Vectorial four-wave mixing field dynamics from individual excitonic transitions

Degenerate four-wave mixing (FWM) from individual localized excitonic transitions is measured using a dual-polarization extension of heterodyne spectral interferometry. A fine-structure splitting (FSS) of the localized excitons of $20–80\mu \text{eV}$ is observed, revealing a spatially anisotropic localization in the quantum well plane, preferentially oriented along the [110] direction. The ratio of the transition dipole moments of the FSS transitions is in the range 0.7–1.4, with no significant correlation to the FSS. Retrieving both polarization components of the FWM in amplitude and phase we measure the time-resolved polarization state of the exciton after pulsed excitation, showing a coherent evolution oscillating between linear and virtually circular polarization. The polarization of the created FWM state is a result of the initial polarization state created by the first excitation pulse, its free evolution up to the arrival of the second pulse, and its conversion into the FWM controlled by the polarization of the second pulse.

Figure 1

(Color online) Spectrally resolved FWM intensity $I_{\text{F}}\left(\tau ,\omega \right)$ versus delay time $\tau$ from a homogenously broadened localized exciton transition. Left: $\left(\to \to \to \right)$ polarization. Bottom: Linear gray scale plot from white (zero) to black, and resonant FWM $I_{\text{F}}\left(\tau ,{\omega }_{\text{X}}\right)$ with fitted exponential decay (solid). Top: $I_{\text{F}}\left(0,\omega \right)$. Right bottom: as left, but $\left(\nearrow \nearrow \nearrow \right)$ polarization. Fit of $I_{\text{F}}\left(\tau ,{\omega }_{\text{X}}\right)$ (see text)—solid line. Right top: $I_{\text{F}}\left(0,\omega \right)$ for $\left(\to \to \to \right)$ (solid) and $\left(\uparrow \uparrow \uparrow \right)$ (dotted) polarization.

Figure 2

(Color online) (a) FWM intensity of the investigated transition (black peak), averaged PL spectrum (gray area) nonresonantly excited at 1.69 eV measured from a sample region of $\left(4\times 5\right)\mu {\text{m}}^2$ size around the probed transition, FWM excitation spectrum ($\sim 200\text{fs}$ long pulse, dashed line), spectral region in (b) is marked by vertical dashed-dotted lines. (b) FSS energies ${\Delta }_{\text{FS}}$ (black circles) and corresponding dipole moment ratios ${\mu }_{\left[1\overline{1}0\right]}/{\mu }_{\left[110\right]}$ (open squares) as a function of ${\omega }_{\left[1\overline{1}0\right]}$ (c)${\mu }_{\left[1\overline{1}0\right]}/{\mu }_{\left[110\right]}$ as a function of ${\Delta }_{\text{FS}}$.

Figure 3

(Color online) Time and polarization resolved FWM field ${\vec{\mathcal{E}}}_{\text{F}}\left(0.25\text{ps},t\right)$ from a FSS exciton transition. (a), (b) $\left(\nearrow \nearrow +\right)$ and (c), (d) $\left(\sigma \sigma +\right)$ configuration. (b), (d) amplitude and (a), (c) relative phase of the field components ${\mathcal{E}}_{\text{F}\to }$ (solid), ${\mathcal{E}}_{\text{F}\uparrow }$ (dotted). The time resolution of the data due to the spectral filtering ${\Delta }_t$ is 7.5 ps [(a), (b)] and 2.9 ps [(c), (d)]—marked by horizontal bars. Dash-dotted line in (b): system response of the HSI setup. Insets: FWM amplitude (black) and spectral filter function (gray).

Figure 4

(Color online) Time-resolved polarization ellipse of the FWM field, expressed by the Stokes angles $\psi$ and $\chi$. The solid and dashed curves correspond to $\left(\nearrow \nearrow +\right)$ and $\left(\sigma \sigma +\right)$ configuration. Thin lines: calculated dynamics for a pair of linearly polarized FSS transitions of equal amplitude. Inset: Poincaré sphere with marked $2\psi$ and $2\chi$ angles. Solid line: time path of the polarization state for $\left(\nearrow \nearrow +\right)$. (c) Polarization ellipses at different times for $\left(\nearrow \nearrow +\right)$.