Abstract
Ultrafast materials science promises optical control of physical properties of solids. Continuouswave circularly polarized laser driving was predicted to induce a lightmatter coupled state with an energy gap and a quantum Hall effect, coined Floquet topological insulator. Whereas the envisioned Floquet topological insulator requires highfrequency pumping to obtain wellseparated Floquet bands, a followup question regards the creation of Floquetlike states in graphene with realistic lowfrequency laser pulses. Here we predict that short optical pulses attainable in experiments can lead to local spectral gaps and novel pseudospin textures in graphene. Pumpprobe photoemission spectroscopy can track these states by measuring sizeable energy gaps and Floquet band formation on femtosecond time scales. Analysing band crossings and pseudospin textures near the Dirac points, we identify new states with optically induced nontrivial changes of sublattice mixing that leads to Berry curvature corrections of electrical transport and magnetization.
Introduction
The ultrafast optical manipulation of materials by femtosecond laser pulses is rapidly becoming a major guiding theme in condensed matter physics^{1,2,3}. At the same time, the quest for novel topological states of matter triggered enormous research activity since the discovery of topological insulators^{4}. Merging both of these vibrant fields, a recent work reported the coupling of short laser pulses to surface Dirac fermions in the topological insulator Bi_{2}Se_{3} (ref. 5). This work demonstrated the creation of Floquetlike sidebands during irradiation as well as the opening of a small band gap at the surface state Dirac point for circular light polarization.
Timereversal symmetry protects massless Dirac fermions on the surface of topological insulators^{6,7} and, in combination with inversion symmetry, also in graphene in the absence of spinorbit coupling^{8}. In a milestone paper, Haldane envisioned that breaking either or both of these symmetries would open a gap at the Dirac points in graphene, allowing one to tune between a trivial insulator and a Chern insulator^{9}. While equilibrium band gap engineering has become a major theme since the first synthesis of monolayer graphene, it was only recently proposed that circularly polarized, highfrequency laser light could turn trivial equilibrium bands into topological nonequilibrium Floquet bands^{10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25}, coined Floquet topological insulator (FTI).
The FTI concept is based on two things: first, in the limit of continuous laser driving at frequency Ω, the temporal periodicity allows one to employ a repeated quasienergy zone scheme with a temporal Brillouin zone of size Ω. Second, in the highfrequency limit, defined by Ω being larger than the electronic bandwidth, these repeated zones contain wellseparated copies of the original electronic bands spaced by integer multiples nΩ, the socalled Floquet sidebands. The effect of the laser on the original n=0 band manifold is perturbative in 1/Ω. If the laser is circularly polarized, timereversal symmetry is broken and an energy gap opens at the Dirac points in graphene due to the fact that photon emission and absorption processes do not commute. The FTI concept then follows from an exact mapping of driven graphene to the Haldane model, leading to a welldefined nonzero Chern number.
Whereas this envisioned highfrequency strong pumping limit that is required for nontrivial topological states is currently experimentally unattainable, a natural followup question regards the engineering of local spectral gaps in realistic pumpprobe experiments. This leads to the question on which time scales the quasisteady Floquet regime can be reached when continuouswave driving is replaced by a short laser pulse. Moreover, it requires the investigation of the lowfrequency regime, in which 1/Ω perturbation theory is not applicable. In this regime, the overlap of different Floquet sidebands prevents a global topological classification of states.
In this work, we address these problems by simulating the realtime development of singleparticle energy gaps in graphene coupled to short laser pulses, using realistic parameters for timeresolved, angleresolved photoemission spectroscopy (trARPES). We show that the trARPES band structure shows welldefined Floquet bands provided that a hierarchy of time scales is fulfilled between the duration of the pumppulse, the duration of the probepulse, and the laser period: σ_{pump}>σ_{probe}≫2πℏ/Ω. We predict the opening of a Dirac point gap and the formation of Floquet sidebands that form on femtosecond time scales. An important difference to the highfrequency limit arises from the overlap of Floquet sidebands. At frequencydependent critical driving field strengths, we find a sequence of level crossings and energy gap closings at the Dirac points. The analysis of snapshots of pseudospin textures near the Dirac points allows us to identify optically induced nontrivial changes of sublattice mixing at these level crossing points, that manifest themselves in Berry curvature corrections of electrical transport and magnetization. Even though a global topology cannot be assigned to the lowfrequency driven states, we show that the analysis of level crossings and energy gap closings leads to a classification scheme in terms of local gaps and Berry curvatures.
