Influence of multi-exciton correlations on nonlinear polariton dynamics in semiconductor microcavities

The correlated dynamics of excitons that are strongly coupled to light, forming exciton–polaritons, is the basis for many intriguing physical phenomena including parametric amplification [1], Bose–Einstein condensation [2] and superfluid behaviour [3]. Correlated Coulomb interactions between pairs of excitations contribute significantly to polariton nonlinear responses [4, 5], but the experimental signatures of Coulomb correlations are usually convolved with those of other nonlinear interactions [6], impeding isolation and understanding of the distinct effects.

Polariton states can be formed by embedding quantum wells (QWs) inside a semiconductor microcavity (SCMC), strongly coupling the QW exciton state and microcavity light mode [7], as shown in figure 1(a). The strong coupling produces upper polariton (UP) and lower polariton (LP) modes. As a consequence of the mixed exciton–photon character, polaritons have extremely low effective mass, giving steeply sloped UP and LP dispersion curves (figures 1(b) and (c)), but they are still influenced by exciton nonlinearities such as Coulomb correlations that arise from interactions between electron and hole constituents of nearby excitons [8, 9]. Besides mean-field repulsive interactions, Coulomb forces can correlate two or more excitons for a finite period of time, resulting in multi-exciton properties and dynamics distinct from those of isolated excitons [10, 11]. In the case of two spin-paired excitons, for example, Coulomb correlations result in a weakly bound biexciton state. Regardless of the spin configuration, two-exciton correlations have been shown to significantly modify polariton nonlinear signal strengths; calculations of polariton scattering magnitudes are incorrect, by orders of magnitude, without including two-exciton correlations [4, 5]. Strong coupling to light modifies two-exciton correlations by altering the resonance frequencies from those of the constituent excitons. Although they play key roles, coherent exciton–polariton pairs or higher-order correlations have never been observed directly.

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Recently, there has been substantial progress detailing multi-exciton correlations in QWs using two-dimensional Fourier transform optical spectroscopy (2D FT OPT) [12–16]. 2D FT OPT is an extension of 1D forms of nonlinear spectroscopy, such as time-resolved and time-integrated four-wave mixing (FWM), that have been applied to study many-body physics in semiconductors for over 20 years. As reviewed by Chemla and Shah [6], experiments using 1D nonlinear spectroscopy clearly demonstrated the importance of multi-exciton correlations, although the signatures of the correlations were convolved with other signals making the interpretation and analysis of high-order exciton correlations difficult. Recent advances in the generation of multiple, fully phase-coherent light fields have enabled direct observation of multiple-quantum coherences that directly reveal the correlations among multiple excitons using 2D FT OPT. For example, two phase-coherent light fields E₁(k₁) and E₂(k₂), where k_i are experimentally specified wave-vectors, can generate a two-quantum coherence at approximately twice the single-exciton frequency during a scanned time interval t_2Q; after this interval, a third phase-coherent light field E₃(k₃) can project the state into a single-quantum coherence which radiates, at a well defined wavevector k_sig = k₁ + k₂ − k₃, the phase-coherent signal field E_sig(k_sig) that is measured as a function of frequency ω_em. The projection depends on the phase of the two-quantum coherence, and therefore the emitted signal shows oscillations at the two-quantum frequency as the time interval t_2Q is scanned. Fourier transformation of the signal with respect to t_2Q reveals the two-quantum coherences from which the signal at ω_em was generated. With additional phase-related fields, higher-order multiple-quantum coherences and multi-exciton correlations can be elucidated, e.g. three fields can be used to generate a three-quantum coherence, and after a scanned time delay t_3Q, two additional fields can be used to project into a single-quantum coherence that radiates the signal field. In GaAs QWs, two-quantum coherences at exactly twice the exciton frequencies were observed from correlated exciton pairs of the same spin [15], in addition to weakly bound biexciton [14] and triexciton [16] correlations among excitons with mixed spins. Here, we report the direct resolution of up to four-polariton coherences contributing to UP and LP nonlinear emission. As will be shown, the spectral separation of various multi-polariton coherences is essential to clearly elucidate the role of multi-exciton correlations in polariton nonlinear interactions.

