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Evidence for a Bose-Einstein condensate ofexcitons

Mathieu Alloing, Mussie Beian, Maciej Lewenstein, David

Fuster, Yolanda Gonzalez, Luisa Gonzalez, Roland

Combescot, Monique Combescot and Francois Dubin

EPL, 107 (2014) 10012

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EPL, 107 (2014) 10012 www.epljournal.org

doi: 10.1209/0295-5075/107/10012

Evidence for a Bose-Einstein condensate of excitons

Mathieu Alloing1, Mussie Beian1, Maciej Lewenstein1,2, David Fuster3, Yolanda Gonzalez3,

Luisa Gonzalez3, Roland Combescot4,5, Monique Combescot6 and Francois Dubin1,6(a)

1 ICFO-The Institute of Photonic Sciences - Av. Carl Friedrich Gauss, num. 3, E-08860 Castelldefels, Spain2 ICREA-Institucio Catalana de Recerca i Estudis Avanats - Lluis Companys 23, E-08010 Barcelona, Spain3 IMM-Instituto de Microelectronica de Madrid (CNM-CSIC) - C. Isaac Newton 8, PTM,E-28760 Tres Cantos, Madrid, Spain4 Laboratoire de Physique Statistique, Ecole Normale Superieure, UPMC Paris 06, Universite Paris Diderot, CNRS24 rue Lhomond, F-75005 Paris, France5 Institut Universitaire de France - 103 boulevard Saint-Michel, F-75005 Paris, France6 Institut des Nanosciences de Paris, UPMC Paris 06, CNRS - 2 pl. Jussieu, F-75005 Paris, France

received 23 May 2014; accepted in final form 19 June 2014published online 9 July 2014

PACS 03.75.Hh – Static properties of condensates; thermodynamical, statistical, and structuralproperties

PACS 78.47.jd – Time resolved luminescencePACS 73.63.Hs – Quantum wells

Abstract – We report compelling evidence for a “gray” condensate of dipolar excitons, electricallypolarised in a 25 nm wide GaAs quantum well. The condensate is composed by a macroscopicpopulation of dark excitons coherently coupled to a lower population of bright excitons. To createthe exciton condensate we use an all-optical approach in order to produce microscopic traps whichconfine a dense exciton gas (∼ 1010 cm−2) that yet exhibits an anomalously weak photoemissionat sub-kelvin temperatures. This is the first fingerprint for the “gray” condensate. It is thenconfirmed by the macroscopic spatial coherence and the linear polarization of the weak excitonicphotoluminescence emitted from the trap, as theoretically predicted.

editor’s choice Copyright c© EPLA, 2014

The demonstration of Bose-Einstein condensation inatomic gases at micro-kelvin temperatures is a strikinglandmark [1] while its evidence for semiconductor exci-tons [2–5] still is a long-awaited milestone. This situ-ation was not foreseen because excitons are light-massboson-like particles with a condensation expected to occurbelow a few kelvins [6,7]. An explanation can be found inthe underlying fermionic nature of excitons which rulestheir condensation [8]. Precisely, it was recently predictedthat, at accessible experimental conditions, the excitoncondensate shall be “gray” with a dominant dark partcoherently coupled to a weak bright component throughfermion exchanges [9]. This counter-intuitive quantumcondensation, since excitons are mostly known for theiroptical activity, directly follows from the excitons inter-nal structure which has an optically inactive, i.e., dark,ground state [8].

(a)E-mail: francois.dubin@insp.jussieu.fr (correspondingauthor)

The Bose-Einstein condensation of semiconductor exci-tons has received a considerable attention since its the-oretical prediction [2–5] in the 1960s. Unlike the mostcommonly studied bosonic atoms [6], the exciton com-posite nature plays a key role in their condensation. Insemiconductor quantum wells, excitons are made of spin(±1/2) electrons and “spin” (±3/2) holes. These carri-ers mainly interact through attractive intraband Coulombprocesses. Weak interband valence-conduction Coulombprocesses also exist, but only for optically active, or“bright”, excitons with total spin (±1) made of (±3/2)holes and (∓1/2) electrons. These (repulsive) interbandprocesses bring the energy of bright excitons above the oneof “dark” excitons with total spin (±2) made of (±3/2)holes and (±1/2) electrons. As a result, Bose-Einsteincondensation of excitons must occur within the low-energydark states [8,10]. This feature, which makes exciton con-densation hard to evidence by conventional optical means,could be the reason why unambiguous signatures of exci-ton condensates have not been given yet, despite several

