Mott insulator of strongly interacting two-dimensional excitons
Camille Lagoin, Stephan Suffit, Kirk Baldwin, Loren Pfeiffer, Francois Dubin
MMott insulating phase of strongly interacting two-dimensional excitons
Camille Lagoin , Stephan Suffit , Kirk Baldwin , Loren Pfeiffer and Fran¸cois Dubin Institut des Nanosciences de Paris, CNRS and Sorbonne Universit´e, 4 pl. Jussieu, 75005 Paris, France Laboratoire de Materiaux et Phenomenes Quantiques, Universite Paris Diderot, 75013 Paris and PRISM, Princeton Institute for the Science and Technology of Materials,Princeton University, Princeton, NJ 08540, USA
In condensed-matter physics, electronic Mott insulators have triggered considerable research dueto their intricate relation with high-temperature superconductors. However, unlike atomic systemsfor which Mott phases were recently shown for both bosonic and fermionic species, the fingerprintof a Mott phase implemented with bosons is yet to be found in the solid state. Here we unveil suchsignature by emulating the Bose-Hubbard hamiltonian with semiconductor excitons confined in atwo-dimensional lattice. The exciton fluid is characterised by strong repulsive dipolar interactions.These mostly constrain the system to behave classically, but nevertheless allow for a very narrowparameter space where two Mott phases are formed, with either one or two excitons per latticesite. Our ability to program the lattice geometry explored by excitons then paves the way towardsquantum simulations of long-sought many-body phases such as supersolids.
Our intuitive understanding is often challengedwhen it is confronted to the physical properties ofstrongly correlated many-body systems. For these,the interplay between the interaction strength and thekinetic energy can lead to intriguing quantum phasesunexpected at first. This is notably the case whenfermions/bosons explore a lattice potential, since anincompressible state with the same integer numberof particles per lattice site is energetically preferredabove a critical interaction strength [1]. Such Mottinsulator (MI) has received a considerable attention,for electrons in a wide class of materials [2–7], andmore recently for both bosonic and fermionic ultra-cold atoms [8–12]. Indeed, a Mott phase provides thebuilding block to study strongly correlated quantummany-body states such as high-temperature supercon-ductors [3] or supersolids [13, 14].For bosonic systems, a MI is specifically capturedby the celebrated Bose-Hubbard (BH) hamiltonian [1].Studies have mostly addressed this model in its sim-plest form, which is restricted to a single state per lat-tice site, i.e. a single Wannier state (WS), includingonly the strength of on-site interactions U togetherwith the tunneling strength between nearest neigh-bouring sites t (Fig.1.a). The underlying reason iscertainly that even in this elementary form the BHmodel is mostly intractable. Its treatment at experi-mentally accessible temperatures T challenges theory[15–17]. In the weak interaction regime, experimentswith ultra-cold atoms [9] have nevertheless shown thatMott insulating phases are accessible for k B T (cid:46) U/ P controls the average densityin the lattice. In the following excitons are studied250 ns after extinction of the laser, corresponding toaround half their optical lifetime (Fig.S4), to ensurethat they are well thermalised [25, 26]. Using nano-patterned metallic electrodes deposited at the surfaceof the field-effect structure embedding the GaAs bi-layer (Fig.1.b and Fig.S1.a), we engineer a spatiallyhomogeneous square lattice potential [27, 28]. Forthat, we exploit the interaction between the excitonslarge permanent electric dipole and the spatially vary-ing electric field defined by the gate electrodes [29–32].For our studies the lattice depth is set to around 1.5meV with 800 nm periodicity. Accordingly, Fig. 1.cshows that theoretically 15 WS are accessible in eachlattice site and separated by around 100 µ eV (see alsoFig.S2)From first principle calculations (Ref. [33] and Fig.S3), we estimate that excitons experience on-site re-pulsive dipolar interactions with a magnitude U ∼ µ eV that can not be neglected compared to the en-ergy splitting between WS. Computing the tunnelingstrength between nearest neighbouring sites, we de-duce the ratio ( U/t ) for each confined state and thenfind that (
