Surface plasmons in graphene may provide an attractive alternative to noble-metal plasmons due to their tighter confinement, peculiar dispersion, and longer propagation distance. We present theoretical studies of the nonlinear difference frequency generation of terahertz surface plasmon modes supported by two-dimensional layers of massless Dirac electrons, which includes graphene and surface states in topological insulators. Our results demonstrate strong enhancement of the DFG efficiency near the plasmon resonance and the feasibility of phase-matched nonlinear generation of plasmons over a broad range of frequencies.PACS numbers: 81.05.ue, 42.65.-k Graphene exhibits many interesting electronic properties because of its chiral symmetry and gapless linear spectrum of free carriers near the Dirac point. In recent years, graphene has also been recognized as a promising broadband optoelectronic material in the infrared (IR) and terahertz (THz) region, especially when utilizing a surface plasmon resonance [1][2][3]. Surface plasmon is a collective mode of coupled charge-density and field oscillations at an interface between a free-carrier system and a dielectric or vacuum. Surface plasmons guided by graphene are expected to have low losses and be highly tunable by gating and doping, making graphene an attractive alternative to metal plasmonics. Surface states in certain topological insulators (TIs) have a massless Dirac-cone electron dispersion at low energies with a slope similar to that in graphene. They provide a potentially even more interesting host medium for surface plasmons due to lower scattering rates of two-dimensional (2D) surface electrons that are topologically protected from scattering on non-magnetic impurities [4]. In particular, Bi 2 Se 3 has a large bulk band gap of about 0.3 eV, suitable for THz and mid-infrared plasmonics, and a tunable Fermi level which can be put within the bulk gap [5]. The combination of highly efficient light-matter interaction, relatively long propagation distances, and tight confinement of surface plasmons in graphene and TIs promises interesting applications including compact room-temperature THz sources for imaging, spectroscopy and telecommunications; integrated photonic circuits; THz modulation of telecom signals, and compact THz sensors. Furthermore, optical methods [6] may provide a better access to characterization and manipulation of massless fermion states than transport measurements that are affected by conductivity in the bulk.Nonlinear optics of massless Dirac fermions has received little attention so far, especially in the THz range where many basic devices and components are lacking. Here we show that the difference frequency generation (DFG) in 2D layers of massless Dirac electrons, e.g. graphene and TIs, is an efficient and controllable way of generating surface plasmons over a broad range of frequencies. Second-order nonlinear processes such as DFG are usually assumed to be forbidden in an isotropic medium [14] such as the plane of a graphene layer. How...

Entangled photon states attract tremendous interest as the most vivid manifestation of nonlocality of quantum mechanics and also for emerging applications in quantum information. Here we propose a mechanism of generation of polarization-entangled photons, which is based on the nonlinear optical interaction (four-wave mixing) in graphene placed in a magnetic field. Unique properties of quantized electron states in a magnetized graphene and optical selection rules near the Dirac point give rise to a giant optical nonlinearity and a high rate of photon production in the mid- or far-infrared range. A similar mechanism of photon entanglement may exist in topological insulators where the surface states have a Dirac-cone dispersion and demonstrate similar properties of magneto-optical absorption.

Although massless Dirac fermions in graphene constitute a centrosymmetric medium for inplane excitations, their second-order nonlinear optical response is nonzero if the effects of spatial dispersion are taken into account. Here we present a rigorous quantum-mechanical theory of the second-order nonlinear response of graphene beyond the electric dipole approximation, which includes both intraband and interband transitions. The resulting nonlinear susceptibility tensor satisfies all symmetry and permutation properties, and can be applied to all three-wave mixing processes. We obtain useful analytic expressions in the limit of a degenerate electron distribution, which reveal quite strong second-order nonlinearity at long wavelengths, Fermi-edge resonances, and unusual polarization properties.

We report on the observation of collective radiative decay, or superradiance, of cyclotron resonance (CR) in high-mobility two-dimensional electron gases in GaAs quantum wells using time-domain terahertz magnetospectroscopy. The decay rate of coherent CR oscillations increases linearly with the electron density in a wide range, which is a hallmark of superradiant damping. Our fully quantum mechanical theory provides a universal formula for the decay rate, which reproduces our experimental data without any adjustable parameter. These results firmly establish the many-body nature of CR decoherence in this system, despite the fact that the CR frequency is immune to electron-electron interactions due to Kohn's theorem.

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