The sample structure was a GaAs SCMC with In_0.06Ga_0.94As QWs embedded inside [1]. The cavity length was 3/2 times the exciton resonance wavelength of 850 nm. A set of three QWs, separated by 10 nm thick GaAs layers, were at the two antinodes. The cavity length was wedged slightly so that the cavity mode could be detuned relative to the exciton mode. The exciton–photon coupling strength, Ω, is 7 meV, as given by the LP and UP splitting at zero detuning. The cavity was surrounded by 17 (20) pairs of distributed Bragg reflectors consisting of GaAs/Al_0.18Ga_0.82As at the top (bottom). Experiments were run with the sample at 10 K.

2D spectra were recorded using 2D spatial and temporal pulse shaping techniques through which the beam geometry and the relative phases and time delays of femtosecond pulses in the separate beams can all be controlled and varied without the need for realignment of mirrors or use of delay stages or interferometers, as previously reported [14, 16, 17]. Nonlinear signals, up to seventh order in the excitation fields, were generated using single or multiple field interactions of the pulses in two excitation beams, with wavevectors k₂ and k₁, pulse widths of 100 fs, and wavelengths centred at 850 nm. The beams were kept less than 2° from normal incidence. Using a pair of identical transmissive diffractive optics, a reference beam was directed around the sample and superimposed with the coherent signal beam that emerged from the sample at the phase-matched wavevector 2k₂ − k₁, 3k₂ − 2k₁ or 4k₂ − 3k₁ for third-, fifth- or seventh-order measurements, respectively. The dispersed frequency components of the signal field were measured through spectral interferometry. For third- and fifth-order spectra, the beam at k₂ was kept horizontally polarized while the beam at k₁ and the detection polarization were either co-circularly or cross-circularly polarized. Assuming that the spins of the polaritons in the final states (k_sig and k₁) preserve the spins of the initially excited polaritons (in k₂), this polarization scheme allows us to selectively detect the signal due to polaritons with the same spin (co-circular) or the signal due to polaritons where one has a spin opposite to the other(s). For seventh-order spectra, all beams were co-circularly polarized.

In figure 2, we show two-quantum 2D spectra, where two-polariton coherences were excited by two time-coincident fields before a third field generates FWM nonlinear emission from a single polariton state. The polarizations of the excitation and reference beams are controlled so that the two initially excited polaritons that contribute to the signal have either the same spin or are spin-paired (see Methods section). The spectra show peaks representing excitation of different combinations of LP and UP: two LPs (LP₂), two UPs (UP₂) and a mixed combination of LP and UP (MP₂). The 2D spectra reveal how LP₂, UP₂ and MP₂ coherences contribute to the nonlinear signals at LP and UP frequencies: LP₂ and UP₂ emit into only LP or UP, respectively, but MP₂ emits into either LP or UP. Although the same two-polariton coherences are observed in same-spin or spin-paired spectra, there are dramatic differences in the relative intensities of the peaks which we characterize using a ratio, R, equal to the integrated intensity of MP₂ over the integrated intensity of LP₂. The integrated intensity is calculated along the two-quantum dimension at the LP emission energy and accounts for both the peak intensity and linewidth of each peak. For the spectra shown in figure 2, R is approximately one (two) for the same-spin (spin-paired) spectrum. The value of R also depends on the detuning of the cavity resonance from the exciton resonance, as shown in figure 2(c): the same-spin R is nearly constant whilst the spin-paired R varies from one to nearly three, taken over a range of detuning values ±3 meV.

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In order to understand the polarization and detuning dependence of R, 2D spectra were calculated following the calculations of 1D FWM spectra by Kwong et al [5, 19], as summarized in appendix A. In the calculations, contributions to the third-order polariton susceptiblity can be clearly identified: mean-field Coulomb repulsion and non-binding two-exciton Coulomb correlations contribute to the same-spin third-order susceptibility and only two-exciton Coulomb correlations contribute to the spin-paired third-order susceptibility [19]. Same-spin and spin-paired two-quantum 2D spectra are calculated for various detuning values, as shown in appendix A, and used to find values of R as a function of detuning, shown in figure 2(c). R is nearly one in the same-spin configuration but varies from one to over two in the spin-paired configuration, over detuning values similar to the experimental values. The calculations confirm the spin-dependent contributions to the 2D spectra in figure 2, including two-exciton correlations.