U/t ) is at least of the order of 10 for the 1 st to the 9 th Wannier states (Fig.1.c). Exciton tunnelingis thus strongly suppressed, and these states are a pri- a r X i v : . [ c ond - m a t . qu a n t - g a s ] F e b ori all candidates to host a MI phase since ( U/t ) lieswell above the critical value (
U/t ) | c where a Mott in-sulator becomes energetically unfavourable. Indeed,one expects that ( U/t ) | c ∼
20 for weak interactionsand low temperatures [17].To characterise the phases realised in the lattice, wehave to extract the occupation of WS. This is achievedby studying the photoluminescence spectrum radiatedby dipolar excitons. Indeed, each WS leads to an in-dividual optical emission line at its corresponding en-ergy, whose amplitude then translates into the fractionof excitons occupying the considered state. Figure 2provides a first example by presenting the photolumi-nescence spectrum spatially resolved along one axis ofthe lattice, for an average filling of around 3 excitonsper site at T = 330 mK (see Supplementary Informa-tions). Adjusting the occupation probability of eachWS to match the photoluminescence intensity profile,Fig. 2.b highlights that we quantitatively reproducethe spectra emitted all along the vertical axis of thelattice. For the three positions highlighted in Fig. 2.b,Fig.2.c displays the inferred occupation probability p of the 15 Wannier states. We note that they are es-sentially all populated in these experiments.In Fig.2 the photoluminescence spectrum stronglyvaries across the lattice. This shows that a classical in-sulator is realised since excitons are well localised butpopulate different WS in neighbouring sites. On onehand, the occupation of various WS in each site cer-tainly results from the strength of on-site interactions,since for an average filling of 3 excitons per site the in-teraction strength (2 U ) exceeds the difference betweenthe energies of successive Wannier states. The excitonpopulation must then be arranged between several WSin each site. Additionally, let us underline that the en-ergy relaxation in the lattice certainly plays anotherimportant role here. Indeed, while excitons efficientlythermalise towards lowest unconfined states thanks tothe phonon bath [25, 26], their relaxation is more te-dious in the lattice since the energy splitting betweenWannier states exceeds the thermal energy by around4-fold. Two or many-body exciton collisions therebyconstitute the only mechanism to populate confinedstates. The occupation of WS then sensitively varieswith the local exciton density. This conclusion is actu-ally directly supported by the difference between thespectra emitted at the center and at the edges of theregion illuminated by our laser excitation, middle andbottom-top panels of Fig.2.b respectively.To possibly implement a Mott insulator, we mustensure that the number of excitons per lattice site n is integer and uniform, and that excitons are all con-fined in the same Wannier state. To meet this strin-gent requirement we thoroughly varied the power P of the loading laser pulse that controls the averagefilling of lattice sites. For the experiments shown inFig.2 we have for instance an average of around 3 ex-citons per site for P = 6 nW. Figure 3 shows thatwe thus found two specific values, namely P = 2 nW(Fig.3.a) and P = 3 nW (Fig.3.b) for which the photo-luminescence spectrum essentially reduces to a single sharp emission line, extending over 3 (Fig. 3.a) and4 (Fig. 3.b) lattice sites