Breakdown of the dipole approximation in strong coupling cavities

Camille Ndebeka-Bandou, Curdin Maissen, Giacomo Scalari, Mattias Beck and Jérôme Faist

Institute for Quantum Electronics, ETH Zurich, Switzerland

The strong light-matter coupling between a cavity mode and an electronic transition is achieved when the vacuum Rabi frequency exceeds the photonic and the electronic mode linewidths. This phenomenon is characterized by the lifting of the degeneracy between the two modes and the creation of two distinct polariton states with a strong light-matter hybridization. This regime has been widely studied in atoms, excitonic systems and quantum electronic circuits based on Josephson junctions. Recently, there is also a strong interest in studying similar effects using terahertz intersubband transitions or cyclotron transitions that exhibit large optical moments. In these systems, the ultra-strong coupling regime can be reached when the vacuum Rabi frequency is a large fraction of the electronic transition frequency [1]. In the present work, we investigate theoretically the scaling of the vacuum Rabi frequency in a terahertz metamaterial where the cyclotron transition of a high-mobility two-dimensional electron gas (2DEG) is strongly coupled to the photonic modes of a split ring resonator [2-4]. In particular, we calculate numerically the optical dipole moment beyond the dipole approximation and study its variation with the relevant parameters of the system, like the filling factor and the cavity length. We find that when the cavity length becomes comparable to the magnetic length of the 2DEG, interesting effects such as a drastic decrease of the dipole moment and the breakdown of the optical selection rule for the cyclotron transitions are observed. This calculation directly leads to the scaling of the coupling strength since this key parameter is proportional to the optical dipole moment.

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