10012-p1

Mathieu Alloing et al.

decades of active experimental research. By contrast, theexcitonic component of a polariton being by constructioncoupled to light, polariton condensation can be studiedthrough photoluminescence, which recently led to remark-able experiments [11–14].

While in the very dilute regime the exciton condensatemust be completely dark, no matter how small the en-ergy difference between dark and bright states is [8], itwas recently shown [9] that at sufficiently large density,carrier exchange between bright and dark excitons bringsa coherent bright component to the condensate which be-comes “gray”. Such a coherence between dark and brightexcitons is very similar to what occurs in the well-knownphases of superfluid 3He (see, e.g., ref. [15]), and in themore recent spinor condensates of ultracold atomic Bosegases [16], where components of these superfluids corre-sponding to different internal degrees of freedom coexistand are coherent. This coherent coupling allows probingthe exciton condensate through the photoluminescence ofits bright part. As the bright component is very small,the photoluminescence signal is very weak. Neverthe-less, it shall unveil the existence of a dense populationof dark excitons, the spatial coherence of the condensateand its internal “spin” structure through the polarisationof the emitted light. In semiconductor quantum wells,the dark nature of exciton Bose-Einstein condensation hasbeen overlooked until very recently [8,10], most probablybecause the splitting between bright and dark states issmall compared to the thermal energy at critical temper-ature. Here, we present compelling experimental evidencefor a “gray” Bose-Einstein condensate of excitons [8,9]through the experimental observation of all its theoreti-cally predicted characteristics.

We study excitons confined in a 25 nm wide GaAs singlequantum well embedded in a field-effect device with anelectrical polarization, set by the potential Vg = −4.7 Vapplied on a surface electrode, keeping electrons and holeswell apart. As the electron and hole wave functions have asmall overlap, these dipolar excitons have a long radiativelifetime (∼ 20 ns) and a rather large energy splitting be-tween bright and dark states (∼ 20 µeV, see refs. [17,18]).The electrical polarisation also ensures a repulsive effectiveexciton-exciton interaction which prevents the formationof biexcitons at the typical density nc where Bose statis-tics becomes dominant (nc ≃ mXkBT/�

2 ∼ 1010 cm−2 for2D excitons with mass mX at 1 K).

We use a pump laser (λpump = 641.5 nm) with an energyabove the AlGaAs barriers of the quantum well in orderto create a dense and well-thermalised exciton gas. Forsuch laser excitation photo-injected electrons and holesare captured by the quantum well with different efficien-cies [19,20]: a region richer in holes is formed aroundthe laser excitation, itself surrounded by an electron-richdomain resulting from both the photo-current passingthrough our device and the modulation doping of thestructure (see fig. 1(A)). In this landscape, dipolar exci-tons are created through the Coulomb interaction between

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Fig. 1: (Color online) (A) Sketch of our field-effect device:a pump beam excites a wide single quantum well (SQW) em-bedded in a field-effect device. This results in a region richin holes (open circles) around the laser spot, surrounded by aregion rich in electrons (filled circles). These excess chargesscreen the electric field imposed by the potential Vg applied onthe top electrode of the device. (B) Schematic time sequenceshowing pump and probe pulses together with the intervalsduring which our experimental results are recorded (green).(C), (D): real photoluminescence image recorded in a 40 ns longtime interval, starting 10 ns after extinction of the pump excita-tion, at 350 mK (C) and at 7K (D). (E), (F): spatial profiles of

the excitonic confinement deduced from the E(probe)X measured

in the “probe 1” pulse (red line) together with the exciton

blueshift deduced from E(pump)X measured during the “probe

2” pulse (blue line). These profiles are taken along the whitestraight lines indicated in (C), (D). The solid black lines in(E), (F) show the profiles of the photoluminescence intensity.In (E) the gray region underlines a large exciton density at350 mK in a region of anomalously weak photoluminescence.It reveals the existence of a “gray” exciton condensate.

photo-injected electrons and holes, the exciton transportbeing somewhat complicated by the ambipolar diffusionof excess carriers which screen the external field appliedthrough our top gate electrode.

