Ultrafast Terahertz Nonlinear Optics of Landau Level Transitions in a Monolayer Graphene

We investigated the ultrafast terahertz (THz) nonlinearity in a monolayer graphene under the strong magnetic field using THz pump-THz probe spectroscopy. An ultrafast suppression of the Faraday rotation associated with inter-Landau level (LL) transitions is observed, reflecting the Dirac electron character of nonequidistant LLs with large transition dipole moments. A drastic modulation of electron distribution in LLs is induced by far off-resonant THz pulse excitation in the transparent region. Numerical simulation based on the density matrix formalism without rotating-wave approximation reproduces the experimental results. Our results indicate that the strong light-matter coupling regime is realized in graphene, with the Rabi frequency exceeding the carrier wave frequency and even the relevant energy scale of the inter-LL transition.

Figure 1

(a) Schematic of THz pump-THz probe setup. WGP: A free-standing wire-grid polarizer. (b) Faraday rotation angle (upper panel) and ellipticity (lower panel) spectra at $B=1$, 3, and 5 T without the pump. The red (blue) dashed line represents the transition energy of $\mathrm{\Delta }{E}_{65}$ at $B=3\mathrm{T}$ ($\mathrm{\Delta }{E}_{43}$ at $B=5\mathrm{T}$). (c) LL energy diagram in the positive energy side. The Fermi energy of $E_F=150\mathrm{meV}$ is indicated by the green line. The arrows show the transitions corresponding to $\mathrm{\Delta }{E}_{65}$ at $B=3\mathrm{T}$ and $\mathrm{\Delta }{E}_{43}$ at $B=5\mathrm{T}$. (d) Pump THz spectrum with respect to $\mathrm{\Delta }{E}_{65}$ at $B=3\mathrm{T}$.

Figure 2

(a) Waveforms of the $y$ component of the transmitted probe $E$ field with (red curve) and without (gray curve) the pump at $t_{\text{p}\text{p}}=0.02\mathrm{ps}$. (b) Pump-induced change of the $y$ component of the transmitted probe peak $E$ field as a function of $t_{\text{p}\text{p}}$ at the indicated pump $E$ fields. A squared waveform of the pump $E$ field is also shown by the orange shaded area. Inset: the pump $E$-field dependence of the pump-induced change around $t_{\text{p}\text{p}}=0\mathrm{ps}$. The dashed line represents a square power law as a guide to the eyes. (c) Faraday rotation angle (left panel) and ellipticity (right panel) spectra at different $t_{\text{p}\text{p}}$. Gray circles represent the spectra without the pump. The spectra are offset for clarity. (d) Calculated Faraday rotation spectra corresponding to those in (c).

Figure 3

(a) Real parts of pump-induced change of the conductivity spectra for right- (${\sigma }_{+}$) and left- (${\sigma }_{-}$) circularly polarized lights represented by open and solid circles, respectively, at different $t_{\text{p}\text{p}}$. The spectra are offset for clarity. (b) Schematic of the population distribution of electrons in the LLs. Before the pump (left panel), only right-circularly polarized light is absorbed by the transitions (thick red arrows) around the Fermi energy (green dashed line). The pump irradiation suppresses the absorption for right-circularly polarized light (thin red arrows) and induces the absorption for left-circularly polarized light (blue arrows).

Figure 4

(a) Calculated population distribution dynamics of ${\mathrm{LL}}_{-5}$ and LLs around the Fermi energy (${\mathrm{LL}}_4$, ${\mathrm{LL}}_5$, and ${\mathrm{LL}}_6$). Squared pump $E$-field waveform is also depicted as the orange shaded area. (b) Population distributions at $t_{\text{p}\text{p}}=-2$, 0, and 2 ps. The gray dotted lines represent the population distribution at $t_{\text{p}\text{p}}=-2\mathrm{ps}$. (c) Simulated real parts of the pump-induced change of the conductivity spectra at the same $t_{\text{p}\text{p}}$ as those in (b). (d) Simulated excitation power dependence of the magnitude of the $\mathrm{Re}[\mathrm{\Delta }{\sigma }_{+}(\omega )]$ with $\hslash \omega =\mathrm{\Delta }{E}_{65}$ at $t_{\text{p}\text{p}}=0\mathrm{ps}$. The gray dashed line represents a square power law for guide to the eyes.