Attractive dipolar coupling between stacked exciton fluids

The interaction between aligned dipoles is long-ranged and highly anisotropic: it changes from repulsive to attractive depending on the relative positions of the dipoles. We report on the observation of the attractive component of the dipolar coupling between excitonic dipoles in stacked semiconductor bilayers. We show that the presence of a dipolar exciton fluid in one bilayer modifies the spatial distribution and increases the binding energy of excitonic dipoles in a vertically remote layer. The binding energy changes are explained by a many-body polaron model describing the deformation of the exciton cloud due to its interaction with a remote dipolar exciton. The results open the way for the observation of theoretically predicted new and exotic collective phases, the realization of interacting dipolar lattices in semiconductor systems as well as for engineering and sensing their collective excitations.


I. INTRODUCTION
The dipolar coupling normally dominates the interaction between charge-neutral species. The characteristic dipolar interaction energy between two dipoles with parallel axes and dipole moments p 1 and p 2 in a medium with dielectric constant can be expressed in the far field as U dd (r) = p 1 p 2 4πεε 0 1 − 3 cos 2 θ r 3 (1) where ε 0 is the vacuum permittivity, θ is the angle between p 1 ||p 2 and r is the vector connecting the dipoles. While sharing the long decay range of the Coulomb interaction, the dipolar interaction is spatially anisotropic and changes from repulsive to attractive at cos(θ)= 1 √ 3 . In natural physical systems containing a large number of dipoles, this anisotropic character gives rise to complex phenomena including self organization, pattern formation, and instabilities in a wide range of dipolar fluids such as in ferromagnetic or electric fluids [1] as well collective effects in dipolar lattices. Fascinating new phases of matter are expected if dipolar interactions are induced into quantum fluids, with an intricate interplay between the attractive and repulsive parts of the interaction and quantum mechanical effects. These new phases may have more than one continuous symmetry simultaneously broken, such as in the prediction of supersolidity. Recent experiments in superfluids of dilute cold atomic species with magnetic dipoles have observed a non-isotropic gas solid-state systems, and in particular spatially indirect dipolar excitons (IXs) in semiconductor bilayers, open up opportunities to explore the complementary phase space of high density, large dipole-moments [6][7][8][9][10][11]. One interesting question is whether the attractive component of the dipolar interaction can be observed and create selfbound states in such solid-state systems. Access to this attractive component has been so far impossible as all IX experiments have been conducted in a single dipolar bilayer of aligned dipoles, where the dipolar interaction is exclusively repulsive.
The stacked DQW structures result in an attractive inter-DQW dipolar component for small lateral separation between the IXs, which has so far escaped experimental detection. Here, by using spatially-resolved spectroscopy, we show that the attractive component of the dipolar interaction induces density correlations between IX fluids in remote DQWs, analogous to the remote dragging [33] observed in solid-state electron-phonon, electron-electron [34], and electron-hole [34,35] systems, but now involving charge-neutral, bosonic species. Interestingly, the energetic changes induced by the remote dipolar coupling exceed the values predicted for formation of dipolar pairs [36], and are non-monotonous in the fluid density. The large coupling energies, which are attributed here to a self-bound, collective many-body fluid excitation identified as a dipolar polaron. The latter is analogous to self-bound three-dimensional entities with compensating attraction and repulsion like atomic nuclei, helium, and cold atom droplets. The experimental findings demonstrate the feasibility of control and manipulation of dipolar species via remote dipolar forces. Furthermore, the sensitivity to the fluid's local correlations opens new ways to study fundamental properties of correlated dipolar fluids.