vertically. Let us then notethat along the horizontal direction we average here 3lattice rows given the optical magnification of our ex-periments. Modelling the photoluminescence spectrawe deduce that in these experiments over 40% of con-fined excitons occupy the 7 th Wannier state whereasthe other 14 accessible states are all very weakly pop-ulated (middle and bottom rows of Fig. 3.a-b). Fur-thermore, the insets in the bottom panels of Fig.3.aand 3.b reveal that for both P = 2 nW and P = 3nW the occupation of the 7 th Wannier level displaysweak spatial variations, which is striking given thenon-uniformity of our laser excitation. Overall, thesecombined observations signal that the filling merelychanges between the lattice sites, and that excitonsall occupy the same Wannier state. The measure-ments shown in Fig.3.a-b are then compatible withtwo quantum insulating phases.Mott insulators with n = 1 and n = 2 particlesper site are energetically separated by U [1]. Remark-ably, between the maximum of the spectra displayedin Fig.3.a and 3.b we measure an energy shift of 80 µ eV well matching the estimated strength of on-siterepulsive dipolar interactions. Moreover, for the ex-periments shown in Fig. 3 the exciton density cannot exceed 1 . · cm − (Supplementary Informa-tions), corresponding to at most 3 excitons per latticesites in average. In the middle row of Fig.3 we ob-serve a ratio of around 2 between the peak intensityof the spectra. Then, we deduce that Fig.3.a signalsa Mott insulator with n = 1 exciton per lattice site,while Fig. 3.b shows a MI phase with n = 2. Fur-thermore, for P (cid:54) = (2 ,
3) nW we do not observe anyother realisation where the occupation probability ofa single Wannier state is dominant. This signals thatMott phases are only found for two particular situa-tions and surrounded by a classical insulator regime.This behaviour is summarised in Fig.3.c, which pro-vides a cut of the BH phase diagram deduced fromour experiments, for t and U fixed while the excitondensity is varied. Strikingly, our measurements qual-itatively reproduce the phases theoretically expectedat finite temperatures for a single band and weaker in-teractions [17] (see inset in Fig.3.c). Nevertheless, themechanism responsible for the buildup of a MI phasein the 7 th Wannier state remains a puzzling question.Each measurement shown in Fig.3 require aroundfive-minute acquisitions so that 255 · realisationsare averaged (see Supplementary Informations). Thisfirst shows that MI phases are well stabilised in ourexperiments. Nevertheless, the mean fraction of ex-citons contributing to them is bound to about 50%for both n = 1 and n = 2 (Fig. 3.c). We attributethis limitation to our lowest accessible bath temper-ature, T = 330 mK. Indeed, it yields U/k B T ∼ n = 2 Mott phase as a functionof the bath temperature. Figure 4.a shows that theoccupation probability of the 7 th Wannier level p (7)is dramatically reduced while the bath temperature isincreased, starting from 55% at 330 mK. Moreover,the occupation of lower energy states is increased,manifesting that exciton relaxation towards deeperconfined levels is more effective. This behaviour wassomewhat expected since the energy splitting betweenWS is around the thermal energy for T ∼ n = 2 Mott phasewe compared p (7) to the summed probability to pop-ulate any lower energy WS, (cid:80) i =1 , p ( i ). Indeed, theMI phase is protected energetically by U [1], from par-ticle/hole excitations that increase the occupation oflower energy states according to Fig.4.a. Theoretically p (7) / (cid:80) i =1 , p ( i ) scales then as e − U/k B T and Fig.4.bshows that our measurements follow this behaviour ifwe set U = 60 ± µ eV. Thus, we confirm the pre-viously deduced magnitude for on-site dipolar repul-sions. Moreover, we verify that the Mott phase meltsvery rapidly since our lowest bath temperature is al-ready relatively high to possibly