Abstract
Superfluidity, first discovered in liquid ^{4}He, is closely related to Bose–Einstein condensation (BEC) phenomenon. However, even at zero temperature, a fraction of the quantum liquid is excited out of the condensate into higher momentum states via interactioninduced fluctuations—the phenomenon of quantum depletion. Quantum depletion of atomic BECs in thermal equilibrium is well understood theoretically but is difficult to measure. This measurement is even more challenging in drivendissipative exciton–polariton condensates, since their nonequilibrium nature is predicted to suppress quantum depletion. Here, we observe quantum depletion of a highdensity exciton–polariton condensate by detecting the spectral branch of elementary excitations populated by this process. Analysis of this excitation branch shows that quantum depletion of exciton–polariton condensates can closely follow or strongly deviate from the equilibrium Bogoliubov theory, depending on the exciton fraction in an exciton polariton. Our results reveal beyond meanfield effects of exciton–polariton interactions and call for a deeper understanding of the relationship between equilibrium and nonequilibrium BECs.
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Introduction
The fundamental understanding of interacting cold bosonic gases was developed by N. Bogoliubov, whose theory predicted the consequences of interparticle interactions for the properties of Bose–Einstein Condensates (BECs)^{1}. According to this theory, an interacting BEC^{2} is characterised by a modified phononlike dispersion of elementary excitations at long wavelengths (or short wavevectors) that explains the superfluid properties of weakly interacting BECs. The crux of the theory is the nonperturbative transformation of Bogoliubov quasiparticles, where a condensate excitation at a given momentum is expressed as a superposition of counterpropagating singleparticle states^{3}. The quantum fluctuations of the interacting particles in the ground state are responsible for the nonzero occupation of these elementary excitations at zero temperature, leading to the socalled quantum depletion of the condensate population. Quantum depletion was observed in weakly interacting BECs^{4,5,6} of ultracold atoms and is the main reason for a low condensed fraction of strongly interacting superfluid ^{4}He^{7}. This effect is predicted to have pronounced experimental signatures, where the occupation of elementary excitation modes exhibits a distribution in momentum space N(k) scaling as k^{−4} for long wavevectors. This behaviour is challenging to observe in weakly interacting atomic BECs^{5,6,8}, because the momentum distribution is not preserved in the timeofflight measurements and is influenced by interactions during the expansion of the condensate^{9}.
Exciton–polariton condensates, which are partlight partmatter bosonic condensates formed in a semiconductor microcavity, allow direct measurement of their momentum space distribution and excitations through the cavity photoluminescence signal. Each exciton polariton (or polariton) is a strongly coupled quantum well exciton and a cavity photon. Due to the finite lifetime of the confined photon state, the polariton eventually decays and a photon escapes the microcavity retaining the energy and momentum of the polariton^{10}. The finite lifetime results in the inherent drivendissipative nature of exciton–polariton condensates, where the system needs constant pumping to maintain the condensate population, hence the steady state is reached based on a balance between driving and dissipation. In addition, the finite lifetime often prevents full thermalisation of the condensate, which is typically manifested by a macroscopic occupation of several singleparticle energy states rather than a single ground state. Furthermore, the constant energy flow in exciton–polariton condensates strongly influences the elementary excitation spectrum, which can possess a gapped mode or exhibits a flat dispersionless Goldstone mode at the low energy limit, gradually recovering the textbook Bogoliubov dispersion at longer wavevectors^{11}. Nonetheless, exciton–polariton condensates preserve superfluid properties^{12,13}, although their features are currently understood as a rigid state under continuous coherent driving^{14}.
The striking feature of the Bogoliubov excitation spectrum is the appearance of two branches which are positive (normal branch—NB) and negative (ghost branch—GB) with respect to the condensate energy. While occupation of the NB occurs via the process of thermal excitation (thermal depletion), the GB is populated solely by the quantum depletion process, thus its appearance in the photoluminescence spectrum can serve as a direct probe of beyond meanfield effects (quantum fluctuations) in an exciton–polariton condensate^{15}. Quite notably, the quantum effects of polariton–polariton interactions have been recently demonstrated, but at the singleparticle level, through correlation experiments on strongly confined exciton polaritons^{16,17,18}. There have been several attempts to measure the full excitation spectrum of exciton–polariton condensates including populating both excitation branches using resonant pump–probe schemes^{19,20,21} and intense incoherent driving^{22,23}. However, none of these approaches created the condensates in a spontaneous, steadystate configuration. More importantly, the population of the GB was forced by scattering on defects^{22} or by an additional resonant laser beam^{19}, hindering the direct observation of interactiondriven manybody quantum effects.