II. EXPERIMENTAL CONCEPT
The two closely spaced (Al,Ga)As DQWs are grown by molecular beam epitaxy (cf. Fig. 1(a,b)) on a GaAs (001) substrate. In order to enable selective optical excitation and detection, the DQWs (DQW L and DQW H ) have QWs of different thicknesses (QW L and QW H ), thus resulting in different resonance energies for their direct (DX i ) and indirect exciton (IX i ) transitions. Here, the subscripts i = L, H denote DQWs with the higher (H) and lower (L) excitonic energy. We will present experimental results recorded at 2 K on two samples (samples A and B, details about both sample structures can be found in Appendix A), both with QW widths of 10 and 12 nm for QW H and QW L , respectively, and inter-QW spacing consisting of a 4 nm-thick Al 0.33 Ga 0.67 As barrier. The 10 nm-thick Al 0.33 Ga 0.67 As spacer layer between the DQWs prevents carrier tunneling, which would effectively result in the annihilation of the IXs. Figure 2(a) shows the intra-and inter-DQW dipolar potentials calculated for these structures using Eq. 1. Note that the latter becomes attractive for small lateral separation between the particles.
The two different QW thicknesses enable selective excitation and detection of IXs in each of the DQWs, as illustrated by the photoluminescence (PL) spectra of Fig. 2(b) and and the excitation diagrams of Fig. 3(a). A laser beam G L tuned to the DX L resonance only excites IX L s in DQW L (throughout the paper superscripts j = L, H, L+H denote excitation by laser beams G L , G H and both, respectively). Since the DX L lies energetically below DX H , a second laser G H tuned to DX H preferen-tially excites IX H s in DQW H but also creates residual IX L s in the neighboring DQW. One can, nevertheless, achieve a high excitation selectivity of IX H s. In fact, from the ratio between the PL intensities we estimated that G H excites DX H densities that are approximately 3.6 times higher than the DX L ones.
The PL experiments were carried out by exciting the sample with laser beams G L and G H with independently adjusted spot sizes and intensities (cf. Fig. 3a). The interaction between the photo-excited exciton clouds was probed by mapping the PL intensities I j i (x, y) with µm spatial resolution. The photo-excited IX densities, typically in the range between 10 9 and 10 11 cm −2 , were determined from the blue-shifts of the emission lines in the uncoupled systems after correction for correlation effects following the procedure depicted in Ref. [7] (cf. Appendix IV).

A. Spatially resolved photoluminescence
The attractive inter-DQW interactions can be directly visualized by detecting intensity changes ∆I i (x,y) in PL maps of a probing excitonic cloud in one of the DQWs induced by a perturbing cloud excited in the other DQW (cf. Fig. 3(a)). ∆I i (x, y) is quantified according to: Here, the term within the brackets on the rhs accounts for the direct generation of IXs in the probing cloud by each of the laser beams. The most sensitive approach to access inter-DQW interactions consists in detecting ∆I H (x, y): since the perturbing laser G L does not directly excite IX H , one obtains ∆I H (x, y) ≈ I L+H H (x, y) − I H H (x, y). Figure 3(b,c) displays a map of the relative changes δI H (x, y) = ∆I H (x, y)/I H H (x, y) in PL intensity of an extended IX H probing cloud induced by a perturbing IX L cloud in sample A. The probing cloud has a diameter of 60 µm (cf. blue dashed circle), while the perturbing G L beam excites a 20 µm-wide IX L cloud with a density of approximately 1.1 × 10 10 cm −2 at its center (cf. red dashed circle). This perturbing IX L cloud induces a local increase in the IX H density. The IX optical crosssection is negligibly small, so that IXs are created by first creating a DX, then converting to an IX. Thus the perturbing laser G L beam effectively does not excite IX H s (cf. Fig. 2b), and the enhanced emission provides a direct evidence for an attractive IX H -IX L inter-DQW coupling. Furthermore, as the IX lifetime within the probing cloud is not expected to change appreciably under the perturbing beam, one can assume the relative density changes δn H (x, y) to be approximately equal to δI PL,H (x, y).
The emission from the probing cloud at the overlapping region of the beams enhances significantly with the IX density. Figure 3d displays a PL map recorded by in-creasing the intensity of G H (note that the density of the perturbing cloud also increases due to the absorption of G H photons in DQW L , cf. Fig. 3). Under the higher IX densities, the PL intensity from the IX H cloud doubles in the region of the perturbing beam.
Further insight into the inter-DQW interaction can be gained from cross-sections of the PL images across the overlap region of the two clouds, as illustrated in Fig. 4. The left panels correspond to the experimental configuration of Figs. 3c-d with a wide IX H and a narrow IX L cloud (cf. diagrams in the upper part of the figure). The changes in the IX H emission in Fig. 4(a) reproduce the density enhancement within the overlap area of the laser beams. The corresponding differential profile ∆I H in Fig. 4(c) shows that the enhanced concentration of IX H within the overlap region is accompanied by a depletion around it. This behavior follows from the fact that the perturbing G L beam does not change the overall IX H density. As a consequence, the enhanced concentration at the overlap area must then arise from the IX H flow from the surrounding areas.
The attractive force leading to the enhanced IX H density should be accompanied by a back-action force on the perturbing IX L cloud (cf. inset of Fig. 4(c))). In order to extract information about this back-action effect on the IX L profiles, one needs to account for the fact that IX L s are also excited by the G H beam (cf. Figs. 3(a) and 4(b)), thus leading to a non-vanishing I H L (x, y) term on the rhs of Eq. 2. The intensity variation ∆I L (x, y) calculated from this equation and displayed in Fig. 4d shows indeed a depletion of the IX L density around the beam overlap region induced by the remote interaction.
The reciprocal of the above effect is expected if the previous experiment is carried out using a narrow G H spot to perturb an IX L cloud excited by an extended G L beam. Qualitatively similar results were indeed obtained in this situation, as illustrated by the right panels of Fig. 4 (here, smaller laser spots relative to the right panel were employed with diameters of 6.5 µm and 5.5 µm for G L and G H , respectively). Since the mobility of IX L is much larger than that of the IX H [37], the density disturbance of the IX L is far more extended than that of the IX H , as is seen from the comparison of Fig. 4(c,g) to Fig. 4(d,h).

