## Abstract

The motions of electrons in solids may be highly correlated by strong, long-range Coulomb interactions. Correlated electron-hole pairs (excitons) are accessed spectroscopically through their allowed single-quantum transitions, but higher-order correlations that may strongly influence electronic and optical properties have been far more elusive to study. Here we report direct observation of bound exciton pairs (biexcitons) that provide incisive signatures of four-body correlations among electrons and holes in gallium arsenide (GaAs) quantum wells. Four distinct, mutually coherent, ultrashort optical pulses were used to create coherent exciton states, transform these successively into coherent biexciton states and then new radiative exciton states, and finally to read out the radiated signals, yielding biexciton binding energies through a technique closely analogous to multiple-quantum two-dimensional Fourier transform (2D FT) nuclear magnetic resonance spectroscopy. A measured variation of the biexciton dephasing rate indicated still higher-order correlations.

Electron correlations play a predominant role in nearly all forms of matter, influencing the distinctive macroscopic properties of superconductors and giant magnetoresistance materials, the collective modes of high-density plasmas, and the orbital structure and energetics of atoms and molecules. Many-body electron correlations strongly influence the electronic and optical properties of confined systems such as semiconductor quantum wells (QWs) and quantum dots at excited-state densities (~10^{10} excitations/cm^{2} in QWs or >1 excitation in single quantum dots) that are reached routinely in applications and experimental studies. Even under conditions of mild irradiation by sunlight, multiple-electron correlations may play important roles in natural photosynthetic antenna complexes, which appear to have evolved rapid relaxation pathways to avert damage under highly energized conditions (*1*), and in semiconductor quantum dots, in which there is hope of extracting extra energy from above-bandgap light absorption through multiple-exciton states (*2*).

Semiconductor QWs are ideally suited for the study of many-electron correlations because of their reduced dimensionality and the fact that the interacting carriers can be generated optically at precisely specified times and densities. Femtosecond laser spectroscopy has been used extensively to study strongly correlated, bound electron-hole pairs (quasiparticles called excitons) in QWs by directly observing exciton coherences induced by ultrashort optical fields and by measuring the dynamics of coherence loss (dephasing) and exciton population lifetimes (*3*). Higher-order correlations were revealed indirectly through measured changes in exciton properties as a function of excitation density, described phenomenologically as local-field effects (*4*, *5*), excitation-induced dephasing (*6*), and excitation-induced energy shifts (*7*). Demanding first-principles calculations that include the full complement of electron-electron interactions and correlations were also performed to understand these correlations (*8*).

The clearest experimental signature of higher-order correlations would be direct observation of the coherences and dynamics of biexcitons [four-particle (or two-quasiparticle) correlations] and higher-order correlated motions. The biexciton is a distinct bound state whose wavefunction is the symmetric combination of the two constituent exciton wavefunctions. The coherent behavior of biexcitons is particularly interesting because they play a key role in coherent control schemes (*9*), quantum information processing (*10*), the manipulation of exciton spin coherences (*11*), and electromagnetically induced transparency in semiconductors (*12*). Biexciton–ground-state coherences generally do not radiate because the corresponding two-quantum transitions are formally forbidden. The exciton-biexciton absorption band that appears at high exciton density can reveal biexciton energetics and some dephasing information (though not that of the biexciton–ground-state two-quantum coherence) if it can be isolated spectrally; that is, if the biexciton binding energy is larger than the low-temperature exciton linewidth (~1 meV). Otherwise, as in the gallium arsenide (GaAs) prototype system, it appears as no more than a shoulder on the much stronger ground-state–exciton absorption band. Measurements in the time domain (*13*, *14*) are just as convolved as in the frequency domain, because coherent signals from both the exciton and biexciton decay because of dephasing within a few picoseconds.

High-order nuclear spin coherences have been isolated through multiple-quantum two-dimensional Fourier transform nuclear magnetic resonance (2D FT NMR) spectroscopy (*15*). As shown in fig. S3 and the supporting online material (SOM) text, dipolar interactions between nearby spins (coherently precessing at frequency ω_{s}) lead to modulations of the local magnetic field at twice the precession frequency and to corresponding two-quantum signals that appear at frequency 2ω_{s} along one of the dimensions of multiple-pulse 2D FT NMR spectra. Signals at such frequencies arise only from interacting spins and are isolated spectrally from one-quantum signals at ω_{s} (*16*). Two-quantum NMR spectra have been used to elucidate the structures of large molecules whose 1D spectra were too crowded with overlapping peaks to allow unambiguous interpretation. Multiple-quantum NMR techniques also permit examination of correlations between spins on different molecules in a liquid mixture, providing information about liquid-state structure and dynamics (*17*, *18*).

