Microscopic observation of magnon bound states and their dynamics

Takeshi Fukuhara1,∗, Peter Schauß1, Manuel Endres1, SebastianHild1, Marc Cheneau1,2, Immanuel Bloch1,3, and Christian Gross1

1Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Str. 1, 85748 Garching, Germany2Laboratoire Charles Fabry, Institut d’optique Graduate School – CNRS – Université Paris Sud, 91127 Palaiseau, France and

3Fakultät für Physik, Ludwig-Maximilians-Universität München, 80799 München, Germany

More than eighty years ago, H. Bethe pointed out the existence of bound states of elementaryspin waves in one-dimensional quantum magnets [1]. To date, identifying signatures of such magnonbound states has remained a subject of intense theoretical research [2–5] while their detection hasproved challenging for experiments. Ultracold atoms offer an ideal setting to reveal such boundstates by tracking the spin dynamics after a local quantum quench [6] with single-spin and single-site resolution [7, 8]. Here we report on the direct observation of two-magnon bound states usingin-situ correlation measurements in a one-dimensional Heisenberg spin chain realized with ultracoldbosonic atoms in an optical lattice. We observe the quantum walk of free and bound magnon statesthrough time-resolved measurements of the two spin impurities. The increased effective mass of thecompound magnon state results in slower spin dynamics as compared to single magnon excitations.In our measurements, we also determine the decay time of bound magnons, which is most likelylimited by scattering on thermal fluctuations in the system. Our results open a new pathway forstudying fundamental properties of quantum magnets and, more generally, properties of interactingimpurities in quantum many-body systems.

The study of non-equilibrium processes in quantumspin models can provide fundamental insight into ele-mentary aspects of magnetism. Magnons are the basicquasiparticle exitations around the ground state of ferro-magnets and govern their low temperature physics [9, 10].Due to the ferromagnetic interaction, two spin excita-tions can remain bound together, forming a so-called two-magnon bound state [1, 9, 11]. In one and two dimen-sions, bound states exist for all center of mass momenta,which prohibits the description of low energy propertiesin terms of free magnon states [9]. In the classical limit,magnon bound states can be regarded as the basic build-ing blocks of magnetic solitons [12, 13]. Next to these fun-damental aspects, the study of non-equilibrium dynamicsin quantum spin chains is also important for a variety ofapplications. The evolution of two localized spin excita-tions realizes an interacting quantum walk [14, 15] in thespin domain, which can be a versatile tool for the studyof complex many-body systems [16]. It is also of import-ance in the context of quantum information [17], wheretransport properties in a one-dimensional chain of qubitscan be strongly influenced by magnon bound states [18].

The spin-1/2 Heisenberg model is one of the founda-tional models for interacting quantum spins. This modelcould be solved analytically in one dimension in the early1930’s by H. Bethe using a systematic Ansatz for the formof the eigenvectors [1]. Later, the Bethe Ansatz provedto be far more general and allowed for solving manymore one-dimensional models, such as the Lieb-Linigeror the fermionic Hubbard model [20], and recent, power-ful extensions include the investigation of the dynamics

∗ Electronic address: takeshi.fukuhara@mpq.mpg.de

a Initial State

b Bound Magnon Hopping

c Free Magnon Propagation

d

Tim

e (ħ

/Jex

)P

rob

abili

ty

10

0

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0.6

-20 0

20

20Position (lattice sites)

Figure 1. Schematic representation of the magnonpropagation. a-c, Initially prepared state with two flippedspins and its decomposition into bound and free magnonspropagating through the lattice. d, Numerical results ob-tained from exact diagonalization showing the probability tofind a flipped spin at a given lattice site following the initialstate preparation. Two different wavefronts corresponding tobound and free magnons can be identified (see insets). Notethat the maximum probability was clipped in the graph forclarity.

of one-dimensional quantum many-body systems. One ofthe first results of Bethe’s analysis was the prediction ofmagnon bound states. Experimentally, infrared scatter-ing experiments provided first evidence for the existenceof such states in materials characterized by a highly an-isotropic, Ising-like Hamiltonian [21, 22]. For ultracoldatoms in optical lattices, high-energy bound states havebeen observed in the density sector in the form of repuls-ively bound atom pairs [23, 24]. Optical lattice systemscan also be used to realize the Heisenberg model with inprinciple tunable anisotropy [25, 26], where bound statesoccur as low energy excitations of the many-body sys-tem. Recent technological advances even allow for thein-situ control and detection of atomic spins in these ex-

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0 2 4 6 80.0

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Figure 2. Quantum state analysis through Bethe Ansatz. a, Overlap of the initial state with the bound (red circles)and free (black dots) magnon states calculated for N = 16 lattice sites [19]. For illustration we show only the states withwave-vectors within the interval k ∈ [0, π/alat]. The inset b shows the corresponding energy spectrum. c, Spin-spin correlationof the magnon bound states as a function of the spin separation d = |j − i| for different wave-vectors k. For k = π/alat thewavefunction of the bound magnon state corresponds to tightly bound spins on neighbouring sites, giving the largest overlapwith our initial state.

periments [7, 8, 27].In our system, we make use of a one-dimensional chain

of bosonic atoms in an optical lattice. Starting from aninitial Mott insulating state and a fully magnetized chain,we flip two neighbouring spins in the center of the chain,thereby realizing a local quantum quench (see Fig. 1 andrefs. [6, 28]). Making use of our single-site and single-spin resolved detection method [8], we are able to directlyobserve individual magnons and their bound states andidentify both of them by correlation measurements afterletting the system evolve.