The linear Bogoliubov spectrum of a spontaneously created highdensity exciton–polariton condensate has been observed, but only the positive energy NB photoluminescence was visible in the momentum space and its population was naturally dominated by thermal excitations of the condensate^{24}, which scaled as k^{−2} in the low wavevector limit. Recently, the problem of reduced visibility of the GB was tackled theoretically^{15,25}, showing that nonequilibrium effects can suppress the quantum depletion and its signatures in the photoluminescence spectrum of an exciton–polariton condensate.
In this work, we create a steadystate highdensity condensate of longlifetime exciton polaritons in an ultrahighquality GaAsbased microcavity, and observe the direct manifestation of quantum depletion in the condensate excitation spectra (see “Methods”). This is achieved by an optical excitation scheme, where a nonresonant pump laser beam with an annular spatial distribution^{26,27} creates a similarly shaped distribution of incoherent excitonic reservoir particles that provides both gain and a trapping potential for the exciton polaritons^{28}. At a sufficiently high pump power, a singlemode condensate in the interactiondominated Thomas–Fermi regime^{27} forms inside the trap away from the pumpinduced potential barrier. The spatial separation of the pump and the condensate allows us to filter out the photoluminescence (see “Methods”) originating from regions with significant overlap with the reservoir and analyse the emission of the condensate and its excitations. In addition, the singlemode condensate ensures that we are dealing with a simple manybody macroscopically occupied ground state of the pumpinduced trap. To test both the equilibrium and nonequilibrium features of the exciton–polariton condensate, we study the system at various values of detuning between the cavity photon and the exciton energies Δ = E_{C}−E_{X}. A wide range of detuning values is accessible via the cavity wedge (see “Methods”), and reflects either a more matterlike (excitonic) or a lightlike (photonic) nature of the polaritons at positive or negative detunings, respectively. In what follows, we present full datasets for two representative values of detuning: positive (Δ = +1.8 meV) and negative (Δ = −3.7 meV). The Hopfield coefficients^{10} determining the exciton (matter) fraction of the polariton at these detunings are X^{2} ≈ 0.56 and X^{2} ≈ 0.39 at k_{∥} = 0, respectively. The excitonic or photonic nature of exciton polaritons affects their ability to achieve thermal equilibrium^{29,30}.
Results
Condensation in an optically induced trap
In our experiment, we utilise a continuous wave (CW) laser and create a steadystate polariton condensate. Nevertheless, condensation features are very similar to those observed in the pulsed excitation regime^{27}. The typical condensate spectra are presented in Fig. 1 together with the dependence of the polariton density and the ground state occupancy on pump power (Fig. 1a). Below the condensation threshold, laserinduced highenergy excitons relax and form exciton polaritons with large wavevectors and energies, which create a local potential barrier and dominate the photoluminescence under these lowpower excitation conditions, Fig. 1b, e. With increasing pump power, exciton polaritons increase in density until they condense into a fragmented condensate with the macroscopic occupation of several modes of the trap, see Fig. 1c, f. This fragmentation is due to the weak polariton–polariton interaction and inefficient phononmediated relaxation processes at intermediate densities^{31,32}. Multimode condensation manifests itself in a smooth nonlinear rise in polariton density at pumping powers slightly before singlemode condensation sets in, see Fig. 1a. After reaching the threshold at large polariton densities, stimulated scattering and efficient energy relaxation drive the condensation towards a singlemode ground state condensate with a spatially homogeneous density distribution and welldefined energy, as seen in Fig. 1d, g and Supplementary Notes 1 and 2. In this interactiondominated Thomas–Fermi regime, we can reliably measure the spectrum of condensate excitations, and perform a direct comparison with the Bogoliubov theory.