In contrast to the NMR case, in which well-understood multiple-quantum spin coherences are exploited to simplify complicated spectra, multiple-quantum optical coherences are of strong interest in their own right because they provide access to spectroscopically “dark” states whose characterization may reveal much about poorly understood high-order correlations in condensed matter systems (*19*). Some of the origins of many-body correlations in semiconductors and the multiple-quantum signals that arise from them in coherent nonlinear spectroscopy measurements are illustrated in fig. S3 and discussed in the SOM text. Two successive optical fields, labeled **E _{A}** and

**E**in Fig. 1, induce coherent responses at the exciton frequency ω

_{B}_{e}; and as in the case of spins, interactions between excitons lead to coherences at twice the exciton frequency, 2ω

_{e}. These interaction-induced coherences, due to local-field effects and excitation-induced dephasing and energy shifts as mentioned above, can be explained by a two-level model of exciton dynamics in which interactions have been added phenomenologically (SOM text). Unlike pairs of coupled nuclear spins, the two-quantum exciton coherences involve four particles that can adopt new time-averaged locations, forming a biexciton state whose energy

*E*

_{b}=

*ħ*ω

_{b}= 2 ε

_{e}– ε

_{B}is minimized at a value lower than twice the exciton energy by a binding energy ε

_{B}. This is reflected directly in the biexciton coherence oscillation frequency ω

_{b}= 2ω

_{e}– ε

_{B}/

*ħ*. The two-quantum interaction-induced and biexciton coherences launched by the first two fields do not radiate, but a third field,

**E**, can induce a radiative one-quantum coherence whose amplitude and phase depend on the phase of the third field relative to the phase of the two-quantum coherence that it encounters. The equivalent of rotating frame detection in NMR (SOM text) can be executed if the phase of a selected carrier frequency component, ω

_{C}_{0}, contained within the spectral bandwidth of the pulses and distinct from the exciton frequency ω

_{e}, can be held equal in all the fields while varying the relative pulse delays. In that case, the phase evolution of the complex signal field

**E**(ω), measured as a function of the time delay τ

_{S}_{2}between the first two pulses and the third, will be slowed to the difference frequency ω

_{e}– ω

_{0}.

At optical frequencies, the emitted signal field **E _{S}** can be determined through its interferometric superposition with a reference field,

**E**. A convenient strategy (

_{R}*20*) is to send the superposed fields into a spectrometer (spectral interferometry) to directly determine the signal field amplitude and phase as a function of emission frequency, ω. This circumvents the need to scan the reference field time delay to determine the

**E**(ω). After Fourier transformation with respect to τ

_{S}_{2}, the results will show distinct two-quantum features that appear along the ω

_{2}axis at frequencies 2ω

_{e}and ω

_{b}and along the ω axis at the one-quantum emission frequency ω

_{e}.

This scenario shows how two-quantum signals could be observed through an optical analog to two-quantum 2D FT NMR; that is, two-quantum 2D FT optical spectroscopy (2D FT OPT). However, this analog faces many experimental challenges not found in magnetic resonance. In NMR, the radiofrequency (rf) wavelengths exceed the sample dimensions, and the sample is surrounded by the coils that deliver the fields and measure the responses. Therefore, the rf field frequencies and polarizations are important, but their propagation directions are not. In contrast, in optical measurements the sample is large compared to the wavelength, and the fields are delivered to the sample and radiated from it in the form of coherent light beams with well-defined propagation directions (wavevectors). This difference presents both opportunities and challenges.

The key opportunities lie in the use of the noncollinear BOXCARS four-wave mixing (FWM) beam geometry, shown in Fig. 1, and the specification of the pulse time-ordering to isolate the emitted signal field as a coherent beam in a well-defined direction with wavevector **k _{S}** =

**k**=

_{R}**k**+

_{A}**k**–

_{B}**k**. Many classes of one-quantum measurements have been conducted with this geometry. For example, time-coincident fields

_{C}**E**and

_{A}**E**produce an excited-state population in a transient grating pattern (with wavevector

_{C}**k**–

_{A}**k**), which yields a coherently scattered signal field

_{C}**E**if probed by field

_{S}**E**(incident at the phase-matching angle for diffraction) within the excited-state lifetime. Alternatively, field

