Ultrastrong coupling cavity QED of the magnetic cyclotron transition in a 2D electron gas: massive versus Dirac fermions

We show that the cyclotron transition of a two-dimensional electron gas can be ultrastrongly coupled to a cavity photon mode. The ratio between the vacuum Rabi frequency $\Omega_0$ and the cyclotron frequency $\omega_0$ can be much larger than $1$ for large filling factor $\nu$ of the Landau levels (the normalized coupling $\Omega_0/\omega_0$ scales as $\sqrt{\alpha \, n_{QW} \nu }$, where $\alpha$ is the fine structure constant and $n_{QW}$ is the number of quantum wells). We present a comprehensive cavity QED theory both for semiconductors with massive electrons\cite{hagenmuller1,scalari} and graphene with Dirac fermions\cite{hagenmuller2}. We show the dramatic impact on the quantum ground state and excitation properties, drawing the comparison between the two different types of 2D electron gas. \\[4pt] [1] D. Hagenm\"{u}ller, S. De Liberato, and C. Ciuti, Phys. Rev. B 81, 235303 (2010) and references therein.\\[0pt] [2] Experiments demonstrating ultrastrong coupling of the cyclotron transition in a GaAs-system in the THz regime have been reported, see G. Scalari \textit{et al}., submitted.\\[0pt] [3] D. Hagenm\"{u}ller and C. Ciuti, submitted.