B. Exciton binding energy
The attraction between the remote IX clouds should be accompanied by changes in the observed IX energies within the overlapping regions of the two beams. The solid lines in steady state densities for the same excitation power [38]).
For all applied fields, the energies of both the IX L and IX H resonances show a pronounced minimum for G L powers between 0 and 10 µW followed by a smooth increase in energy for higher IX L excitation powers. Note that for a given applied field, the IX H density remains constant as the IX L density changes. Strikingly, the minima only appear when both species are present and have similar amplitudes for IX L and IX H . In fact, the energy profiles for the IX L species recorded under resonant excitation by solely G L (dashed lines) show only the characteristic energy increase associated with the repulsive intra-DQW IX-IX interactions. The reduction in the excitonic resonance energies is attributed to the attractive inter-DQW interactions, which display a non-monotonic density dependence. They appear for G L laser powers within a relatively small range and essentially vanishes at high IX L densities, where the IX energy becomes equal to the uncoupled case (dashed lines).

C. The dipolar-polaron model
The experiments described above provide evidence for an attractive dipolar interaction between IX clouds located in stacked DQWs.
The inter-DQW interaction also induces density-dependent energetic shifts (cf. Fig. 5), which will be quantified by an inter-DQW binding energy ∆E IX defined as the difference between the IX energies with and without inter-DQW interactions, both referenced at the same IX density.
The dependence of ∆E IX for the IX H cloud on the perturbing IX L density n IX are summarized in Fig. 6. The n IX values for the different G L laser powers and applied fields were extracted from the data in Fig. 5 following the procedure delineated in Appendix B.
The three sets of experimental data points in  Surprisingly, the maximal observed energy shifts are very large reaching up to 7 meV. Such large energies are not expected if one considers only the mutual attractive interaction and binding of a pair of IXs, one from each DQW layer. The formation of such bound pairs ("vertical IX molecules") was recently investigated theoretically by Cohen and co-workers [36]. The inter-DQW dipolar potential calculated for the structures investigated here is illustrated in Fig. 1(c). This attractive potential binds the two IX species into an IX "molecule" with a binding energy ∆E IX of only a few tenths of a meV (dashed line in Fig. 1). ∆E IX is much smaller than the depth of the potential due to the large zero-point energy corrections arising from the small (reduced) mass of the particles and short spatial extent of the potential. The measured IX energy shifts in Fig. 6 are over an order of magnitude larger than the estimated IX molecular binding energy. These shifts are also significantly larger than the depth of the attractive inter-DQW potential of Fig. 1(c), which is indicated by the horizontal dashed line in Fig. 6 (see a more detailed analysis in Appendix C).
This disagreement between the calculated molecular IX binding energies and the experimental values is not unexpected, since the large energetic shifts appear for rather high IX densities, for which the average lateral inter-particle separation within each layer (L x ) becomes comparable to the vertical separation (L z ) between the DQWs. Under these conditions many-body interactions can no longer be neglected. We therefore consider the mutual deformation of the exciton clouds induced by inter-DQW interactions, which may lead to the formation of an IX dipolar-polaron. For simplicity, we consider the case where the density in one of the layers is low, so that we can approach the problem as an "impurity problem": a single IX in DQW 2 interacting with an exciton fluid in DQW 1 (cf. inset of Fig. 6). This approximation, which is described in detail in the Sec. SM4, might still qualitatively capture the case of large IX densities in both layers. We start from a Fröhlich-type polaron Hamiltonian [39]: FIG. 6. Interaction-induced energy shifts ∆EIX of IXH excitons induced by a remote IXL cloud with different densities nIX. Results are shown for probing IXH densities of 8 × 10 10 cm −2 (dots), 9 × 10 10 cm −2 (triangles) and 9.7 × 10 10 cm −2 (diamonds). The error bar only shown for nIX = 5 × 10 10 cm −2 applies for all data points. The vertical dashed arrow marks the density for which Lx = √ 3Lz. The thick dotted and solid lines display the prediction of the polaron model in the limit of fully correlated (corr., cf. Eq. 4) and uncorrelated (uncorr., cf. Eq. 5) IX gases, respectively. This model is sketched in the inset, where the thick arrows schematically represent the distortion of the IX cloud. The horizontal dashed line marks the minimum of the inter-DQW interaction potential U dd (Lzêz + r ||ê|| ) given by Eq. (1).
