Nonlinear optics in the fractional quantum Hall regime

TL;DR

This paper demonstrates that polariton-polariton interactions are significantly enhanced in a fractional quantum Hall state, revealing new insights into strongly correlated electronic systems through nonlinear optical measurements.

Contribution

It shows that fractional quantum Hall states can strongly enhance polariton interactions, opening new avenues for photonic quantum systems and probing quantum Hall states via nonlinear optics.

Findings

Enhanced polariton-polariton interactions in fractional quantum Hall states

Role of exciton-electron interactions and polaron polaritons in nonlinear response

Potential for nonlinear optics to reveal fractional quantum Hall properties

Abstract

Engineering strong interactions between optical photons is a great challenge for quantum science. Envisioned applications range from the realization of photonic gates for quantum information processing to synthesis of photonic quantum materials for investigation of strongly-correlated driven-dissipative systems. Polaritonics, based on the strong coupling of photons to atomic or electronic excitations in an optical resonator, has emerged as a promising approach to implement those tasks. Recent experiments demonstrated the onset of quantum correlations in the exciton-polariton system, showing that strong polariton blockade could be achieved if interactions were an order of magnitude stronger. Here, we report time resolved four-wave mixing experiments on a two-dimensional electron system embedded in an optical cavity, demonstrating that polariton-polariton interactions are strongly enhanced when the electrons are initially in a fractional quantum Hall state. Our experiments indicate that in addition to strong correlations in the electronic ground state, exciton-electron interactions leading to the formation of polaron polaritons play a key role in enhancing the nonlinear optical response. Besides potential applications in realization of strongly interacting photonic systems, our findings suggest that nonlinear optical measurements could provide information about fractional quantum Hall states that is not accessible in linear optical response.

Equations

$$I_{\det }(t,\tau )\propto {ϵ}_0{\left|P_1^{(1)}(t)+P_2^{(1)}(t,\tau )+P^{(3)}(t,\tau )\right|}^2,$$

$${\left|P_1^{(1)}(t)\sin ({\omega }_{\mathrm{m}}t)\right|}^2+{\left|P_2^{(1)}(t,\tau )\right|}^2+2\Re \left({P_1^{(1)}(t)}^{\ast }\sin ({\omega }_{\mathrm{m}}t)P_2^{(1)}(t,\tau )\right),$$

$$2\Re \left(\left({P_1^{(1)}}^{\ast }\sin ({\omega }_{\mathrm{m}}t)+{P_2^{(1)}}^{\ast }\right)\left(P_{111}^{(3)}\sin {({\omega }_{\mathrm{m}}t)}^3+P_{112}^{(3)}\sin {({\omega }_{\mathrm{m}}t)}^2+P_{122}^{(3)}\sin ({\omega }_{\mathrm{m}}t)+P_{222}^{(3)}\right)\right).$$

$$N={\eta }_s{\eta }_c{n}_{\mathrm{ph}},$$

$$\frac{d\psi (t)}{dt}=-\frac{\gamma }{2}\psi (t)-ig|\psi (t){|}^2\psi (t)+F(t,\tau ,t_{\mathrm{mod}}),$$

$$F(t,\tau ,t_{\mathrm{mod}})=A_1(t_{\mathrm{mod}})G(t)+A_{2}G(t-\tau )e^{i{\omega }_{\mathrm{L}}\tau },$$

$$U_X=A\cdot g/|X{|}^4=13_{-9}^{+18}\mu \mathrm{eV}\mu {\mathrm{m}}^2.$$

Full text

Nonlinear optics in the fractional quantum Hall regime

Patrick Knüppel

Institute of Quantum Electroncis, ETH Zürich,

CH-8093 Zürich, Switzerland

  

Sylvain Ravets

Institute of Quantum Electroncis, ETH Zürich,

CH-8093 Zürich, Switzerland

CNRS Centre for Nanoscience and Nanotechnology,

Université Paris-Sud, Université Paris-Saclay, 91120 Palaiseau, France

  

Martin Kroner

Institute of Quantum Electroncis, ETH Zürich,

CH-8093 Zürich, Switzerland

  

Stefan Fält

Institute of Quantum Electroncis, ETH Zürich,

CH-8093 Zürich, Switzerland

Solid State Physics Laboratory, ETH Zürich,

CH-8093 Zürich, Switzerland

*These authors contributed equally to this work.

