Abstract
Synthetic crystal lattices provide ideal environments for simulating and exploring the band structure of solidstate materials in clean and controlled experimental settings. Physical realisations have, so far, dominantly focused on implementing irreversible patterning of the system, or interference techniques such as optical lattices of cold atoms. Here, we realise reprogrammable synthetic bandstructure engineering in an all optical excitonpolariton lattice. We demonstrate polariton condensation into excited states of linear onedimensional lattices, periodic rings, dimerised nontrivial topological phases, and defect modes utilising malleable optically imprinted nonHermitian potential landscapes. The stable excited nature of the condensate lattice with strong interactions between sites results in an actively tuneable nonHermitian analogue of the SuSchriefferHeeger system.
Introduction
Particles subjected to potential landscapes with discrete translational symmetries, whether natural or artificially made, exhibit bands of allowed energies corresponding to the quasimomentum of the crystal’s Bloch states^{1}. For instance, electronic band theory explains the difference between insulating and conducting phases of materials, as well as their optical properties. With advances in energy band synthesis in atomic systems (optical lattices) or photonic crystals, complicated yet meticulous lattice investigations are now possible including superfluidtoMott insulator phase transitions^{2}, networks of Josephson junctions^{3}, and solitonic excitations^{4,5}. When the symmetry of a periodic structure is broken and/or boundaries are engineered in a desired way, there can arise defect states, surface states, and bound states in the continuum that do not dissipate energy into the surrounding environment. Advancements in photonics have allowed for the design and study of nearly lossless waveguides, filters, and splitters^{6}, with applications in communications and biomedicine. Recent developments have led to the study of topological states of matter in photonics^{7} and separately in cold atoms^{8,9}.
Onedimensional (1D) crystals provide the simplest platform to study nontrivial topological phases, the prime example being the Su–Schrieffer–Heeger (SSH) model^{10,11}. Today, the Zak phase (or the 1D topological winding number)^{12} has been measured in a system of cold atoms^{13}, followed by the demonstration of adiabatic Thouless pumping^{14}, and an electronic topological superlattice^{15}. Recently, nonHermitian solidstate and photonic systems have attracted a huge interest in the study of outofequilibrium topological phases^{16,17,18,19,20,21}, dissipative quantum physics^{22,23,24}, and the advantageous effects of unbroken parity–time symmetry^{25}.
In the optical regime, a rapidly developing platform for the study of the abovementioned phenomena are exciton–polaritons (from here on polaritons), realised in semiconductor microcavities. These hybrid light–matter quasiparticles are formed by the strong coupling of light confined in Fabry–Pérot microcavities and electronic transitions in embedded semiconductor slabs^{26}. Their dissipative and outofequilibrium nature permits condensation into excited states^{27,28,29} that still presents a nontrivial task for cold atoms in thermal equilibrium^{30}.
In polaritonic systems, there are two processes available to sculpt a crystal lattice. The most commonly applied process is through periodically patterning of the cavity mode and/or the intracavity quantum wells (QWs). This is typically achieved through patterned metallic deposition on top of the sample^{27,29}, etch and overgrowth patterning techniques^{31}, surface acoustic waves^{32}, or microstructuring a sample into arrays of micropillars^{33,34,35}. Linear features such as Dirac cones and flat bands have been demonstrated with polaritons utilising etched lattices in Lieb^{34} and honeycomb^{36} geometries with topological transport recently reported^{35,37}, as well as nonlinear dynamics of bright gap solitons^{38,39}. The other process utilises the matter component of polaritons to produce periodic potentials through manybody interactions. Similar to dipole momentinduced optical traps for cold atoms^{40}, or photorefractive crystals^{41}, one can design an alloptical potential landscape for polaritons by using nonresonant optical excitation beams to create reservoirs of excitons, which result in effective repulsive potentials due to polariton–exciton interactions^{42,43,44,45,46,47}.