Condensate excitations and polariton–polariton interaction
In order to investigate the excitations of the steadystate exciton–polariton condensate, we measure the energyresolved farfield emission of the condensate, which corresponds to the momentum and energyresolved distribution of exciton polaritons. Figure 2a presents the excitation spectrum of a singlemode condensate at intermediate densities, where the excitation spectrum is indistinguishable from the singleparticle dispersion, i.e. the condensate interaction energy is too small to modify considerably the Bogoliubov spectrum within the linewidth of the photoluminescence signal. The result of the momentum space integration is presented in Fig. 2b in doublelogarithmic scale. One can distinguish the condensate profile (the strongest signal located near k_{} = 0, where k_{} is the polariton momentum in the plane of the quantum well) from the thermal excitations, which are visible at longer wavevectors and are fitted with a thermal distribution, Fig. 2b (see “Methods”). The condensate at k_{} = 0 has a finite width in momentum space, which is a consequence of the spatial confinement in an optical trap^{24,28}.
The momentum spectra differ significantly in the highdensity regime around n ~ 10^{3} μm^{−2}, where an exceptionally strong signal from the ground state exceeds all other contributions in the spectrum within the dynamical range of the CCD camera, see Fig. 3a. The condensate emission also reveals a characteristic Airy pattern due to diffraction of the condensate photoluminescence on a circular aperture we use as a real space filter to block the emission from the trap barrier (see “Methods” and Supplementary Note 3). To reveal the much weaker signal of the excitation branches, we impose an edge filter in momentum space, blocking the strongest contribution to the signal up to about 0.55 μm^{−1}, Fig. 3b, c. Remarkably, the signal from the two branches becomes clear and distinguishable, despite the strong contribution of the diffracted condensate photoluminescence. The appearance of the GB signal is evidence of quantum depletion of the exciton–polariton condensate, being a consequence of quantum fluctuations in the steady state.
Fitting the excitation branches dispersions, see Fig. 3b, c, with Eq. (2) (see “Methods” and Supplementary Notes 4 and 5) allows us to extract the interaction energy (chemical potential) μ of the condensate as a function of the condensate density, which follows closely the expected linear dependence μ = gn, where g is the polariton–polariton interaction strength and n is the measured polariton density, see Fig. 4a. This fitting procedure can be applied to the experimental data at various detunings, where the GB signal and the dispersion change of the excitation branches are observable. The extracted values of the polariton–polariton interaction strengths, normalised to the number of quantum wells, is presented in Fig. 4b, together with the values obtained under pulsed excitation and at larger densities (n ~ 10^{4} μm^{−2}) in the Thomas–Fermi regime^{27}. One observes an excellent agreement for all experimental values showing the theoretically predicted quadratic dependence on the exciton fraction: g ∝ X^{4}g_{X}, where g_{X} is the exciton–exciton interaction constant^{10,27}, which in our case is g_{X} = 13.5 ± 0.6 μeVμm^{2}. We note that, in contrast to ref. ^{27}, the direct measurement of the interaction constant from the blueshift of the Thomas–Fermi condensate energy in the CW regime is not possible because of the incomplete depletion of the reservoir, which effectively lifts the zeropoint energy of the optically induced potential (see Supplementary Note 5). The measurement of the interaction strength presented here is therefore methodologically different and independent from that in ref. ^{27}.
Occupation of the excitation branches in momentum space
The essential information on the mechanisms populating the elementary excitation branches, summarised in Fig. 5, is contained in the characteristic momentum occupation distributions N(k_{}). The employed filtering technique allows us to measure these distributions in the long wavevector range, i.e. k_{}ξ > 1, where \(\xi = \frac{\hbar }{{\sqrt {2mgn} }}\) is the healing length and m is the polariton effective mass (see Supplementary Note 6). In this wavevector range, the NB occupation is characterised by nonequilibrium features at both probed values of the exciton–photon detuning, see Fig. 5b, c. At densities n < 10^{3} μm^{−2}, N_{NB} displays thermallike distributions at higher momenta for the excitonic detuning. However, at the largest probed densities, thermal excitations are reduced, leading to a population of excited trap states for both photonic and excitonic polaritons, Fig. 5b, c. This is caused by the interplay of the different mechanisms populating the NB, as schematically depicted in Fig. 5a. In addition to thermal and quantum depletion of the condensate, the inefficient energy relaxation of polaritons naturally leads to an occupation of highk_{} states (see Supplementary Note 1). These highenergy polaritons originate from the potential barrier region (the reservoir) and are more prevalent for more photonic polaritons where the interactions and thermalisation are much weaker^{10,29,30}. The peak around k_{} = 2 μm^{−1} observed at the excitonic detuning in Fig. 5b originates from highenergy states on top of the pump region, which spread all over the area of the trap, and are similar to a highenergy state observed in a single spot excitation of a highquality sample^{33}. In addition, polariton buildup and accumulation at highenergy states may be linked to the polaritontoreservoir upconversion mechanism, which has been observed at strong coherent pumping fields in multiple independent experiments and is a source of the incoherent reservoir generation within the trap^{17,18,34,35,36}.