stabilise it deeply [9].We have shown that dipolar excitons offer a new av-enue to explore the Bose-Hubbard physics for stronglyinteracting two-dimensional fluids. For that, our solid-state technology is very attractive, first because itis a priori scalable to a few hundreds of excitons,but also since excitons experience dipolar interactions with a magnitude greatly exceeding the one accessi-ble to other systems. Evidencing nearest-neighbourdipolar interactions between the lattice sites then con-stitutes an outstanding challenge for future experi-ments. We estimate that this requires lattice peri-odicities around 300 nm, which are at experimentalreach. For such short lattice period, decreasing theexciton temperature to a few 10 mK shall allow us toexplore the long-sought many-body phases accessibleto dipolar gases [34] and that spontaneously break thelattice symmetry, such as stripes, checkerboard [35] orsupersolid phases [36–40]. Acknowledgments
We would like to thank M. Lewenstein, M. Holz-mann, M. Polini and A. Reserbat-Plantey for a criti-cal reading of our manuscript, together with S. Gas-paretto for graphical works. Our research has beenfinancially supported by the Labex Matisse and by IX-TASE from the French Agency for Research (ANR).The work at Princeton University was funded bythe Gordon and Betty Moore Foundation throughthe EPiQS initiative Grant GBMF4420, and by theNational Science Foundation MRSEC Grant DMR1420541. [1] ”Many-Body Physics with Ultracold Gases”, LectureNotes of the Les Houches Summer School (Eds. C.Salomon, G. V. Shlyapnikov, and L. F. Cugliandolo,2010)[2] P. A. Lee, N. Nagaosa, and X. G. Wen, Rev. Mod.Phys. , 17 (2006)[3] F. Gebhard ”The Mott Metal-Insulator Transition”(Ed. Springer, Berlin, 1997)[4] Y. Cao et al., Nature , 80 (2018)[5] Y Shimazaki, I Schwartz, K Watanabe, T Taniguchi,M Kroner, A Imamoglu, Nature , 472 (2020)[6] Y. Tang, L. Li, T. Li, Y. Xu, S. Liu, K. Barmak, K.Watanabe, T. Taniguchi, A. H. MacDonald, J. Shanand K. F. Mak, Nature , 353 (2020)[7] E. C. Regan, D. Wang, C. Jin, M. I. B. Utama, B.Gao, X. Wei, S. Zhao, W. Zhao, Z. Zhang, K. Yu-migeta, et al., Nature , 359 (2020)[8] W.S. Bakr et al., Science , 547 (2010)[9] Sherson, J., Weitenberg, C., Endres, M. et al., Nature , 68 (2010)[10] R. Jordens, N. Strohmaier, K Guenter, H. Moritz, T.Esslinger, Nature , 204 (2008)[11] U. Schneider et al., Science , 1520 (2010)[12] D. Greif et al., Science , 953 (2016)[13] K. Goral, L. Santos, and M. Lewenstein, Phys. Rev.Lett. , 170406 (2002)[14] A. J. Leggett, Phys. Rev. Lett. , 1543 (1970)[15] F. Gerbier, Phys. Rev. Lett. , 120405 (2007)[16] B. DeMarco, C. Lannert, S. Vishveshwara, and T.-C.Wei, Phys. Rev. A , 063601 (2005)[17] K. W. Mahmud, E. N. Duchon, Y. Kato, N. Kawashima, R. T. Scalettar, and N. Trivedi, Phys.Rev. B , 054302 (2011)[18] J. Larson, A. Collin, and J.-P. Martikainen, Phys.Rev. A , 033603 (2009)[19] K. L. Seyler, P. Rivera, H. Yu, N. P. Wilson, E. L.Ray, D. G. Mandrus, J. Yan, W. Yao and X. Xu,Nature , 66 (2019)[20] K Tran, G. Moody, F. Wu, X. Lu, J. Choi, K. Kim, A.Rai, D.l A. Sanchez, J. Quan, A. Singh et al., Nature , 71 (2019)[21] C. Jin, E. C. Regan, A. Yan, M. I. B. Utama, D.Wang, S. Zhao, Y. Qin, S. Yang, Z. Zheng, S. Shi etal., Nature , 76 (2019)[22] E. M. Alexeev, D. A. Ruiz-Tijerina, M. Danovich, M.J. Hamer, D. J. Terry, P. K. Nayak, S. Ahn, S. Pak7,J. Lee, J. I. Sohn et al., Nature , 81 (2019)[23] S. Baier, M. J. Mark, D. Petter, K. Aikawa, L.Chomaz, Z. Cai, M. Baranov, P. Zoller, F. Ferlaino,Science , 251 (2016)[24] M. Combescot, R. Combescot, F. Dubin, Rep. Prog.Phys. , 066401 (2017).