"impurity" (i.e., the single IX in DQW 2 ) with momentum p and mass M while the second term gives the kinetic energy of the bosonic bath (e.g. phonons in the exciton liquid formed in DQW 1 ), parametrized by the dispersion relation ω(k). The last term gives the impurity-boson interactions. Here, U (k) = f (k)V (k), where V (k) is the Fourier transform of the two-body interaction potential U dd in Eq. 1 and f (k) is a function that depends on the correlation state of the IX gas (cf. Eq. 10 of SM).
If we consider a static impurity (an "infinite-mass polaron", M = ∞, located at r = 0), the Hamiltonian in Eq. 3 can be diagonalized using a coherent-state transformation (see details in Sec. SM4), yielding a negative "deformation energy" ∆E IX , as depicted in Fig. 7. In order to quantitatively estimate ∆E IX , we analyze two limiting solutions of Eq. 3 depending on the correlation state of the IX fluid. We first consider a gas of non-interacting IXs with dispersion relation given bȳ hω(k) ≡ ε(k) =h 2 k 2 /(2m), where m = m e + m hh is the exciton mass (we take m e = 0.067 and m hh = 0.23 for the electron and in-plane heavy-hole effective masses in GaAs). In this case, the energy shift becomes: where µ i = p i / √ 4πεε 0 . The magenta solid line in Fig. 6 compares the predictions of Eq. 4 with the experimental results for ∆E IX . The presence of this exciton causes changes in the density distribution of the IX fluid, which can be described as coupled acousto-electric waves, or polarons. The breadth of the polaron, Lp, is determined by the strength of the interlayer dipolar coupling, which itself is highly dependent on the separation between layers, Lz.
The model reproduces reasonably well the measured magnitude and density dependence of the shifts in the regime of low to moderate IX fluid densities (i.e., for IX L densities below 4 − 8 × 10 10 cm −2 ). This agreement is quite surprising: Eq. 4 yields large red-shifts because the expression used for ω(k) neglects the additional intra-DQW repulsive interactions arising from the polaron density fluctuation, while the IX fluid at this density range (n IX > 10 10 cm −2 ) is known to be in a correlated state, where the repulsive interactions play an important role [7,8,27].
The increasing role of intra-layer repulsion and dipolar particle correlations within the IX L fluid [7,40] expresses itself in Fig. 5 as a significant reduction of the energy shifts when the IX L densities exceed ∼ 8 × 10 10 cm −2 . At this density range, the IX L fluid is expected to be a highly correlated liquid [7,10] with a linear dispersion relation ω(k) ≈ c(n IX )k determined by a speed of sound c(n IX ), which in turn depends on the density n IX [40]. Under such a linear dispersion, the energy shift becomes: Numerical computations by Lozovik et al. [40] revealed that the speed of sound for an IX liquid is given by c(n IX ) ∼ c 0 n 0.7 IX (cf. Fig. 3b of Ref. [40]). In this case, ∆E IX ∼ n −0.4 IX reduces with increasing density. This behavior is reproduced by the thick dotted line in Fig. 6, which was determined from Eq. (5) using the sound velocities from Ref. [40]. It can be shown that the polaron cloud has a gaussian spatial profile with a gaussian width L p = 2L z / √ 35 (cf. Sec. 13). The decreasing energy shifts with increasing n IX can also be understood by the increase stiffness of the IX liquid, which results in a smaller polaron density deformation amplitude.
The polaron binding energies given by Eq. 5 coincides with the reduction of the emission energy of a recombining IX only in the adiabatic approximation, i.e., for interaction processes on a time scale longer than the typical polaron response time, τ p ≈ L p /c(n IX ) = 3 ps for n IX = 10 10 cm −2 and 0.3 ps for 10 11 cm −2 . This is a good approximation in view of the long IX lifetimes. If, in contrast, the bound single IX recombines within a time shorter then τ p , it will leave the IX fluid in a deformation state described by a Poissonian superposition of an integer number n ph = 0, 1, 2, . . . of deformation quanta ("phonons"). The characteristic phonon energy can be determined from the Gaussian polaron profile to be |∆E na IX | = √ πhc(n IX )/(2L p ) = 5.5 meV for n IX = 10 11 cm −2 thus leading to a red-shift n ph |∆E na IX | for each recombination event (cf. Appendix D). In this case the red-shift energy of each IX recombination event will be given by n ph |∆E na IX |. owever, for many such recombination events, and if the linewidth is larger than |∆E na IX |, the measured red-shift will be given by their average: n ph |∆E na IX |. Calculating n ph within the liquid approximation yields an average red-shift energy that differs from that predicted by Eq. 5, up to a numerical factor of order unity (cf. Appendix F). This shows the robust relation between the polaron binding and the red-shift of the IX emission energies.
The cross-over from an uncorrelated to a correlated regime should thus significantly reduce the energy shifts at high IX densities. Since the fraction of particles in a correlated state increases with density, one also expects a reduction of ∆E IX at high densities. This behavior agrees with the reduction of the binding energy observed in cw experiments for densities beyond approximately 8 × 10 10 cm −2 . The polaron model can thus qualitatively reproduce the energy red-shifts over a wide density regime.

