The interaction of matter with light, i.e., electromagnetic fields, is able to generate a coherent superposition of excited quantum states in the material.
Macroscopically, the superposition state of the material results in an optical polarization, i.e., a rapidly oscillating dipole density.
The optical polarization is a genuine non-equilibrium quantity that decays to zero when the excited system relaxes to its equilibrium state after the electromagnetic pulse is switched off.
Due to this decay which is called dephasing, coherent effects are observable only for a certain temporal duration after pulsed photoexcitation.
Various materials such as atoms, molecules, metals, insulators, semiconductors are studied using coherent optical spectroscopy and such experiments and their theoretical analysis has revealed a wealth of insights on the involved matter states and their dynamical evolution.
Then, a few prominent examples for coherent effects in semiconductor optics are described all of which can be understood theoretically on the basis of the SBEs.
Macroscopically, Maxwell's equations show that in the absence of free charges and currents an electromagnetic field interacts with matter via the optical polarization
Microscopically, the optical polarization arises from quantum mechanical transitions between different states of the material system.
are the energies of the conduction and valence band states, their dynamic quantum mechanical evolution is according to the Schrödinger equation given by phase factors
is given by a summation over the microscopic transition dipoles which all oscillate with frequencies corresponding to the energy differences between the involved quantum states.
[6] These equations are named after Felix Bloch who formulated them in order to analyze the dynamics of spin systems in nuclear magnetic resonance.
They are, however, only well suited for systems with optical transitions between isolated levels in which many-body interactions are of minor importance as is sometimes the case in atoms or small molecules.
A prominent and important result of the Coulomb interaction among the photoexcitations is the appearance of strongly absorbing discrete excitonic resonances which show up in the absorption spectra of semiconductors spectrally below the fundamental band gap frequency.
The many-body Coulomb interaction leads to significant complications since it results in an infinite hierarchy of dynamic equations for the microscopic correlation functions that describe the nonlinear optical response.
Whereas this level is sufficient to describe excitonic resonances, there are several further effects, e.g., excitation-induced dephasing, contributions from higher-order correlations like excitonic populations and biexcitonic resonances, which require one to treat so-called many-body correlation effects that are by definition beyond the Hartree–Fock level.
The systematic truncation of the many-body hierarchy and the development and the analysis of controlled approximations schemes is an important topic in the microscopic theory of the optical processes in condensed matter systems.
For highly excited systems, it is often sufficient to describe many-body Coulomb correlations using the second order Born approximation.
In the limit of weak light intensities, signature of exciton complexes, in particular, biexcitons, in the coherent nonlinear response have been analyzed using the dynamics controlled truncation scheme.
[8][9] These two approaches and several other approximation schemes can be viewed as special cases of the so-called cluster expansion[10] in which the nonlinear optical response is classified by correlation functions which explicitly take into account interactions between a certain maximum number of particles and factorize larger correlation functions into products of lower order ones.
By nonlinear optical spectroscopy using ultrafast laser pulses with durations on the order of ten to hundreds of femtoseconds, several coherent effects have been observed and interpreted.
Such studies and their proper theoretical analysis have revealed a wealth of information on the nature of the photoexcited quantum states, the coupling among them, and their dynamical evolution on ultrashort time scales.
In particular, the consequences of many-body effects which depending on the excitation conditions may lead to, e.g., a coupling among different excitonic resonances via biexcitons and other Coulomb correlation contributions and to a decay of the coherent dynamics by scattering and dephasing processes, has been explored in many pump-probe and four-wave-mixing measurements.
[1][2][3] In nonlinear optics it is possible to reverse the destructive interference of so-called inhomogeneously broadened systems which contain a distribution of uncoupled subsystems with different resonance frequencies.
When photon echo experiments are performed in semiconductors with exciton resonances,[14][15][16] it is essential to include many-body effects in the theoretical analysis since they may qualitatively alter the dynamics.
For example, numerical solutions of the SBEs have demonstrated that the dynamical reduction of the band gap which originates from the Coulomb interaction among the photoexcited electrons and holes is able to generate a photon echo even for resonant excitation of a single discrete exciton resonance with a pulse of sufficient intensity.
[17] Besides the rather simple effect of inhomogeneous broadening, spatial fluctuations of the energy, i.e., disorder, which in semiconductor nanostructure may, e.g., arise from imperfection of the interfaces between different materials, can also lead to a decay of the photon echo amplitude with increasing time delay.
To consistently treat this phenomenon of disorder induced dephasing the SBEs need to be solved including biexciton correlations.
[18] such a microscopic theoretical approach is able to describe disorder induced dephasing in good agreement with experimental results.
In contrast to the optical Bloch equations, the SBEs including coherent biexcitonic correlations were able to properly describe the experiments performed on semiconductor quantum wells.
This effect of superradiance[23] has been demonstrated by monitoring the decay of the exciton polarization in suitably arranged semiconductor multiple quantum wells.