  

Werner Wegscheider

Solid State Physics Laboratory, ETH Zürich,

CH-8093 Zürich, Switzerland

*These authors contributed equally to this work.

  

Atac Imamoglu

Institute of Quantum Electroncis, ETH Zürich,

CH-8093 Zürich, Switzerland

**Engineering strong interactions between optical photons is a great challenge for quantum science. Envisioned applications range from the realization of photonic gates for quantum information processing O’Brien et al. (2009) to synthesis of photonic quantum materials for investigation of strongly-correlated driven-dissipative systems Carusotto and Ciuti (2013). Polaritonics, based on the strong coupling of photons to atomic or electronic excitations in an optical resonator, has emerged as a promising approach to implement those tasks Sanvitto and Kéna-Cohen (2016). Recent experiments demonstrated the onset of quantum correlations in the exciton-polariton system Muñoz-Matutano et al. (2019); Delteil et al. (2019), showing that strong polariton blockade Verger et al. (2006) could be achieved if interactions were an order of magnitude stronger. Here, we report time resolved four-wave mixing experiments on a two-dimensional electron system embedded in an optical cavity Smolka et al. (2014), demonstrating that polariton-polariton interactions are strongly enhanced when the electrons are initially in a fractional quantum Hall state. Our experiments indicate that in addition to strong correlations in the electronic ground state, exciton-electron interactions leading to the formation of polaron polaritons Schmidt et al. (2012); Sidler et al. (2017); Efimkin and MacDonald (2018); Ravets et al. (2018) play a key role in enhancing the nonlinear optical response. Besides potential applications in realization of strongly interacting photonic systems, our findings suggest that nonlinear optical measurements could provide information about fractional quantum Hall states that is not accessible in linear optical response. **

Polaritons have recently attracted considerable interest, motivated by the fact that their interactions can be engineered almost at will through the tunability of their matter component. For example, strongly interacting Rydberg polaritons have recently been obtained using the nonlinear behavior of Rydberg excitations in an ensemble of atoms Peyronel et al. (2012), which led to the demonstration of Rydberg polariton blockade Jia et al. (2018) where the presence of a single polariton in a well-delimited region of space prevents the resonant injection of other polaritons. In parallel, efforts are being made to realize polariton blockade in condensed matter systems that hold great potential for realizing compact and integrated synthetic quantum materials Sanvitto and Kéna-Cohen (2016). Exciton polaritons in semiconductor materials are part light part matter particles that arise from the strong coupling of a quantum well exciton and a cavity photon Deng et al. (2010). These photonic particles inherit a nonlinear behavior from exciton-exciton interactions Carusotto and Ciuti (2013); Ciuti et al. (1998); Tassone and Yamamoto (1999). For efficient blockade to be obtained, the nonlinearity $U$ needs to be greater than the inverse lifetime $\gamma$ of the polaritons Verger et al. (2006). Recent state-of-the art experiments based on photon correlation measurements in semi-integrated microcavities attained optimized values of the ratio $U/\gamma\simeq 0.1$ in a photonic dot with about $3~{}{\rm\mu m}^{2}$ area Muñoz-Matutano et al. (2019); Delteil et al. (2019). These experiments represent the culmination of decade long technological developments aimed at increasing $U/\gamma$ through reducing the photonic mode area El Daïf et al. (2006); Ferrier et al. (2011); Besga et al. (2015) as well as increasing the lifetime Sun et al. (2017). Recently, several possibilities have been explored for enhancing $U$ through an increase of exciton-exciton interactions, focusing either on biexciton Feshbach resonance Carusotto et al. (2010); Takemura et al. (2014) or on excitons with a permanent dipole moment Cristofolini et al. (2012); Byrnes et al. (2014); Nalitov et al. (2014); Rosenberg et al. (2018); Togan et al. (2018). The experiments we report here reveal a hitherto unexplored mechanism for optical nonlinearity emerging for polaritonic excitations out of a two dimensional electron system (2DES) in the fractional quantum Hall (FQHE) regime. More specifically, using time resolved four-wave mixing (FWM) experiments Boyd (2008); Smallwood and Cundiff (2018), we find that polariton-polariton interactions $U$ can be enhanced by more than an order of magnitude around the fractional state at filling factor $\nu=2/5$ as compared to other neighboring compressible states. The interplay between photonic excitations and a 2DES is an exciting field Kukushkin et al. (1999); Byszewski et al. (2006); Groshaus et al. (2007); Bar-Joseph (2005); Hayakawa et al. (2013) with open problems, among others, concerning the relation between transport and optics Bartolo and Ciuti (2018); Paravicini-Bagliani et al. (2019); Cotlet et al. (2018) and the description of exciton-electron interactions in a magnetic field Efimkin and MacDonald (2018).