Results
Haldane model
To set the stage for our results, we briefly outline the basic ingredients for the lowenergy physics of Haldane’s equilibrium model. We start from two Dirac cones with effective Hamiltonian v_{D}(q_{x}σ_{x}⊗τ_{z}+q_{y}σ_{y}⊗I). Here v_{D} is the Dirac point velocity, the Pauli matrices σ label pseudospin arising from the graphene sublattices and , τ labels the valley degree of freedom corresponding to the Dirac cones around K and K′, and I is the 2 × 2 identity matrix. Electron spin can be neglected in the absence of spinorbit coupling. Momentum q is measured from the respective Dirac points. The pseudospin content P(q) essentially measures orbital band content (see Supplementary Note 1). For instance, a pseudospin pointing along the +z (up) direction means that the band is predominantly of sublattice character, while a pseudospin pointing along the −z (down) direction indicates mainly sublattice character. Together with the winding of the P_{x} and P_{y} inplane pseudospin components around the Dirac points, P_{z} determines the local Berry curvature ^{26}.
In Haldane’s model, an effective mass term mσ_{z}⊗τ_{κ} leads to an energy gap Δ=2m at the Dirac points (Fig. 1a). Its relative sign between K and K′ is determined by τ_{κ} and depends on its origin: If the gap is induced by introducing a staggered sublattice potential breaking inversion symmetry, τ_{κ}=τ_{0}, implying that the effective mass term has the same sign at K and K′, and the outofplane pseudospin component P_{z} is the same at both Dirac points (Fig. 1b). By contrast, if the gap originates from breaking timereversal symmetry, τ_{κ}=τ_{z}, hence P_{z} points in opposite directions (Fig. 1c).
Timeresolved photoemission spectra of driven graphene
We now come to the discussion of our nonequilibrium results. We start from the minimal honeycomblattice tightbinding model of graphene. We drive this system by coupling to a timedependent, spatially homogeneous electric field modelled as a timedependent vector potential A(t), which couples to the electrons via Peierls substitution. The relativistic magnetic component of the light field is neglected. The pumppulse has a temporal width σ_{pump}=165 fs, photon frequency Ω=1.5 eV (laser period of 2.8 fs), with linear or circular light polarization, corresponding to a femtosecond pumppulse. The field strength is given by A_{max}, which is measured in units of the inverse carbon–carbon distance. For graphene, the conversion to the peak electric field strength is E_{max}=A_{max} × 1,060 mV Å^{−1} for Ω=1.5 eV. We track the time and momentumresolved singleparticle spectrum of the pumpdriven electrons using a short 26 fs probepulse that emits photoelectrons and thereby generates a photocurrent^{27,28}, as measured experimentally with trARPES (see Methods).
We first characterize the nonequilibrium band structures using trARPES spectra. Fig. 2 shows the trARPES spectra on a momentum cut along the Γ−K−K′ direction near K at peak field (Δt=0 fs). We first perform a calculation using pump pulses with linear polarization along the k_{x} direction for two different field strengths (Fig. 2a,b). One can see the formation of Floquet sidebands, but since the pump preserves timereversal symmetry, the spectrum remains gapless at the Dirac point energy. The main effect of the linearly polarized pump is a shift of the Dirac point location from K towards K′, which increases with increasing field strength. This Dirac point shift is due to the nonlinearity of bands and does not happen for perfect Dirac cones.
Floquet spectrum and level crossings
Next, we turn to circular light polarization, thereby breaking timereversal symmetry. In Floquet theory, the quasistatic eigenvalue spectrum at finite driving field A shows copies of the original bands shifted by integer multiples of Ω, the socalled Floquet sidebands. Energy gaps of nth order in the field open at avoided level crossings of sidebands which differ by n photon energies. For circular light, an energy gap of second order in the field opens at the Dirac point. In our trARPES simulation, for a moderate field strength and 1.5 eV photons, an energy gap exceeding 100 meV at K is induced, accompanied by avoided level crossing gaps nearby (Fig. 2c). Due to the aforementioned hierarchy of time scales, we observe an excellent agreement of trARPES spectra and the quasistatic Floquet band structure obtained by diagonalizing the Floquet Hamiltonian involving large numbers of sidebands (solid lines, see Supplementary Material).