IV. CONCLUSIONS
We have experimental evidence for the attractive component of the dipolar interaction between IX dipoles in stacked DQWs by spatially-resolved PL spectroscopy. We have shown that the remote interaction between IX fluids located in stacked DQWs leads to changes in the IX spatial distribution as well as to an increase in the IX-IX inter-layer energy ∆E IX . Surprisingly, |∆E IX | values far exceed those expected from the binding of two IXs in a molecule. The magnitude and qualitative density dependence of |∆E IX | is well accounted for by a many-body dipolar-polaron model. The presented results are expected to challenge state-of-the-art theoretical models of dipolar quantum liquids, however further work will be required to quantify the detailed dependence of the polaron binding energy on IX densities. In particular, it is still not understood why we observe large binding energies, which are qualitatively reproduced by the non-interacting polaron picture of Eq. 4, in a density regime where strong intra-layer repulsive interactions are expected to suppress the polaron deformations and hence its binding energies. We also note that in the current experiments, the densities of the IX H fluid were not negligible, therefore the single impurity model used here should be extended in order to get a more quantitative comparison to the ex-perimental data.
The strong attractive inter-DQW coupling opens up possibilities to observe new complex many-body phenomena of dipolar quantum fluids in solid-state systems, that now involve the full anisotropic nature of the dipoledipole interactions. Since IX systems can probe density and interaction strengths currently unavailable in atomic realizations, it is expected to reveal new collective effects, the attractive dipolar-polaron being a good such example. The sensitivity of the inter-layer coupling to intralayer fluid correlations demonstrated here can be used as a sensitive tool to probe intricate particle correlations in interacting quantum condensates. These experiments also demonstrate the feasibility of dipolar control of interlayer flow in excitonic devices based on stacked dipolar structures. Concepts for the control of IX flows based on repulsive interactions have previously been put forward [36]. The results presented here enable their extension to attractive potentials, which can be realized using stacked DQW structures. Finally, the present investigations also open the way for the realization of dipolar lattices in the solid state. One-dimensional lattices can be realized by simply stacking DQWs. These lattices can be extended to three dimensions by introducing a lateral modulation via electrostatic gates [13,41] or acoustic fields [42].