The dramatic detuning dependence of the spin-paired R can be attributed to biexciton correlations. As the cavity detuning is varied, LP₂ and MP₂ move in or out of resonance with the biexciton energy and become more or less strongly correlated through a biexciton interaction. Specifically, as the cavity energy is lowered and the LP becomes more light-like, the LP₂ energy moves out of resonance with the biexciton until biexciton interactions no longer correlate two LPs. Because the UP becomes exciton-like as the cavity energy is lowered, biexciton interactions can correlate MP₂ much more than LP₂ for negative detunings. In fact, for negative detunings in the spin-paired configuration, correlations between LP and UP are the main source of nonlinear signal. Although the same-spin Coulomb correlation strength is also frequency dependent, the correlation strength is not sharply peaked (note that there is no bound biexciton state for two same-spin excitons) but increases relatively gradually with negative detuning frequency [19]. At least for the detuning values in figure 2(c), the frequency dependence of the same-spin Coulomb correlations does not cause strong frequency dependence in the strength of polariton nonlinear interactions. Regardless of the spin configuration, the relative nonlinear signal strengths at LP or UP depend on the strengths of the LP₂, UP₂ and MP₂ correlations which are normally integrated over in typical FWM experiments [18]. By clearly separating the LP₂ and MP₂ coherences, we are able to directly resolve the role of the biexciton correlation in polariton nonlinear dynamics.

Polariton nonlinear dynamics may also be mediated by higher-order correlations. In bare GaAs QWs, three excitons have recently been shown to correlate into a bound triexciton state if the spin of one of the excitons is opposite to the other two [16]. Coulomb correlations among three excitons of the same spin or four excitons of any spin combination were not observed. Other work studying QWs using six-wave mixing has shown that the biexciton and exciton may mix, and even correlate [22, 23]. As far as the authors are aware, there have been no reported observations of correlations or coherent mixing between three or more excitons of the same spin in bare exciton systems. If polariton interactions were mediated only by exciton nonlinearities, then triexciton correlations and biexciton–exciton mixing should mediate three-polariton coherences and no coherences of three or more polaritons with the same spin should be observed. However, as shown in figures 3 and 4, we can detect three-quantum and four-quantum coherences in the six-wave mixing and eight-wave mixing directions, respectively, including coherences between polaritons of the same spin. The 2D spectra show peaks representing various combinations of LP and UP with the same spin (figures 3(a) and 4) and with one spin opposite the others (figure 3(b)). In the three-quantum spectra, mixed polariton coherences (LP₂UP) are again stronger relative to the pure LP coherence (LP₃) if one spin is opposite to the others.

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In addition to high-order Coulomb correlations, the high-order multiple polariton coherences we observe may occur through multiple third-order exciton nonlinearities. A 2k₂ − k₁ polariton generated from third-order exciton nonlinearities can interact with two polaritons that are directly excited by the k₂ and k₁ input fields to generate a 3k₂ − 2k₁ polariton with a fifth-order dependence on the input fields [20, 21]. The fifth-order polariton can interact with two more first-order polaritons in yet another third-order process, yielding a seventh-order 4k₂ − 3k₁ signal and so on. The coupling of two-exciton Coulomb correlations with a third polariton through the cavity mode is shown on the left side of figure 1(d). Such 'cascaded' third-order signals were not observed in bare QWs but have been observed in bulk samples [16, 24, 25]. The signals represent sequential interactions mediated by emitted fields propagating through the sample, with no influence on the earlier exciton participants by the later ones, rather than interactions that mutually influence and correlate all the excitons involved. In microcavities, a coherent exciton–polariton field can play a similar role in generating a new high-order coherence, but instead of leaving the sample, the field components are confined and continue to interact with the electronic coherences, mutually correlating them. In other words, the two coherences are mutually influenced and correlated through their associated fields, and the signal that emerges from the sample represents a true multi-exciton–polariton correlation that we will denote as 'field-coupled' (see appendix B for more discussion).

