  We report on experimentally measured light shifts of superconducting flux
qubits deep-strongly coupled to LC oscillators, where the coupling constants
are comparable to the qubit and oscillator resonance frequencies. By using
two-tone spectroscopy, the energies of the six lowest levels of each circuit
are determined. We find huge Lamb shifts that exceed 90% of the bare qubit
frequencies and inversions of the qubits' ground and excited states when there
are a finite number of photons in the oscillator. Our experimental results
agree with theoretical predictions based on the quantum Rabi model.