Figures 1(C) and (D) show the photoluminescence emis-sion 10 ns after the pump pulse at 350 mK and 7 K, re-spectively. Both reveal a pattern characteristic of thecharge separation existing in the quantum well, namelya macroscopic exciton ring formed a few tens of micronsaway from the pump excitation [21,22]. Figures 1(E)and (F) show the exciton confinement potential togetherwith the profile of the exciton density for these measure-ments. They are both deduced from a weak probe pulsewhich injects a very dilute exciton cloud after the pump

10012-p2

Evidence for a Bose-Einstein condensate of excitons

pulse. Note that this probe beam is tuned well belowthe bandgap of the AlGaAs barriers (λprobe = 790 nm) inorder to bring essentially no perturbation [23]. Excitonsinjected by the probe beam emit a photoluminescence at

an energy EX ≃ (−→d ·

−→F screen+u0nX). The first term corre-

sponds to the energy increase of a single exciton resultingfrom the screening field Fscreen induced by excess carriers,d ≈ e · 15 nm being the excitonic dipole moment [24]. Thesecond term, where nX is the exciton density and u0 a pa-rameter given by the geometry of our heterostructure [25],corresponds to the energy increase resulting from repulsiveexciton-exciton interactions. By sending the probe pulse100 ns after extinction of the pump, i.e. when all excitonsinjected by the pump have recombined, we directly mapthe excitons confinement as for very small exciton densities

EX reduces to E(probe)X =

−→d ·

−→F screen. We can estimate the

density profile of excitons produced by the pump excita-tion by sending a probe pulse 10 ns after extinction of the

pump as EX reduces then to E(pump)X = (E

(probe)X + u0nX).

This pump-probe spectroscopy then allows us to extractthe whole exciton density injected by the pump pulse, nX ,including dark and bright excitons, a crucial ingredient tosignal a “gray” condensate.

Figures 1(E), (F) show the profile of the exciton con-

finement through E(probe)X . It displays a maximum at the

pump laser spot, then slowly decreases with the distanceto the laser excitation before an abrupt decrease just be-fore the region where the ring is formed. This is physicallyreasonable because the ring is expected to be located atthe interface between the electron-rich and hole-rich do-mains where the near absence of excess carriers leads toa minimum screening. Most importantly, our measure-ments reveal at 350 mK that an electrostatic trap, i.e. a

local minimum of E(probe)X , is spontaneously formed in the

vicinity of the ring (see fig. 1(E)). This potential confinesdipolar excitons which are high-field seekers, i.e., attractedby strong field regions. We note that the exciton trapis not homogeneous along the circumference of the ring,i.e., it is not identical on opposite sides (see fig. 1(E)). Itsdepth can be as large as ≃ 1.2 meV at 350 mK, an increaseof the bath temperature leading to a reduction or a totalsuppression of the trap (fig. 1(F)).

We evaluate the exciton density by comparing E(pump)X

to E(probe)X . Figure 1(E) shows at 350 mK that across the

emission E(pump)X is blueshifted compared to E

(probe)X . This

reveals that the pump pulse creates a dense exciton gas

(nX ∼ 1010 cm−2 for a blueshift (E(pump)X − E

(probe)X ) ∼

1 meV). More strikingly, we also note that E(pump)X is al-

most constant in the region of the ring (fig. 1(E)), unlikeat 7 K (fig. 1(F)). This reveals that dipolar excitons com-pletely fill the trap spontaneously formed in the vicinity ofthe ring. The experimental blueshift (1.2 meV around theposition −25 µm in fig. 1(E)) leads to an exciton density∼ 2 ·1010 cm−2 across the entire trap [25–27]. At the sametime, the photoluminescence intensity is reduced by more

