Fundamentally fastest optical processes at the surface of a topological insulator

We predict that a single oscillation of a strong optical pulse can significantly populate the surface conduction band of a three-dimensional topological insulator, ${\mathrm{Bi}}_2{\mathrm{Se}}_3$. Both linearly- and circularly-polarized pulses generate chiral textures of interference fringes of population in the surface Brillouin zones. These fringes constitute a self-referenced electron hologram carrying information on the topology of the surface Bloch bands, in particular, on the effect of the warping term of the low-energy Hamiltonian. These electron-interference phenomena are in sharp contrast to graphene where there are no chiral textures for a linearly-polarized pulse and no interference fringes for circularly-polarized pulse. These predicted reciprocal space electron-population textures can be measured experimentally by time resolved angle resolved photoelectron spectroscopy (TR-ARPES) to gain direct access to non-Abelian Berry curvature at topological insulator surfaces.

Figure 1

(a) Low-energy electron dispersion of the surface state around the Dirac point, i.e., the $\mathrm{\Gamma }$ point at $\mathbf{k}=(0,0)$, as a function of wave vector $\mathbf{k}$. The Dirac cone at the Gamma point is clearly visible. (b) Energy dispersion in a wide range where the sixth-order warping is manifest.

Figure 2

Complex vector of non-Abelian Berry connection $\mathcal{A}={\mathcal{A}}_{cv}$. (a) Magnitude of $x$-component $\left|{\mathcal{A}}_x\left(\mathbf{k}\right)\right|$ (in units of $\AA$) as a function of crystal momentum $\mathbf{k}$. (b) Phase ${\mathrm{\Phi }}_x\left(\mathbf{k}\right)=arg\left({\mathcal{A}}_x\left(\mathbf{k}\right)\right)$ as a function of crystal momentum $\mathbf{k}$. (c) Same as (a) but for ${\mathcal{A}}_y\left(\mathbf{k}\right)$. (d) Same as (b) but for ${\mathrm{\Phi }}_y\left(\mathbf{k}\right)=arg\left({\mathcal{A}}_y\left(\mathbf{k}\right)\right)$.

Figure 3

Residual CB population, $N^{\left(\mathrm{res}\right)}$, on the surface of ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ as a function of wave vector $\mathbf{k}$ for different values of angle $\theta$. Amplitude of the electric field is $F_0=0.10\mathrm{V}/\AA$. The angle is (a) $\theta =0$, (b) $\theta =\pi /4$, (c) $\theta =\pi /3$, and (d) $\theta =\pi /2$.

Figure 4

Residual CB population, $N_{\mathrm{CB}}^{\left(\mathrm{res}\right)}$, of graphene as a function of wave vector $\mathbf{k}$ for different values of angle $\theta$. Amplitude of the electric field is $F_0=0.10\mathrm{V}/\AA$. The angle is (a) $\theta =0$, (b) $\theta =\pi /4$, (c) $\theta =\pi /3$, and (d) $\theta =\pi /2$. The dynamics of the graphene system is modeled within the low energy effective Dirac model.

Figure 5

Mean residual CB population $n_c$ as a function of field amplitude $F_0$ for different values of $\theta$: $\theta =0$ (blue solid line), $\theta =\pi /4$ (red solid line), $\theta =\pi /3$ (green dashed line), and $\theta =\pi /2$ (black solid line). Inset: total residual CB population, $n_c$, as a function of $\theta$ for the amplitude of the electric field of $F_0=0.18\mathrm{V}/\AA$.

Figure 6

Residual CB population as a function of crystal momentum. Excitation pulse is right-handed circularly polarized; its waveform is shown in panel (a). The amplitude of the pulse is (b) $F_0=0.05\mathrm{V}/\AA$, (c) $F_0=0.10\mathrm{V}/\AA$, (d) $F_0=0.15\mathrm{V}/\AA$, and (e) $F_0=0.20\mathrm{V}/\AA$. The solid closed black lines display the separatrices (see the text) for the corresponding pulses.

Figure 7

Residual CB population in the vicinity of a Dirac point for graphene for circularly-polarized right-handed one-cycle pulse with amplitude of (a) $F_0=0.05\mathrm{V}/\AA$, (b) $F_0=0.10\mathrm{V}/\AA$, (c) $F_0=0.15\mathrm{V}/\AA$, and (d) $F_0=0.20\mathrm{V}/\AA$. The solid closed black lines display the separatrices (see the text) for the corresponding pulses.

Figure 8

Residual CB population for circularly-polarized left-handed one-cycle pulse [waveform is shown on panel (a)] with amplitude of (b) $F_0=0.05\mathrm{V}/\AA$, (c) $F_0=0.10\mathrm{V}/\AA$, (d) $F_0=0.15\mathrm{V}/\AA$, and (e) $F_0=0.20\mathrm{V}/\AA$. The solid closed black lines display the separatrices (see the text) for the corresponding pulses.

Figure 9

Residual CB population for two-cycle circularly polarized pulse with the amplitude of 0.1 $\mathrm{V}/\AA$. The two-cycle pulses are 2R (a) and 1R+1L (b). The solid closed black lines display the separatrices (see the text).

Figure 10

Residual electron population after 1R single-oscillation pulse with amplitude $F_0=0.1\mathrm{V}/\AA$. The corresponding carrier-envelope phase (i.e., the angle between the maximum field, ${\mathrm{F}}_0$, and the positive $x$ axis) is indicated in the corresponding panels.