  The central theme of cavity quantum electrodynamics is the coupling of a
single optical mode with a single matter excitation, leading to a doublet of
cavity polaritons which govern the optical properties of the coupled structure.
Especially in the ultrastrong coupling regime, where the ratio of the vacuum
Rabi frequency and the quasi-resonant carrier frequency of light,
$\Omega_{\mathrm R}/\omega_{\mathrm c}$, approaches unity, the polariton
doublet bridges a large spectral bandwidth $2\Omega_{\mathrm R}$, and further
interactions with off-resonant light and matter modes may occur. The resulting
multi-mode coupling has recently attracted attention owing to the additional
degrees of freedom for designing light-matter coupled resonances, despite added
complexity. Here, we experimentally implement a novel strategy to sculpt
ultrastrong multi-mode coupling by tailoring the spatial overlap of multiple
modes of planar metallic THz resonators and the cyclotron resonances of
Landau-quantized two-dimensional electrons, on subwavelength scales. We show
that similarly to the selection rules of classical optics, this allows us to
suppress or enhance certain coupling pathways and to control the number of
light-matter coupled modes, their octave-spanning frequency spectra, and their
response to magnetic tuning. This offers novel pathways for controlling
dissipation, tailoring quantum light sources, nonlinearities, correlations as
well as entanglement in quantum information processing.
