Interference of stimulated electronic Raman scattering and linear absorption in coherent control

We consider quantum interference effects in carrier and photocurrent excitation in graphene using coherent electromagnetic field components at frequencies $\omega$ and $2\omega$. The response of the material at the fundamental frequency $\omega$ is presented, and it is shown that one-photon absorption at $\omega$ interferes with stimulated electronic Raman scattering (combined $2\omega$ absorption and $\omega$ emission) to result in a net contribution to the current injection. This interference occurs with a net energy absorption of $\hslash \omega$ and exists in addition to the previously studied interference occurring with a net energy absorption of $2\hslash \omega$ under the same irradiation conditions. Due to the absence of a band gap and the possibility to block photon absorption by tuning the Fermi level, graphene is the perfect material to study this contribution. We calculate the polarization dependence of this all-optical effect for intrinsic graphene and show that the combined response of the material at both $\omega$ and $2\omega$ leads to an anisotropic photocurrent injection, whereas the magnitude of the injection current in doped graphene, when transitions at $\omega$ are Pauli blocked, is isotropic. By considering the contribution to coherent current control from stimulated electronic Raman scattering, we find that graphene offers tunable, polarization sensitive applications. Coherent control due to the interference of stimulated electronic Raman scattering and linear absorption is relevant not only for graphene but also for narrow-gap semiconductors, topological insulators, and metals.

Figure 1

Feynman diagrams of first- and second-order absorption processes contributing to coherent control in the presence of a two-color, $\omega$ and $2\omega$ field. The time axis points to the right, a plain arrow pointing to the right (left) represents an electron (hole), and vertices represent the dipole interaction. (a) Processes where the net energy absorption amounts to $2\hslash \omega$: one-photon absorption (1PA) at $2\omega$ and two-photon absorption (2PA) at $\omega$; the final state is an electron-hole pair of energy $2\hslash \omega$.(b) Processes where the net energy absorption amounts to $\hslash \omega$: 1PA at $\omega$ and stimulated electronic Raman scattering (ERS) at $\omega$ $(2\omega$ absorption and $\omega$ emission); the final state is an electron-hole pair of energy $\hslash \omega$. The crossed processes, where light is emitted before absorption, are also included in Eq. (5) but their diagrams omitted for brevity. Second-order processes where the net energy absorption amounts to $3\hslash \omega$ or $4\hslash \omega$ are not included since they do not contribute to coherent control unless third- and higher-order absorption processes are also included.

Figure 2

The conventional excitation scheme (right) employs interference between two-photon absorption at $\omega$ (red arrows) and one-photon absorption at $2\omega$ (blue arrows). The additional contribution at $\hslash \omega$ (left) occurs when one-photon absorption at $\omega$ interferes with stimulated electronic Raman scattering (ERS, $2\omega$ absorption and $\omega$ emission). Depending on the chemical potential $\mu$, the ERS contribution is allowed $\left(2\left|\mu \right|<\omega \right)$ or blocked $(2\left|\mu \right|>\omega ,$ not shown).

Figure 3

Reciprocal space distribution of injected carriers in intrinsic graphene under two-color, $\omega$ and $2\omega$ field. Absorption processes up to second order in the field are considered, with net energy absorption of $\hslash \omega$ (red, inner distribution) and $2\hslash \omega$ (blue, outer distribution), respectively due to ERS-1PA interference and 1PA-2PA interference. The carrier injection rates [Eqs. (13) and (14)] are indicated by the thickness of the distributions; the current resulting from each distribution is indicated by a correspondingly colored arrow, and the overall current injection is indicated by a thick, black arrow. Photoresponse for circularly-polarized $\omega$ and $2\omega$ beams: (a) opposite polarization, yielding no net current injection, and (b) co-circular polarization, yielding a net current injection. Photoresponse for linearly-polarized $\omega$ and $2\omega$ beams: (c) $\theta =0$, i.e., parallel polarization axes, (d) $\theta =\pi /4$, (e) $\theta =\pi /2$, i.e., perpendicular polarization axes, and (f) $\theta =3\pi /4$; all yielding current injection.

Figure 4

Contour plot of the parallel-perpendicular polarization disparity parameter $d={\eta }_I^{xyyx}/{\eta }_I^{xxxx}$ in $p$-doped graphene as a function of the electron population $f_{-\omega }$ and $f_{-\omega /2}$ in the valence band $\left(f_{-\omega /2}\le f_{-\omega }\right)$. With $f_{-\omega /2}=0$, stimulated ERS and 1PA at $\omega$ is blocked and interference of 2PA and 1PA at $2\omega$ yields $d=-1$ characterizing an injection current that is isotropic with respect to $\theta ={cos}^{-1}({\widehat{\mathbf{e}}}_{\omega }·{\widehat{\mathbf{e}}}_{2\omega })$. With $f_{-\omega /2}\ne 0$, the additional ERS contribution yields $d>-1$ characterizing an anisotropic current. In particular, for $d=0$ (thick contour line) the injection current vanishes for perpendicular polarization axes, $\theta =\pi /2$.

Figure 5

Angular plots of the current injection strength as a function of the angle $\theta$ between the linear polarization axes ${\widehat{\mathbf{e}}}_{\omega }$ and ${\widehat{\mathbf{e}}}_{2\omega }$, for the ratio $d={\eta }_I^{xyyx}/{\eta }_I^{xxxx}$ taking values of $0.14,0.24$, and $0.33$. The current components along ${\widehat{\mathbf{e}}}_{2\omega }$ (plain red line) and ${\widehat{\mathbf{e}}}_{2\omega }^{\perp }$ (plain/dashed black line) are plotted; a dashed line indicates a negative current along that axis.