Matched infrared soliton pairs in graphene under Landau quantization via four-wave mixing

We investigate a type of matched infrared soliton pairs based on four-wave mixng (FWM) in Landau-quantized graphene by using density-matrix method and perturbation theory. The linear and nonlinear dynamical properties of the graphene system are first discussed, and, in particular, we focus on the signatures of nonlinear optical response. Then we present analytical solutions for the fundamental bright and dark solitons, as well as bright two-soliton, which are in good agreement with the results of numerical simulations. Moreover, due to the unusual dispersion relation and chiral character of electron states, we find that the matched spatial soliton pairs can propagate through a two-dimensional crystal of graphene and their carrier frequencies are adjustable within the infrared frequency regimes. Our proposed scheme may provide a route to explore the applications of matched infrared soliton pairs in telecommunication and optical information processing.

Figure 1

(a) Sketch of Landau levels (LLs) near the Dirac point superimposed on the electronic energy dispersion without the use of a magnetic field $E=\pm v_F\left|p\right|$. The magnetic field “condenses” the original states in the Dirac cone into discrete energies. LLs in graphene are unequally spaced: $\propto \sqrt{B}$. (b) Energy level diagram and optical transitions involved in the formation of matched soliton pairs in Landau-quantized graphene interacting with the continuous-wave (CW) control field 2 (with carrier frequency ${\omega }_2$) and a weak pulsed probe field 1 with carrier frequency ${\omega }_1$ to generate a FWM field 3 with carrier frequency ${\omega }_3$. The states $|1\rangle$, $|2\rangle$, $|3\rangle$, and $|4\rangle$ correspond to the LLs with energy quantum numbers $n=-2,-1,0,1$, respectively. Graphene monolayer is a one-atom-thick monolayer of carbon atoms arranged in a hexagonal lattice, which we will treat as a perfect two-dimensional (2D) crystal structure in the $x-y$ plane.

Figure 2

The ratio of the absorption coefficients ${\alpha }_{-}/{\alpha }_{+}$ for the two different modes as a function of the amplitude $|{\Omega }_2^{-}|$ of the control field (a) for several different amplitudes $|{\Omega }_2^{+}|$ with ${\Delta }_1=5{\gamma }_2$, ${\Delta }_2=7{\gamma }_2$, ${\Delta }_3=9{\gamma }_2$, ${\gamma }_4=1.0{\gamma }_2$ and (b) for the different decay rates ${\gamma }_4$ with ${\Delta }_1=10{\gamma }_2$, ${\Delta }_2=8{\gamma }_2$, ${\Delta }_3=6{\gamma }_2$, $|{\Omega }_2^{+}|=60{\gamma }_2$. The other parameter values for the simulations are ${\kappa }_{12}={\kappa }_{14}=0.2{\gamma }_2$ $\mu {\text{m}}^{-1}$, ${\gamma }_3={\gamma }_2$, and ${\gamma }_2=3\times {10}^{13}$ ${\text{s}}^{-1}$.

Figure 3

(a) The ratios $K_{+}^{\left(2i\right)}/K_{+}^{\left(2r\right)}$ (dashed curve) and $S_i/S_r$ (solid curve), as well as (b) the absorption coefficient ${\alpha }_{+}$ as functions of the control-field amplitude $|{\Omega }_2^{-}|/{\gamma }_2$. The system parameters used for the simulations are $|{\Omega }_2^{+}|=60{\gamma }_2$, ${\kappa }_{12}={\kappa }_{14}=0.2{\gamma }_2$ $\mu {\text{m}}^{-1}$, ${\gamma }_3={\gamma }_4={\gamma }_2$, ${\Delta }_1=5{\gamma }_2$, ${\Delta }_2=7{\gamma }_2$, ${\Delta }_3=9{\gamma }_2$, and ${\gamma }_2=3\times {10}^{13}$ ${\text{s}}^{-1}$. This figure corresponds to the case of bright solitons because of $K_{+}^{\left(2r\right)}{S}_r>0$.

Figure 4

Surface plots of the relative intensities of pulsed probe and FWM fields versus dimensionless time $\eta /\tau$ and distance $\xi /L$ for (a) the fundamental bright soliton and (b) the bright soliton of second order, which is obtained by numerically solving Eq. (36) without ignoring the imaginary part of coefficients with $L=0.1$ cm, $\tau =3.33\times {10}^{-14}$ s, and other parameters are explained in the text.