The interaction between light and matter is one of the oldest research areas
of quantum mechanics, and a field that just keeps on delivering new insights
and applications. With the arrival of cavity and circuit quantum
electrodynamics we can now achieve strong light-matter couplings which form the
basis of most implementations of quantum technology. But quantum information
processing also has high demands requiring total error rates of fractions of
percentage in order to be scalable (fault-tolerant) to useful applications.
Since errors can also arise from modelling, this has brought into center stage
one of the key approximations of quantum theory, the Rotating Wave
Approximation (RWA) of the quantum Rabi model, leading to the Jaynes-Cummings
Hamiltonian. While the RWA is often very good and incredibly useful to
understand light-matter interactions, there is also growing experimental
evidence of regimes where it is a bad approximation. Here, we ask and answer a
harder question: for which experimental parameters is the RWA, although perhaps
qualitatively adequate, already not good enough to match the demands of
scalable quantum technology? For example, when is the error at least, and when
at most, 1\%? To answer this, we develop rigorous non-perturbative bounds
taming the RWA.

We find that these bounds not only depend, as expected, on the ratio of the
coupling strength and the oscillator frequency, but also on the average number
of photons in the initial state. This confirms recent experiments on
photon-dressed Bloch-Siegert shifts. We argue that with experiments reporting
controllable cavity states with hundreds of photons and with quantum error
correcting codes exploring more and more of Fock space, this state-dependency
of the RWA is increasingly relevant for the field of quantum computation, and
our results pave the way towards a better understanding of those experiments.

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