We present experimental coherent two-dimensional Fourier-transform spectra of Wannier exciton resonances in semiconductor quantum wells generated by a pulse sequence that isolates two-quantum coherences. By measuring the real part of the signals, we determine that the spectra are dominated by two-quantum coherences due to mean-field many-body interactions, rather than bound biexcitons. Simulations performed using dynamics controlled truncation agree with the experiments.

We propose a three-pulse coherent ultrafast optical technique that is particularly sensitive to twoexciton correlations. Two Liouville-space pathways for the density matrix contribute to this signal which reveals double quantum coherences when displayed as a two-dimensional correlation plot. Two-exciton couplings spread the cross peaks along both axes, creating a characteristic highly resolved pattern. This level of detail is not available from conventional one-dimensional four-wave mixing or other twodimensional correlation spectroscopy signals such as the photo echo, in which two-exciton couplings show up along a single axis and are highly congested.Investigating the correlations of multiple excitons in semiconductors is a challenging many-body problem that had drawn considerable theoretical [1-5] and experimental [6,7] attention. Correlations of two excitons beyond the Hartree-Fock (HF) approximation may result in either a redshift [two-exciton binding energy (TBE)] or a blueshift [two-exciton scattering energy (TSE)]. In GaAs semiconductor quantum wells both couplings are a few meV s and may not be easily resolved. Two-exciton formation has been reviewed in Refs. [6,8]. Coherent ultrafast fourwave mixing (FWM) [6] provides a direct probe for twoexciton correlations in quantum wells. The best evidence for bound two excitons and the most accurate TBE in GaAs quantum wells are obtained by time integrated FWM (TIFWM), where signals are displayed as a function of a single (time or frequency) variable [6,9,10]. Quantum beats were observed in the negative-delay two-pulse signal along 2k b ÿ k a , where k b arrives first, and in the positivedelay three-pulse signal along k a k b ÿ k c (k c comes after k b ). Some attempts have been made to enhance the spectral resolution by displaying FWM signals versus two time variables [11,12].Multidimensional analysis of coherent signals is commonly used in NMR to study correlations between spins [13]. These techniques were recently extended to the femtosecond regime [14] and applied to several chemical and biological systems [15,16]. Three ultrashort laser pulses generate a signal which is heterodyne detected by a fourth pulse in a chosen phase-matching direction (Fig. 1, top left). Three time delays (t 1 , t 2 , and t 3 ) can be controlled between the chronologically ordered pulses, k 1 , k 2 , and k 3 , and the heterodyne pulse k s . For an excitonic system the signal can be generated along the phase-matching directions, ÿk 1 k 2 k 3 , k 1 ÿ k 2 k 3 , and k 1 k 2 ÿ k 3 . We denote these three techniques as S I , S II , and S III [14,17] respectively. The time-domain nonlinear response is given by combinations of multitime correlation functions which depend on the time delays t 1 , t 2 , and t 3 . Displaying the signal as a function of two time delays (or their conjugate frequencies) while holding the third fixed gives the two-dimensional correlation spectroscopy (2DCS) signals. Note that t 1 , t 2 , and t 3 are positive. This is different from conventional FWM where there is ...

IntroductionLinear-spectroscopy is one-dimensional (1D); the absorption spectrum provides information about excitation energies and transition dipoles as projected into a single frequency axis. In contrast, multidimensional optical spectroscopy uses sequences of laser pulses to perturb or label the electronic degrees of freedom and watch for correlated events taking place during several controlled time intervals. The resulting correlation plots can be interpreted in terms of multipoint correlation functions that carry considerably more detailed information on dynamical events than the two-point functions provided by 1D techniques [1][2][3][4][5][6][7] . Correlations between spins have been routinely used in NMR to study complex molecules. The Nobel prize was awarded to Richard Ernst 8 for inventing the technique and to Kurt Wüthrich 9 for developing pulse sequences suitable for large proteins. Optical analogues of 2D NMR techniques first designed to study vibrational dynamics by Raman or infrared pulses 1 and later extended to resonant electronic excitations in chromophore aggregates 10 have been made possible thanks to the development of stable femtosecond laser sources with controlled phases 11 . In an ideal heterodyne-detected 2D experiment ( Fig. 1) 3 laser pulses with wavevectors k 1 , k 2 , k 3 interact sequentially with the molecules in the sample to create a polarization with wavevector k 4 given by one of the linear combinations ±k 1 ±k 2 ±k 3 . In all other directors the polarization vanishes due to the random phases of contributions from different molecules. The coherent signal is generated in directions close to the various possible k 4 . The missmatch caused by frequency variation of the index of refraction is optimized ("phase matched") to generate an intense signal detected by interference with a 4th pulse at the desired wavevector k 4 . When the radiation field is described quantum mechanically the entire process can be viewed as a concerted 4 photon process. The signal S(t 3 ,t 2 ,t 1 ) depends parametrically on the time intervals between pulses which constitute the primary control-parameters. Other parameters include the direction k 4 , pulse polarizations, envelope shapes, and even the phases.We shall illustrate the power of 2D techniques and how they work using the three-band model system shown in Fig. 1 which has a ground state (g), a singly excited manifold (e) and a doubly excited manifold (f ). The dipole operator can induce transitions between g to e and e to f . All transitions in the system are stimulated: spontaneous emission is neglected. This three-band model represents electronic excitations in the various physical systems covered in the this article. Multidimensional signals monitor the dynamics of the system's density matrix during the time intervals between pulses. Diagonal elements of this matrix ρ nn represent populations of various states, while the off diagonal elements ρ nm (n ≠ m), known as coherences, carry additional valuable phase information. These signals can be descr...

