Abstract
A quantum simulator is a wellcontrolled quantum system that can follow the evolution of a prescribed model whose behaviour may be difficult to determine. A good example is the simulation of a set of interacting spins, where phase transitions between various spin orders can underlie poorly understood concepts such as spin liquids. Here we simulate the emergence of magnetism by implementing a fully connected nonuniform ferromagnetic quantum Ising model using up to 9 trapped ^{171}Yb^{+} ions. By increasing the Ising coupling strengths compared with the transverse field, the crossover from paramagnetism to ferromagnetic order sharpens as the system is scaled up, prefacing the expected quantum phase transition in the thermodynamic limit. We measure scalable order parameters appropriate for large systems, such as various moments of the magnetization. As the results are theoretically tractable, this work provides a critical benchmark for the simulation of intractable arbitrary fully connected Ising models in larger systems.
Introduction
Cold atomic systems provide an ideal standard for quantum simulation^{1,2}, by virtue of their ability to support many classes of interactions as well as their excellent quantum coherence and readout properties. Spin chains with nearestneighbour interactions have been simulated in neutral atoms stored in an optical lattice^{3,4}, whereas longrange Ising models have been implemented with small numbers of trapped atomic ions^{5,6,7,8,9,10}. The quantum coherence in such atomic systems should allow the observation of quantum phase transitions (QPTs)^{11} that are driven by nonthermal parameters, like the transverse magnetic field in the longrange quantum Ising model.
In the experiment reported here, we find the ground state of the transverse field Ising Hamiltonian for N interacting spin1/2 systems:
where σ^{i}_{α} is the Pauli matrix for the i^{th} spin (α=x,y,z and i=1,2, ..., N), J_{i,j}>0 is the ferromagnetic (FM) Ising coupling matrix, with J=〈J_{i,j}〉 and B is an external effective magnetic field. Here we set Planck's constant h to unity, and use x,y,z for the Bloch sphere coordinates and X,Y,Z for the spatial coordinates throughout the paper. Our experiment is performed according to adiabatic quantum simulation protocol^{12}, where the dimensionless coupling B/J is tuned slowly enough so that the system follows instantaneous eigenstates of the changing Hamiltonian^{8,9,10}. As B/J→∞, the ground state has all spins polarized along the magnetic field, or is paramagnetic, along the Ising directionx. In the other limit B/J=0, the spins order according to the Ising couplings and the ground state is a superposition of FM states ↑↑...↑〉 and ↓↓...↓〉 where ↑〉 and ↓〉 are eigenstates of σ_{x}.
We characterize the magnetic order in the system by measuring various correlation functions between all N spins, including the probability of FM occupation and the second and fourth moments of the total magnetization. We compare the results with theory, which itself may become intractable for nonuniform Ising couplings as the number of spins grows beyond 20–30 (ref. 13), and even NP complete for a fully connected frustrated Ising model^{14}. This experiment is thus an important benchmark for largescale quantum simulation.
Results
Engineering the quantum Ising Hamiltonian
We represent each spin1/2 system by the hyperfine clock states ^{2}S_{1/2}F=0,m_{F}=0〉 and F=1,m_{F}=0〉 of ^{171}Yb^{+} separated by v_{HF}=12.642819 GHz (in a real magnetic field of ∼4 Gauss defining the quantization axis), which are denoted by the eigenstates ↓_{z} and ↑_{z} of σ_{z}, respectively. These states are detected by standard spindependent resonant fluorescence on the cycling ^{2}S_{1/2} to ^{2}P_{1/2} transition at 369.5 nm using a photomultiplier tube^{15}. The ions are trapped along the Z axis of a threelayer linear Paul trap (Fig. 1a) with centre of mass (CM) vibrational frequencies of v_{X}=4.748, v_{Y}=4.300, and v_{Z}=1.002 MHz along the X,Y (transverse) and Z (axial) directions, respectively^{16}. The modes of motion along X are cooled to near their vibrational ground states and within the Lamb–Dicke regime.