_{B}**E**can arrive first to generate an electronic coherence. A transient population grating forms upon the arrival of field

_{C}**E**, and field

_{A}**E**generates a new coherence whose phase evolution is in reverse of the first, resulting in rephasing of different frequency components (that may originate from within a single inhomogeneously broadened transition or from distinct transitions) to generate a photon echo signal,

_{B}**E**, that is free of inhomogeneous dephasing. If two distinct transitions interact, then the signal field components at each frequency will be modulated at the other frequency. Interferometric measurement and subsequent Fourier transformation of the signal field yield a one-quantum 2D spectrum that shows diagonal peaks due to each individual rephased coherence and off-diagonal cross-peaks that reveal the interacting coherences. One-quantum 2D FT OPT measurements have revealed key insights into coupled chromophores in photosynthetic antenna systems (

_{S}*21*) and semiconductor QWs (

*22*). Unlike one-quantum 2D FT OPT measurements using collinear phase-controlled pulses (

*23*), in which only the absorptive part of the signal is detected, it is through isolation and detection of the full signal field that the full optical analog to 2D FT NMR is realized (

*24*).

In terms of the photon echo and transient grating measurements described above, our measurements in which **E _{A}** and

**E**arrive first at the sample should not yield any signal at all. Unlike the previous cases, a population grating (with difference wavevector

_{B}**k**–

_{A}**k**) formed by the first two fields does not give rise to coherent scattering of the third field,

_{B}**E**, in the signal direction. However, generation of a two-quantum coherence at the sum wavevector

_{C}**k**+

_{A}**k**followed by a transition induced by field

_{B}**E**to a one-quantum coherence at wavevector

_{C}**k**+

_{A}**k**–

_{B}**k**does yield coherent emission in the signal direction. This could only result from a nonlinear response due to interactions between excitons generated by the first two fields, as described above. Thus, the BOXCARS geometry with proper time-ordering of the fields isolates two-quantum signals and eliminates the stronger one-quantum signals.

_{C}Two-quantum signals were first observed in measurements with just two incident beams, with fields **E _{A}** and

**E**combined. In addition to the expected photon echo signal that was observed when

_{B}**E**arrived at the sample first, a signal appeared at “negative” delay times; that is, when the combined

_{C}**E**and

_{A}**E**fields were incident at the sample first (

_{B}*25*). The many-body origin of this signal was quickly recognized. However, in this 1D measurement of signal intensity versus

**E**delay time, the various two-quantum contributions were mixed. In addition, the inability to independently control the polarizations of fields

_{C}**E**and

_{A}**E**imposed severe limitations on two-quantum signal selectivity, as we will see below. The present observations are distinct from those associated with one-quantum exciton-biexciton coherences observed in earlier one-quantum 2D FT OPT measurements as partially resolved shoulders on the excitonic peaks (

_{B}*22*,

*26*). Our measurements not only track the two-quantum signal phase evolutions at optical frequencies but also correlate them to optical one-quantum coherences, unlike previous frequency-domain FWM experiments on single quantum dots (

*27*) and time-integrated FWM experiments on QWs (

*28*,

*29*). Our experiments separate the two-quantum coherences that arise from multi-exciton interactions, allowing the phenomena to be studied even when their signatures cannot be separated spectrally.

The main experimental challenge presented by wavevector definition in the optical regime lies in the difficulty of producing multiple beams of light with pulses whose optical phases are specified and maintained even when the pulses are variably delayed in time-resolved measurements. None of the one-quantum measurements described above required all four optical fields to be phase-coherent, because after the first two field interactions, the system had electronic excited-state population but no optical-frequency coherent superposition between the ground and excited states. The first two fields needed to be phase-related, and the third field and the reference field needed to be as well, but no well-defined phase relationship was needed between the two pulse pairs. Even this partial phase stability among the four fields presents challenges, and only a handful of research groups worldwide have conducted optical 2D FT OPT measurements of this sort with reflective or diffractive beam-splitting optics in order to produce the pulse pairs, and with two interferometers to measure the required phase relationships, which change each time one of the pulse delays is varied. In contrast, a larger community has studied coupled vibrational transitions through the infrared (IR) analog, 2D FTIR, because it is easier to maintain the phase relationships at the longer IR wavelengths. Although two-quantum 2D FTIR of molecular vibrational overtones has been demonstrated (*30*, *31*), a comparable measurement in the visible region is far more demanding.