APPENDIX A: SAMPLES
The studies were carried out in two (Al,Ga)As layer structures (samples A and B) grown by molecular beam epitaxy on GaAs (001) at the Paul-Drude-Institut (sample A) and at Princeton University (sample B). Both samples have DQWs with the same layer structure, as described in the main text.
For sample A, the DQW stack was placed approximately 500 nm away from the semi-transparent top gate and only 100 nm above the bottom electrode. The electric field responsible for IX formation was applied between the top gate and this bottom electrode. The short distance between the DQWs and the bottom electrode minimizes coplanar stray electric fields at the edges of the top gate. This electrode consists of a n-type doped distributed Bragg reflector (DBR) consisting of four Al 0.15 Ga 0.85 As and AlAs layer stacks designed for a central wave length λ c = 820 nm. The DBR enhances the IX emission by back-reflecting the photons emitted towards the substrate. In addition, it suppresses the PL from the substrate (most notably the lines related to the GaAs exciton (around 818 nm) and GaAs:C (830 nm) transitions, which spectrally overlap with the IX PL line.
In Sample B, the DQW stack was also placed ≈520 nm away from the top gate, but was situated 250 nm away from the bottom electrode. The substrate was n-doped and used as the back contact in a Schottky-type diode, with the DQWs being again situated in the intrinsic region. The QWs are GaAs, while the intra-DQW barriers are Al 0.3 Ga 0.7 As. The barriers between substrate and top contact are also Al 0.3 Ga 0.7 As.
The main difference between the two samples is the addition of a Bragg mirror in sample A, as well as smaller radial electric fields, due to the placement of the DQWs closer to the (semi-infinite) ground plane. The Bragg mirror allows for the operation of the device at higher electric fields (limited by the breakdown voltage, instead of the photoluminescence flux) so that the IX energies are 30-40 meV lower than the direct excitons.

Appendix B: DETERMINATION OF IX DENSITY
The data shown in Fig. 5 is processed from several individually recorded spectra. The FWHM linewidth of the recorded IX spectra are typically around 2-5 meV, mainly dependent on the density and the integration area. The spectra shown in Fig. 8 (recorded from sample A) is typical raw PL data demonstrating the energy shifts induced by the inter-DQW interactions. The energies used in Fig. 5 (recorded from sample B) were determined from the peak energies and intensities obtained from such spectra using the procedure described below.
In instances where the diffusion of the IX clouds resulted in pronounced energy shifts, the energy at the highest density was used.
The exact calibration of the exciton density is such systems is a well-known challenge. Here we use the following procedure: for every experiment with a given applied bias, we use the experiment with only the G L laser as a reference. Since this laser creates only a population of IX L , the interactions in this case are only repulsive, leading to a blue shift of the energy with increasing laser power (increasing density). We then choose a point that has an interaction energy well within the range expected for a correlated liquid regime, described in detail in Ref. [7]. We then use Eq. 5.5 in that reference to estimate the density of IX L s for this experimental point. The IX densities of both IX L and IX H can then be induced relative to this reference density by comparing the relative emission intensities of each of the IX species to the emission intensity of the reference point, using the procedure developed in Refs. [8,38]. Here, it was shown that the emission intensity of the IX, I i ∝ n i /τ i where n i is the IX i density and τ i is the IX lifetime. This lifetime was shown in Ref. [38] to be related to the energy difference between the IX emission and the DX emission energies: where ∆E DX−IX = E DX − E IX and the proportionality factor c d depend on the layer structure of the sample and the applied bias, but does not dependent on the density over a rather wide range of densities. Thus, for every two points with the same applied bias but different laser excitation powers, the ratio between their corresponding IX densities can be found using: This ratio was used to calibrate the absolute density of all experimental points in any given experiment with a fixed applied bias to the reference point in that experiment.

APPENDIX C: ELECTROSTATIC CONTRIBUTIONS
The inter-DQW potential in Fig. 1c of the main text applies for the inter-DQW interaction between two aligned dipoles, each in one of the DQWs. In this section, we have estimated the dependence of the inter-DQW potential V lat on the density of particles. For that purpose, we calculate the dipolar potential experienced by a single IX in DQW 2 due to the coupling to an excitonic cloud in DQW 1 (cf. inset of Fig. 9) by (i) neglecting kinetic effects and (b) assuming that the IXs within the cloud of DQW 1 are arranged in a closed-packed triangular lattice with lattice constant L x [43]. L x , as well as the associated particle density in the triangular lattice n IX = 2 , are determined by the spot size and intensity of the excitation laser as well as by recombination and expansion rates of the excitonic cloud. V lat (r) was determined by summing the two-particle contributions U dd (r) (cf. Eq. 1) over a large number of lattice lattices.
The shaded region in Fig. 9 marks the range of energies spanned by V lat (r) as the single IX (with coordinate r) moves relative to the lattice, calculated for different lattice densities. For densities yielding L x >> L z , the lattice potential around each site resembles the one for U dd in Fig. 1(c) of the main text and indicated by the dashed horizontal line in Fig. 6. As L x decreases to values comparable to L x , the minima of V lat (r) remain aligned with the lattice sites. In the opposite limit L x << L z V lat (r) → 0, thus reproducing the fact that the electric field generated by an infinite sheet of dipoles vanishes at large distances. The minimum values for V lat (r) are always larger than the minimum for the IX-molecule interaction potential U dd . This simple model for the interaction underestimates the measured binding energies |∆E IX | indicated by the symbols in Fig. 6 on the main text.