Offresonant laser beams address the ions globally, driving stimulated Raman transitions between the spin states and also imparting spindependent forces exclusively in the Xdirection, as depicted in Figure 1b^{9,10,17}. The Raman beams contain a 'carrier' beatnote at frequency v_{HF}, which provides an effective uniform transverse magnetic field B. Raman beatnotes at frequencies v_{HF}±μ are near motional sidebands and generate a spindependent force. The radio frequency phase difference between the carrier beatnote and the average beatnote of the sidebands is set to π/2 so that the magnetic field is transverse to the Ising couplings (equation (1))^{9,10}. We suppress direct sideband (phonon) excitation by ensuring that the beatnote detuning μ is sufficiently far from any normal mode frequency^{17}. This requires that μ−v_{m}≫η_{i,m}Ω_{i}, where η_{i,m} is the Lamb–Dicke parameter of the ith ion and mth normal mode at frequency v_{m} (with v_{1}=vX denoting the CM mode), and Ω_{i}=g^{2}_{i}/Δ is the carrier Rabi frequency on the ith ion. Here g_{i} is the single photon Rabi frequency of ith ion and Δ is the detuning of the Raman beams from the ^{2}S_{1/2}−^{2}P_{1/2} transition.
This results in an Ising interaction between the spins with control parameter μ that dictates the form of the coupling matrix^{17}.
In the experiment Δ∼2.7 THz, Ω_{i}∼370 kHz and we expect J_{i,j}/N∼1 kHz for the beatnote detuning μ such that μ−v_{1}≈4η_{i,1}Ω_{i}, as shown in Figure 2a. This beatnote corresponds to 63 KHz blue of the CM mode frequency for 2 ions and 30 KHz for 9 ions, as the Lamb–Dicke parameter η_{i,m}∼1/√N. This maintains roughly the same level of virtual phonon excitation for any number of ions. The expected Ising coupling pattern for a uniformly illuminated ion chain is shown in Figure 2b for N=9 ions and the couplings are dominated by uniform contribution of the CM mode. The nonuniformity in the Ising couplings arises from other vibrational modes and variation in Ω_{i} across the ion chain for gaussian Raman beams with a waist of ∼70 μm along the ion chain and ∼6 μm perpendicular to the ion chain used in the experiment. For N=9 ions the chain is ∼14 μm long, and the variation in Ω_{i} is ∼2%.
Experimental protocol
In the experiment, we follow the highest excited state of the HamiltonianH (refs 8,17), which is formally equivalent to the ground state of Hamiltonian H (equation (1)). It proceeds as follows (Fig. 1c). We cool all the Xtransverse modes of vibration to near their ground states, and deep within the Lamb–Dicke regime by standard Doppler and Raman sideband cooling procedures. We initialize the spins to be aligned to the ydirection of the Bloch sphere by optically pumping to ↓_{z}↓_{z}...↓_{z}〉 and then coherently rotating the spins through π/2 about the Bloch xaxis with a carrier Raman transition. Next, we switch on the Hamiltonian H with an effective magnetic field B_{0}∼5J so that the spins are prepared predominantly in the ground state. Then, we exponentially ramp down the effective magnetic field with a time constant of 80 μs to a final value B, keeping the Ising couplings fixed. We finally measure the spins along the Ising (x) direction by coherently rotating the spins through π/2 about the Bloch yaxis before fluorescence detection. We repeat the experiment ∼1,000×N times for a system of N spins and generate a histogram of fluorescence counts and fit to a weighted sum of basis functions to obtain the probability distribution P(s) of the number of spins in state (↑〉), where s=0,1, ..., N, as described in the Methods section.
