3.2 Solids and Nanostructures: Electrons, Spins, and Phonons

We generate short pulses with wavelengths longer than about 10 μm (corresponding to frequencies below 30 THz) and use them to study transitions in solids in this frequency range (e.g., intersubband transitions in semiconductor nanostructures, interlevel transitions in impurities, interband transitions in narrow-gap materials, and intraband transitions). Compared to the visible and near-infrared spectral range, the THz and mid-infrared range poses special problems. Since essentially no lasers exist that can directly generate short pulses in this spectral range, the pulses have to be generated by nonlinear conversion (often not very efficient) from near-infrared pulses. Regarding the detection, the sensitivity of photon detectors generally decreases for longer wavelengths, so that in practice they are used only for wavelengths shorter than about 25 μm. On the other hand, at these wavelengths it is possible to use electro-optic sampling, which yields directly the electric field of the pulses as a function of time, i.e., the full information about the pulse.

In most cases we are interested in the time dependence of the processes we study. This can be determined by using two or three pulses and measuring the transmitted electric fields through the sample as a function of the delays between the incident pulses. Below we show an example for this type of experiment, where two-quantum oscillations of atoms in a semiconductor are analyzed using the terahertz waves radiated from the moving atoms.

In this experiment we use three pulses A, B, and C with center frequencies around 22 THz and pulse lengths between 100 and 200 fs. To determine the nonlinear response, we measure the transient when all three pulses are incident on the sample, an InSb crystal, the three transients for two pulses and the three transients for each pulse alone. From these data the nonlinear signal is obtained as E_NL = E_ABC − E_AB − E_BC − E_AC + E_A + E_B + E_C, it is a function of the real time t, the delay τ between pulses A and B, and the delay T_W between pulses B and C. The figure above shows in (a) E_ABC(t, τ) and in (b) E_NL(t, τ), both for T_W = 827 fs.

To be able to separate different nonlinear processes one can transform E_NL into the frequency domain, see (c) and (d). Here the most interesting signals are those at ν_t = 10 THz [marked with circles in (c)]. Where does this frequency come from? On the one hand, the incident pulses have negligible intensity at this frequency. On the other hand, the sample does not have transitions at this frequency, the highest phonon frequency is 5 THz and the lowest electronic transition is 44 THz. It turns out that this frequency results from a two-quantum excitation of longitudinal optical phonons at double the frequency of a single phonon. Such a two-quantum excitation is a nonclassical state, it requires that the atoms move to the left and to the right at the same time. The amplitude of this two-quantum excitation is much larger than expected from the transition dipole moment of the phonon. This is caused by excitation via higher-order processes involving (much larger) electronic transition dipole moments. A calculation taking these higher-order processes into account agrees very well with the experimental results (e).