We calculated fifth-order 2D spectra for the same-spin and mixed-spin configurations, figures 3(c) and (d). The 2D spectra were calculated by coupling five polaritons with two third-order exciton interactions, as described in appendix A. The calculated 2D spectra qualitatively reproduce the experimental results. In particular, the relative strengths of LP₃ and LP₂UP for the two spin configurations are well reproduced, indicating that two coupled third-order interactions may explain the polarization dependence of the fifth-order 2D spectra. In the mixed-spin spectra, for example, a LP and UP interact strongly through biexciton attraction, and additional Coulomb interactions with a second LP coherence gives the relatively strong peak observed at LP₂UP. The seventh-order 2D spectra for the same-spin configuration are calculated by field-coupling of seven polaritons through three third-order exciton interactions. The results are shown in figure 4(b). Again the calculations qualitatively reproduce the experimental 2D spectrum: the emission from UP is diminished and correlations involving at least three LPs are the strongest features.

Based on these calculations, we cannot exclude contributions from high-order Coulomb correlations to high-order polariton correlations. As previously mentioned, triexciton interactions should mediate correlations between three polaritons with mixed spins. Considering a total binding energy of approximately 3 meV [16], triexciton interactions would be the strongest for LP₂UP at the detuning shown in figure 3. Triexciton interactions would therefore give qualitatively similar correlation strengths as field-coupled third-order exciton–polariton nonlinear interactions. Careful calculations of the triexciton binding energy should help separate the field-coupled and high-order Coulomb contributions to the three-quantum coherences, but such calculations remain at or beyond the limit of current theoretical capabilities [26]. Additionally, although unbound three-exciton or four-exciton correlations were not observed in bare QWs, unbound three-exciton or four-exciton correlations may still play a role between polaritons because the high-order Coulomb and field-coupled correlations should be frequency dependent, just as the two-exciton correlation is frequency-dependent. Full understanding of the origins of these high-order multi-polariton coherences will require further theoretical work describing the high-order correlation functions. Additional experiments could be conducted with the k₂ and/or k₁ field interactions separated into different beams so that wavevector combinations that are resonant with both biexciton and high-order correlations could be examined. Five-quanta and higher-order multiple-quantum coherences also should be explored.

Using 2D spectroscopy we directly resolve how multipolariton correlations contribute to nonlinear signals. We are able to clearly demonstrate that the strong coupling to light modifies two-exciton correlations by tuning the oscillation frequencies of the constituent polaritons: depending on the spectral overlap with the biexciton resonance, two spin-paired polaritons can become strongly correlated through biexciton Coulomb interactions. In particular, for negative detunings and spin-paired polaritons, correlations between the LP and UP are the dominant sources of nonlinear emission from the LP state. The ability to control the nonlinear LP signal by exploiting the correlations with the UP could be an interesting mechanism for nonlinear devices. Additionally, correlations among up to four polaritons are observed, a higher order than observed in bare QWs, demonstrating that high-order correlations can be mediated through the polariton fields. The maximum order of the Coulomb correlations cannot be clearly elucidated: multiple third-order Coulomb interactions, coupled through the cavity light field, give calculated correlation signals that are qualitatively similar to the experimental spectra, but high-order Coulomb correlations may still contribute to the observed multi-exciton–polariton signals. Further theoretical work detailing the exact nature of these correlations should inform future experiments and devices exploiting polariton nonlinearities.

The authors thank John Roberts for sample fabrication. GC and JBB were supported by UK EPSRC grants EP/G060659/1 and EP/F011393 and the EU grants CLERMONT4, PITNGA-2009-235114 and INDEX FP7-2011-289968. PW was supported as part of the Center for Excitonics, an Energy Frontier Research Center funded by the US Department of Energy, Office of Science, Office of Basic Energy Sciences under Award Number DE-SC0001088.

In this section, we describe in detail our method to calculate third-order and field-coupled fifth-order 2D spectra for a sample structure consisting of a single QW embedded inside a SCMC. Although the sample structure is slightly different than the structure used in the experiments, the calculations qualitatively reproduce key features in the 2D spectra. We first briefly review how to calculate FWM signals in a model structure and show how to calculate third-order 2D spectra. We then describe and calculate the fifth-order and seventh-order polariton correlations.