Extraction of order parameters from measured probabilities
We can generate several magnetic order parameters of interest from the distribution P(s), showing transitions between different spin orders. One order parameter is the average absolute magnetization (per site) along the Ising direction, . Hamiltonian 1 has a global timereversal symmetry of {σ^{i}_{x}→−σ^{i}_{x,} σ^{i}_{z}→−σ^{i}_{z,} σ^{i}_{y}→σ^{i}_{y}}, and this does not spontaneously break for a finite system, necessitating the use of average absolute value of the magnetization per site along the Ising direction as the relevant order parameter. For a large system, this parameter shows a secondorder phase transition, or a discontinuity in its derivative with respect to B/J. On the other hand, the fourthorder moment of the magnetization or Binder cumulant (refs 18,19) becomes a step function at the QPT and should therefore be more sensitive to the phase transition. We illustrate this point by plotting the exact groundstate order in the simple case of uniform Ising couplings for a moderately large system (N=100) in Figure 3a. Here we scale the twoorder parameters properly to account for trivial finite size effects, as described in the Methods section. The scaled magnetization and Binder cumulant are denoted by _{x} and ḡ respectively. In Figure 3a, we also plot the exact groundstate order parameters for N=2 and N=9 spins. In Figure 3bd, we present data for these twoorder parameters, as B/J is varied in the adiabatic quantum simulation. Figure 3b shows the scaled magnetization, _{x} for N=2 to N=9 spins that depict the sharpening of the crossover curves from paramagnetic to ferromagnetic spin order with increasing system size. The linear time scale indicates the exponential ramping profile of the (logarithmic) B/J scale. Figure 3c,d compares the two extreme system sizes in the experiment, N=2 and N=9, and clearly shows the increased steepness for larger system size. The scaled magnetization _{x} is suppressed by ∼25% (Fig. 3b,c) and the scaled Binder cumulant ḡ is suppressed by ∼10% (Fig. 3d) from unity at B/J=0, predominantly due to decoherence from offresonant spontaneous emission and additional dephasing due to intensity fluctuations in Raman beams, during the simulation.
We compare the data shown in Figure 3c,d with the theoretical evolution, taking into account experimental imperfections and errors discussed below, including spontaneous emission to the spin states and states outside the Hilbert space, and additional decoherence. The evolution is calculated by averaging 10,000 quantum trajectories. This takes only 1 minute on a single computing node for N=2 spins and approximately 7 h, on a single node, for N=9 spins. Extrapolating from this calculation suggests that averaging 10,000 trajectories for N=15 spins would require 24 hours on a 40node cluster, indicating the inefficiency of classical computers to simulate even a small quantum system.
Discussion
A faithful quantum simulation requires an excellent understanding of errors and their scaling, especially when the underlying problem is otherwise intractable. We characterize errors in the current simulation by plotting the observed parameter P(FM)=P(0)+P(N) for N=2 to N=9 spins in Figure 4. Theoretically P(FM)=2/2^{N} when there is no ferromagnetic order that is, P(FM)=0.5 for N=2 spins and exponentially goes down to 0.004 for N=9 spins, and unity, when there is perfect ferromagnetic order. Because P(FM) involves only two of the 2^{N} basis states, it is more sensitive to errors compared with the order parameters _{x} and ḡ. For instance, at B/J=0 in Figure 3bd and Figure 4ad, we find that _{x} and ḡ do not change appreciably with system size, but P(FM) degrades to ∼0.55 for N=9 spins from ∼0.9 for N=2. In Figure 4 we compare the data with theory results that include experimental sources of diabatic errors.
There are several primary sources of experimental error. Diabaticity due to finite ramping speed and error in initialization is estimated to suppress P(FM) by ∼3% for N=2 to ∼8% for N=9. This also gives rise to oscillations seen in the data (Figs 3b–d, 4a–d). A major source of error is the spontaneous emission from Raman beams that amounts to a ∼10% spontaneous emission probability per spin in 1 ms. Spontaneous emission dephases and randomizes the spin state and loosely behaves like a 'spin temperature' in this system, though the spins do not fully equilibrate with the 'bath' and the total probability of spontaneous emission increases linearly during the quantum simulation. In addition, each spontaneous emission event populates other states outside of the Hilbert space of each spin with a probability of 1/3. Spontaneous emission errors grow with increasing system size, which also suppresses P(FM) order with increasing N, as seen in Figure 4ad. We theoretically estimate the suppression of P(FM) due to diabaticity and spontaneous emission, together by averaging over quantum trajectories to be ∼7% for N=2 spins and ∼26% for N=9 spins. Intensity fluctuations on the Raman beams during the simulation modulate the AC Stark shift on the spins, and dephase the spin states, which causes additional diabaticity and degrades the final ferromagnetic order. When we introduce a theoretical dephasing rate of 0.3 per ms per ion (Methods) in the quantum trajectory computation the predicted suppression of P(FM) increases to ∼9% for N=2 and ∼50% for N=9.