Two-body interactions.
We consider two layers of excitons with dipole moments µ 1,2 = p 1,2 / √ 4πεε 0 = ed 1,2 / √ 4πεε 0 , separated by a distance L z . We will treat excitons as point dipoles, which is a good approximation only for d 1,2 L z . In our setup d 1,2 /L z ∼ 0.3, however, it should still provide a reasonable estimate. The dipole-dipole interaction between the dipole µ 1 (located at ρ = 0 in DQW 1 ) and the dipole µ 2 (located in DQW 2 at a lateral separation ρ) can be written as (cf. Eq. 1): This interaction is sign-changing, so a net mean-field interaction of a dipole with a dipolar plane vanishes: In order to solve Eq. 3 of the main text we first note that the Fourier transform of the two-body interaction potential of Eq. (8) can be expressed as: V (k) = d 2 ρV (ρ)e −ikρ = −µ 1 µ 2 2πk e −kLz (10) In addition, f (k) = [n IX ε(k)/(hω(k))] 1/2 is a function that depends on the density n IX , single-particle energy ε(k) =h 2 k 2 /(2m) as well as on the correlation state of the IX gas expressed in terms of its dispersion relation hω(k).
If we consider a static impurity (an "infinite-mass polaron", M = ∞, located at r = 0), the Hamiltonian (3) can be diagonalized using a coherent-state transforma-tionŜ which gives the following ground-state energy shift: (the ground state is given by |ψ =Ŝ |0 ).
One can see that ∆E is always negative: this is a general property of Hamiltonians with linear coupling, such as Eq. (3). The energy shifts for a gas of non-interacting excitons expressed by Eq. EqE02 of the main text was obtained by integrating Eq. 12 using the dispersion relation is given byhω(k) ≡ ε(k). The corresponding expression for an interacting exciton gas (Eq. 5 of the main text) was determined in the same way using a dispersion relation ω(k) ≈ c(n IX )k, where c(n IX ) is the density dependent speed of sound.

APPENDIX E: POLARON DENSITY PROFILES
In real space, the density deformation of DQW 1 is given by ∆n IX (ρ) = ψ|b † rbr |ψ , whereb † r = d 2 k/(2π) 2b † k e ikρ . In the case of a correlated excitons in DQW 1 , the density deformation can be approximated by a Gaussian at small values of ρ: where L p = 2L z / √ 35. By integrating ∆n IX over the DQW plane one obtains a total density excess corresponding to approx. 0.1 particles for n IX = 10 10 cm −2 .

APPENDIX F: NON-ADIABATIC ENERGY SHIFTS
The polaron wavefunction |ψ = q F (q)b † r |0 = q F (q) |q , where F (q) is the Fourier transform of the gaussian real space profile with width L p (cf. Eq. 13). For a state having a single polaron quantum ("phonon" ), the normalization condition q ,q q | F * (q )F(q ) |q yields F (q) = 8mL 2 p e − ρ 2 2L 2 p . The single phonon energy can be determined by replacing ω(q) ≈ c(n IX )q in the following expression: This expression yields |∆E na IX | = 1.1 meV for n IX = 10 10 cm −2 and 5.5 meV for n IX = 10 11 cm −2 .
The average phonon energy, which corresponds to the red-shift and broadening of a bound IX in the nonadiabatic approximation, can then be calculated according to: = −f na n IX µ 2 1 µ 2 2 3π 8L 4 z mc 2 (n IX ) .
This expression is similar to Eq. 5 of the main text, exception for a pre-factor f na = 3π 3/2 /4 ∼ 1.4. The densitydependent shifts are comparable to the ones determined in the adiabatic approximation and, thus, much smaller than the measured ones.