The interaction of matter with light, i.e., electromagnetic fields, is able to generate a coherent superposition of excited quantum states in the material. Coherent denotes the fact that the material excitations have a well defined phase relation which originates from the phase of the incident electromagnetic wave. 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.
This article focusses on coherent optical effects in semiconductors and semiconductor nanostructures. After an introduction into the basic principles, the semiconductor Bloch equations (abbreviated as SBEs)Schäfer, W.; Wegener, M. (2002). Semiconductor Optics and Transport Phenomena. Springer. .Haug, H.; Koch, S. W. (2009). Quantum Theory of the Optical and Electronic Properties of Semiconductors (5th ed.). World Scientific. .Meier, T.; Thomas, P.; Koch, S. W. (2007). Coherent Semiconductor Optics: From Basic Concepts to Nanostructure Applications (1st ed.). Springer. .Lindberg, M.; Koch, S. (1988). "Effective Bloch equations for semiconductors". Physical Review B 38 (5): 3342–3350. doi:10.1103/PhysRevB.38.3342Schmitt-Rink, S.; Chemla, D.; Haug, H. (1988). "Nonequilibrium theory of the optical Stark effect and spectral hole burning in semiconductors". Physical Review B 37 (2): 941–955. doi:10.1103/PhysRevB.37.941 which are able to theoretically describe coherent semiconductor optics on the basis of a fully microscopic many-body quantum theory are introduced. 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.
Starting point
Macroscopically, Maxwell's equations show that in the absence of free charges and currents an electromagnetic field interacts with matter via the optical polarization . The wave equation for the electric field reads and shows that the second derivative with respect to time of , i.e., , appears as a source term in the wave equation for the electric field . Thus, for optically thin samples and measurements performed in the far-field, i.e., at distances significantly exceeding the optical wavelength , the emitted electric field resulting from the polarization is proportional to its second time derivative, i.e., . Therefore, measuring the dynamics of the emitted field provides direct information on the temporal evolution of the optical material polarization .
Microscopically, the optical polarization arises from quantum mechanical transitions between different states of the material system. For the case of semiconductors, electromagnetic radiation with optical frequencies is able to move electrons from the valence () to the conduction () band. The macroscopic polarization is computed by summing over all microscopic transition dipoles via , where is the dipole matrix element which determines the strength of individual transitions between the states and , denotes the complex conjugate, and is the appropriately chosen system's volume. If and 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 and , respectively. The superposition state described by is evolving in time according to . Assuming that we start at with , we have for the optical polarization
.
Thus, 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. Clearly, the optical polarization is a coherent quantity which is characterized by an amplitude and a phase. Depending on the phase relationships of the microscopic transition dipoles, one may obtain constructive or destructive interference, in which the microscopic dipoles are in or out of phase, respectively, and temporal interference phenomena like quantum beats, in which the modulus of varies as function of time.
Ignoring many-body effects and the coupling to other quasi particles and to reservoirs, the dynamics of photoexcited two-level systems can be described by a set of two equations, the so-called optical Bloch equations.Allen, L.; Eberly, J. H. (1987). Optical Resonance and Two-Level Atoms. Dover Publications. . These equations are named after Felix Bloch who formulated them in order to analyze the dynamics of spin systems in nuclear magnetic resonance. The two-level Bloch equations read
and
Here, denotes the energy difference between the two states and is the inversion, i.e., the difference in the occupations of the upper and the lower states. The electric field couples the microscopic polarization to the product of the Rabi energy and the inversion . In the absence of the driving electric field, i.e., for , the Bloch equation for describes an oscillation, i.e., .
The optical Bloch equations enable a transparent analysis of several nonlinear optical experiments. 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. In solid state systems, such as semiconductors and semiconductor nanostructures, an adequate description of the many-body Coulomb interaction and the coupling to additional degrees of freedom is essential and thus the optical Bloch equations are not applicable.
The semiconductor Bloch equations (SBEs)
For a realistic description of optical processes in solid materials, it is essential to go beyond the simple picture of the optical Bloch equations and to treat many-body interactions that describe the coupling among the elementary material excitations by, e.g., the see article Coulomb interaction between the electrons and the coupling to other degrees of freedom, such as lattice vibrations, i.e., the electron-phonon coupling. Within a semiclassical approach, where the light field is treated as a classical electromagnetic field and the material excitations are described quantum mechanically, all above mentioned effects can be treated microscopically on the basis of a many-body quantum theory. For semiconductors the resulting system of equations are known as the semiconductor Bloch equations. For the simplest case of a two-band model of a semiconductor, the SBEs can be written schematically as
Here is the microscopic polarization and and are the electron occupations in the conduction and valence bands ( and ), respectively, and denotes the crystal momentum. As a result of the many-body Coulomb interaction and possibly further interaction processes, the transition energy and the Rabi energy both depend on the state of the excited system, i.e., they are functions of the time-dependent polarizations and occupations and , respectively, at all crystal momenta .
Due to this coupling among the excitations for all values of the crystal momentum , the optical excitations in semiconductor cannot be described on the level of isolated optical transitions but have to be treated as an interacting many-body quantum system.
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. Since an exciton consists of a negatively charged conduction band electron and a positively charged valence band hole (i.e., an electron missing in the valence band) which attract each other via the Coulomb interaction, excitons have a hydrogenic series of discrete absorption lines. Due to the optical selection rules of typical III-V semiconductors such as Galliumarsenide (GaAs) only the s-states, i.e., 1s, 2s, etc., can be optically excited and detected, see article on Wannier equation.
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. The terms given explicitly in the SBEs above arise from a treatment of the Coulomb interaction in the time-dependent Hartree–Fock approximation. 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. These contributions are formally included in the SBEs given above in the terms denoted by .
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. Depending on the particular system and the excitation conditions several approximations schemes have been developed and applied. For highly excited systems, it is often sufficient to describe many-body Coulomb correlations using the second order Born approximation.Jahnke, F.; Kira, M.; Koch, S. W.; Tai, K. (1996). "Excitonic Nonlinearities of Semiconductor Microcavities in the Nonperturbative Regime". Physical Review Letters 77 (26): 5257–5260. doi:10.1103/PhysRevLett.77.5257 Such calculations were, in particular, able to successfully describe the spectra of semiconductor lasers, see article on semiconductor laser theory. 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.Lindberg, M.; Hu, Y.; Binder, R.; Koch, S. (1994). "χ(3) formalism in optically excited semiconductors and its applications in four-wave-mixing spectroscopy". Physical Review B 50 (24): 18060–18072. doi:10.1103/PhysRevB.50.18060Axt, V. M.; Stahl, A. (1994). "The role of the biexciton in a dynamic density matrix theory of the semiconductor band edge". Zeitschrift für Physik B Condensed Matter 93 (2): 205–211. doi:10.1007/BF01316964 These two approaches and several other approximation schemes can be viewed as special cases of the so-called cluster expansionKira, M.; Koch, S. W. (2011). Semiconductor Quantum Optics. Cambridge University Press. . 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.
Selected coherent effects
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 the following, a few important effects are briefly described.
Quantum beats involving excitons and exciton complexes