We study a semiconductor heterostructure that features, at the center of an optical microcavity, a GaAs QW containing an electron system of density $n_{e}=3\times 10^{10}\rm{cm}^{-2}$ (see Methods). In the presence of a 2DES, electron-exciton interactions modify the excitation spectrum and pioneering studies showed the existence of the strong coupling regime Rapaport et al. (2000, 2001). A consistent description of the new excitonic excitations that emerge in the presence of a 2DES was first provided by R. Suris Suris (2003), and were later termed exciton-polarons Schmidt et al. (2012); Sidler et al. (2017); Efimkin and MacDonald (2018). Strong coupling to an optical mode in turn results in the formation of polaron-polaritons as the elementary excitations of the 2DES-cavity system Sidler et al. (2017); Ravets et al. (2018). Under an external magnetic field $B$ orthogonal to the 2DES surface, discrete Landau levels ${\rm LL}n_{e}$ (${\rm LL}n_{\rm hh}$) form out of the conduction (heavy-hole valence) band. We focus, in this article, on resonant optical excitations from the lowest Landau level ${\rm LL}0$. As we increase $B$, the filling factor $\nu$ of ${\rm LL}0$ decreases, allowing us to reach the fractional quantum Hall regime. We address transitions to the $\left|\uparrow\right>$ ($\left|\downarrow\right>$) ${\rm LL}0$ subband using $\sigma^{-}$ ($\sigma^{+}$) circularly-polarized light.

We first characterize our sample using optical spectroscopy in the low-power (linear) regime. The sample is mounted inside a dilution refrigerator with fibered optical access, as shown in Fig. 1a. We record reflectivity spectra for several values of $\nu$ using circularly-polarized light from a broadband light source. Fig. 1b plots an overview of the polaron-polariton lines for our system, obtained by calculating the difference between the spectra measured using $\sigma^{-}$ (red) and $\sigma^{+}$ (blue) polarized light. We observe generic strong dispersion of the polariton energies with magnetic field around integer and fractional values of $\nu$. This striking behavior of the linear optical spectrum stems from strong modification of electron-exciton interactions in and around gapped quantum Hall states Ravets et al. (2018), which in turn leads to a $\nu$-dependent modification of the cavity-polaron coupling strength. Especially striking are the energy shifts experienced by the lower polariton in $\sigma^{-}$ polarization (${\rm LP}_{\sigma^{-}}$) at filling factors $\nu=1$, $2/3$ and $2/5$. Since the optical spectrum strongly depends on $\nu$, we refer to optical excitations in this system as quantum Hall polaritons.

Figure 2a shows the principle of the time-resolved experiment we use to characterize our sample in the nonlinear regime. We use a pulsed Ti:Sapphire laser with a $T_{\rm pulse}=4\,{\rm ps}$ pulse duration, a 76 MHz repetition rate and center frequency tuned to the ${\rm LP}_{\sigma^{-}}$ resonance. We split the laser into two paths and introduce a variable time delay $\tau$ between the two pulses. For optical excitation, we recombine both beams onto a beam splitter and couple the linearly-polarized light into an optical fiber routed to the sample. The excitation light is then focused onto the sample surface using a low NA objective. The total field incident on the sample is given by $E(t,\tau)=E_{1}(t)+E_{2}(t,\tau)$, where the average intensities of the two beams are chosen to be equal. For detection, we collect the generated resonance fluorescence using the same fiber as the one used for excitation, and we filter out the laser background light by detecting along the cross-polarized axis. The collected light is finally sent onto an avalanche photodiode (APD) for detection. Modeling our system as a third order nonlinear medium, we can expand the total intensity reaching the photodetector $I_{\rm det}$ as:

