Ultrastrong coupling between a cavity resonator and the cyclotron transition of a two-dimensional electron gas in the case of an integer filling factor

David Hagenmüller, Simone De Liberato, and Cristiano Ciuti

Phys. Rev. B 81, 235303 – Published 2 June 2010

Abstract

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 $ν ⪢ 1$ , the latter condition being typically realized for cyclotron transitions in the microwave range. Implications of such ultrastrong coupling regime are discussed.

Received 10 March 2010

DOI:https://doi.org/10.1103/PhysRevB.81.235303

Authors & Affiliations

David Hagenmüller¹, Simone De Liberato^1,2, and Cristiano Ciuti^1,*

¹Laboratoire Matériaux et Phénomènes Quantiques, Université Paris Diderot-Paris 7 and CNRS, Bâtiment Condorcet, 10 rue Alice Domont et Léonie Duquet, 75205 Paris Cedex 13, France
²Department of Physics, University of Tokyo, Hongo, Bunkyo-Ku, Tokyo 113-8656, Japan

^*cristiano.ciuti@univ-paris-diderot.fr

Article Text (Subscription Required)

Click to Expand

References (Subscription Required)

Click to Expand

Issue

Vol. 81, Iss. 23 — 15 June 2010

Reuse & Permissions

Access Options

Author publication services for translation and copyediting assistance advertisement

Images

Figure 1
(Color online) Sketch of a cavity resonator embedding $n_{QW}$ identical QWs, each containing a two-dimensional electron gas (2DEG) in the $x y$ plane. An uniform and static magnetic field $B$ is applied along the $z$ axis.Reuse & Permissions
Figure 2
(Color online) Sketch of Landau levels with an integer filling factor $ν$ ( $ν ⪢ 1$ is the regime considered in this paper). The cyclotron transition between levels $ν - 1$ and $ν$ is resonantly coupled to cavity mode quantum field.Reuse & Permissions
Figure 3
(Color online) The dimensionless vacuum Rabi coupling $Ω_{r e s} / ω_{0}$ versus the filling factor $ν$ for different values of the number $n_{QW}$ of quantum wells. We consider only the photonic mode ${q_{x} = 0, n_{z} = 1}$ at resonance with the transition. Precisely, the resonance is defined such that $ω_{q_{x} = 0, n_{z} = 1} = ω_{0}$ . Other parameters are given in the text.Reuse & Permissions
Figure 4
(Color online) Thick solid colored lines: normalized frequency of the first four magnetopolariton branches as a function of the 2D electron gas density $ρ_{2 DEG}$ in the case where the energy of the first cavity mode ${q_{x} = 0, n_{z} = 1}$ is resonant to the cyclotron transition. Thin black solid lines depict the energies of the photonic branches, which are not coupled to the cyclotron transition [for the selection rules, see description in the text, in particular Eqs. (13, 14)]. The two vertical straight lines indicate the densities $ρ_{2 DEG} = 2 \times 10^{10} {cm}^{- 2}$ and $ρ_{2 DEG} = 2 \times 10^{11} {cm}^{- 2}$ (giving a dimensionless vacuum Rabi coupling $Ω_{r e s} / ω_{0} ≃ 0.8$ and $Ω_{r e s} / ω_{0} ≃ 2.5$ , respectively): the corresponding frequency dispersions as a function of the wave vector $q_{x}$ are plotted in Figs. 5, 6. Parameters: $m^{*} = 0.068 m_{0}$ (GaAs effective mass); $B = 40 mT$ (giving a cyclotron frequency $ω_{0} = 100 GHz rad$ ); $n_{QW} = 50$ and $L_{z} = 0.25 cm$ which is chosen in such a way that $ω_{q_{x} = 0, n_{z} = 1} = ω_{0}$ . With these parameters, a good numerical convergence is obtained for a cutoff number of photonic modes $Λ > 7$ .Reuse & Permissions
Figure 5
(Color online) Thick solid colored lines: normalized frequencies of the first four normal mode branches as a function of the wave vector $q_{x}$ for $ρ_{2 DEG} = 2 \times 10^{10} {cm}^{- 2}$ and $B = 40 mT$ which correspond to a filling factor $ν ≃ 10$ . The dashed color lines represent the dispersions of the photonic modes with odd values of $n_{z}$ that are coupled to the cyclotron transition. The red dashed-dotted line depicts the bare cyclotron frequency; the thin black solid lines represent the photonic modes with even values of $n_{z}$ , which are not coupled to the electronic system. Other parameters are the same as in Fig. 4.Reuse & Permissions
Figure 6
(Color online) Same as Fig. 5 but with an electron density $ρ_{2 DEG} = 2 \times 10^{11} {cm}^{- 2}$ (corresponding to a filling factor $ν ≃ 100$ ).Reuse & Permissions

Physical Review B

covering condensed matter and materials physics