At larger field strength (Fig. 2d), the Floquet bands move closer to each other and cross. They separate again and the Dirac point energy gap decreases (Fig. 2e). At even larger fields, there is another crossing between Floquet bands (Fig. 2f) before the Dirac point gap closes (Fig. 2g) and eventually reopens (Fig. 2h). At the largest field strength shown here (A_{max}=1.00), the bands are almost flat, indicating that the ac Wannier–Stark limit is approached. This creation of flat Wannier–Stark bands in the strong driving limit impedes the continuous growth of the gap with increasing field strength.
To analyse the pump photon frequency dependence of the Dirac point level crossings and gap closing in more detail, we show in Fig. 3 the first two negative Floquet eigenvalues tracking the position of the first two Floquet bands below E_{D}. Fig. 3a shows the initial gap opening at Ω=1.5 eV, which is quadratic in the field at small A_{max}, followed by two level crossings between the two Floquet bands indicated by two arrows. The gap at the Dirac point is then closed, indicated by the third arrow. For Ω=3.0 eV, the field range between the two Floquet band level crossings increases (Fig. 3b), then decreases at Ω=4.5 eV, and finally vanishes for Ω=5.5 eV. The initial quadratic gap opening is the same for all photon frequencies due to the linearity of the graphene bands near the Dirac points. The differences between different photon frequencies at larger fields then arise from the nonlinearity of the bands further away from the Dirac points, which is specific to graphene.
Local pseudospin textures near Dirac points
We now turn to the discussion of local pseudospin content. Fig. 4a–c present false color plots of the momentumresolved pseudospin contents near the Dirac points for the driven system at Ω=1.5 eV. At small field before the first level crossing (Fig. 4a), the P_{x} and P_{y} components have two sign changes along a path around K, as expected for weakly driven graphene. This nodal structure is directly related to the q_{x}σ_{x}+q_{y}σ_{y} term in the effective lowenergy Hamiltonian introduced above, which shows that the P_{x} component transforms like q_{x} and the P_{y} component transforms like q_{y}. We coin this state S_{1}, with one nodal line and therefore a single pseudospin winding in the vicinity of the Dirac points. Importantly, the P_{z} component changes sign between K and K′, which is consistent with breaking timereversal symmetry.
When the field is increased through the first level crossing (Fig. 4b), the character of the local pseudospin textures changes. The effective mass term still changes sign between K and K′. Remarkably, the P_{x} and P_{y} components double their winding number, changing sign four times along a path around K. We call this state S_{2}, since it has two nodal lines in the vicinity of the Dirac point, which cross at the Dirac point. Although the corresponding ARPES spectrum for the same parameters (Fig. 2f) has only little spectral weight near K in the Floquet bands close to E_{D}, the pseudospin texture is welldefined at all momenta considered here. Also, other sidebands have higher spectral weight, and each of the Floquet sidebands carries the same pseudospin information.
A similar texture is also obtained for Bernal stacked bilayer graphene^{29} in a perpendicular electric field^{30}. In this analogy, we stress that the doubled winding in bilayer graphene persists even when the energy gap induced by the electric field goes to zero. By contrast, in our case the doubled winding state S_{2} vanishes when the Floquet sidebands cross, and gives way to a single winding state S_{1} in the lowfield limit. Also the effective perpendicular electric field generated by the circularly polarized light pulse in our work points in opposite directions at K and K′, in contrast to the static electric field applied in ref. 30, which corresponds to the Haldane mass term. In any case, the observation of state S_{2} suggests the possibility to dynamically engineer effective models with higher pseudospin winding numbers, similarly to higherorder spinorbital textures in topological insulators^{31}.
When the field is increased further through the second level crossing and the gap closing, one obtains the pseudospin textures shown in Fig. 4c. Here all the pseudospin components are flipped compared to the ones in Fig. 4a, and we coin this flipped state with single pseudospin winding S'_{1}.
