[(a)–(c)] Various Liouville pathways contributing to the third-order FWM signal, i.e., the two-pulse photon echo, shown in two different representations: as vector chains in our 2D spectra (left panels) or as the commonly used double-sided Feynman diagrams (right panels). (d) One Feynman diagram contributing to the fifth-order nonlinear polarization (six-wave mixing) occurring at and .
Simulated electric-field transients transmitted through the sample. (a) for the delay time . (b) and . (c) Subtracting the transients in (b) from the transient in (a) yields the nonlinear signal . (d) 2D transients in the time domain. (e) 2D nonlinear signal in the time domain.
2D Fourier transform of the simulation results for a resonantly driven two-level system. The correspondence to the usual wavevector space for selected optical nonlinearities is shown by arrows, in (a) for signals at the fundamental frequency , in (b) for the third harmonic . (a) Solid arrows indicate pump-probe signals, dashed arrows are the FWM signals, and dotted arrows are the six-wave mixing signal. (b) Solid arrows indicate the generation of the third harmonic.
(a) Fourier transformation of the nonlinear polarization at lower field amplitudes than in Fig. 3. [(b)–(d)] For a detailed analysis the individual peaks can be transformed back into the time domain. (b) shows the FWM signal, [(c) and (d)] are the two pump-probe signals.
Schematic of the experimental setup. The output of an amplified Ti:sapphire laser system is split into two pulses in a Michelson interferometer. The time interval between the two pulses is controlled by a delay stage. The pulses are sent onto a thick GaSe crystal for difference frequency generation. The resulting midinfrared pulses propagate through the sample and are detected by electro-optic sampling, using a part of the 12 fs oscillator output. Choppers in both beams synchronized to the amplifier repetition rate allow for measuring the individual pulses and both pulses together.
Experimental data. (a) Measured transients . (b) Resulting nonlinear signal . (c) Fourier transform of . The pump-probe signals are indicated with solid arrows and the FWM signals with dashed arrows. (d) FWM signal in the time domain, i.e., the back transformation of the signal at and .
Position in real time of the center of gravity of the two pulses and of the FWM signal as a function of the delay time . The four-wave mixing signal appears at a constant time after the second of the two pulses. The present results confirm earlier results (Ref. 21) on the same sample, namely, that the intersubband transition is predominantly homogenously broadened. An inhomogeneously broadened transition would give rise to a photon echo, occurring at the delay after the second pulse (dashed lines).
Experimental data: Contour plots (normalized to the respective signal maximum) of measured 2D spectra for various electric-field strengths of the phase-locked pulse pair as indicated. The field strength of 18 kV/cm corresponds approximately to a pulse area of .
Theoretical simulation: Calculated 2D spectra for various electric-field strengths of the phase-locked pulse pair. The indicated pulse areas at the top of the panels correspond to a single driving pulse. Obviously, an approximative description with third-order response functions is only valid for very small pulse areas .
Correspondence between the wavevector space and the two-dimensional frequency space for selected optical nonlinearities (Ref. 27).
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