  We report experimental and theoretical results on the extremely large Lamb
shift in a multimode circuit quantum electrodynamics (QED) system in the
deep-strong coupling (DSC) regime, where the qubit-resonator coupling strength
is comparable to or larger than the qubit and resonator frequencies. The system
comprises a superconducting flux qubit (FQ) and a quarter-wavelength coplanar
waveguide resonator ($\lambda/4$ CPWR) that are coupled inductively through a
shared edge that contains a Josephson junction to achieve the DSC regime.
Spectroscopy is performed around the frequency of the fundamental mode of the
CPWR, and the spectrum is fitted by the single-mode quantum Rabi Hamiltonian to
obtain the system parameters. Since the qubit is also coupled to a large number
of higher modes in the resonator, the single-mode fitting does not provide the
bare qubit energy but a value that incorporates the renormalization from all
the other modes. We derive theoretical formulas for the Lamb shift in the
multimode resonator system. As shown in previous studies, there is a cut-off
frequency $\omega_{\rm{cutoff}}$ for the coupling between the FQ and the modes
in the CPWR, where the coupling grows as $\sqrt{\omega_n}$ for
$\omega_n/\omega_{\rm{cutoff}}\ll 1$ and decreases as $1/\sqrt{\omega_n}$ for
$\omega_n/\omega_{\rm{cutoff}}\gg 1$. Here $\omega_n$ is the frequency of the
$n$th mode. The cut-off effect occurs because the qubit acts as an obstacle for
the current in the resonator, which suppresses the current of the modes above
$\omega_{\rm{cutoff}}$ at the location of the qubit and results in a reduced
coupling strength. Using our observed spectrum and theoretical formulas, we
estimate that the Lamb shift from the fundamental mode is 82.3\% and the total
Lamb shift from all the modes is 96.5\%.
