(b) Write out the equations for the time dependence of ρ11, ρ22, ρ12 and ρ21 assuming that a light field interaction V = -μ12Ee iωt + c.c. couples only levels |1> and |2>, and that the excited levels exhibit spontaneous decay. (8 marks) (c) Under steady-state conditions, find the ratio of populations in states |2> and |3>. (3 marks) (d) Find the slowly varying amplitude ̃ ρ 12 of the polarization ρ12 = ̃ ρ 12e iωt . (6 marks) (e) In the limiting case that no decay is possible from intermediate level |3>, what is the ground state population ρ11(∞)? (2 marks) 2. (15 marks total) In a 2-level atom system subjected to a strong field, dressed states are created in the form |D1(n)> = sin θ |1,n> + cos θ |2,n-1> |D2(n)> = cos θ |1,n> sin θ |2,n-1>

The Quantum Theory of Light

Predicting the electronic structure of extended organic molecules constitutes an important fundamental task of modern chemistry. Studies of electronic excitations, charge-transfer, energy-transfer, and isomerization of conjugated systems form the basis for our understanding of the photophysics and photochemistry of complex molecules1-3 as well as organic nanostructures and supramolecular assemblies.4,5 Photosynthesis and other photochemical biological processes that constitute the basis of life on Earth involve assemblies of conjugated chromophores such as porphyrins, chlorophylls, and carotenoids.6-8 Apart from the fundamental interest, these studies are also closely connected to numerous important technological applications.9 Conjugated polymers are primary candidates for new organic optical materials with large nonlinear polarizabilities.10-19 Potential applications include electroluminescence, light emitting diodes, ultrafast switches, photodetectors, biosensors, and optical limiting materials.20-27 Optical spectroscopy which allows chemists and physicists to probe the dynamics of vibrations and electronic excitations of molecules and solids is a powerful tool for the study of molecular electronic structure. The theoretical techniques used for describing spectra of isolated small molecules are usually quite different from those of molecular crystals, and many intermediate size systems, such as clusters and polymers, are not readily described by the methods developed for either of these limiting cases.28

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Density Matrix Analysis and Simulation of Electronic Excitations in Conjugated and Aggregated Molecules

The authors review the physics of ultrafast dynamics in semiconductors and their heterostructures, including both the observed experimental phenomena and the theoretical description of the processes. These are probed by ultrafast optical excitation, generating nonequilibrium states that can be monitored by time-resolved spectroscopy. Light pulses create coherent superpositions of states, and the dynamics of the associated phase relationships can be directly investigated by means of many-pulse experiments. The commonly used experimental techniques are briefly reviewed. A variety of different phenomena can be described within a common theoretical framework based on the density-matrix formalism. The important interactions of the carriers included in the theoretical description are the phonon interactions, the interactions with classical and quantum light fields, and the Coulomb interaction among the carriers themselves. These interactions give rise to a strong interplay between phase coherence and relaxation, which strongly affects the non equilibrium dynamics. Based on the general theory, the authors review the physical phenomena in various semiconductor structures including superlattices, quantum wells, quantum wires, and bulk media. Particular results which have played a central role in understanding the microscopic origins of the relaxation processes are discussed in detail.

Theory of ultrafast phenomena in photoexcited semiconductors

Molecular aggregates are abundant in nature; they form spontaneously in concentrated solutions and on surfaces and can be synthesized by supramolecular chemistry techniques.1-3 Assemblies of chromophores play important roles in many biological processes such as light-harvesting and primary * To whom correspondence should be addressed. E-mail: smukamel@uci.edu. † University of California Irvine. ‡ Universität Würzburg. § Charles University. Chem. Rev. 2009, 109, 2350–2408 2350

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Coherent Multidimensional Optical Spectroscopy of Excitons in Molecular Aggregates; Quasiparticle versus Supermolecule Perspectives

Low-dimensional (LD) materials demonstrate intriguing optical properties, which lead to applications in diverse fields, such as photonics, biomedicine and energy. Due to modulation of electronic structure by the reduced structural dimensionality, LD versions of metal, semiconductor and topological insulators (TIs) at the same time bear distinct nonlinear optical (NLO) properties as compared with their bulk counterparts. Their interaction with short pulse laser excitation exhibits a strong nonlinear character manifested by NLO absorption, giving rise to optical limiting or saturated absorption associated with excited state absorption and Pauli blocking in different materials. In particular, the saturable absorption of these emerging LD materials including two-dimensional semiconductors as well as colloidal TI nanoparticles has recently been utilized for Q-switching and mode-locking ultra-short pulse generation across the visible, near infrared and middle infrared wavelength regions. Beside the large operation bandwidth, these ultrafast photonics applications are especially benefit from the high recovery rate as well as the facile processibility of these LD materials. The prominent NLO response of these LD materials have also provided new avenues for the development of novel NLO and photonics devices for all-optical control as well as optical circuits beyond ultrafast lasers.