In this article, we realise an alloptical, actively tunable bandstructure engineering platform harnessing reprogrammable nonHermitian potential landscapes that result from interparticle interactions. The platform is actively tunable due to the use of a spatial light modulator to spatially sculpt the nonresonant excitation beam and the resulting potential. The sample used is a nonpatterned planar 2λ GaAsbased cavity containing eight 6nm InGaAs QWs^{48} (for more details, please see ‘Methods’). Utilising this platform, we demonstrate a variety of band structure features including polariton condensation into highsymmetry points in arbitrarily excited energy bands of the resulting Bloch states. By dimerising the potential landscape, we experimentally realise an analogue of the topologically nontrivial SSH system, resulting in the formation of split energy band states. We determine through theoretical investigations that there is a π change in the Zak phase (1D Berry phase) of the bands between the two choices of inversion symmetry points in the dimerised lattice. This confirms that our system experimentally provides a platform for studying nontrivial topology in nonHermitian systems. Finally, by introducing local defects in the potentials periodicity, we demonstrate controllable highly localised defectstate condensation opening up possibilities to investigate analogues of bright and dark solitonic gap modes in strongly nonHermitian lattices.
Results
Uniform 1D chains
We start by considering 1D chains of narrow nonresonant Gaussian pumps (fullwidth at halfmaximum ≈ 2 μm) exciting colocalised polariton condensates, where the intercondensate separation is kept constant along the chain (see Fig. 1). The band structure along the lattice can be characterised via a single image of the dispersion (energy resolved kspace) providing that the chain is parallel to the entrance slit of the spectrometer. In Fig. 1, we show the experimental realspace and kspace photoluminescence (PL) distributions along with the corresponding dispersions for linear chains of eight polariton condensates with a lattice constant (a) of approximately 13 μm for Fig. 1b–d and 8.6 μm for Fig. 1e–g. It can be seen in Fig. 1d, g that condensate chains exhibit clear band structure in the their dispersions with dominant occupation at the highsymmetry points of their reduced Brillouin zone and all the repeated zones within the free polariton dispersion. These results evidence that polaritons, generated at the pump spots, sense the periodic nature of the potential, resulting in macroscopic coherent Bloch states and thus qualifying the technique even for relatively few pump cells. Furthermore, the energy band wherein the system condenses can be controlled by changing the separation between neighbouring condensates as is demonstrated in Fig. 1d, g, where we realise access to nonlinear condensate dynamics in arbitrarily excited states through alloptical control.
We note the intricate Talbot interference patterns observed experimentally in the regions perpendicularly away from the chains, e.g. in Fig. 1b. Such patterns were previously demonstrated for polariton condensates using a chain of etched mesa traps^{49} and demonstrate the ability of optically imprinted condensates with the concomitant potentials to achieve effects of etched/patterned systems. Moreover, polaritons condensing into the highsymmetry points of the lattice, observed also in refs. ^{27,29,31}, can be intuitively understood from the fact that these Bloch modes have the strongest overlap with the gain (pump) region. The results are verified both through diagonalisation of the nonHermitian Bloch problem and by numerically solving the drivendissipative Gross–Pitaevskii equation describing a coherent macroscopic field of polaritons under pumping and dissipation (see Supplementary Notes 1 and 2).
Topologically nontrivial band gap opening in 1D chains
Figure 2 shows the experimental dispersions in Fig. 2a–e and realspace PL distributions in Fig. 2g–k for chains of eight condensates, demonstrating the splitting and periodic doubling of the band as the difference between the long (a_{l}) and short separation (a_{s}) is increased (panels a → e and g → l). For marginal differences in separation distance, δ = a_{l} − a_{s}, the band gap formed is smaller than or comparable to the linewidths of the condensate polaritons and thus not fully resolvable. Increasing δ leads to an increased band splitting and the gaps become clearly visible when they exceed the polariton linewidth. In Fig. 2, the newly opened gap in the dominantly occupied energy band is indicated by the red arrows. Eventually for large enough δ, the band splitting becomes significant enough that adjacent energy bands mix; see Fig. 2e. By increasing the number of unit cells in the experimental crystal potential, the splitting approximates the ideal infinite scenario (see the ‘Methods’ section for a discussion around the limits of the current experimental setup). As a result, the finesse of the band structure features becomes enhanced; this can be seen clearly in Fig. 2f, l, which show the dispersion and realspace distribution, respectively, of the PL from a chain of 12 condensates with a_{l} = 10.2 μm and a_{s} = 9.2 μm. We point out that in coldatom systems topologically nontrivial band structures can be engineered by generating artificial gauge potentials using laser beams, where the hopping amplitude between adjacent lattice sites picks up a controllable phase factor (Peierls substitution) from the laser amplitudes^{50,51,52} or from periodic modulation^{53,54}. Here we have engineered an alternating pattern of tunnelling amplitudes between neighbouring polariton condensates by utilising the variation of the condensate hopping amplitude with the laser separation distance, such that interference of condensate polaritons between neighbouring sites is staggered.