Recently, we showed that full phase coherence among all the fields could be accomplished through spatiotemporal pulse shaping, in which the optical fields of multiple beams of light are specified (*32*). The pulse shaper (Fig. 1 and fig. S1) controls the timing and optical phases of all pulses in each beam, so that coherent multiple-field excitation of a sample and interferometric readout of the emitted signal field are possible. The phase of a selected reference frequency, ω_{0}, can be kept constant among all the fields, permitting rotating frame detection of the signal (SOM text and fig. S2). In the present work, we exploited these capabilities to conduct two-quantum 2D FT OPT measurements (*33*), which revealed biexciton coherences and their dynamics in GaAs QWs.

The origin and energy-level diagram of the different exciton states, labeled heavy-hole exciton (HX), light-hole exciton (LX), and biexciton (X_{2}) states in GaAs QWs are shown in Fig. 2, A to C. The optical selection rules for the two excitons differ with respect to circular polarization of the light (*34*). Because biexcitons are formed by the correlations of opposite-spin excitons, cross-circular polarization of fields **E _{A}** and

**E**will excite pure HX

_{B}_{2}and LX

_{2}biexciton coherences, whereas co-circular polarization will excite mixed (MX

_{2}) biexciton coherences. The sample response of primary interest is illustrated by the double-sided Feynman diagram in Fig. 2D(a), in which the sequence of coherences and the fields that induce them are shown. For instance, the HX

_{2}biexciton coherence was observed when the first circularly polarized resonant field

**E**produced an HX–ground-state coherence, with wavevector

_{A}**k**, that evolved during τ

_{A}_{1}, and the second oppositely circularly polarized field

**E**converted the coherence to a two-quantum HX

_{B}_{2}–ground-state coherence, with wavevector

**k**+

_{A}**k**, that underwent coherent oscillations during τ

_{B}_{2}. A third circularly polarized field,

**E**, regenerated an HX–ground-state coherence with wavevector

_{C}**k**+

_{A}**k**–

_{B}**k**, which during time

_{C}*t*radiated the signal field

**E**that was overlapped spatially with the oppositely circularly polarized reference field

_{S}**E**and whose amplitude and phase were determined through spectral interferometry, as described above. Another signal contribution expected from biexciton–ground-state coherences induced by the first two fields is indicated in Fig. 2D(b). In this case, the third field generated a new exciton coherence from the ground state. Signal was radiated through an exciton-biexciton (HX-HX

_{R}_{2}) coherence whose frequency ω

_{eb}is expected to be red-shifted from the exciton coherence frequency ω

_{e}because of the biexciton binding energy, ω

_{eb}= ω

_{e}– ε

^{eb}_{B}/

*ħ*, where we have superscripted the binding energy for reasons that will become evident.

The interaction-induced two-quantum coherences at frequency 2ω_{e} (with no binding energy) cannot be described by standard double-sided Feynman diagrams, because the diagrams indicate distinct states and optical transitions among them but do not explicitly indicate interactions or correlations among states. A recently developed formalism has been advanced to include many-body correlations in QWs (*35*, *36*). These signals are discussed in more detail in the SOM text.

The emission spectrum measured in the phase-matched direction with time-coincident excitation fields is plotted in Fig. 3A with the excitation laser spectrum. The HX and LX exciton transition frequencies are 372.2 and 373.8 THz (1.539 and 1.546 eV), respectively. The 2D surface obtained with cross-circular excitation (Fig. 3B) shows the HX_{2} biexciton coherence. (The LX_{2} biexciton feature is too weak to be seen here because the LX transition dipole is one-third that of the HX.) Detection by spectral interferometry gives the signal field as a function of the absolute emission frequency, so we have subtracted the carrier frequency (368.00 ± 0.04 THz) to represent both frequency axes in the rotating frame. The HX_{2} peak (*a*) appears at the HX emission frequency along the ω axis (4.270 ± 0.003 THz higher than the carrier frequency), but at 8.32 ± 0.01 THz along the ω_{2} axis. This ω_{2} value is less than twice the HX frequency, yielding a biexciton binding energy of ε_{B} = 0.9 ± 0.2 meV (*37*). The exciton-biexciton coherence (*b*) appears as a shoulder of the biexciton peak red-shifted slightly in emission frequency ω. A two-Lorentzian least-squares fit to peaks *a* and *b* as functions of ω gives ω_{e} and ω_{eb}, respectively. The difference yields a biexciton binding energy of ε_{B}* ^{eb}* = 1.5 ± 0.1 meV. The binding energy obtained in this manner is in closer agreement with previously observed values (

*38*–

*40*) that also were obtained from exciton-biexciton coherences. The values ε

_{B}and ε

_{B}

^{eb}differ because they are derived from different correlation terms. A simple definition of the binding energy is the energy difference between uncorrelated and correlated four-particle states, which corresponds to ε

_{B}. However, the value ε

_{B}

^{eb}depends on the exciton-biexciton transition, which weights subsets of the correlated two-particle (exciton) and four-particle (biexciton) states that may not reflect the precise energies of either relative to the ground state.