[25] A L Ivanov 2004 J. Phys.: Condens. Matter , S3629(2004)[26] M. Beian et al., EuroPhys. Lett. , 37004 (2017)[27] C. Lagoin et al., Phys. Rev. B , 245428 (2020)[28] C. Lagoin et al., Phys. Rev. Lett , 067404 (2021)[29] A.A. High et al., Phys. Rev. Lett. , 087403 (2009)[30] G. Grosso, Nat. Phot. , 577 (2009)[31] A. G. Winbow et al., Phys. Rev. Lett. , 196806(2011)[32] I. Rosenberg, Y. Mazuz-Harpaz, R. Rapaport, K. West, and L. Pfeiffer, Phys. Rev. B ,195151 (2016)[33] G. J. Schinner, J. Repp, E. Schubert, A. K. Rai, D.Reuter, A. D. Wieck, A. O. Govorov, A. W. Holleit-ner, and J. P. Kotthaus, Phys. Rev. Lett. , 127403(2013)[34] M. A. Baranov, M. Dalmonte, G. Pupillo, P. Zoller,Chem. Rev. , 5012 (2012)[35] C. Trefzger, C. Menotti, B. Capogrosso-Sansone andM. Lewenstein, Jour. of Phys. B , 193001 (2010)[36] J. L´eonard, A. Morales, P. Zupancic et al., Nature , 87 (2017)[37] J. R. Li, J. Lee, W. Huang et al., Nature , 91(2017)[38] M. Guo, F. B¨ottcher, J. Hertkorn et al., Nature ,386 (2019)[39] Tanzi, L., Roccuzzo, S.M., Lucioni, E. et al., Nature , 382 (2019)[40] P. Ilzh¨ofer, M. Sohmen, G. Durastante, C. Politi, A.Trautmann, G. Natale, G. Morpurgo, T. Giamarchi,L. Chomaz, M. J. Mark, F. Ferlaino, Nat. Phys.(2021) E ne r g y [ m e V ] Position [ μ m]-0.8 0 0.801.5 1715 Le v e l nu m be r a) b) c) FIG. 1:
Bose-Hubbard physics. a) Within the BH model, bosonic particles (red) explore a two-dimensional lattice(blue), experiencing on-site interactions with a strength U while the tunneling strength between nearest neighbouringsites has an amplitude t . b) Our semiconductor device relies on two GaAs quantum wells, each confining electrons orholes that form dipolar excitons (red ball). These are confined in a two-dimensional electrostatic lattice due to theinteraction between their permanent electric dipole (arrow) and the electric field imposed by the array of surface gateelectrodes (gold). Electronic carriers are optically injected in the lattice using a laser beam focussed to about 4 µ m atthe surface (red), thus covering 5x5 sites. c) For our 1.5 meV deep lattice with 800 nm period 15 Wannier states (red)coexist, separated by around 100 ueV. Energy [meV]0-33 P o s i t i on [ μ m ] I n t en s i t y [ a . u .] p ( n )
71 15WS number (n)0.20.100.1000.1 a) b) c)
FIG. 2:
Exciton classical insulator. a) Spatially and spectrally resolved photoluminescence emitted at T =330 mKwhen we impose an average filling around 3 excitons per lattice site ( P = 6 nW). b) Spectra measured at the positionsunderlined by the dashed lines in the panel a), from top to bottom. Experimental data are displayed by gray points,error bars representing the poissonian uncertainty, while the solid red lines show the modelled photoluminescence spectra,assigning a resolution limited 100 µ eV linewidth for the emission of each WS. c) Occupation probability p of all 15 WSused to reproduce the experiments shown in b). a) b) c) FIG. 3:
Exciton quantum insulator. a) Spatially and spectrally resolved photoluminescence for P = 2 nW at T =330mK (top). The middle panel displays the spectrum measured in the 1.5 µ m central region (gray points) together withthe fitted profile (red) from which the occupation probability p of all WS is deduced (bottom panel, where the insetshows the normalized spatial variation of p (7)). b) Same measurements as in a) but for P = 3 nW. The vertical dashedline in the middle panel marks the maximum of the spectrum for P = 2 nW. c) Occupation probability p (7) as a functionof P . The axis on the left side provides a scale converted in units of µ/U . The inset depicts schematically the phasediagram predicted for the BH model at finite temperature and for weak interactions, with the first n = 1 and n = 2 MIsurrounded by a classical phase. The dashed line illustrates the vertical cut corresponding to our experiments. Temperature [K]1 1.50.3521 Σ p i p
550 mK330 mK p ( n ) WS number (n)71 150 a) b)
FIG. 4:
Melting of the exciton Mott phase. a) Occupation probability of the 15 Wannier states as a functionof the bath temperature, from 330 mK to 1.5 K from top to bottom. In every case P =3 nW so that the system isinitially prepared in the n = 2 MI phase. b) Ratio between the occupation probability of the 7 th WS p (7) and thesummed probabilities to occupy any lower energy state (cid:80) i =1 − p ( i ). The red shaded area marks an exponential decrease e − U/k B T setting U = 60 ± µ eV. I. SUPPLEMENTARY INFORMATIONSA. Sample structure and experimentalprocedure