Imperfect spin detection efficiency contributes ∼5–10% error in P(FM). Fluorescence histograms for P(0) and P(1) have a ∼1% overlap (in detection time of 0.8 ms) owing to offresonant coupling of the spin states to the ^{2}P_{1/2} level. This prevents us from increasing detection beam power or photon collection time to separate the histograms. Detection error in the data include uncertainty in fitting the observed fluorescence histograms to determine P(s), intensity fluctuations and finite width of the detection beam.
The role of phonons in the results of the quantum simulation is investigated both experimentally and numerically. The average number of centre of mass phonons excited during the simulation is numerically found to be always under 1.5 for N=9 spins, and even lower for N<9. We perform another set of experiments with μ−v_{1}=63 KHz for all N, which amounts to reducing the phonon excitation as the number of spins N is increased, as the Lamb–Dicke parameter η_{i,m}∼1/√N. We do not note any appreciable difference (beyond the margin of experimental errors) in spin population with the results reported here. In the presence of an effective magnetic field, phonon modes are coherently populated and generally exhibit spinmotional entanglement. However, these phonons do not alter the spin ordering and hence preserve spinspin correlation, even if the entanglement between spin states is partly destroyed when tracing over the phonon states. This will hence degrade entanglement characterization beyond a few spins using the standard GHZ type witness operators.
In this experiment, we have qualitatively observed sharpening of crossover curves that indicates the onset of a QPT as the system size increases. This scheme can be scaled up to a larger number of spins where it is possible to quantitatively estimate finite size effects, for example, scaling in critical exponents near the phase transition point^{20}. The primary challenges in experimenting with larger system sizes include the spontaneous emission as described above, the requirement of larger optical power to maintain the same level of Ising couplings, and nonadiabatic effects due to shrinking gap between ground and first excited state^{21,22}. One solution is to implement a high power laser with a detuning far from the ^{2}P energy levels, which would minimize spontaneous emission while maintaining the same level of Ising couplings. This would also allow versatility in varying the Ising interaction (together with the effective external field) during the simulation, as the differential AC stark shift between spin states is negligible for a sufficiently large detuning. The coherence time increases in the absence of spontaneous emission, allowing for a longer simulation time necessary to preserve adiabaticity as the system grows in size. Recently Raman transitions have been driven using a modelocked high power pulsed laser at a wavelength of 355 nm, which is optimum for ^{171}Yb^{+} wherein the ratio of differential AC Stark shift to Rabi frequency is minimized and spontaneous emission probabilities per Rabi cycle are <10^{−5} per spin^{23}.
With this system, it is possible to engineer different Ising coupling patterns by controlling the Raman beatnote detuning μ and observe interesting spin ordering such as with antiferromagnetic longrange couplings leading to frustration^{9,17}, and phase transitions some of which can be very sharp, or of first order^{24}. Longrange interactions in this spatially onedimensional system allow for simulating multidimensional spin models by selectively exciting vibrational modes using multiple Raman beatnote detunings. With additional laser beams, this scheme can potentially simulate more complicated and higher spindimensional Hamiltonians like xy and xyz models, which map onto nontrivial quantum Hamiltonians such as the BoseHubbard Hamiltonian.
Methods
Detection of spin states
The spin states are detected by spindependent fluorescence signals collected through f/2.1 optics by a photomultiplier tube. Spin state ↑_{z}〉 is resonantly excited by the 369.5 nm detection beam and fluoresces from ^{2}P_{1/2} states, emitting Poissondistributed photons with mean ∼12 in 0.8 ms. This state appears as 'bright' to PMT. The detection light is far offresonant to spin state ↓_{z}〉 and this state appears 'dark' to the PMT. However, due to weak offresonant excitation bright state leaks onto dark state, altering the photon distribution^{25}. Unwanted scattered light from optics and trap electrodes also alter the photon distribution. We construct the basis function for s bright ions by convolution techniques, and include a 5% fluctuation in the intensity of detection beam, which is representative of our typical experimental conditions. We then fit the experimental data to these basis functions, and obtain probabilities P(s) at each time step t_{i} in the experiment. Mean photon counts for dark (m_{D}) and bright (mB) states are used as fitting parameters so as to minimize the error residues.