Emerging Low-Dimensional Materials for Nonlinear Optics and Ultrafast Photonics.

A unified microscopic theoretical framework for the calculation of optical excitations in molecular and semiconductor materials is presented. The hierarchy of many-body density matrices for a pair-conserving many-electron model and the Frenkel exciton model is rigorously truncated to a given order in the radiation field. Closed equations of motion are derived for five generating functions representing the dynamics up to third order in the laser field including phonon degrees of freedom as well as all direct and exchange-type contributions to the Coulomb interaction. By eliminating the phonons perturbatively the authors obtain equations that, in the case of the many-electron system, generalize the semiconductor Bloch equations, are particularly suited for the analysis of the interplay between coherent and incoherent dynamics including many-body correlations, and lead to thermalized exciton (rather than single-particle) distributions at long times. A complete structural equivalence with the Frenkel exciton model of molecular materials is established. [S0034-6861(98)00201-3]

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Nonlinear optics of semiconductor and molecular nanostructures; a common perspective

Four-wave-mixing signals from excitons under linear-circular polarized excitation exhibit an elliptical polarization, depending on both the pulse delay and the spectral position. Besides the resonances corresponding to excitons and exciton-biexciton transitions, a breakup of the exciton line is found reflecting the influence of correlations on the four-point level. An analysis accounting for the exciton density, the bound biexciton, and the exciton-exciton scattering continuum reveals that these features are not due to an antibound two-exciton state. Instead, they result from an interference of the correlated exciton continuum with the exciton density. In addition, the modeling shows that the signals ellipticity is highly sensitive to the influences of different correlations and, therefore, allows for a discrimination of their contributions.

Identification of Higher-Order Electronic Coherences in Semiconductors by their Signature in Four-Wave-Mixing Signals

The Exciton–Exciton Continuum and Its Contribution to Four‐Wave Mixing Signals

The polarization state of the diffracted beam in degenerate four-wave mixing is experimentally and theoretically investigated, resolved in the time as well as in the frequency domain. The use of thin ZnSe quantum wells having large (bi)exciton binding energies makes it possible to restrict the optical excitation by broadband 110 fs pulses to quasi-two-dimensional heavy-hole excitons and biexcitons only. Despite these most simple experimental conditions, an extremely complex dynamics of the polarization state is observed when one circularly and one linearly polarized beam are used for excitation. The detailed behavior sensitively depends on the excitation density, showing the importance of nonlinearities of higher than third order even at exciton densities less than 0.01 times the Mott density. A comparison of the experimental data with theoretical results obtained by a microscopic density-matrix approach yields very close agreement. It turns out that a dynamical object describing fluctuations of the exciton amplitude has to be included in the theory in addition to biexcitonic contributions and correlated parts of the exciton-exciton scattering continuum.

Manifestation of exciton-amplitude fluctuations in the transient polarization state of four-wave-mixing signals

A generating function algorithm that allows the calculation of the optical response of coupled exciton-phonon systems is developed. For a model of assemblies of three-level molecules coupled via dipole interaction and interacting linearly with nuclear degrees of freedom, we derive a closed set of equations of motion for five generating functions representing the exact response to third order in the external field. These are equivalent to an infinite hierarchy of equations of motion for phonon-assisted variables. Starting with the equations for the generating functions, several reduction schemes are derived. By eliminating the phonon degrees of freedom in favor of self-energies, the Haken-Strobl model of relaxation is recovered as a limiting case. A set of time-local equations is presented extending the Haken-Strobl treatment by keeping the temperature dependence as well as the excitonic signatures of the phonon self-energies. Finally, we derive equations that interpolate between the coherent and incoherent limits of exciton propagation and properly include the two exciton dynamics.

Vollrath M. Axt

Papers

Nonlinear optics of semiconductor and molecular nanostructures; a common perspective

Identification of Higher-Order Electronic Coherences in Semiconductors by their Signature in Four-Wave-Mixing Signals

The Exciton–Exciton Continuum and Its Contribution to Four‐Wave Mixing Signals

Manifestation of exciton-amplitude fluctuations in the transient polarization state of four-wave-mixing signals

Influence of a Phonon Bath on Electronic Correlations and Optical Response in Molecular Aggregates