[TABLE]

where the linear $P^{(1)}_{1,2}(t)$ (nonlinear $P^{(3)}(t)$) polarizations are the inverse Fourier transforms of $P^{(1)}_{1,2}(\omega)$ ($P^{(3)}(\omega)$) Boyd (2008). To isolate weaker nonlinear terms $\propto{P^{(1)*}_{i}}P^{(3)}$ from the dominant linear contributions $\propto{P^{(1)*}_{i}}{P^{(1)}_{j}}$ $(i,j=1,2)$, we modulate the field amplitude $E_{1}(t)$ sinusoidally at frequency $\omega_{\rm m}$. By calculating $\mathcal{I}(\omega,\tau)$, the Fourier transform of $I_{\rm det}(t,\tau)$, we can separate different terms: the (mostly) linear term $\mathcal{I}(\omega_{\rm m},\tau)$ and the nonlinear term $\mathcal{I}(3\omega_{\rm m},\tau)$ (see Methods). In the following, we use these two terms to quantify the nonlinearity of the system.

We now focus on pump-probe measurements around $\nu=2/5$ ($B=3.15\,{\rm T}$). We observe, in Fig. 2b, that $\mathcal{I}(\omega_{\rm m},\tau)$ features a fast oscillation modulated by an exponential envelope. This is the expected waveform since $\mathcal{I}(\omega_{\rm m},\tau)$ is, to lowest order, the autocorrelation signal of the resonance fluorescence emitted by the sample: the carrier frequency of the fast oscillation corresponds to the (undersampled) optical frequency and the characteristic decay time is the polariton coherence time $T_{\rm LP}=24\pm 1\,{\rm ps}$ (dashed black line). The nonlinear contribution $\mathcal{I}(3\omega_{\rm m},\tau)$, depicted in Fig. 2c, also exhibits fast oscillations but its envelope has a more complex structure as a consequence of the interplay between several interfering nonlinear terms, with characteristic decay times $T_{\rm LP}$ and $T_{\rm LP}/3$ that compensate at short delays. Figure 2d shows a logarithmic plot of the integrals $\left<\mathcal{I}(\omega_{\rm m},\tau)\right>_{\tau}=\int{\mathcal{I}(\omega_{\mathrm{m}},\tau)d\tau}$ and $\left<\mathcal{I}(3\omega_{\rm m},\tau)\right>_{\tau}=\int{\mathcal{I}(3\omega_{\mathrm{m}},\tau)d\tau}$ as a function of the average incident power. We observe that the former exhibits a power law with exponent $1.3\pm 0.3$, which is consistent with the expected linear behavior. By contrast, $\left<\mathcal{I}(3\omega_{\rm m},\tau)\right>_{\tau}$ shows a power law with exponent $2.2\pm 0.3$ that is consistent with the anticipated dependence of third-order nonlinear response, validating that $\left<\mathcal{I}(3\omega_{\rm m},\tau)\right>_{\tau}$ is indeed a good measure of the nonlinearity. The observed deviation of the power law exponents from the expected values 1.0 and 2.0 is most likely due to systematic errors on the input power calibration. We emphasize that the measured nonlinearity occurs on timescales that are comparable to the polariton lifetime, which demonstrates that our method allows us to access (fast) polariton-polariton interactions. We also note that the nonlinear response saturates at high optical powers (Fig. 2c). The saturation behaviour of $\left<\mathcal{I}(3\omega_{\rm m},\tau)\right>_{\tau}$ at high optical powers may be attributed to the saturation of the ${\rm LP}_{\sigma^{-}}$ red shift induced by a change in $\nu$ (Fig. 1b). Saturation could also be a consequence of (slow) light-induced modifications of $n_{e}$, which may start to play a role at the highest powers investigated Ravets et al. (2018).