In contrast to the NB, the GB is populated solely by quantum fluctuations of the highdensity polariton condensate. Therefore, one expects to observe clear signatures in the momentum space distribution as predicted by the Bogoliubov theory. Indeed, the occupation distribution shows a k^{−4} decay for the excitonic detuning, see Fig. 5d and Supplementary Notes 9 and 10 (henceforth k denotes the inplane momentum k_{}).
This is in agreement with the Bogoliubov theory prediction of the asymptotic behaviour at large wavevectors and suggests the equilibriumlike character of the condensate quantum depletion despite the deviations caused by nonequilibrium effects in the NB distributions. In addition, we observe a predicted quadratic increase of the GB occupation as a function of the condensate density (see “Methods” and Supplementary Note 11). The most remarkable behaviour of the GB occupation distributions is observed at the photonic detuning, as seen in Fig. 5e and Supplementary Note 9. Here, in spite of the rising population density (and pump power), we observe a gradual transition from the k^{−4} distribution to a plateau approaching k^{−1}. The deviations from the equilibrium distributions have been predicted by nonequilibrium theories^{37,38,39}, indicating that at photonic detunings polariton condensate fluctuations are mediated mostly by reservoir–condensate interactions, and polariton–polariton interactions do not play a major role. This assertion is supported by the fact that the polariton–polariton interaction is weaker for more photonic polaritons (Fig. 4b), and that the reservoir density, n_{R}, extracted from our measurements grows with pump power, with the condensate fraction, defined as ρ = n/(n + n_{R}), reaching maximum values of ρ ≈ 0.5 (see Supplementary Note 5). Furthermore, as discussed in the Supplementary Note 12, condensates at photonic detunings might be subject to large reservoirdriven fluctuations, which could lead to departure from the Bogoliubov prediction for the GB occupation^{39}.
It is important to note that the unusual momentum distributions N_{GB}(k) observed at photonic detunings do not originate from the nonparabolic dispersion of polaritons. Taking into account the full polariton dispersion rather than the parabolic effective mass approximation, leads to a deviation from the k^{−4} behaviour of the population numbers, which is small within the experimentally accessed range of momenta (see “Methods” and Supplementary Note 13).
Discussion
To summarise, our experiment represents a direct probe of quantum fluctuations and interparticle interactions in a manybody drivendissipative BEC. In our exciton–polariton system, we observe a crossover from the nearequilibrium regime of quantum fluctuations at the excitonic detunings to a fully nonequilibrium regime at the photonic detunings, where reservoir fluctuations might play a critical role. Our experimental findings are beyond the current theoretical understanding of nonequilibrium condensates. Thus, they call for further development of a theory describing the quantum depletion of BECs in the crossover from thermal equilibrium to farbeyond equilibrium conditions, as well as detailed mechanisms responsible for population of both normal and ghost branches of Bogoliubov excitations in the nonequilibrium regime. Furthermore, in the regime where the equilibrium Bogoliubov theory of quantum depletion does apply, our experiment paves the way for the measurement of the Tanʼs contact^{5,40} for exciton–polariton condensates (see “Methods”). This, in turn, will allow us to test whether exciton–polariton condensates exhibit universal thermodynamical properties of a system with contact interactions^{41}.