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Fig. 2: (Color online) (A), (B): photoluminescence taken fromthe same position of the ring at 370 mK (A) and at 7K (B).(C), (D): map of the interference contrast for δx = 1.5µmat 370 mK (C) and at 7K (D). The shaded white area masksthe region where the interference contrast cannot be accuratelymeasured because the photoluminescence intensity is too weak.(E), (F): variation of the interference contrast, at 370 mK (E)and at 7 K (F), as a function of the interferometer phase. Dataare taken at the position of the ring (black) and 5 µm outsidethe ring (red). The white circles in (A) and (B) show thesetwo positions.

than ten fold between the positions (−25) and (−31)µm.In the outer rim of the ring, the excitonic population thendominantly consists of optically inactive, i.e. dark, exci-tons. The dense but nearly dark exciton gas, highlightedby the gray area in fig. 1(E), can only be explained by theformation of a condensate which acts as a sink and cap-tures most of the excitons, this condensate being nearlydark, or “gray”. Indeed, in the absence of a condensate,i.e. in the classical regime, the dark and bright popu-lations should be very similar, because the dark-brightenergy splitting (∼20 µeV) is smaller than the thermalenergy (∼ 30 µeV at our lowest bath temperature). Thiswould lead to a much stronger photoluminescence thanthe one we observe. Such a conclusion is further sup-ported by the spectral profile of the photoluminescencewhich is essentially identical on the brightest part of thering and 6 µm outside [24], fully coherent with the fact

10012-p3

Mathieu Alloing et al.

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Fig. 3: (Color online) (A) Intensity (black) and interference contrast (red) of the photoemission across the fragmented ring at370 mK. The red area shows the relative error in our measurements. (B) Interference contrast for δx = 1.5 µm as a functionof the bath temperature Tb, taken 5 µm outside the ring (black) and at the position of the ring (red). (C) Coherence lengthas a function of the bath temperature measured 5 µm outside of the ring. The gray area shows the limit of our experimentalresolution.

that the total exciton density stays unchanged through-out this region. Finally, fig. 1(F) shows that at highertemperatures we no longer find the spectral signature ofa “gray” condensate. We wish to note that the specificposition where the condensate is formed certainly is theresult of a complex hydrodynamical diffusion, dipolar ex-citons experiencing a chute from the pump laser spot tothe trapping region, while cooling at the same time.

To unambiguously confirm the formation of a “gray”condensate at sub-kelvin bath temperatures, we furtherstudy the first-order spatial coherence |g(1)| of the pho-toluminescence [24] since the light emitted by the brightpart of the condensate reflects its long-range order. Theclassical regime is distinguished from the quantum regimethrough its spatial coherence [28,29], the classical coher-ence length being of the order of the de Broglie wavelength(λdB ∼ 100 nm at sub-kelvin temperatures). We experi-mentally assess the coherence length of bright excitons byusing an actively stabilized Mach-Zehnder interferometer.One arm of the interferometer displaces the photolumi-nescence laterally by δx = 1.5 µm with respect to thesecond arm; it also tilts it vertically, so that output in-terference fringes end up aligned horizontally [24,30]. Byscanning the phase of the interferometer, we reconstructpoint by point the amplitude of the interference contrast.This allows us to draw the map of the emission first-orderspatial coherence from which we deduce [24] the excitoncoherence length ξ.

Figure 2 shows |g(1)| in the ring region. At high temper-ature (Tb = 7 K in fig. 2(D)), the interference contrast doesnot significantly vary across the emission, staying approx-imately equal to 10–15% which is the background value ofour interference contrast [24]. This leads to a coherencelength ξ � 200 nm. At lower temperature (Tb = 370 mKin fig. 2(C)), the interference contrast exhibits a patterncorrelated with the spatial profile of the photoemissionin a way which again reveals a “gray” condensate. In-deed, fig. 3(A) shows that the interference contrast is min-imal (∼10%) in the brightest part of the ring while, inthe outer region where the photoluminescence intensity

is strongly decreased, |g(1)| can reach ≈ 40% which isabove half the auto-correlation value (70% for δx = 0).This shows that the observed bright excitons with a co-herence length ξ ∼ 1.5 µm one order of magnitude largerthan the de Broglie wavelength, belong to the condensate.The variation of the excitonic coherence as a function ofthe bath temperature moreover shows that non-classicalcorrelations become dominant below a critical tempera-ture ∼2 K, the coherence length abruptly increasing fromnear classical to non-classical values (see fig. 3(C)).