The gainlocalised nature of the condensate polaritons at their respective excitation spots permits description through discretised set of coherent polariton equations of motion. In particular, if the distances between adjacent condensates are weakly staggered the hopping amplitudes follow suit due to both differences in polariton travel times (i.e. the condensate envelope decays rapidly outward from its respective pump spot) and interference coming from their large outflow kvector. The dimerised system, characterised by two distinct complex hopping amplitudes J_{±}, for long and short distance between the condensates, respectively, mimics a singleparticle twoband problem representing a nonHermitian version of the SSH model^{10} (see ‘Methods’ and Fig. 3b). The single polariton Hamiltonian describing the twosublattice chain in reciprocal space is written as,
where q is the crystal (Bloch) momentum and Ω is the onsite energy of polaritons at their pump spots. We note that J_{±} are complex valued (see Eq. (6) in ‘Methods’), but their conjugate is not taken in the lower offdiagonal element of the above Hamiltonian. This is due to the nonHermitian nature of our system, which, in the context of topologically nontrivial phases, has taken a surge of interest^{19,20,21,55,56,57,58,59,60,61}. In a ringshaped lattice that forms periodic boundary conditions that we discuss later, the Bloch waves are exact eigenstates and the description of the Zak phase also becomes exact.
The Bloch eigenstates belonging to Eq. (1) are written \({b}^{(\pm )}\rangle ={(\!\pm 1,{{\rm{e}}}^{i\phi (q)})}^{T}/\sqrt{2}\), where (±) denotes the upper (conduction) and lower (valence) band of the system. The energies belonging to these two bands are plotted as red curves in Fig. 3a in the first Brillouin zone. The standard procedure to validate the presence of topologically nontrivial phase transitions in 1D lattices is through the definition of the Zak phase^{12}, which can be regarded as the 1D parameter space extension of the geometric Berry phase,
The Zak phase can only take values 0 or π (modulo 2π) when the origin is chosen at an inversion centre of the system. By solving the eigenvalue problem posed by Eq. (1), the Zak phase can be calculated straightforwardly by integration over the Brillouin zone.
In Fig. 3, we present numerical results reproducing the experimental gap opening shown in Fig. 2f. Figure 3a shows the fitted gapped bulk dispersion from Eq. (1) (red curves) in the lattice Brillouin zone. The curves are plotted on top of a blackandwhite colourmap showing the numerically timeresolved singleparticle dispersion based on a Monte Carlo technique (see Supplementary Notes 1). Figure 3b shows a schematic of the staggered lattice. In Fig. 3c, we plot ϕ(q) across the Brillouin zone corresponding to the two distinct centres of inversion symmetry in the dimerised lattice, which is the same as interchanging the values of J_{±}. Integrating ϕ(q) across the Brillouin zone reveals a π change in the Zak phase between the dimerisations, marking the existence of two topologically distinct phases. The findings are corroborated through firstprinciple calculations on the polariton system Schrödinger equation (see Supplementary Notes 2). We point out that our system is very different from that of hybridised orbitals in micropillar chains^{33}, where in the current case, the opening of the gap arises from the staggered interference between adjacent polariton condensate ‘antennas’ (see Eq. (6)). Experimentally, the gap opening observed in Fig. 2 implies a topological phase transition due to the localisation of polariton modes at each pump spot. This is in analogy to deep periodic potentials where the particles occupy a single mode at each site in the lowest band (i.e. the wavefunction can be described as a superposition of localised Wannier functions). The strong nonHermitian nature of our hybrid light–matter system instead opens new avenues towards topological physics where the localisation of the particles is not dictated by the potential minima of the lattice with evanescent tunnelling.
