Quantum Interference of Virtual and Real Amplitudes in a Semiconductor Exciton System

By two-color pulse shaping, we simultaneously create virtual and real amplitudes for excitons in GaAs quantum wells, and monitor population and amplitude by pump-probe and four-wave mixing spectroscopies. Excited-state probability amplitude can be induced by the off-resonant, virtual excitations as well as by the resonant, real excitations. Population modulation in time-domain results from the interference between the virtual and real amplitudes, and the modulation depth reveals the relative contributions of these two amplitudes. The fact that virtual and real amplitudes have a phase difference of 90° is demonstrated directly in time-domain.

Figure 1

(a) Schematic diagram of the experiments for a simple two-level system whose natural frequency and damping constant are ${\omega }_0$ and $\gamma$, respectively. Real amplitude at ${\omega }_0$ and a virtual one at ${\omega }_1$ are simultaneously excited by two-color picosecond pulse. (b) Schematics of the experimental setup describing picosecond/femtosecond PP and FWM. Spectrally broad 30 fs pulses are used as a probe. SP stands for spectrometer.

Figure 2

(a) TI-FWM (solid lines) versus delay for four different relative strengths of $I_1/I_0$. The pronounced beatings are observed, whose periods of oscillations (0.85 ps) correspond to the inverse of the detuning (4.8 meV). The depth of oscillation increases to a certain point (top three), after which it gets weaker again (bottom). Cross-correlation traces are shown for comparison (dashed lines) (b) Numerical calculation of TI-FWM for a two-level system with pulse parameters that match the experiment.

Figure 3

Differential transmission for three different $I_1/I_0$.

Figure 4

(a) The oscillatory parts in FWM are shown for the three different $\delta \omega$’s. $\Delta \omega$ is fixed at 7.1 meV and the magnitude of $\gamma$ is estimated to be 1 meV. The overall phase changes are observed from $90°$(up) to $0°$(bottom) for the increasing value of $\delta \omega$. (b) Numerical calculation for the same conditions.

Figure 5

The oscillatory parts in the FWM signal are shown for various detunings from 3.6 meV (top) to 40.7 meV (bottom). The frequency of the beating follows exactly $\Delta \omega$, while their depth of oscillation decreases with $\Delta \omega$. (inset) The plot for the magnitude of virtual amplitude versus $\Delta \omega$, normalized to the resonant real amplitude.