Strong, long-range Coulomb interactions can lead to correlated motions of multiple charged particles, which can induce important many-body effects in semiconductors. The exciton states formed from correlated electron–hole pairs have been studied extensively1, 2, but basic properties of multiple-exciton correlations—such as coherence times, population lifetimes, binding energies and the number of particles that can be correlated—are largely unknown because they are not spectroscopically accessible from the ground state. Here we present direct observations of high-order coherences in gallium arsenide quantum wells, achieved using two-dimensional multiple-quantum spectroscopy methods in which up to seven successive light fields were used. The measurements were made possible by the combination of a reconfigurable spatial beam-shaper that formed multiple beams in specified geometries and a spatiotemporal pulse-shaper that controlled the relative optical phases and temporal delays among pulses in all the beams. The results reveal triexciton coherences (correlations of three excitons or six particles), whose existence was not obvious because the third exciton spin is unpaired, and the values of their coherence times and binding energies. Rephasing of biexcitons, triexcitons and unbound two-exciton coherences was demonstrated. We also determined that there are no significant unbound correlations of three excitons and no bound or unbound four-exciton (eight-particle) correlations. Thus, the limits, as well as the properties, of many-body correlations in this system were revealed. The measurement methods open a new window into high-order many-body interactions in materials and molecules3, and the present results should guide ongoing work on first-principles calculations of electronic interactions in semiconductor nanostructures4.
Figures at a glance
Figure 1: GaAs exciton and multiexciton states observed or sought in this work. 
a, Exciton ladder illustrating the ground state (|0
), the two exciton states H and L (|1), the three biexciton states HH, HL and LL (|2), the four triexciton states HHH, HHL, HLL and LLL (|3), and the five quadexciton states (|4). b, The spectrum of our laser field (solid line) and the emission spectrum of the two excitons (dashed line), with peaks at 1,539.98 ± 0.01 meV and 1,547.47 ± 0.02 meV for the H and L excitons, respectively.
Figure 2: Four-particle correlations. 
Fifth-order collinear-polarization rephasing 3k2 − 2k1 spectrum showing the three biexcitons—HH, HL and LL at two-quantum energies near 3,080, 3,090 and 3,100 meV, respectively—observed through emission at both H and L exciton energies. An interaction-induced (II) unbound HH two-exciton feature with energy equal to exactly twice the H exciton energy (dashed line) is visible near the HH biexciton peak. The Feynman pathways ((i)–(iii)) illustrate how the fields create each main feature: the three biexciton coherences are represented by |0
2| and the two single-exciton/ground state coherences that emit signal are represented in (i) by |10|. In pathways (ii) and (iii), the last three field interactions produce radiative biexciton–exciton (|21|) and triexciton–biexciton (|32|) coherences that appear as redshifted wings of the main peaks.
Figure 3: Six-particle correlations. 
a, Three-quantum fifth-order collinear-polarization spectrum showing four triexciton coherences near 4,618.2 ± 0.2 meV (HHH), 4,625.6 ± 0.2 meV (HHL), 4,632.5 ± 0.3 meV (HLL) and 4,640.0 ± 0.03 meV (LLL). The HHH triexciton binding energy is determined by the location of the peak below the diagonal line drawn along E3Q = 3Eemit, and it is measured to be 1.7 ± 0.2 meV. The coherence pathway on the left (at bottom) illustrates how the main peaks are created: |3
0| represents the four triexciton coherences and |10| represents the two single-exciton coherences that emit the signal. As in Fig. 2, the redshifted shoulders of the main peaks are due to biexciton–exciton and triexciton–biexciton emission. b, Seventh-order collinear-polarization rephasing spectrum showing HHH and perhaps HHL triexciton coherences, represented by the seventh-order pathway (bottom right), and a large exciton-continuum scattering feature. c, Seventh-order co-circular-polarization rephasing spectrum showing lack of three-quantum interaction-induced features. Only the exciton-continuum scattering feature is visible, and the redshifted shoulders are gone because neither biexciton–exciton emission nor triexciton–biexciton emission is possible.
Figure 4: Four-quantum spectroscopy. 
A four-quantum seventh-order collinear-polarization measurement indicates the absence of eight-particle correlations. If HHHH quadexciton coherences were produced by the first four (k2) field interactions, then three interactions with the variably delayed (−k1) field would project them onto single-quantum coherences and a peak would appear below the diagonal line drawn along E4Q = 4Eemit, as illustrated by the pathway shown. The large vertical features are due to exciton-continuum scattering.
