We study many-body interactions between excitons in semiconductors by applying the powerful technique of optical two-dimensional Fourier transform spectroscopy. A two-dimensional spectrum correlates the phase (frequency) evolution of the nonlinear polarization field during the initial evolution and the final detection period. A single two-dimensional spectrum can identify couplings between resonances, separate quantum mechanical pathways, and distinguish among microscopic many-body interactions.

Optical 2D Fourier transform spectroscopy (2DFTS) provides insight into the many-body interactions in direct gap semiconductors by separating the contributions to the coherent nonlinear optical response. We demonstrate these features of optical 2DFTS by studying the heavy-hole and light-hole excitonic resonances in a gallium arsenide quantum well at low temperature. Varying the polarization of the incident beams exploits selection rules to achieve further separation. Calculations using a full many-body theory agree well with experimental results and unambiguously demonstrate the dominance of many-body physics.excitons ͉ many-body effects ͉ ultrafast O ptical excitation of a direct gap semiconductor, such as gallium arsenide (GaAs), produces electron-hole pairs. The Coulomb attraction between the electron and hole can result in a bound state, known as an exciton, with a hydrogenic wavefunction for the relative coordinate. Excitons have a large oscillator strength because of the proximity of the electron and hole and thus can dominate the absorption spectrum close to the fundamental band gap. In GaAs heterostructures, the exciton binding energy is of order 10 meV; thus, excitonic resonances appear only at low temperatures. Excitons and unbound electron-hole pairs exhibit dynamics on a femtosecond-to-picosecond time scale. These timescales, combined with the strong interaction with light, make ultrafast spectroscopy an ideal tool for studying carrier dynamics in semiconductors.Over the last two decades, excitonic resonances in semiconductors have been studied extensively by using ultrafast spectroscopy, primarily transient four-wave-mixing (TFWM) (1, 2). The measurements clearly showed signatures of many-body effects. The first and most prominent was a signal for the ''wrong'' time delay in a two-pulse TFWM experiment. Theoretically, such signals could arise from several effects including local fields (3, 4), biexcitons (5), excitation-induced dephasing (6, 7), or excitation-induced shift (8). Time resolving the signal also provided evidence for many-body contributions (9, 10), although it did not resolve the ambiguity regarding the underlying phenomena.Recent results using optical 2D Fourier transform spectroscopy (2DFTS) to study the exciton resonances have shown that much more information is obtained, promising a more stringent test of the theory (11). 2DFTS traces its roots to NMR (12). Recently, there has been significant progress in translating multidimensional NMR techniques into the infrared and optical domains for the study of vibrations (13) and electronic excitations (14-16) in molecules. Although the usefulness of adding a second dimension was recognized in TWFM studies of semiconductors (17-20), only the intensity of the emitted signal, not the phase-resolved electric field, was measured. The transient absorption experiments clearly show that detecting only the real part of the emitted field is advantageous (21, 22), but the effects of inhomogeneous broadening cannot be removed. 2DFTS combines the best...