We further confirm that $\left<\mathcal{I}(3\omega_{\rm m},\tau)\right>_{\tau}$ provides a good measure of polariton-polariton interactions, by applying our measurement procedure to a different sample featuring a neutral quantum well ($n_{e}=0$). Based on a numerical analysis relying on solving the Gross-Pitaevskii equation (see Supplementary), we quantify the nonlinearity of exciton-polaritons for the neutral quantum well sample. We obtain a value of $13_{-9}^{+18}\,\mathrm{\mu eV}\,\mathrm{\mu m}^{2}$ for the exciton-exciton interaction strength, which is consistent with values reported elsewhere Muñoz-Matutano et al. (2019); Delteil et al. (2019); Amo et al. (2009); Ferrier et al. (2011); Rodriguez et al. (2016); Brichkin et al. (2011); Walker et al. (2017).

Having established that our measurements can be used to reliably determine the polariton-polariton interaction strength, we analyze the evolution of $\left<\mathcal{I}(\omega_{\rm m},\tau)\right>_{\tau}$ and $\left<\mathcal{I}(3\omega_{\rm m},\tau)\right>_{\tau}$ as a function of $\nu$ by measuring $I(t,\tau)$ for different values of $B$. The data consists of three sets centered around filling factors $1$, $2/3$, and $2/5$ that exhibit clear signatures of optical coupling to quantum Hall states, as demonstrated in Fig. 1b. For each of the three sets, we tune the cavity to resonance with the singlet attractive polaron channel (polaron content $55\pm 5\%$). Then, we tune $B$ to access neighboring filling factors while carefully adjusting the laser frequency to resonantly excite ${\rm LP}_{\sigma^{-}}$ for every datapoint. The main result of this letter is the remarkable $\nu$-dependence of the nonlinear signal $\mathcal{I}(3\omega_{\mathrm{m}},\tau)$ shown in Fig. 3a. We observe a strong increase of the nonlinearity at fractional filling factors $\nu=2/3$ ($B\simeq 1.95\,{\rm T}$) and $\nu=2/5$ ($B\simeq 3.15\,{\rm T}$), as compared to neighboring filling factors. Away from these states, e.g. for $B=3.5\,{\rm T}$, the nonlinearity becomes weaker and eventually comparable to the noise level of our apparatus. This gives clear evidence that polariton-polariton interactions are enhanced around the fractional quantum Hall states $\nu=2/3$ and $\nu=2/5$. In stark contrast, we observe that $\mathcal{I}(3\omega_{\rm m},\tau)$ is only marginally modified around the integer filling factor $\nu=1$. Also, another interesting feature is manifest in the delay dependence of $\mathcal{I}(3\omega_{\rm m},\tau)$ at the fractional filling factors ($\nu=2/5$ & $2/3$), which show envelopes with a qualitatively different shape than observed for integer filling factors ($\nu=1$) or expected based on numerical solutions of the driven-dissipative Gross-Pitaevskii equation (see Supplementary section II). This suggests that the nonlinear optical response close to fractional filling factors cannot be captured by a simple Kerr nonlinearity.

We note that, since polaritons interact through their matter part, a change in the polaronic content of the polaritons as a function of $B$ could also lead to a modification of polariton-polariton interactions: polaritons with a higher polaron content will indeed show larger interactions. Since we use a slightly blue-detuned cavity mode, we expect polaron-polaritons to be more polaronic and less photon-like at $\nu=2/5$ due to the reduction of the normal-mode splitting around this filling factor (see Fig. 1b). Measurements of the evolution of the polariton effective mass around $\nu=2/5$ have shown that the polaron fraction varies by about a factor of two Ravets et al. (2018) between $B=3.15~{}{\rm T}$ and $B=3.5~{}{\rm T}$. We therefore cannot exclude that part of the enhancement at $\nu=2/5$ stems from the increase in polaron content of the lower polariton branch. However, for $\nu=2/3$, the polariton splitting (see Fig. 1b) and thus the polaron content of the polaritons is only marginally changed and yet we observe a factor of $\sim 6$ enhancement of the nonlinearity at this filling factor: this indicates that the increase of interaction around fractional filling factors only partially stems from an increase in polaron content of the ${\rm LP}_{\sigma^{-}}$ mode.