The linewidth of a Lorentzian least-squares fit to peak *a* along ω_{2} yields an HX_{2} biexciton dephasing time of 2.23 ± 0.07 ps. Figure 4 shows a previously undiscovered dependence of the biexciton dephasing time on the excitation density. The observed excitation-induced biexciton dephasing effects can be ascribed to six-particle correlations (interactions between biexcitons and excitons). Biexciton interactions with free electrons and holes should be negligible because the excitation pulses are detuned from the free carrier absorption band. Correlations of this order have been observed in previous nonlinear spectroscopy measurements (*41*), but only in a highly convolved manner without isolation of their distinct effects. The biexciton binding energy was found to be essentially independent of excitation density in the present study.

Measurements conducted at the same excitation density with co-circularly polarized excitation are shown in Fig. 3C with features denoted *c* to *f*. The MX_{2} mixed biexciton ground-state coherence (*c*), which represents correlations between the HX and LX excitons, is observed directly. Its position on ω_{2} is 9.72 ± 0.02 THz, indicating a mixed biexciton binding energy of 1.42 ± 0.41 meV. This is larger than the HX_{2} binding energy, consistent with the expectation that the binding energy should increase with decreasing electron-hole effective mass ratio (*42*) because the LX has a larger effective mass in the unconfined dimension of the QW. The MX_{2} dephasing time is 1.46 ± 0.07 ps and, like the HX_{2}, it decreases with increasing excitation density (Fig. 4). A red-shifted exciton-biexciton emission (*d*) appears, analogous to *b* in Fig. 3B, but it is not sufficiently distinct to yield an accurate frequency or binding energy.

The mixed biexcitons are found to emit only at the LX frequency, although emission at the HX frequency is also expected. This discrepancy is likely due to interference with emission at the same frequency (*43*) from feature *e*, centered at 8.57 ± 0.02 THz (twice the HX frequency) in ω_{2}, which is solely the result of interaction-induced correlations that cannot be described in terms of double-sided Feynman diagrams, as discussed earlier. There is also a weak interaction-induced coherence (*f*) at twice the LX transition frequency. Deconvolved two-quantum coherence transients for the HX_{2}, MX_{2}, and HX interaction-induced effects can be extracted from Fig. 3 through Fourier filtering (fig. S4). In contrast to conventional FWM experiments (fig. S5), the phase sensitivity and spectral separation of two-quantum 2D FT OPT provide a clear picture of the higher-order Coulomb correlations involved.

The two-quantum 2D FT OPT measurements presented here represent a decisive step in the isolation and elaboration of many-body interactions in the prototype GaAs system that cannot be treated using a mean-field approximation. Comprehensive analysis of the real and imaginary parts of the complex spectra will provide further guidance for first-principles calculations of the correlated electronic responses. In quantum dots, observation of multiple-quantum coherences and populations from biexcitons and higher-lying states will have additional importance in laser gain, solar energy conversion, and other applications. Our present results demonstrate the ability of fully coherent multidimensional optical spectroscopy to access “dark” states using multiple-photon transitions and the simplicity of the spatiotemporal pulse shaping approach for the execution of otherwise daunting measurements. The same apparatus and measurements can be used for feedback-directed quantum control over the coherences.

## Supporting Online Material

www.sciencemag.org/cgi/content/full/324/5931/1169/DC1

Materials and Methods

SOM Text

Figs. S1 to S5

References

## References and Notes

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Materials and methods are available as supporting material on
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- ↵ The uncertanity in the binding energy includes the 95% confidence interval for the Lorentzian least-squares fitting parameters and the uncertainty in the frequency resolution of the scanned coordinate, which is related to the spectral dispersion over one pixel of the 2D spatial light modulator (SLM).
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- This work was supported in part by NSF grant CHE-0616939. The authors thank M. Kira for helpful discussions. D.T. thanks the National Defense Science and Engineering Graduate Fellowship Program for financial support. X.L. acknowledges support from the Army Research Office and the Welch Foundation. A patent application for the optical setup presented here is currently being filed.