We demonstrate how dynamic correlations of heavy-hole and light-hole excitons in semiconductor quantum wells may be investigated by two dimensional correlation spectroscopy (2DCS). The coherent response to three femtosecond optical pulses is predicted to yield cross (off-diagonal) peaks that contain direct signatures of manybody two-exciton correlations. Signals generated at various phase-matching directions are compared. I. INTRODUCTIONUnderstanding the signatures of many-body interactions in the nonlinear optical response of semiconductors is an important fundamental problem with implications to all-optical and electro-optical device applications. 1 The linear response to a weak optical field is well described by a model of non-interacting quasiparticles. However, residual interactions, not accounted for by these quasiparticles, can considerably affect the nonlinear response.Similar to Frenkel excitons in molecular crystals and aggregates, 2 Coulomb correlations among quasiparticles can dominate the nonlinear optical response of semiconductors, in marked contrast to the behavior of atomic systems. 3,4,5,6,7,8,9,10,11,12,13 The coherent ultrafast response and many-body correlations in semiconductor heterostructures have been studied extensively in the past two decades. 5,14,15,16,17,18,19,20,21,22,23,24,25 Due to various dephasing and relaxation mechanisms, the coherent response usually persists only on the tens of picosecond time scale.Optical spectra such as the linear absorption, pump-probe and Four-wave mixing (FWM) are commonly displayed as a function of a single (time or frequency) variable, and hence provide a one-dimensional (1D) projection of the microscopic information. 1D spectra are hard to interpret in systems with many congested energy levels. The spectroscopic signatures of complex many-body dynamics projected on a 1D spectral plot strongly overlap and may not be easily identified. For example, when 1D techniques are employed in III-V semiconductor quantum wells (QWs), it is difficult to pinpoint the signatures of. (11) Eq. (11) has various diagonal peaks (Ω 1 = Ω 3 ) and cross-peaks (Ω 1 = Ω 3 ). The relative contributions of different terms may be controlled by the carrier frequencies, ω 1 , ω 2 , and ω 3 .Spreading the signal in an extra frequency dimension enhances the resolving power of the 2DCS, compared to 1D techniques. 2 We can further improve the resolution by controlling other parameters such as the pulse polarization directions, carrier frequencies and envelopes.Other 2D techniques generated in different phase-matching directions and using different pairs of time variables (e.g. t 2 and t 3 ) provide complementary information 2,34,57 through different projections of the response, as will be discussed in Section III. Closed expressions for the other 2D signals S II and S III are given in Appendix A.

The correlated behavior of electrons determines the structure and optical properties of molecules, semiconductors, and other systems. Valuable information on these correlations is provided by measuring the response to femtosecond laser pulses, which probe the very short time period during which the excited particles remain correlated. The interpretation of four-wave-mixing techniques, commonly used to study the energy levels and dynamics of many-electron systems, is complicated by many competing effects and overlapping resonances. Here we propose a coherent optical technique, specifically designed to provide a background-free probe for electronic correlations in many-electron systems. The proposed signal pulse is generated only when the electrons are correlated, which gives rise to an extraordinary sensitivity. The peak pattern in two-dimensional plots, obtained by displaying the signal versus two frequencies conjugated to two pulse delays, provides a direct visualization and specific signatures of the many-electron wave functions.