Phase diagram at lowfrequency driving
We are now in a position to discuss the phase diagram of local Berry curvatures, which follow from the pseudospin textures around the Dirac points, as a function of field strength and driving frequency. Fig. 4d shows the positions of the Floquet band level crossings indicating the transition to a pseudospin texture with a doubled number of nodal lines, as well as the Dirac energy gap closing leading to a state with inverted P_{z} component. There is an upper frequency limit for the former state in the range of field strengths shown here. This is consistent with the fact that in the infinitefrequency limit, only states with pwave pseudospin textures corresponding to S_{1} and S'_{1} were found, which can be understood from the exact mapping to the static Haldane model in this limit^{32}.
On one hand, the characterization of nonequilibrium states in terms of local pseudospin textures is restricted to momenta near the Dirac points by Floquet sideband level crossings. Such level crossings generically appear at low driving frequency Ω because different Floquet sidebands overlap if Ω is smaller than the electronic bandwidth. On the other hand, sideband level crossings at the Dirac point are the root cause of the appearance of the exotic pseudospin textures in Fig. 4b. The lowfrequency behaviour of driven graphene is therefore more complicated, but also contains new states that are absent in the highfrequency limit. The complete evolution of the trARPES spectra and pseudospin textures across the boundaries of the S_{2} state is shown in the Supplementary Figs 1, 2 and 3 for driving frequencies of 1.5, 3.0 and 4.5 eV, respectively, and discussed in Supplementary Note 2.
Discussion
Our combined results show that band gaps induced by breaking timereversal symmetry in graphene are within reach under realistic experimental conditions. In particular, the achievable energy resolution for probephoton energies which are sufficiently high to reach the Dirac points^{33,34} should allow for the detection of photoemission gap sizes exceeding 100 meV. The change in pseudospin texture near critical driving offers the exciting opportunity of optical manipulation of local Berry curvatures near Dirac points on ultrafast time scales. Moreover, the combination of broken inversion symmetry and broken timereversal symmetry opens up the possibility of controlling the valley degree of freedom and inducing different energy gaps at the two Dirac points^{35,36,37,38}.
The spectroscopic detection of pseudospin textures requires access to orbital band content. To this end, hexagonal structures with inequivalent orbitals on the and sublattices having different photoemission probeenergy crosssections could be examined. A candidate material for this purpose is hexagonal boron nitride. The demonstration of pseudospin imbalance at the two Dirac points by circularly polarized light in boron nitride would be intriguing. Alternatively, artificial hexagonal lattices with sublattice potentials have already been demonstrated with cold atoms^{39}. Thus the proposed pseudospin textures could in principle also be realized in driven ultracold quantum gases^{40,41}.
Methods
Methods summary
The simulations presented here start from a minimal tightbinding model of spinless electrons with nearestneighbour hopping on the honeycomb lattice^{8,42}. The pumppulse drives the electrons via minimal coupling to a gauge field , with a Gaussian shape function for a pulse of width σ_{pump} centred around time t_{p}, and photon frequency Ω. The phase shift of π/2 between the x and y components, represented by the unit vectors e_{x} and e_{y}, describes circular light polarization. For comparison we also study linearly polarized light with vanishing y component. Throughout this work we use units where e=ℏ=c=1. The electric field is E(t)=−∂A(t)/∂t, and we neglect the relativistic magnetic field of the laser pulse. This means that the electronic spin degree of freedom maintains its full degeneracy, and it is therefore not explicitly included in our calculations. In particular, both the photoemission spectra and the pseudospin textures are identical for both physical spin species.
The trARPES is computed from the trace of the nonequilibrium lesser Green function by a postprocessing step involving the probelaserpulse shape s_{W}(t) with time resolution W, which leads to an effective trARPES energy resolution ∝ 1/σ_{probe}^{27,28}. The delay time of a pump peaked at t_{p} and probe peaked at t_{pr} is given by Δt≡t_{pr}−t_{p}.