The best fitting at time step t_{i} is obtained for the parameters {m_{D,i},m_{B,i}}. These parameters fluctuate at different time steps of the quantum simulation, primarily due to fluctuations in the intensity of detection beam and background scatter, and also due to uncertainties in a multivariate fitting. The fitting errors are propagated to the spinstate probabilities P(s) using Monte Carlo method of error analysis, as follows. We extract P(s) and compute the order parameters at time step t_{i} with mean dark and bright state counts chosen randomly from a Gaussian distribution with mean {_{D},_{B}} and standard deviations {δm_{D},δm_{B}} respectively. Here _{D} and ,_{B} are averages of m_{D,i} and m_{B,i} respectively over different time steps t_{i}. Similarly δm_{D} and δm_{B} are standard deviations of m_{D,i} and m_{B,i} respectively. By repeating this process ∼400 times we generate a histogram of each order parameter and fit the histograms to a Gaussian distribution. The standard deviation of the distribution is chosen to represent the random error due to fitting in that order parameter. The uncertainty in amount of fluctuation of the detection beam power during the experiment is conservatively included in the error analysis by repeating the fitting process for a range of fluctuations. The finite width of the detection beam is taken care of by modelling the Gaussian beam having a threestep intensity profile with appropriate intensity ratios.
Scaled order parameters
To characterize the spin orders we use different order parameters in the experiment, namely the average absolute magnetization per site (m_{x}) and the Binder cumulant (g). When the spins are polarized along the ydirection of the Bloch sphere, the distribution of total spin along xdirection is Binomial and approaches a Gaussian (with zero mean) in the limit of N→∞. For system size of N, m_{x} takes on theoretical value of in the perfect paramagnetic phase (B/J→∞) and unity in the other limit of B/J=0. In Figure 3b,c, we rescale m_{x} to which should ideally be zero in perfect paramagnetic phase and unity in perfect ferromagnetic phase for any N. This accounts for the 'trivial' finite size effect due to the difference between Binomial and Gaussian distribution. Similarly the Binder Cumulant g is scaled to in Figure 3d, where is the theoretical value of g for B/J→∞.
Quantum MonteCarlo simulations
Quantum trajectories are generated by numerically integrating the Schrödinger equation, with Hamiltonian (1), while simultaneously executing quantum jumps to account for spontaneous emission and decoherence. Spontaneous emission from ion i either localizes the spin of the ion, projecting it into 2S_{1/2}F=0,m_{F}=0〉 (spin state ↓_{z}〉) or 2S_{1/2}F=1,m_{F}=0〉 (spin state ↑_{z}〉), or projects the ion into 2S_{1/2}F=1,m_{F}=1〉, in which case ion i is factored out of the Schrödinger evolution, though it is counted as spin up at the time of measurement. Decoherence (dephasing) is modelled by the quantum jump operator σ_{x}; thus a jump for ion i, ψψ〉→σ^{i}_{x}ψψ〉, introduces a π phase shift between the spin states ↑〉 and ↓〉 (in xbasis). Jump rates are taken to be fixed and equal for all ions. Note that a decoherence jump rate of Γ_{decoh} leads to decay of the spin coherence at rate 2Γ_{decoh}. To determine the entangled state of the spin ensemble after a spontaneous emission, for example, from ion i, we assume that the groundstate configuration before emission,
is mapped, by the fardetuned Raman beams, into a very small excitedstate contribution to the overall system entangled state,
with λ≪1 proportional to the amplitude of the Raman beams and inversely proportional to their detuning. The (unnormalized) state after the emission is
where ?〉_{i} is ↑_{z}〉_{i}, ↓_{z}〉_{i}, or the factored state 2S_{1/2}F=1,m_{F}=1〉.
Additional information
How to cite this article: Islam, R. et al. Onset of a quantum phase transition with a trapped ion quantum simulator. Nat. Commun. 2:377 doi: 10.1038/ncomms1374 (2011).
References
 1
Feynman, R. Simulating physics with computers. Int. J. Theor. Phys. 21, 467–488 (1982).
 2
Lloyd, S. Universal quantum simulators. Science 273, 1073 (1996).
 3
Simon, J. et al. Quantum simulation of an antiferromagnetic spin chains in an optical lattice. Nature 472, 307–312 (2011).
 4
Struck, J. et al. Quantum simulation of frustrated magnetism in triangular optical lattices arXiv:1103.5944.
 5
Porras, D. & Cirac, J. I. Effective quantum spin systems with trapped ions. Phys. Rev. Lett. 92: 207901 (2004).