Figure 3b shows the evolution of the (mostly linear) term $\mathcal{I}(\omega_{\rm m},\tau)$ at $\nu=1$, $2/3$, $2/5$, where we observe another striking feature that coincides with the enhancement of the nonlinearity. For the intermediate pump power used in Fig. 3, we find that the characteristic decay time of $\mathcal{I}(\omega_{\rm m},\tau)$ (i.e. the lower polariton coherence time $T_{\mathrm{LP}}$) is prolonged at $\nu=2/3$ and $\nu=2/5$. A detailed study of this effect as a function of input power (see Supplementary section III) shows that $T_{\mathrm{LP}}$ is multiplied by a factor two to three as we increase the input power. The origin of this power dependent enhancement of $T_{\mathrm{LP}}$ remains unclear.

We summarize our results in Fig. 3c, where we provide values of the enhancement of polariton-polariton interactions close to fractional filling factors, obtained by calculating the ratio of the areas $R_{\rm{a}}=\left<\mathcal{I}(3\omega_{\rm m},\tau)\right>_{\tau}/\left<\mathcal{I}(\omega_{\rm m},\tau)\right>_{\tau}$ and the ratio of the signal peak-to-peak values $R_{\rm{pp}}$ 111$R_{\rm{pp}}=\left(\max(\mathcal{I}(3\omega_{\mathrm{m}},\tau))-\min(\mathcal{I}(3\omega_{\mathrm{m}},\tau))\right)/\left(\max(\mathcal{I}(\omega_{\mathrm{m}},\tau))-\min(\mathcal{I}(\omega_{\mathrm{m}},\tau))\right)$.. At $\nu=2/5$, both measures show a significant enhancement of the interaction, of the order of $10^{1}$, as compared to neighboring filling factors. At a first glance, a possible explanation for the enhancement of the nonlinear response may lie in the filling factor-dependent polariton energy shifts observed in Fig. 1b: increasing the incident power changes the local electron density and modifies the polariton Rabi splitting, leading to power-dependent modifications of the polariton energy. However, this explanation is inconsistent with our observation of an increase of the nonlinearity around $\nu=2/3$ despite the fact that the lower polariton energy only changes marginally with $B$ (Fig. 1b). Moreover, we do not measure a significant enhancement of the nonlinearity at $\nu=1$, despite strong variations in polariton splitting as we vary $B$ or the pump power.

We also repeated the experiment around the $\nu=1/3$ state ($B=3.9~{}\rm{T}$), where we did not observe an enhanced nonlinearity. We speculate that this is due to the suppression of $\sigma^{-}$ polarized polaron formation at $\nu\leq 1/3$ due perfect spin polarization of the 2DEG, leaving the polariton mode mostly photonic Ravets et al. (2018). Moreover, we repeated the experiments with the ${\rm LP}_{\sigma^{+}}$ resonance at $\nu=2/5$. Remarkably, the nonlinearity was not enhanced. We tentatively explain this observation by the fact that the FQH state is less sensitive to an electron introduced into a higher LL in the case of ${\rm LP}_{\sigma^{+}}$ resonance, as compared to the lowest LL in the case of ${\rm LP}_{\sigma^{-}}$ resonance. Last but not least, we measured qualitatively similar results on a second sample with higher electron density ($n_{e}=1.4\times 10^{11}\,\mathrm{cm}^{-2}$) for the (spin-polarized) $\nu=2/3$ state at $B\simeq 8.5\,{\rm T}$. There, we did observe a comparable enhancement with $\nu$ (see Supplementary section IV).

Strong enhancement of polariton-polariton interactions around FQHE states opens up new perspectives for the study of strongly correlated electron as well as for photonic systems. In light of recent photon correlation measurements in confined exciton-polariton systems Muñoz-Matutano et al. (2019); Delteil et al. (2019), the enhancement of interactions by an order of magnitude that we found here shows great potential for reaching the strong polariton blockade regime. Understanding the physical mechanism for enhanced nonlinear response and prolonged polariton coherence times for FQHE states constitutes a very interesting open problem. Our experiments show that despite their qualitatively similar linear optical response, fractional and integer QHE states show strikingly different nonlinear optical signatures: this suggests that nonlinear spectroscopy could reveal signatures of strongly correlated electronic systems that are not accessible by linear optical or transport measurements.