Simulations are performed for an initial equilibrium sample temperature T=116 K. We typically use 500,000−1,500,000 time steps for the computation of the time evolution operator, depending on the field parameters. This corresponds to a maximal timestep size of 0.0016, fs. Green functions for trARPES measurements are sampled on a grid with 5,000−15,000 realtime steps and a maximal step size of 0.16 fs. The Dirac point velocity is given by v_{D}=4.2 eV a_{C−C}, where a_{C−C}=1.42 Å is the carbon–carbon distance^{8}. We choose a chemical potential μ=0.5 eV, which sets the Dirac point energy E_{D}=−0.5 eV relative to μ. This choice is motivated by the fact that typical graphene samples on substrates are doped, and that states both below and above the Dirac point energy are occupied in the initial equilibrium states and therefore nicely visible in the trARPES spectra.
The simulation parameters for the pumpprobe setup are as follows: The pump laser field has frequency Ω=1.5 eV, unless denoted otherwise, implying oscillation periods of 2.58 fs. Its temporal width is σ_{pump}=165 fs. We vary the peak vector potential A_{max}=0.10 … 1.00 in units of a_{C–C}^{−1}. This corresponds to peak electric field strengths E_{max}=ΩA_{max} of 106 … 1,060 mV Å^{−1} for Ω=1.5 eV and the graphene lattice parameters. The photoemission probepulse has a width σ_{probe}=26 fs. This choice of parameters is motivated by the hierarchy of time scales in the system: The oscillation period for the pump laser light, the temporal width of the probepulse, which controls the time and energy resolution for the trARPES signal, and the temporal width of the pumppulse, which controls the nonequilibrium state and ensures a welldefined center frequency for the pumppulse.
Model and time evolution
Our goal is to obtain the lesser Green function matrix in 2 × 2 orbital space (see below) with matrix elements
where is a creation (annihilation) operator for a fermion at momentum k in orbital α (β) ∈{a,b}. As shown below, the photocurrent and pseudospin contents are computed from these lesser Green functions.
Including the field via Peierls substitution, the timedependent Hamiltonian for and sublattices with corresponding orbitals a and b reads
with the Hamiltonian matrix elements
where V=2.8 eV is the nearestneighbour hopping matrix element matching the graphene bandwidth and Dirac point velocity. In equilibrium, the Hamiltonian has two Dirac points at momenta K and K′ given by , where momenta and the vector field A(t) are measured in multiples of the inverse of the carbon–carbon distance a_{C–C}^{8}.
It is convenient to define the Hamiltonian matrix for momentum k in orbital basis,
In the absence of a driving field, this Hamiltonian is diagonalized by a rotation at t=0, which is a time before the pumppulse is turned on. Note that t=0 is used here as a notation for the earliest real time we consider, not to be confused with zero delay time Δt≡t_{pr}−t_{p}=0, which refers to the time where the Gaussian pumppulse envelope is maximal. For later times, the given rotation does not diagonalize the Hamiltonian except for accidental cases where the gauge field is an integer multiple of a reciprocal lattice vector.
The computation of doubletime propagators requires the evaluation of the time evolution operators
Since at different times do not commute with each other, the time ordering in is taken into account by discretization of the realtime axis and multiplication of the resulting timestep evolution operators. We then obtain the time evolution operator as 2 × 2 matrices in band basis,
where N_{t,t′} is the number of fine time steps of size δt between t and t′, and the product is understood as time ordered with later times to the left.
The lesser Green function matrix results from
where time 0 refers to an initial time where the system is in equilibrium before the pumppulse is turned on, and is the timeindependent diagonal matrix of initial equilibrium band occupation with diagonal elements and corresponding to Fermi function filling for the two energy eigenvalues.
Floquet spectra
The Floquet spectra shown in Figs 2 and 3 are calculated from the Floquet Hamiltonian corresponding to equation (3):
where g_{m−n}(k) are the Fourier series expansion coefficients of g(k−A(t)):
Here J_{n}(A) is the Bessel function of the first kind. As the pump frequencies considered in this work are small with respect to the electronic bandwidth, the corresponding spectrum must be evaluated numerically via truncation of the full Floquet Hamiltonian. In practice, we achieve convergence for m≤40.
