Terahertz Coherent Control of Optically Dark Paraexcitons in ${\mathrm{Cu}}_2\mathbf{O}$

Intense multiterahertz fields of order megavolts per centimeter are used to coherently promote optically dark and dense paraexcitons in ${\mathrm{Cu}}_2\mathrm{O}$ from the $1s$ into the $2p$ state. The nonlinear field response of the intraexcitonic degrees of freedom is directly monitored in the time domain via ultrabroadband electro-optic sampling. The experimental results are analyzed with a microscopic many-body theory, identifying up to two internal Rabi cycles. The effects of population inversion and ponderomotive contributions are disentangled.

Figure 1

(a) Formation and cooling dynamics of $1s$ excitons in ${\mathrm{Cu}}_2\mathrm{O}$ probed by multi-THz spectroscopy: Pump-induced changes of absorption $\Delta \alpha$ and refractive index $\Delta n$ (shaded area with dots) for various delay times $t_D$ after two-photon absorption of 12-fs pulses centered at 1.55 eV. Vertical lines: $1s\mathrm{\text{−}}2p$ resonances at vanishing center-of-mass momentum. Solid line: Theoretical result with $n_{1s}^{\mathrm{para}}=1\times {10}^{16}{\mathrm{cm}}^{-3}$, $n_{1s}^{\mathrm{ortho}}=0.5\times {10}^{16}{\mathrm{cm}}^{-3}$, and $T_{1s}=10\mathrm{K}$. Dashed line: Theoretical $\Delta n$ without ponderomotive contribution. (b) Because of different effective masses of $1s$ and $2p$ excitons, THz transitions (arrows) depend on the center-of-mass momentum.

Figure 2

Nonlinear THz response of $1s$ paraexcitons ($t_D=1\mathrm{ns}$): quantitative comparison of (a) experiment and (b) microscopic theory. Upper panels: Real-time profile of the exciting THz pulse $E_{\mathrm{THz}}$. Lower panels: Reemitted THz field $\Delta E$ (vertically offset) for six values of the peak driving field, with $E_{\mathrm{THz}}^{\mathrm{peak}}=0.13$, 0.24, 0.45, 0.67, and 0.92, $1.0\times E_0$ for curves (i)–(vi), respectively ($E_0=0.5\mathrm{MV}/\mathrm{cm}$ in experiment and $E_0=0.3\mathrm{MV}/\mathrm{cm}$ in theory). (c) Theoretical transient for the highest intensity. The envelopes of $\Delta E_A$ (dashed curve) and $\Delta E$ (solid curve) at the spot center are shown together with the scaled excitation pulse (shaded area). (d) Inversion (solid curve) and polarization (shaded area) of the two-level system consisting of $1s$ and $2p$ paraexciton states as well as ionized exciton fraction (dashed curve) at the center of the THz spot.

Figure 3

(a) Reemitted THz amplitude versus maximum $E_{\mathrm{THz}}^{\mathrm{peak}}$ of the driving field at four delay times $T$, extracted from Fig. 2; circles and shaded areas: experiment; connecting thin curves: guides to the eye; dashed curves: simulation ($E_0=0.5\mathrm{MV}/\mathrm{cm}$ in experiment and $E_0=0.3\mathrm{MV}/\mathrm{cm}$ in theory). (b) Phase offset between pump-induced and driving fields at a preset time $T=-0.25\mathrm{ps}$ obtained from Fig. 2. Experiment (circles) and theory with (solid curve) and without (dashed curve) a ponderomotive part are shown.