FWM signals are calculated following the work of Kwong et al [5, 19]. The calculations have been described in detail and we only provide a brief summary here. Assuming classical light fields, propagation of light through the SCMC structure can be solved using the following wave equation:

$$\begin{equation} \left[n^2(z,\omega)\frac{\omega^2}{c^2}+\frac{\partial^2}{\partial z}\right]\mathbf{E}(z,\omega) = -4\pi\frac{\omega^2}{c^2}\mathbf{P}(z,\omega), \end{equation} \tag{ A.1 }$$

where n(z,ω), E(z,ω) and P(z,ω) are the refractive index, electric field and polarization at each position along the z-axis, normal to the QW plane. Except for the QW layer, the propagation through the SCMC can easily be solved using standard transfer matrix calculations. The key challenge is calculation of the nonlinear polarization generated in the QW, which acts as a source term on right-hand side of (A.1).

The third-order polarization is calculated by standard perturbation theory. In the frequency domain, the third-order polarization at the position of the QW is given by [27]:

$$\begin{eqnarray} &&\mathbf{P}^{(3)}_l(\omega) = \frac{1}{(2\pi)^2}\sum_{m,n,p}\int \mathrm{d}\omega_1 \,\mathrm{d}\omega_2 \,\mathrm{d}\omega_3 \delta(\omega_1+\omega_2-\omega_3-\omega)\chi^{(3)}_{lmnp}(\omega_1,\omega_2,\omega_3)\nonumber\\ &&E_{Qm}(\omega_1)E_{Qn}(\omega_2)E^*_{Qp}(\omega_3), \end{eqnarray} \tag{ A.2 }$$

where χ⁽³⁾_lmnp is the third-order exciton susceptibility describing interactions between excitons in the m, n and p spin states contributing to the polarization in the spin state l. The electric fields in (A.2) are not the applied fields external to the SCMC, but are the electric fields at the position of the QW, which have peaks at the LP and UP resonances.

Microscopic calculation of the third-order susceptibility is extremely challenging because of the many-body Coulomb interactions. Kwong et al [19] calculated the susceptibility using the dynamics-controlled truncation approach giving the result:

$$\begin{eqnarray} &&\chi^{(3)}_{mn}(\omega_1,\omega_2,\omega_3)=-\frac{\chi^{(1)}(\omega_1)\chi^{(1)}(\omega_2)\chi^{(1)*}(\omega_3)\chi^{(1)}(\omega_1+\omega_2-\omega_3)}{q^4_{\mathrm{e}}|\langle d \rangle|^4|\phi(0)|^4}\nonumber\\ &&\left[\delta_{mn}G^{\mathrm{PSF}}(\omega_1,\omega_2)+T_{mn}(\omega_1+\omega_2)\right], \end{eqnarray} \tag{ A.3 }$$

where χ⁽¹⁾(ω) is the first-order exciton susceptibility, q_e is the magnitude of the charge of an electron, 〈d〉 is the transition dipole matrix element and ϕ(0) is the exciton wavefunction at k = 0. G^PSF(ω₁,ω₂) is the phase-space filling contribution, i.e. the decrease in absorption due to occupation of the exciton state. In all calculations, we use values of 〈d〉ϕ(0) = 0.035, consistent with [19]. The last term, T_mn, gives the Coulomb interactions and depends on the spin configuration, mn:

$$\begin{equation} T_{++}(\Omega) = V^{\rm HF}+2G_+(\Omega), \end{equation} \tag{ A.4 }$$

$$\begin{equation} T_{+-}(\Omega) =G_+(\Omega)+G_-(\Omega), \end{equation} \tag{ A.5 }$$

where + and − denote spin up and spin down, V^HF is the frequency-independent mean-field Coulomb interaction and G_m(Ω) is the frequency-dependent two-exciton Coulomb correlation.

The frequency dependence of T_nm depends significantly on the spin configuration, as shown by Kwong et al [19]. There is a sharp peak in T₊₋ at the biexciton resonance, approximately 2 meV below twice the bare exciton resonance for a GaAs sample [19]. Although T₊₊ increases with frequency beginning at roughly twice the bare exciton energy due to the increasing density of states, there is no similar resonance in its spectrum. Based on equations (A.3)–(A.5), χ⁽³⁾ has contributions from phase-space filling, mean-field Coulomb repulsion and two-exciton Coulomb correlations in the same-spin configuration but only contributions from two-exciton Coulomb correlations in the spin-paired configuration. By inserting equations (A.2)–(A.5) into (A.1), the transmitted third-order electric field can be computed for different spin configurations. The frequency-dependent values of T_nm and G^PSF(ω₁,ω₂) used in all calculations are given by [19].