Timeresolved ARPES formalism
The computation of the timeresolved photocurrent involves normalized Gaussian probepulse shape functions s_{σprobe}(t) of width σ_{probe} centred around time t. In the Hamiltonian gauge, the photocurrent (trARPES intensity) at momentum k, binding energy ω and pumpprobe delay time Δt≡t_{pr}−t_{p} is then obtained from^{27}
In the main text (Fig. 2) we show trARPES spectra at peak field strength with false colour plots of the trARPES intensity I(k,ω,Δt=0), that is, intensity variations as a function of binding energy ω along selected momentum cuts k. The location of energy bands E(k) can be obtained from the maxima in the ARPES intensity as a function of binding energy at constant momentum, the socalled energy distribution curves. As seen in Fig. 2 of the main text, these bands are in excellent agreement with quasistatic Floquet bands, whose calculation is described below.
The photocurrent as defined in equation (12) is computed from the lesser Green function in a fixed gauge. We would like to point out that this quantity is not gaugeinvariant. In fact, the general definition of a gaugeinvariant photocurrent that fulfills the positivity criterion I(k,ω,Δt)≥0 for all k,ω,Δt in the presence of a field is an outstanding research problem^{43}. The problem likely lies in the neglect of photoemission matrix elements, which can be momentum and field dependent. The photocurrent according to equation (12) manifestly fulfills the positivity criterion. In addition, it also matches the Floquet band structure, as shown in the present work. The Floquet band structure is not gaugeinvariant either. Importantly, general conclusions drawn from the analysis of Floquet sidebands, level crossings and gap closings are valid even in a fixedgauge calculation. This is due to the fact that the timeresolved, momentumintegrated photoemission spectrum (trPES) is always manifestly gaugeinvariant and positive. The trPES signal is obtained by integrating the trARPES spectrum in any gauge over all momenta. Hence, conclusions about the presence or absence of energy gaps can be drawn even in a gaugevariant formalism.
Additional information
How to cite this article: Sentef, M.A. et al. Theory of Floquet band formation and local pseudospin textures in pumpprobe photoemission of graphene. Nat. Commun. 6:7047 doi: 10.1038/ncomms8047 (2015).
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Acknowledgements
We acknowledge helpful discussions with Patrick Kirchmann, Sri Raghu, XiaoLiang Qi, ShouCheng Zhang, and Bruce Normand. This work was supported by the Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering under Contract Nos. DEAC0276SF00515 (Stanford/SIMES), DEFG0208ER46542 (Georgetown) and DESC0007091 (for the collaboration). Computational resources were provided by the National Energy Research Scientific Computing Center supported by the Department of Energy, Office of Science, under Contract No. DE AC0205CH11231. J.K.F. was also supported by the McDevitt bequest at Georgetown. A.F.K. was supported by the Laboratory Directed Research and Development Program of Lawrence Berkeley National Laboratory under U.S. Department of Energy Contract No. DEAC0205CH11231.
Author information
Affiliations
Stanford Institute for Materials and Energy Sciences (SIMES), Stanford University and SLAC National Accelerator Laboratory, Menlo Park, California 94025, USA
 M.A. Sentef
 , B. Moritz
 & T.P. Devereaux
HISKP University of Bonn, Bonn 53115, Germany
 M.A. Sentef
Geballe Laboratory for Advanced Materials, Stanford University, Stanford, California 94305, USA
 M. Claassen
 & T.P. Devereaux
Lawrence Berkeley National Lab, 1 Cyclotron Road, Berkeley, California 94720, USA
 A.F. Kemper
Department of Physics and Astrophysics, University of North Dakota, Grand Forks, North Dakota 58202, USA
 B. Moritz
Department of Applied Physics, University of Tokyo, Tokyo, 1138656, Japan
 T. Oka
Department of Physics, Georgetown University, Washington, DC 20057, USA
 J.K. Freericks
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Contributions
M.A.S. developed the computer code and performed the ARPES and pseudospin calculations. A.F.K. assisted in developing the computer code for the ARPES calculations. M.C. developed the computer code and performed the Floquet calculations, and T.O. assisted in developing the computer code for the Floquet calculations. M.A.S., M.C., A.F.K., B.M., J.K.F. and T.P.D. wrote the manuscript. T.P.D. and J.K.F. are responsible for project planning.
Competing interests
The authors declare no competing financial interests.
Corresponding authors
Correspondence to M.A. Sentef or T.P. Devereaux.
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Supplementary Information
Supplementary Figures 13 and Supplementary Notes 12
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