold rubidium atoms in optical potentials. With
this method we achieve a Rabi coupling strength of 6.5 times the field mode
frequency, which is far in the deep strong coupling regime, and observe a
subcycle timescale raise in bosonic field mode excitations. In a measurement
recorded in the basis of the coupling term of the quantum Rabi Hamiltonian, a
freezing of dynamics is revealed for small frequency splittings of the
two-level system, as expected when the coupling term dominates over all other
energy scales, and a revival for larger splittings. Our work demonstrates a
route to realize quantum-engineering applications in yet unexplored parameter
regimes.