Our device relies on two 8 nm wide GaAs quantumwells, separated by a 4 nm Al . Ga . As barrier. Thequantum wells are positioned 450 nm below the sur-face and 150 nm above a n -doped GaAs substrate. Atthe surface we deposited the metallic electrodes shownin Fig. S1.a. We apply an electric bias V = − E z has an amplitude inaverage equal to 4.3 V/ µ m, modulated by ∆ E z ∼ µ m between the regions lying under the electrodesand under the interstices between them that are unpo-larised. The interaction between the modulated fieldand the excitons electric dipole d leads to an electro-static lattice with a depth V = ( d. ∆ E z ) =1.5 meV,where d = e ·
12 nm with e the electron charge. Notethat our device is around 15x15 µ m , with 18 sitesalong both horizontal and vertical axis.Fig. 1.b illustrates that excitons are optically in-jected in the lattice, using a laser excitation resonantwith direct exciton transition (DX) of the two GaAsquantum wells (Fig.S1.b). Thus we minimise thedensity of photo-induced free carriers, as confirmedby the photocurrent bound to 10-20 pA in ourstudies. Fig.S1.c illustrates that our laser excitationlasts 100 ns and is repeated at 850 kHz. The excitonphotoluminescence, emitted at around 1.528 eV, isthen analysed at variable delays to the laser pulse, ina time window set to 50 ns long. In the measurementsdiscussed in the main text the delay is fixed to 250ns. B. Lattice potential and Wannier states
To evaluate the distribution of Wannier states wesolved the Schr¨odinger equation for an exciton ex-ploring our electrostatic lattice free from interaction.This turns into solving a second order differentialMathieu equation, which solutions are known asMathieu functions corresponding to the Bloch func-tions, associated to eigen-energies providing thelattice energy bands. Fig. S2.a shows these Blochenergy bands, in the first Brillouin zone. We observethat Bloch bands are energetically separated byaround 100 µ eV, most of them exhibiting a rather flatdispersion characteristic of a deep confining potential.We then deduce that the tunneling strength betweennearest neighbouring sites, given by the width of theBloch energy bands, is bound to about 10 − µ eV,except for the energy bands closest to the top of theconfining potential. Bloch functions provide an orthogonal set of eigensolutions of the Sch¨odinger equation, which by def-inition exhibit the lattice periodicity. It is usuallymore convenient to study physical properties in an-other basis, namely the one of Wannier functions thatare exponentially localised around the lattice sites.Fig. S2.b shows that Wannier states lie at energiescorresponding to the Bloch bands energies averagedover the first Brillouin zone. Moreover, Fig.S2.c illus-trates the spatial profile of a few Wannier functions. C. On-site interaction strength U To estimate U we followed a semi-classical ap-proach [33]. First, we approximate the profile ofthe trapping potential in the lattice sites by anharmonic dispersion characterised by its frequencyΩ = 2 √ V · E r / (cid:126) , V being the potential depth while E r = π (cid:126) / (2 ma ) with a the lattice period and m ∼ . m the exciton effective mass, m being thefree electron mass. Then, we look for the distancebetween 2 excitons confined in the same lattice site,2 r , that minimises the sum of the potential energyand the dipolar repulsion between them. Thus, wefind r = (cid:16) (cid:15) d m Ω (cid:17) / where (cid:15) denotes the dielectricconstant of GaAs. Accordingly we deduce that theon-site interaction strength reads U = d / (cid:15)r . Fig.S3 displays the scaling of U as a function of thelattice period. We note that for our 0.8 µ m periodlattice U ∼ µ eV. D. Average exciton density