Methods

Sample structure. Our sample structure, detailed in Ravets et al. (2018), features a 20 nm modulation doped gallium arsenide (GaAs) quantum well located at the center of a $2\lambda$ $\rm Al_{0.19}Ga_{0.81}As$ microcavity. The two distributed Bragg reflectors consist of 25 (19) pairs of $\rm AlAs/Al_{0.20}Ga_{0.80}As$ layers. The cavity quality factor, measured by white light reflectivity, is $Q\simeq(5.5\pm 0.1)\times 10^{3}$. The 2DES shows an electron density $n_{e}\simeq 0.33\times 10^{11}\,{\rm cm}^{-2}$ and a mobility $\mu\simeq 1.6\times 10^{6}\,{\rm cm}^{2}{\rm V}^{-1}{\rm s}^{-1}$, as measured by magneto-transport. Note that with these parameters, we conveniently access various integer and fractional quantum Hall states for relatively low magnetic fields $B\leq 5\,{\rm T}$.

Optical characterization. We perform an initial characterization of our sample by polarization-resolved white light reflectivity as a function of $B$. We couple light from a broadband light emitting diode into an optical fiber and shine photons onto the sample placed inside a dilution refrigerator with a base temperature of 30 mK (see Fig. 1a). The light is focused on the sample surface using a low numerical aperture lens (${\rm N.A.=0.15}$) in confocal configuration. Reflected light is then collected by the same fiber and analyzed with a spectrometer. More details on the optical setup can be found in the Supplementary section I.

Time-resolved measurements. One standard method for evaluating interactions between exciton-polaritons in 2D uses a resonant continuous wave excitation laser to monitor the blue-shift experienced by the lower polariton line due to the (Kerr-like) nonlinearity as the polariton population increases. In these experiments, however, one cannot differentiate between the contribution due to fast ($\sim 10\,{\rm ps}$) polariton-polariton interactions, and other unwanted contributions due to the slow ($>100\,{\rm ps}$) buildup of an excitonic reservoir Stepanov et al. (2018). This issue is critical in the context of quantum Hall polaritons since the 2DES electron density is particularly sensitive to optical power due to possible photoionization of DX centers when illuminating the sample: increasing the optical power density may lead to unwanted modifications of $n_{e}$ and therefore to slow variations of the ($\nu$-dependent) polariton energies, which in turn may prevent us from properly evaluating the interactions. In order to isolate pure polariton-polariton interactions, we use a carefully designed sample structure with reduced sensitivity of $n_{e}$ to light Ravets et al. (2018), and perform time-resolved experiments in the pulsed-excitation regime in which the pulse duration ($\sim 4\,{\rm ps}$) is shorter than the polariton lifetime ($>12\,{\rm ps}$). A traditional approach to isolate the nonlinear contribution in four-wave mixing experiments consists in introducing an angle between the two exciting beams in order to generate a background-free nonlinear response at a different angle. However, the requirement for ultra-low temperatures render standard FWM experiments technically challenging to implement in our experimental geometry that uses a fiber coupled scanning confocal microscope in a dilution refrigerator. Instead, we use an electro-optic modulator to sinusoidally modulate the field amplitude $E_{1}$ in one arm of the interferometer, with angular frequency $\omega_{\rm m}/2\pi=8\,\mathrm{kHz}$ (see Fig. 2a). As a consequence, the power spectral density $\mathcal{I}(\omega,\tau)$ contains terms that oscillate at multiples of the modulation frequency $\omega_{\rm m}$. Expanding the first order terms in Equation (1)

[TABLE]

we find that a field autocorrelation term appears at $\omega_{\mathrm{m}}$. The next order terms are the cross-products between linear and nonlinear polarizations with subscripts denoting fields originating from optical paths 1 and 2

[TABLE]

In turns out that $3\omega_{\mathrm{m}}$ is the first frequency for which $P^{(3)}$ contributes to all terms with no background from $P^{(1)}$, so $\mathcal{I}(3\omega_{\mathrm{m}},\tau)$ is used to monitor the nonlinear response. Similar techniques have been used for pump probe experiments in collinear geometry Hall et al. (1992); Mecozzi and Mørk (1998); Nardin et al. (2013); Smallwood and Cundiff (2018).

Data availability. The data that support the plots within this paper and other findings of this study are available from the corresponding authors upon reasonable request.