Defectstate condensation
Moreover, by optically engineering a defect state in the lattice, one can mimic the behaviour of solitons in the polyacetylene polymers of the original SSH model^{10,11}. Such a defected system is depicted as …BABABAABABAB… where one site is adjacent to either two shortdistance or two longdistance neighbours. The generation of the SSH dimerisation and defect states here is analogous to the engineering of a controllable phase factor (Peierls substitution) for the hopping amplitudes between adjacent sites in coldatom systems using laserassisted tunnelling^{62}. Solving the complex eigenenergies of a finite system (see Eq. (7)) including such a defect (e.g. one site linked by two J_{+} couplings) one can observe in Fig. 3d–f that a defect (midgap) state forms in the system, clearly distinguished from the bulk as it lies at zero energy.
Broken translational symmetry in a uniform chain also results in gap (defect) states appearing. These manifest as dispersionless states in the band structure (indicated by the blue arrows in Figs. 4 and 5), showing strong spatial localisation around the position of the defect in the pump geometry. Figure 4 shows the experimental realspace PL distribution from a chain of 12 condensates with separation distances of a ≈ 10.2 μm except between the central two pump spots where the separation is reduced to a_{d} ≈ 9.0 μm, creating a defect in the potentials periodicity. A corresponding gap mode is visible in the dispersion (indicated with the blue arrow in Fig. 4b) and the energyresolved strip of real space (Fig. 4c) demonstrates strong spatial localisation of the condensate for the defect energy (Fig. 4e). Such strongly localised states could permit investigation into optically generated analogue of polariton bright gap solitons observed previously for polariton condensates in photonic lattices^{39}. On the other hand, the delocalised band energetically above the defect state suffers significant suppression in condensate occupation spatially around the defect, representing a dark solitonlike mode (see Fig. 4d). This suppression is a consequence of the bulk energy bands vanishing around the defect and thus inhibiting energy flow between the left and the right bulk region of the optical polariton crystal. We present simulations on such defect states in Supplementary Notes 4.
Optically imprinting the potential landscape affords the ability to finely tune the spectral position of the defect state, within the gap, by only changing the defect length (a_{d}) in the excitation geometry. The PL dispersions for chains of 12 condensates with a = 10 μm for five defect lengths between a_{d} = 8.9 μm and a_{d} = 7.1 μm are shown in Fig. 5a–e. As the defect separation distance is reduced, the gap mode (indicated by blue arrows) blueshifts from the bottom of the gap to the top, at which point it begins to mix with neighbouring energy bands. For all excitation geometries shown in Fig. 5, the spatial distribution of the condensate occupying the defect state, and the energy band above it, have features comparable to those shown in Fig. 4d, e. We note that there also exists a dispersionless state in the next lower energy band gap that demonstrates the same blueshift behaviour with reducing defect length.
Beyond finite 1D systems
While the chains we investigate above show clear band formation with exquisite alloptical control over many band features including band splitting, dispersionless defectstate condensation, and arbitrarily excited band condensation, they remain finite systems. As shown in Fig. 2f, increasing the number of unit cells brings the system closer to the ideal infinite system and increases the fidelity of the band features. However, there are technical limitations to the size of chains that can be created, for example, due to the field of view of the objective or available power of the pump laser. In Fig. 6, we demonstrate polariton condensation in geometries of uniform and staggered octagons. Such a system implements a periodic boundary condition and provides a platform to avoid effects originating due to finite lattice sizes. Indeed, in ideal realisations of synthetic crystal lattices, one would like to achieve a welldefined crystal momentum for energy bands that follows from periodic boundary conditions. In optical lattices of cold atoms, such a system is difficult to create; the typical lattices have a finite length and they are additionally also superposed with a harmonic trapping potential. In a finite chain that we have considered until now, the description of eigenmodes in terms of their momenta is only approximate. To overcome this limitation, the ringshaped lattice can be engineered for the polariton condensates in which case the Bloch waves of Eq. (1) form exact eigenstates of the corresponding tightbinding Hamiltonian Eq. (7) of the system. The presence of very weak radial modes clearly seen in logarithmic colour scale in Fig. 6 can be minimised by increasing the polygon’s size. As long as the general features of the couplings between the condensates can be approximated by the tightbinding model, the assumption of the periodic boundary conditions remains valid. We point out that for our detection setup the extraction of polariton band features along the polygons circumcircle in Fig. 6 is currently not possible.
