Nonlinear optical response of graphene in time domain

We study nonlinear optical response of electron dynamics in graphene to an intense light pulse within the model of massless Dirac fermions. The time-dependent Dirac equation can be cast into a physically transparent form of extended optical Bloch equations that consistently describe the coupling of light-field-induced intraband dynamics and interband transitions. We show that the nonlinear optical response is not sufficiently described neither by pure intraband nor by pure interband dynamics but their interplay has to be taken into account. When the component of the instantaneous momentum parallel to the field changes its sign, the interband transition is strongly enhanced and considerably influences the intraband dynamics. This counteracts anharmonic response expected from purely intraband dynamics and relaxes nonlinearly. Nevertheless, graphene is still expected to exhibit nonlinear optical response in the terahertz regime such as harmonic generation.

Figure 1

(Color online) Schematic representation of the dynamics of a single Dirac fermion with $p_y=0$, whose field-free state is marked by a filled circle, under an intense optical-field polarized along the $x$ axis. The bold double-sided arrow corresponds to the variation in the kinetic momentum. $A_0$ denotes the amplitude of the vector potential. When the electron arrives at the Dirac point, it does not stay in the same band but moves to the other band. The transition is instantaneous and total.

Figure 2

(Color online) Temporal evolution of the population difference $n\left(t\right)$ between the upper and lower states for $p_x/e{A}_0=-3/4$, $\hslash \omega /v_{F}e{A}_0=9.46\times {10}^{-3}$ and different values of $p_y/e{A}_0$ calculated using the EBOEs, except for the case of $p_y=0$, where extracted from Eq. (2). Also, $\left[p_x+eA\left(t\right)\right]/e{A}_0$ is plotted in a thin solid line (right axis).

Figure 3

(Color online) Temporal evolution of the normalized single-electron current $j_x\left(t\right)/v_F$ for the same parameters as in Fig. 2 and three different values of $p_y/e{A}_0$. Thick solid line: from Eq. (4) for (a) and using the EBOE Eqs. (8, 9, 10) for (b) and (c), thin solid line: calculated using the TDDE, and dashed line: calculated by switching off the interband transitions, i.e., setting $n=-1$ and $\rho =0$ in Eq. (10) all the time.

Figure 4

(Color online) (a) Normalized vector potential $v_{F}eA\left(t\right)/\hslash \omega$ of the incident optical pulse. (b) Temporal evolution of the normalized integrated electric current $J_x/e{n}_s{v}_F$. Thick solid line: total current calculated with Eqs. (15, 16), thin solid line: contribution from the interband polarization, and thick dotted line: calculated by switching off the interband transitions, i.e., keeping the initial values of $n$ and $\rho \left(=0\right)$ in Eq. (16) all the time. Labels A and B are referred to in Fig. 5.

Figure 5

(Color online) Carrier occupation distribution calculated by switching [(a) and (b)] off and [(c) and (d)] on the interband transitions. (a) and (c) are for the moment labeled as $A$ in Fig. 4 and (b) and (d) for $B$.

Figure 6

(Color online) Harmonic intensity spectra for the case of Fig. 4. Thick solid line: total spectrum calculated with Eq. (18), thin solid line: contribution from the interband polarization, and thick dotted line: calculated by switching off the interband transitions.