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Ultrastrong coupling between a cavity resonator and the cyclotron transition of a 2D electron gas in the case of integer filling factor

We investigate theoretically the coupling between a cavity resonator and the cyclotron transition of a two dimensional electron gas under an applied perpendicular magnetic field. We derive and diagonalize an effective quantum Hamiltonian describing the magnetopolariton excitations of the two dimensional electron gas for the case of integer filling factors. The limits of validity of the present approach are critically discussed. The dimensionless vacuum Rabi frequency $Ω_0/ω_0$ (i.e., normalized to the cyclotron frequency $ω_0$) is shown to scale as $\sqrt{α\: n_{QW} ν}$, where $α$ is the fine structure constant, $n_{QW}$ is the number of quantum wells and $ν$ is the filling factor in each well. We show that with realistic parameters of a high-mobility semiconductor two dimensional electron gas, the dimensionless coupling $Ω_0/ω_0$ can be much larger than 1 in the case of $ν\gg 1$, the latter condition being typically realized for cyclotron transitions in the microwave range. Implications of such ultrastrong coupling regime are discussed.