Nonperturbative Interband Response of a Bulk InSb Semiconductor Driven Off Resonantly by Terahertz Electromagnetic Few-Cycle Pulses

Intense multiterahertz pulses are used to study the coherent nonlinear response of bulk InSb by means of field-resolved four-wave mixing spectroscopy. At amplitudes above $5\mathrm{MV}/\mathrm{cm}$ the signals show a clear temporal substructure which is unexpected in perturbative nonlinear optics. Simulations based on a model of a two-level quantum system demonstrate that in spite of the strongly off-resonant character of the excitation the high-field few-cycle pulses drive the interband resonances into a nonperturbative regime of Rabi flopping. The rotating wave approximation breaks down in this case and the system reaches a complete population inversion.

Figure 1

(a) Schematic band structure of InSb with conduction (CB) and valence bands (VB). The vertical arrow illustrates a nonresonant excitation by a terahertz pulse. (b) Terahertz transient generated by a $370\mathrm{\text{−}}\mu \mathrm{m}$-thick GaSe crystal. (c) Amplitude spectrum of the terahertz transient in panel (b). (d) Scheme of the FWM setup: two phase-locked terahertz pulse trains are generated by a Ge beam splitter (BS). Both branches are modulated by mechanical choppers (CH) and delayed with respect to each other via a translation stage (DS). The terahertz transients are noncollinearly focused on the sample and the transmitted signals are detected by electro-optic sampling (EOS).

Figure 2

(a) Electric field of two terahertz transients traversing InSb plotted as a function of delay time $\tau$ and sampling time $t$. (b) Nonlinear signal $E_{\mathrm{NL}}$. (c) Two-dimensional FT of panel (b) revealing pump-probe signals at wave vector positions ${\mathbf{k}}_1$ and ${\mathbf{k}}_2$ and the FWM signature at ${\mathbf{k}}_4$. (d) Selective inverse FT of the pump-probe signal emerging in the direction of ${\mathbf{k}}_1$. (e) Inverse FT of the pump-probe signature at ${\mathbf{k}}_2$ only. (f) Inverse FT of the FWM signal in the direction of ${\mathbf{k}}_4$ only.

Figure 3

(a) Oval FWM signal by applying an external field of $2\mathrm{MV}/\mathrm{cm}$ per pulse. (b) $S$-shaped FWM signature driven by an external field of $3.5\mathrm{MV}/\mathrm{cm}$. (c) Splitted FWM signal at an external exciting field strength of $5.3\mathrm{MV}/\mathrm{cm}$. (d)–(f) Calculated FWM signatures reproducing the main features of the corresponding measured signals on the left side of each image.

Figure 4

Simulated response of the two-level system for $\tau =0\mathrm{ps}$: (a) Normalized polarization of the two-level system at the fundamental frequency of the driving field as a function of the delay time $t$ and the peak electric field $E_m$. (b) Maximum inversion $w_{\max }$ as a function of $E_m$ (red solid line) and of the amplitude of a continuous wave excitation (blue dashed line) (c) Driving field of the terahertz transients. (d)–(e) Corresponding pathways of the Bloch vector for a moderate ($E_m=2\mathrm{MV}/\mathrm{cm}$) and a high ($E_m=5.3\mathrm{MV}/\mathrm{cm}$) peak electric field.