Phase relaxation of optical excitations in condensed matter can be studied by coherent four—wave—mixing experiments in the time domain without requiring assumptions on the spectral properties of the material. For inhomogeneously broadened transitions, three laser pulses at times t = 0, t = t21 and t = T with wave vectors k1, k2 and k3, respectively, generate a stimulated photon echo pulse, emitted time—delayed at t = T+t21 in the momentum—matching direction according to k3+k2−k1. As a function of t21 the echo amplitude decays due to scattering events, described by the dephasing time constant T2 [1].
The phase relaxation of optically excited electronic states can be studied in fourwave mixing experiments where three excitation pulses at t = 0, t =t21, and t = T, respectively, interact via the third-order polarization of the medium to generate a nonlinear signal whose intensity depends on the dephasing time T2. For inhomogeneously broadened optical transitions the signal will be emitted as a time delayed photon echo pulse at t = T + t21 provided the pulse duration is short compared with T2. The echo amplitude decays as exp(−4t21/T2).1
Time-resolved demonstration of picosecond photon echoes due to intrinsic optical excitations of a semiconductor is presented. Phase relaxation times up to several hundred picoseconds are found for excitons localized by chemical disorder in
CdSx${\mathrm{Se}}_{1\mathrm{\ensuremath{-}}\mathrm{x}}$ mixed crystals. These long dephasing times are attributed to reduced scattering of the localized excitons as compared to free excitons. Dephasing is identified as due to energy relaxation of the excitons.
A new method of picosecond and femtosecond transient spectroscopy yielding information about ultrafast relaxation processes without requiring ultrashort light pulses is presented. It is based on the present theoretical analysis of the resonant degenerate four-wave-mixing process excited by two temporally incoherent light beams with wave vectors ${\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}}_{1}$ and ${\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}}_{2}$ which are originated from a single beam at frequency $\ensuremath{\omega}$ but have mutual time delay $\ensuremath{\tau}$. Under the assumption that the incoherent light field has Gaussian random complex amplitude and that the resonant material consists of the usual two-level atoms, the statistically averaged intensity of the output light field with ${\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}}_{3}=2{\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}}_{2}\ensuremath{-}{\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}}_{1}$ at $\ensuremath{\omega}$ is calculated as a function of $\ensuremath{\tau}$. Even with the light having a much longer duration than both
T1 (the longitudinal relaxation time) and
T2 (the transverse relaxation time), the correlation trace, i.e., output intensity versus $\ensuremath{\tau}$, represents a decay profile determined mainly by
T2 for both homogeneously and inhomogeneously broadened transitions, as long as the light correlation time ${\ensuremath{\tau}}_{c}$ is much shorter than the relaxation times. The correlation trace does not always represent a single-exponential decay but is sometimes slightly deformed by the
T1 effect. However, it does not cause a significant error in the determination of
T2. Moreover, as $\frac{{T}_{2}}{{T}_{1}}\ensuremath{\rightarrow}0$, the trace becomes a single-exponential decay curve determined only by
T2. The feature of the results obtained by the present method is similar to that obtained by the conventional coherent transient spectroscopy with short pulses, such as the photon echo. The time resolution in the present method, however, is limited only by ${\ensuremath{\tau}}_{c}$ much shorter than the light duration. By regarding the incoherent light as a series of random ultrashort pulses, the present four-wave-mixing process is also interpreted as the ensemble of numerous transient four-wave-mixing processes caused by various combinations of these pulses.
The influence of disorder and localization on optical dephasing of excitons in the semiconductor mixed crystals ${\mathrm{CdS}}_{1\mathrm{\ensuremath{-}}\mathit{x}}$
Sex and
Alx${\mathrm{Ga}}_{1\mathrm{\ensuremath{-}}\mathit{x}}$As has been investigated by means of time-resolved four-wave mixing and photon echo experiments. A dephasing time of several hundreds of picoseconds is found for resonantly excited localized excitons in ${\mathrm{CdS}}_{1\mathrm{\ensuremath{-}}\mathit{x}}$
Sex while the dephasing time in
Alx${\mathrm{Ga}}_{1\mathrm{\ensuremath{-}}\mathit{x}}$As amounts to only a few picoseconds. In ${\mathrm{CdS}}_{1\mathrm{\ensuremath{-}}\mathit{x}}$
Sex dephasing results mainly from hopping processes, i.e., exciton-phonon interaction. The contribution of disorder is negligible in terms of phase relaxation in ${\mathrm{CdS}}_{1\mathrm{\ensuremath{-}}\mathit{x}}$
Sex. In contrast, in
Alx${\mathrm{Ga}}_{1\mathrm{\ensuremath{-}}\mathit{x}}$As elastic disorder scattering yields an essential contribution to the dephasing rate. We present a theoretical model, which treats dephasing of optical excitations in a disordered semiconductor, including the influence of disorder as well as exciton-phonon interaction. On the base of this model, the experimentally observed differences in the dephasing behavior of excitons in ${\mathrm{CdS}}_{1\mathrm{\ensuremath{-}}\mathit{x}}$
Sex and
Alx${\mathrm{Ga}}_{1\mathrm{\ensuremath{-}}\mathit{x}}$As are related to the microscopic structure of the disorder potential and the mechanism of exciton localization.
The results of the femtosecond four-wave mixing measurements performed on the CdS crystal at room temperature are reported. The dephasing time T d = 180 fs was obtained. The anti-Stokes shift of the primary photon echo spectrum with respect to the excitation spectrum was observed. The some peculiarities of the coherent responses decay curves indicate that this optical dephasing is due to the exciton-exciton interaction. The stimulated photon echo decay curves appear to have a fast decay component (the time constant of approximately 4.5 ps) in case the energy of exciting pulses exceeds the threshold value of 0.4 μJ.