To evaluate the average exciton density in the lat-tice we measured the decay of the photoluminescenceenergy after extinction of the loading laser pulse. Pre-cisely, we extracted the photoluminescence blueshift,i.e. the difference between the photoluminescenceenergy 250 ns after extinction of the laser pulse, tothe one for much longer delays when the density isvanishingly small. This allows us to directly deducethe average exciton density [26, 27]. Nevertheless,such measurement is difficult to interpret whenexcitons explore the lattice potential, since at lowdensities excitons are localised in the Wannier statesof the lattice sites and possibly even dynamicallyrelax between them. Therefore we studied thedynamics of the photoluminescence energy when ex-citons explore a spatially homogeneous confinementlandscape, for the same excitation conditions as forthe experiments discussed in Fig. 2-4. Precisely, weused a continuous gate electrode deposited at thesurface of our heterostructure. Fig. S4.a shows thenthat the blueshift of the photoluminescence energydoes not exceed 100 µ eV for a delay set to 250 nsand for P = 6 . cm − in these measurements. The number of excitonsper lattice site n can then not exceed 3 to 4 for this”high” excitation power. In fact, lower densities arehardly inferred from the photoluminescence blueshift,since the latter becomes then comparable to the50-100 µ eV precision of our measurements, as forinstance for delays between 350 and 500 ns in Fig.S4.a. Nevertheless, a decay of the exciton populationis resolved studying the integrated intensity of thephotoluminescence (Fig. S4.b) from which we deducethat dipolar excitons exhibit a radiative lifetime ofaround 600 ns.0 Time [ns]
LASER excitation
50 ns detection
Laser at 787 nmDXIXDX
200 nm a) b)c)
Fig. S1 : a) Electron microscope image of the surface electrodes engineering the lattice potential explored bydipolar excitons. b) Our experiments rely on a laser excitation set at resonance with the direct exciton (DX)absorption of each quantum well of the GaAs bilayer. Dipolar excitons (IX) are formed once photo-injectedelectrons and holes, filled and open circles, have tunnelled towards their minimum energy states that lie in adistinct layer. c) Our measurements are all performed dynamically, relying on a 100 ns long laser excitationrepeated at 850 kHz, while the excitons’ photoluminescence is detected in a 50 ns long time window at a variabledelay after extinction of the laser excitation. In the experiments discussed in the main text the delay is alwaysset to 250 ns and the power of the loading laser pulse P is varied. E ne r g y [ m e V ] π a) 0.501.01.510.50 x/a0 0.6-0.6 | W S n ( x ) | a) b)c) Fig. S2 : a) Dispersion of Bloch bands in the first Brillouin zone of our lattice potential. The momentum p is shown in units of (cid:126) π/a , where a=0.8 µ m is the period of the lattice. b) Energy of the Wannier states deducedfrom the Bloch bands. c) Spatial profile of the Wannier functions for the 1 st (violet), the 7 th (blue) and the 15 th (gray) confined states. The horizontal axis marks the coordinate along one direction of the lattice, normalisedby the period a . Lattice period [ m] U [ m e V ] Fig. S3 : Estimated on-site interaction strength U between 2 excitons confined in the same lattice site as afunction of the period of the lattice potential. E ne r g y [ m e V ] I n t eg r a t ed i n t en s i t y [ a . u .] .2.4.6.81 a) b) Fig. S4 : a) Energy and b) integrated intensity of the photoluminescence emission as a function of the delayto the end of the loading laser pulse. The latter is set to a power P ∼ . nW which corresponds to the highestexcitation in Fig. 3.c. For a delay set to 250 ns (gray region), as in the experiments shown in Fig. 2-4, thepanel a) shows that the blueshift of the photoluminescence energy does not exceed 100 µµ