The imaginary part of two-dimensional Fourier-transform spectra in the rephasing and nonrephasing modes is used to analyze the homogeneous and inhomogeneous broadening of excitonic resonances in semiconductor nanostructures. Microscopic calculations that include heavy-and light-hole excitons as well as coherent biexcitonic many-body correlations reveal distinct differences between the rephasing and nonrephasing spectra. A procedure is proposed that allows separation of disorder-induced broadening in complex systems that show several coupled resonances.In the past decades, various optical techniques have been used to investigate and unravel the structure of electronic states in semiconductor nanostructures and other material systems. 1-5 Spatially resolved linear optical measurements give information about homogeneous and inhomogeneous broadening separately. Typically, however, they provide only general information, e.g., the total linewidth. On the other hand, nonlinear experiments have been applied successfully to obtain much more detailed information about the nature of excited states, the coupling among them, and many-body effects. In addition, different nonlinear optical techniques were used to investigate the amounts of homogeneous and inhomogeneous broadening ͑see Ref. 5 and references therein͒.Pump-probe measurements provide one-dimensional spectral information that cannot distinguish between homogeneous and inhomogeneous broadening. Hole burning can find the homogeneous contribution to the optical linewidth and, by comparing to the linear spectrum, provides an estimate of the inhomogeneous contribution. Four-wave-mixing ͑FWM͒ experiments show photon echoes in the timeresolved ͑TR͒ traces. 5,6 Their temporal width is determined by the inhomogeneous linewidth. However, for systems where more than a single resonance is simultaneously excited, the width of the echo is ill defined due to beating, 5,7,8 in particular, for small inhomogeneous broadening see ͑Fig.1͒. The time-integrated ͑TI͒ trace yields the homogeneous width, i.e., the dephasing rate. However, in the presence of more than just a single optical resonance, the decay parameter cannot uniquely be determined and a fitting procedure is needed.In semiconductor nanostructures, many-body Coulomb interaction strongly alters the nonlinear optical response. [3][4][5]9,10 Even at the Hartree-Fock level, e.g., the line shape of time-resolved FWM is significantly modified and signals for the wrong time ordering appear. [3][4][5]11,12 Additionally, already in the low-intensity third-order ͓ ͑3͒ ͔ limit, characteristic dependencies of the nonlinear transients and spectra on the polarization directions of the incident pulses and couplings among optically isolated resonances appear due to many-body correlations. [3][4][5]9,[13][14][15][16][17] A detailed microscopic description of interacting excitons in the presence of disorder is a formidable task. Thus, wellestablished knowledge is lacking on this topic. It was, however, shown that Hartree-Fock renormalizations...

On the basis of a microscopic theory, the signatures of many-particle correlations in Two-Dimensional Fourier-Transform Spectra (2D-FTS) of semiconductor nanostructures are identified and compared to experimental data. Spectra in the photon energy range of the heavy-hole and light-hole excitonic resonances show characteristic features due to correlations, which depend on the relative polarization directions of the excitation pulses.Recent reports illustrate the potential of a novel method, known as "Two-Dimensional Fourier-Transform Spectroscopy" (2D-FTS), for the investigation of many-particle induced correlations in semiconductor structures [1,2,3]. 2D-FTS is based on a four-wave-mixing experiment, where three excitation pulses are separated in time and is heterodyne detected to fully characterize its phase. The signal is transformed into frequency domain ω t (emission frequency) and ω τ (excitation frequency), both with respect to real time t and time separation τ of the first two pulses, respectively. The third pulse is delayed with respect to the second one by T . 2D-FTS is widely used to study vibrational [4,5,6] and electronic excitations [7,8,9] in molecules. Applying 2D-FTS in the optical regime we investigate the complex interplay between exciton, biexciton and continuum excitations, in semiconductor nanostructures.In previous publications demonstrating 2D-FTS of semiconductors [1, 2, 3], the experimental results were compared to a phenomenological theory based on extending the Optical Bloch Equations to include terms that describe excitation induced dephasing (EID) [10,11] and excitation induced shift (EIS) [12]. Here, we use a microscopic theory to calculate 2D-FTS. Furthermore, the model predicts that the 2D-FTS qualitatively depend on the polarizations directions of the excitation pulses. Our results demonstrate that for 2D-FTS, the influence of many-particle correlations on the spectra can clearly be identified and are in agreement with the experimental findings.For the qualitative modeling of 2D-FTS in such systems and in order to keep the numerical requirement within reasonable limits we use a microscopic many-body theory and apply it to a one-dimensional tight-binding model. Therefore, a quantitative agreement between theory and experiment can not be expected. It has, however, been shown that many important signatures of nonlinear optical experiments performed on quantum wells can qualitatively well be reproduced by this model [13,14,15,16,17,18,19].The optical nonlinearities are treated up to third order in the coherent χ (3) -limit beyond the Hartree-Fock level. In order to separate the correlation effects from those due to the first-order Coulomb interaction (i.e., Hartree-Fock) we use a set of equations for the interband coherence P and two-exciton amplitudeB which in symbolic form reads [14,15,20,21] In fact, all terms in the microscopic version of the equations depend on the spatial indices of the tight-binding model as well as band indices, referring to electron, heavy-hole (h) and light-...

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