A full 2D spectrum can be calculated by variably delaying the excitation fields in (A.2). A time delay, τ, of the electric fields is given by a linear phase sweep in the frequency domain:

$$\begin{equation} E(\omega)=g(\omega)\times\exp[\mathrm{i}(\omega-\omega_{\mathrm{c}})\tau], \end{equation} \tag{ A.6 }$$

where ω_c is a carrier frequency that sets the centre of the phase sweep and g(ω) is the Gaussian envelope of the pulse. Conveniently, time delays in the experiment are generated in the frequency domain by a pulse shaper, exactly as indicated in (A.6), with an experimentally chosen carrier frequency [17]. (A.6) is inserted into (A.2) and the transmitted third-order electric field is calculated for time delays between the two time-coincident non-conjugate fields, E₁ and E₂, and the third, conjugate field, E₃. Fourier transformation along the scanned time delay dimension gives the 2D spectra shown in figure A.1.

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There are two mechanisms for fifth-order signal to be generated in exciton systems: two coupled third-order exciton correlations or a fifth-order exciton correlation. For the case of two field-coupled third-order exciton interactions, the fifth-order polarization is given by:

$$\begin{eqnarray} &&\mathbf{P}^{(5)}_{\mathrm{field},l}(\omega) = \frac{1}{(2\pi)^2}\sum_{m,n,p}\int \mathrm{d}\omega_1\,\mathrm{d}\omega_2 \,\mathrm{d}\omega_3 \delta(\omega_1+\omega_2-\omega_3-\omega)\chi^{(3)}_{lmnp}(\omega_1,\omega_2,\omega_3)\nonumber \\ &&E^{(3)}_{Qm}(\omega_1)E^{(1)}_{Qn}(\omega_2)E^{(1)*}_{Qp}(\omega_3). \end{eqnarray} \tag{ A.7 }$$

The fifth-order polarization in (A.7) is generated by two third-order exciton interactions; one third-order interaction generates the polariton given by the E⁽³⁾_Qm term, the other third-order interaction couples this polariton with two other polaritons directly excited by the external fields. Crucially, all interactions described in (A.7) occur at the position of the QW. Normally, first-order coherences in thin QWs do not couple to third-order fields emitted from the same QW samples [16], but in a microcavity structure the fields are confined and can continue to interact at the QW layer. In bulk samples, sequential cascaded interactions also generate fifth-order signal similar to the signal described in (A.7) but at different positions in the sample [20, 21]; the third-order field generated at one position of the sample propagates along the sample, interacting with the first-order excitation fields as it arrives at different positions on the sample.

The field-coupled signals emitted from microcavities are fundamentally different from sequential cascades from bulk samples. In the microcavity case, the field components are not just the incident fields but dynamic polariton fields. The correlated nature of the coupled third-order exciton interactions may not be obvious from (A.7). This is because the electric fields in (A.7) are treated semiclassically [5, 19]; the electric field operators and the exciton operators have been factorized and the former written as a classical field. The consequence of this formulation is that the electric fields are not actually influenced by the exciton correlations. However, considering the confinement by the microcavity, the quantum nature of the electric field should be important in nonlinear experiments. The effects of quantum correlation and semiclassical factorization were first elucidated by Savasta and Girlanda [9] and more recently extended to higher-orders by Portolan et al [28]. In these works, the electric field is described by its own dynamic equation and is correlated to the exciton through Pauli-blocking interaction [9]. This correlation, in turn, is mutually influenced by both multi-exciton correlations and multi-photon cavity correlations. Two-exciton–photon correlations have been demonstrated by measuring correlated intensity fluctuations in twin beams generated by FWM in a SCMC [29]. Despite the role of quantum correlations, the application of the semiclassical approximation in (A.7) should not detrimentally affect our ability to compare qualitative characteristics (e.g. relative peak intensities) between the fifth-order signal calculated by (A.7) and the experimental fifth-order signal since such characteristics are usually given by the exciton correlations described in χ⁽³⁾. For certain experimental configurations (e.g. if the Rabi-splitting is equal to the biexciton binding energy) [30, 31], the multi-exciton correlations become strongly dressed by the multi-photon states, significantly altering the lineshapes and energies of the multi-polariton correlations, but the sample used in our experiments does not appear to show these effects (see next section for more discussion).