Methods
Experiment
The sample used in the experiment is a highquality factor GaAsbased microcavity characterised by a long cavity photon lifetime exceeding 100 ps^{42}. The 3λ/2 cavity consists of distributed Bragg reflectors with 32 (top) and 40 (bottom) pairs of alternating Al_{0.2}Ga_{0.8}As/AlAs layers and an active region of 12 GaAs/AlAs quantum wells of 7 nm nominal thickness positioned in three groups at the maxima of the confined photon field. The normal mode anticrossing (Rabi splitting) is measured to be about ћΩ = 15.9 ± 0.1 meV^{43}, the exciton resonance energy is E_{X} = 1.6062 eV, and the cavity photon effective mass is about 3.6 × 10^{−5}m_{0}, where m_{0} is the free electron mass^{27,44}. In all experiments the microcavity is kept in a continuous flow helium cryostat, ensuring the sample temperature of 7–8 K.
The nonresonant excitation in the experiment was provided by a singlemode CW Ti:Sapphire laser tuned to a reflectivity minimum of the microcavity (around 719 nm) for efficient photon absorption in the quantum wells. To minimise the thermal heating of the sample, the laser pump was chopped with an acoustic optical modulator at 10 kHz and 5% duty cycle. The ringshaped excitation profile is created by utilising an axicon lens in a confocal configuration between two imaging lenses that produces a hollow beam^{45} when reimaged onto the sample surface via a microscope objective of NA = 0.5. The same objective collects the photoluminescence from the sample. The imaging setup is composed of four lenses in confocal configuration for measurements of near and farfield emission image planes^{26}. The image filtering is performed in the intermediate conjugate planes with an optical iris in real space and a movable razorblade edge in momentum space. The filtered signal is then imaged onto the monochromator slit and dispersed by a grating, allowing measurement of spatial and momentum spectra of exciton–polariton photoluminescence. The signal is recorded by a highefficiency EMCCD camera.
Extraction of the occupation numbers in momentum space
Calculation of the momentum distribution is obtained based on the integration of the signal and taking into account the local density of states in momentum space^{26}. The collection efficiency of the experimental setup was calibrated with a reference laser tuned to the emission wavelength of polaritons and is expressed as η. The mean number of polaritons recorded within a single pixel row on a CCD camera representing a wavevector k_{i} is calculated from the photon count rate \(\frac{{dN_{{\mathrm{ph}}}(k_i)}}{{dt}}\):
where τ_{LP} is the polariton lifetime calculated based on the Hopfield coefficients and cavity photon lifetime^{10,26,46}. The occupation number of polaritons at a given k_{i} state is calculated taking into account the number of states subtended by a pixel at k_{i} position in cylindrical coordinates \(N_{{\mathrm{st}}}\left( {k_i} \right) = k_i{\mathrm{\Delta }}k_i{\mathrm{\Delta }}\varphi _i \cdot \left( {\frac{{4{\uppi}^2}}{A}} \right)^{  1}:\)
The formula takes into account the volume of a single state in momentum space, spin degeneracy of 2, and the momentum space volume subtended by a single pixel. The duty cycle of the acoustooptic modulator is denoted by D, and the real space filter area is denoted by A. A detailed derivation of the integration formulas is given elsewhere^{26}.
Analysis of the Bogoliubov excitation branches
One can diagonalise a simplified Hamiltonian of a BEC of weakly interacting bosons using quasiparticle operators at a given wavevector^{47} k: \(\hat b_k = u_k\hat a_k + v_{  k}\hat a_{  k}^\dagger\), which is a linear combination of creation and annihilation singleparticle operators for a boson at given wavevector, \(\hat a_k^\dagger,\, \hat{a}_{k}\), and the amplitudes are expressed as:
where \({\it{\epsilon }}(k)\)is the Bogoliubov dispersion in equilibrium
and \({{E}}(k)=\hbar^{2}k^{2}{\!}/{\!}(2m)\). Occupation of the singleparticle excited states is expressed as \(\langle \hat a_k^\dagger \hat a_k \rangle = (\left {u_k} \right^2 + \left {v_{  k}} \right^2)\, \langle\hat b_k^\dagger \hat b_k \rangle + \left {v_{  k}} \right^2\). Here, the first term, proportional to \(\langle \hat b_k^\dagger \hat b_k \rangle = 1/[\exp \left( {{\it{\epsilon }}\left( k \right)/k_bT} \right)  1]\), describes the thermal depletion of the condensate. At zero temperature, T = 0, there is no thermal occupation of Bogoliubov quasiparticles \(\langle \hat b_k^\dagger \hat b_k\rangle = 0\), and the nonzero occupation of singleparticle states is due to quantum depletion \(\langle \hat a_k^\dagger \hat a_k \rangle = \left {v_{  k}} \right^2\). The GB states are populated due to quantum depletion, therefore the GB occupation can be expressed as N_{GB}(k) = v_{−k}^{2}.