Discussion
Our study advances the emulation of many different lattice structures using a recyclable, and optically reprogrammable, multipurpose platform in the strong light–matter coupling regime. The controllable condensation into arbitrarily excited Bloch states of the system gives access to excited orbital manyparticle dynamics, which previously have been difficult to reach in solidstate systems. In particular, we address the challenge of realising a condensate lattice with periodic boundary conditions, which, in general, is attractive for analytical considerations (Bose–Hubbard model on a ring), and more closely resembles classic bandstructure models of solidstate physics. In finite chains, the description of polariton Bloch eigenmodes in terms of their momenta is only approximate. Ringshaped lattices, however, overcome such limitations where the definition of topological quantities like the Zak phase in the tightbinding limit becomes exact.
The observed defectstate condensation paves the way towards strong nonlinear lattice physics, with application in polaritonic devices such as information routing and fine tunable emission wavelength lasers. In addition, we expect that topological defect lasing can be realised by controlled defect preparation. We point out that the current study is performed in the scalar polariton regime but can be easily extended to include its spin degree of freedom by changing the polarisation of the pump, which creates different spin populations of the excitonic reservoirs feeding the condensates. Working with a horizontally polarised excitation, the system is chiral symmetric and each pump spot results in a randomly linearly polarised condensate. If interactions between the condensates, or onsite energies, are made spin dependent through typical photonic TETM microcavity splitting^{63}, or sample birefringence, then one gains access to spindependent band structures. This broadens the impact of nonresonantly generated artificial polariton lattices and, in principle, permits design of optical Chern insulators given the inherent spin–orbit coupling of polaritons in conjunction with applied magnetic fields^{35,37}. Another exciting area for future research is expanding to topologically protected transport states with investigation into robustness against engineered imperfections.
Methods
Sample and experimental techniques
We use a planar distributed Bragg reflector microcavity with a 2λ GaAsbased cavity containing eight 6nm InGaAs QWs organised in pairs at the three antinodal positions of the confined field, with an additional QW at the final node either side of the cavity^{48}. The sample is cooled to ~6 K using a cold finger flow cryostat and is excited with a monomode continuous wave laser blue detuned energetically above the stop band to maximise coupling in efficiency. The laser is modulated in time into square wave packets with a frequency of 10 kHz and a duty cycle <5% to prevent sample heating, and we operate at ~50% above the excitation density required for formation of a macroscopic coherent singleparticle state. The sample has a vacuum Rabi splitting ~8 meV^{48} and the regions of the sample utilised have an exciton–photon detuning of ~−3.5 meV.
The spatial profile of the excitation beam is sculpted using a phaseonly spatial light modulator to imprint a phase map so that, when the beam is focused via a 0.4 numerical aperture microscope objective lens, the desired realspace is projected onto the sample surface. The same objective lens is used to collect the PL, which is then directed into the detection setup. By controlling the spatial intensity distribution of the nonresonant excitation beam, we imprint a reprogrammable potential landscape^{43,44,46} without the need of irreversible engineering. In the relaxation process from a nonresonant optical injection of free charge carriers to the polariton condensate, an incoherent ‘hot’ excitonic reservoir is produced that feeds the condensate. This reservoir is colocalised with the nonresonant excitation beam(s) and due to the strong polariton–exciton interaction results in a potential hill for polaritons where the excitation density is high^{42}. This method additionally enables the elimination of large inhomogeneities since each element of the potential lattice can be adjusted through the power or shape of its respective pump element, such that the system achieves a homogeneous crystal structure.
In the current experimental setup, when using similar lattice constants to those used throughout the manuscript, the upper limit of condensates in a 1D chain is approximately 14. However, we highlight that this is not a fundamental limit of the experimental technique. By replacing a few optical components, such as the microscope objective lens, this number could be increased. Equally by reducing the lattice constant, one can fit more nonresonant excitation beams. We note here that the lower limit of the lattice constant is determined by the width of the condensate bright centres, which approximately coincide with the Gaussian form of the nonresonant beam. In order to avoid strong overlap between the condensate centres, they should be separated by more than the FWHM of the pump beam.