In order to obtain further insight into the internal struc-ture of a gray condensate, we filter the polarization ofthe photoluminescence. In a “gray” condensate, dark andbright states are coupled by carrier exchanges [9] or othercoupling processes as suggested recently [30–33]. Sincethe lowest energy for degenerate (±1) or (±2) states isobtained for a linear polarisation, due again to carrier ex-changes [8], a “gray” condensate must exhibit a linearlypolarized photoluminescence. And indeed, fig. 4(B) showsat 370 mK that the photoluminescence is mostly polarizedlinearly in the outer rim of the ring where macroscopiccoherence is also observed (fig. 3(C)) —the degree of cir-cular polarization being not as significant (see fig. 4(C)).In the inner region of the ring, we also observe a linearpolarization, but along the orthogonal direction. Theseobservations contrast with recent studies performed indouble quantum well heterostructures where dipolar (spa-tially indirect) excitons exhibit correlated patterns havingboth linear and circular polarizations [34]. These patternshave been interpreted in terms of coherent exciton trans-port and spin-orbit coupling [30,32,33]. Our experiments,which do not reveal these patterns, are performed in a sin-gle quantum well with a dark-bright energy splitting muchlarger than in bilayer heterostructures. Since this splittingplays a key role in selecting the specific condensate whichis formed, it is reasonable to think that experiments per-formed in single and double quantum wells probe differentregimes. On the other hand, experiments [35] in a doublequantum well, performed essentially at the same time asours, display, when temperature is lowered below a few

10012-p4

Evidence for a Bose-Einstein condensate of excitons

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Fig. 4: (Color online) Exciton ring at 370 mK (A). (B), (C): pattern of linear (B) and circular (C) polarizations at 370 mK. In(C), the blue and red colors are associated to σ+ and σ− polarized light, respectively. In (B), we see that the photoemissionis mostly linearly polarized in the outer region of the ring where extended spatial coherence is observed. (D) Degree of linear(red) and circular (blue) polarization of the photoluminescence at 370 mK. The solid black line shows the intensity profile ofthe emission.

kelvins, a marked darkening of the excitonic fluid, whichis fully consistent with our results.

As a final remark, one might wonder if our essentiallytwo-dimensional geometry would not dramatically affectBose-Einstein condensation. This is not so because con-densation occurs in small regions which stabilizes the con-densate [1]. In addition, phase fluctuations, responsible forthe main qualitative differences between 2D and 3D sys-tems [1], are then quenched. Hence the condensate shouldbe fairly similar to a 3D condensate.

To summarise we have reported the signatures theoret-ically predicted for a “gray” Bose-Einstein condensate ofdipolar excitons. These were highlighted in microscopicand dark regions where at sub-kelvin temperatures ex-citons obey a quantum statistical distribution, with adominant (∼ 90%) fraction of dark excitons, and ex-hibit a macroscopically large spatial coherence. In ourexperiments the gray condensate is created in a complexparameter space, where most notably we tune both theelectric polarisation of our field-effect device and the in-tensity of the optical excitation. Thus we produce micro-scopic traps, through the screening of the internal field byoptically injected excess charges, where the condensate isspontaneously formed.

∗ ∗ ∗

This work has been supported by the Europeanprojects (EU) ITN-INDEX, EU-CIG XBEC, EU-ERCOSYRIS, EU-IP SIQS, EU-STREP EQUAM, by theSpanish MINECO (Grant TEC2011-29120-C05-04), CAM(Grant S2009ESP-1503).

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