Taking the asymptotic behaviour of N_{GB}, one finds that in the low wavenumber regime, which was inaccessible in our experiment, N_{GB}\(\left( k \right)\mathop {\longrightarrow }\limits^{k\xi \ll 1} \frac{{\sqrt {mgn} }}{{2\hbar k}} \propto k^{  1}\), and in the opposite regime \(N_{\text{GB}}\left( k \right)\mathop{ \longrightarrow }\limits^{k \xi \gg 1} \frac{m^2 g^2 n^2}{{\hbar}^4 k^4} \propto k^{  4}\) (see details in Supplementary Note 13). The asymptotic value \(\mathop {\mathrm{lim }}\nolimits_{k \to \infty } N\left( k \right)k^4 = C\) is referred to as Tan’s contact^{40}, a universal quantity relating contact interactions to the thermodynamics of a manybody system. The contact in the equilibrium theory depends quadratically on peak density \(C\propto n^{2},\) in agreement with the values extracted from the experiment (see Supplementary Note 11).
In the case of drivendissipative systems, the Bogoliubov dispersion is modified and, after adiabatic elimination of the reservoir modes, the dispersion can be expressed in the analytical form:^{11}
where the nonequilibrium relaxation parameter \(\Gamma \mathop{\longrightarrow}\limits^{n\to \infty}\gamma_{\mathrm{LP}}\) and γ_{LP} represents the polariton decay rate. In our case, the polariton lifetime exceeds 100 ps^{42}, so ћΓ ≈ 3–5 μeV, which has a negligible contribution to the fitted dispersion at large wavevectors measured in this work. Therefore, for the purpose of the analysis, we use an approximation of equilibrium dispersion \({\it{\epsilon }}(k)\), Eq. (2). The fitting of the dispersions of elementary excitations is performed using a single fitting parameter, the condensate interaction energy μ = gn, and by taking into account the experimentally measured polariton dispersion E(k) = E_{LP}(k) − E_{LP}(0), where, \(E_{{\mathrm{LP}}}(k) = \frac{1}{2}\left( {E_{\mathrm{X}} + E_{\mathrm{C}}(k)  \sqrt {\left( {\hbar {\mathrm{\Omega }}} \right)^2 + {\mathrm{\Delta }}^2(k)} } \right)\).
Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.
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Acknowledgements
This work was supported by the Australian Research Council (ARC) through the Centre of Excellence Grant CE170100039. The work at Pittsburgh was funded by the Army Research Office (Grant No. W911NF1510466). The work of sample fabrication at Princeton was funded by the Gordon and Betty Moore Foundation (GBMF4420) and by the National Science Foundation MRSEC programme through the Princeton Center for Complex Materials (Grant No. DMR0819860). J.L. was supported through the Australian Research Council Future Fellowship FT160100244. M.P. would like to acknowledge useful discussions with Ryo Hanai.
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E.A.O. and A.G.T. conceived and guided this research. M.P., E.E. and M.B. conducted the experiments. M.P. analysed the data and together with E.E., E.A.O. and A.G.T. interpreted the data. O.B., J.L. and M.M.P. contributed to the theoretical analysis and discussion of results. K.W. and L.P. fabricated the sample, which was designed and characterised by M.S. and D.W.S. M.P. and E.A.O. wrote the manuscript with the input from all authors.
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Pieczarka, M., Estrecho, E., Boozarjmehr, M. et al. Observation of quantum depletion in a nonequilibrium exciton–polariton condensate. Nat Commun 11, 429 (2020). https://doi.org/10.1038/s41467019142436
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DOI: https://doi.org/10.1038/s41467019142436
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