 6
Deng, X. L., Porras, D. & Cirac, J. I. Effective spin quantum phases in systems of trapped ions. Phys. Rev. A 72 (2005).
 7
Taylor, J. M. & Calarco, T. Wigner crystals of ions as quantum hard drives. Phys. Rev. A 78: 062331 (2008).
 8
Friedenauer, A., Schmitz, H., Glueckert, J. T., Porras, D. & Schaetz, T. Simulating a quantum magnet with trapped ions. Nature Phys. 4, 757–761 (2008).
 9
Kim, K. et al. Quantum simulation of frustrated ising spins with trapped ions. Nature 465, 590–593 (2010).
 10
Edwards, E. E. et al. Quantum simulation and phase diagram of the transversefield ising model with three atomic spins. Phys. Rev. B 82: 060412 (2010).
 11
Sachdev, S. Quantum magnetism and criticality. Nature Phys. 4, 173–185 (2008).
 12
Farhi, E. et al. A quantum adiabatic evolution algorithm applied to random instances of an npcomplete problem. Science 292, 472 (2001).
 13
Lanczos, C. An iteration method for the solution of the eigenvalue problem of linear differential and integral operators. J. Res. Nat. Bur. Standards 45, 255–282 (1950).
 14
Barahona, F. On the computational complexity of Ising spin glass models. J. Phys. A: Math. Gen. 15, 3241–3253 (1982).
 15
Olmschenk, S. et al. Manipulation and detection of a trapped Yb^{+} hyperfine qubit. Phys. Rev. A 76: 052314 (2007).
 16
Deslauriers, L. et al. Zeropoint cooling and low heating of trapped ^{111}Cd^{+} ions. Phys. Rev. A 70: 043408 (2004).
 17
Kim, K. et al. Entanglement and tunable spinspin couplings between trapped ions using multiple transverse modes. Phys. Rev. Lett. 103: 120502 (2009).
 18
Binder, K. Finite size scaling analysis of Ising model block distribution functions. Physik B 43, 119 (1981).
 19
Binder, K. Critical properties from monte carlo coarse graining and renormalization. Phys. Rev. Lett. 47, 693–696 (1981).
 20
Fisher, M. E. & Barber, M. N. Scaling theory for finitesize effects in the critical region. Phys. Rev. Lett. 28, 1516–1519 (1972).
 21
Dusuel, S. & Vidal, J. Continuous unitary transformations and finitesize scaling exponents in the lipkinmeshkovglick model. Phys. Rev. B 71: 224420 (2005).
 22
Caneva, T., Fazio, R. & Santoro, G. E. Adiabatic quantum dynamics of the lipkinmeshkovglick model. Phys. Rev. B 78: 104426 (2008).
 23
Campbell, W. C. et al. Ultrafast gates for single atomic qubits. Phys. Rev. Lett. 105: 090502 (2010).
 24
Lin, G. D., Monroe, C. & Duan, L. M. Sharp phase transitions in a small frustrated network of trapped ion spins ArXiv 1011. 5885.
 25
Acton, M. Detection and Control of Individual Trapped Ions and Neutral Atoms Ph.D. thesis (University of Michigan, 2008).
 26
Zhu, S. L., Monroe, C. & Duan, L. M. Trapped ion quantum computation with transverse phonon modes. Phys. Rev. Lett. 97: 050505 (2006).
Acknowledgements
We thank D. Huse for help with theoretical understanding of the quantum Ising model and finitesize scaling. This work is supported under Army Research Office under Award No. 911NF0710576 with funds from the DARPA Optical Lattice Emulator Program, IARPA under ARO contract, the NSF Physics at the Information Frontier Program, the European Program on Atomic Quantum Technologies, and the NSF Physics Frontier Center at JQI.
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R.I., E.E.E., K.K., S.K. and C.M. designed and performed the experiments, and analysed the data. C.N., H.C., G.D.L., L.M.D., C.C.J.W and J.K.F provided theoretical support. C.N. and H.C. performed the quantum trajectory calculations. All the authors contributed to the manuscript.
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Islam, R., Edwards, E., Kim, K. et al. Onset of a quantum phase transition with a trapped ion quantum simulator. Nat Commun 2, 377 (2011). https://doi.org/10.1038/ncomms1374
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