Theory
The single particle dynamics of planar cavity polaritons, occupying the lower polariton dispersion curve, can be described by a twodimensional Schrödinger equation^{26},
Here μ is the polariton mass, γ is their lifetime, and V(r) is the pumpinduced complex potential. For the nonHermitian lattice of Gaussian potentials, the interaction between polariton wavefunctions, gainlocalised at their respective potentials, and separated by a distance ∣r_{n} − r_{m}∣ = d_{nm}, we can project the system onto an appropriate basis of wavefunctions. Omitting the diffusion of polaritons perpendicular from the chain, we consider a 1D system with the ansatz \({\phi }_{n}(x)=\sqrt{\kappa }{{\rm{e}}}^{ik x{x}_{n} }\), where k = k_{c} + iκ. The condensate wavefunction is then written,
Here k_{c}, κ > 0 represents the outflow momentum and decaying envelope of the polaritons generated at each potential. Given the narrow width of the pumps, we have approximated them as delta potentials, which, by direct integration, gives the following discretised singleparticle equations of motion (details given in Supplementary Notes 2),
Here J_{nm} denotes the condensate hopping amplitudes, Ω is the complexvalued potential energy of polaritons generated at their respective pump spots, and k_{c} is the outflow momentum of the polaritons from their pump spot, which depends on exciton–photon detuning, excitation beam waist, and excitation density^{42}, \({H}_{0}^{(1)}\) is the zeroth order Hankel function of the first kind that accounts for the twodimensional envelope of the propagating polaritons, \({V}_{0}\in {\mathbb{C}}\) is the strength of the complexvalued pumpinduced potential, and η a fitting parameter. The physical meaning of Eqs. (5) and (6) is that condensate polaritons do not tunnel from one site to the next (evanescent coupling) but rather ballistically exchange energy. The term ballistically refers to the nonnegligible polariton phase gradient away from the potentials determined by their strong outflow momentum k_{c}, which gives rise to the interferometric hopping dependence (sine and cosine functions).
In particular, in a distance staggered system (see Fig. 2) the condensates become linked by interchanging long and short distance d_{±} = d ± δ, respectively, where we assume d ≫ δ. For only nearest neighbour interactions, it leads to dimerisation of Eq. (5), which becomes characterised by two hopping amplitudes J_{±}. As a consequence, one obtains an approximate singleparticle twoband problem representing a nonHermitian version of the SSH model^{10}. In the picture of second quantisation, Eq. (5) can be written as (see Supplementary Notes 2),
Here \(\leftm,\alpha \right\rangle\) are state vectors of unit cell m on sublattice α ∈ {A, B}. With periodic boundary conditions, Eq. (7) can be diagonalised by standard Fourier transformation to the basis of crystal momentum \(\leftq\right\rangle ={M}^{1/2}\mathop{\sum }\nolimits_{m = 1}^{M}{{\rm{e}}}^{imq}\leftm,\alpha \right\rangle\), where q ∈ {δ_{q}, 2δ_{q}, 3δ_{q}, …} and δ_{q} = 2π/M. It then follows that \({\mathcal{H}}(q)=\langle q {\mathcal{H}} q\rangle\) giving Eq. (1).
Parameters used for the calculations presented in Fig. 3 are: d = 9.5 μm, k_{c} = 1.5 μm^{−1}, μ = 0.32 meV ps^{2} μm^{−2}, η = 0.24, Ω = 1.315 meV, and V_{0} = 1.44 + i0.5 meV.
Data availability
The data supporting the findings of this study are openly available from the University of Southampton repository at https://doi.org/10.5258/SOTON/D1194^{64}.
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Acknowledgements
We acknowledge the support of the UK’s Engineering and Physical Sciences Research Council (grant EP/M025330/1 on Hybrid Polaritonics) and the RFBR project No. 205212026 (jointly with DFG) and No. 200200919. J.R. acknowledges the support of the UK’s Engineering and Physical Sciences Research Council (grants EP/S002952/1 and EP/P026133/1).
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P.G.L. led the research project. P.G.L. and L.P designed the experiment. L.P. carried out the experiments and analysed the data. H.S. and J.R. developed the theoretical modelling. H.S performed numerical simulations. All authors contributed to the writing of the manuscript.
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Pickup, L., Sigurdsson, H., Ruostekoski, J. et al. Synthetic bandstructure engineering in polariton crystals with nonHermitian topological phases. Nat Commun 11, 4431 (2020). https://doi.org/10.1038/s41467020182131
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