Fifth-order polarizations may also be generated by fifth-order Coulomb interactions. In this case, the polarization is given by:

$$\begin{eqnarray} &&\mathbf{P}^{(5)}_{\mathrm{coul},l}(\omega) = \frac{1}{(2\pi)^2}\sum_{j,k,l,m,n}\int \mathrm{d}\omega_1\,\mathrm{d}\omega_2 \,\mathrm{d}\omega_3 \delta(\omega_1+\omega_2+\omega_3-\omega_4-\omega_5-\omega)\nonumber \\ &&\chi^{(5)}_{lmnpqr}(\omega_1,\omega_2,\omega_3,\omega_4,\omega_5) E_{Qm}(\omega_1)E_{Qn}(\omega_2)E_{Qp}(\omega_3)E^*_{Qq}(\omega_4)E^*_{Qr}(\omega_5). \end{eqnarray} \noindent \tag{ A.8 }$$

The fifth-order exciton susceptibility in (A.8) accounts for three-exciton Coulomb correlations such as bound triexciton interactions. Fifth-order electric fields will be generated by both P_field and P_coul and propagate outside the sample using (A.1). Unfortunately, calculation of the fifth-order exciton susceptibility, including all Coulomb correlations, is extremely challenging and outside the scope of this paper so only P_field is included in the calculations. Full 2D spectra are calculated as described above and shown in figures 3(c) and (d).

Seventh-order nonlinear signals can be generated through similar mechanisms: three coupled third-order exciton interactions, a third-order exciton interaction coupled to a fifth-order interaction, and a seventh-order exciton interaction can all contribute signal in the seventh-order direction. Without a calculation of the fifth-order or seventh-order exciton susceptibility, however, only the contribution from three coupled third-order exciton interactions can be calculated:

$$\begin{eqnarray} &&\mathbf{P}^{(7)}_{\mathrm{field},l}(\omega) = \frac{1}{(2\pi)^2}\sum_{m,n,p}\int \mathrm{d}\omega_1\,\mathrm{d}\omega_2\,\mathrm{d}\omega_3\delta(\omega_1+\omega_2-\omega_3-\omega)\chi^{(3)}_{lmnp}(\omega_1,\omega_2,\omega_3)\nonumber \\ &&E^{(5)}_{\mathrm{field},Qm}(\omega_1)E_{Qn}(\omega_2)E^*_{Qp}(\omega_3), \end{eqnarray} \tag{ A.9 }$$

where E⁽⁵⁾_field,Qm is the field at the QW due to the polarization from (A.7). The two coupled third-order exciton interactions calculated in (A.7) are coupled to two other polaritons through another third-order exciton interaction. Full 2D spectra are calculated using (A.9) and shown in figure A.1(b).

There has been some work detailing how biexcitons may become strongly coupled to the cavity light field. Although the biexciton itself does not directly couple to light, FWM experiments have shown that the exciton-to-biexciton transition may become strongly coupled to the light mode when the exciton transition becomes saturated [32, 33]. Since we work at the perturbative limit, keeping our excitation powers at least an order of magnitude lower than the saturation threshold, such effects are not observed in our results. The highest powers used in our experiments, for the seventh-order spectra, show a seventh-order dependence on the field strengths, confirming that we are below the saturation point.

Additionally, the possibility exists that the biexciton can couple to two-photon cavity modes, forming what is often described as a bipolariton. Normally, such coupling does not lead to significant changes in biexciton energy or linewidth since the biexciton interaction occurs over a much larger k-space range than that within which the cavity light field is coupled to the exciton [34, 35]. However, some work has demonstrated the existence of a bipolariton state when the biexciton binding energy is equal to the Rabi splitting of the cavity [30, 31], as mentioned above. Light fields can also couple to the biexciton when considering the bulk (or zero-order) photon modes [36], although such effects are not unique to microcavities. A two-qauntum 2D spectrum should provide a sensitive probe to the bipolariton dispersion or detuning dependence. However, for our specific sample, the Rabi-splitting (7 meV) is most likely several times larger than the biexciton binding energy (2 meV in GaAs) [14] and we do not observe shifts in energy in the positions of LP₂, MP₂ or UP₂ that we can attribute to bipolaritons.