The system is described by the one-dimensional two-species single-band Bose-Hubbard Hamiltonian at unityfilling. In the strong coupling limit, where the on-site in-teraction energy is much larger than the tunnelling mat-rix element, this Hamiltonian can be mapped onto theferromagnetic spin-1/2 Heisenberg chain (also known asXXZ spin-1/2 chain) [25, 26]:

H = −Jex∑i

[1

2

(S+i S−i+1 + S−i S

+i+1

)+∆Szi S

zi+1

],

(1)where Jex is the superexchange coupling and ∆ is theanisotropy between the transversal and longitudinal spincoupling. The pseudo-spin operators are defined in termsof creation a†σ,i and annihilation aσ,i operators for a bosonon site i with spin σ =↑, ↓: S+

i = a†↑,ia↓,i, S−i = a†↓,ia↑,i

and Szi = (n↑,i − n↓,i) /2, where the number operatorsnσ,i count the bosons of the respective spin state on eachlattice site. The transversal coupling (the first term ofequation (1)) corresponds to the spin exchange betweentwo neighbouring sites and results in the propagationof spin excitations, or magnons [28, 29]. The longitud-inal coupling describes a nearest-neighbour interactionbetween the spins, which favours ferromagnetic order forJex∆ > 0. This term is the origin of the magnon boundstates: two flipped spins can lower their energy when

located on neighbouring sites. For our scattering para-meters, the Heisenberg interactions are almost isotropic,that is ∆ ' 1 (see Supplementary Information). We notethat the Heisenberg model above can be mapped onto aspinless Fermi system with nearest-neighbour attractiveinteractions via the Jordan-Wigner transformation [30].In the noninteracting case (∆ = 0) magnons thereforebehave as free fermions.

Starting from a general wavefunction for the case oftwo flipped spins of the form

|Ψ〉 =∑

1≤i<j≤N

a(i, j)|i, j〉, (2)

with |i, j〉 = S−i S−j | . . . ↑↑↑↑ . . .〉 and N denoting the

length of the chain, we use the Bethe Ansatz to obtainthe eigenvalues and eigenvectors of the system (see Fig. 2and ref. [19]). The bound states can be identified fromthe corresponding energy spectrum through their separa-tion from the scattering states (see Fig. 2b). The spatialextension of the spin-spin correlations

∑i |a(i, i+ d)|2 for

each bound state can be calculated (see Fig. 2c). Thisanalysis reveals that our initial state | . . . ↑↑↓↓↑↑ . . . 〉 haslarge overlap (∼ 50%) with two-magnon bound states,the rest being shared among free magnon scatteringstates. We therefore expect both bound and free magnondynamics to appear in the subsequent dynamical evolu-tion after flipping two neighbouring spins (see Fig. 1).

The experiment started with the preparation of a two-dimensional degenerate gas of Rubidium-87 atoms in the|↑〉 state in a single antinode of a vertical optical lat-tice (lattice spacing alat = 532 nm). The spin degree offreedom was encoded in two hyperfine states with |↑〉 ≡|F = 1,mF = −1〉 and |↓〉 ≡ |2,−2〉. By ramping up onehorizontal lattice, the gas was then split into approxim-ately ten decoupled one-dimensional tubes of comparablelength. The splitting was carried out in 120ms with afinal lattice depth of 30Er, where Er = h2/(8ma2lat) de-

3

-10 0 10 -10 0 10 -10 0 10 -10 0 10 -10 0 10 Lattice site i

Latt

ice

site

j

0

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-1

1

0-10

0

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-10

0

10

-10

0

10

Pi,j

Ci,j

a

b

c

0 ms 40 ms 60 ms 80 ms 120 ms

Figure 3. Spatial correlations after dynamical evolution. a, Measured joint probability distributions Pi,j of the positionof the two spins for different evolution times as indicated. The bound magnon signal and its spreading is visible on the diagonalsj = i±1 (arrow). Color scales are normalized for each image to the measured peak value. b, Corresponding correlation functionsCi,j = Pi,j − PiPj of the measured data. The subtraction of uncorrelated detection events caused by finite temperature effectsand finite preparation fidelity gives better access to the correlation signal of the zero-temperature two-magnon evolution. Forexample, anti-bunching for the free magnons becomes visible, which is reflected in the outward propagating signal along theorthogonal diagonal. c, Numerical results for the correlations using exact diagonalization. Color scales are normalized analogto a. Note that the symmetry around the j = i diagonal in all plots is given by construction.

notes the recoil energy and m is the atomic mass. Sim-ultaneously, the lattice along the tubes was increased toV = 20Er, driving the system into the Mott insulatingphase. In the next step, we applied a microwave drivenspin flip to the state |↓〉 of two neighbouring atoms at thecenter of each chain. For this spin flip, we used a line-shaped laser beam generated with a spatial light modu-lator that selectively shifted the addressed sites in reson-ance with the microwave radiation [28]. The addressinglight was chosen at a wavelength and polarization suchthat the |↑〉 states were unaffected while the |↓〉 stateswere lowered in energy, and thus pinned at their posi-tions. We then ramped down the lattice along the tubesto V = 10Er in 50ms and subsequently switched off theaddressing beam within 1ms. This marked the startingpoint of the dynamics. At this final lattice depth, thedynamics is sufficiently fast (Jex/~ = 54Hz) comparedto the typical heating time of several 100ms. After avariable evolution time, we rapidly ramped up all lat-tices to approximately 80Er in order to freeze out thedynamics. For state selective detection, we applied a mi-crowave sweep to invert the spin population followed bya resonant laser pulse on the closed cycling transition inorder to push out the |2,−2〉 majority component. Wefinally detected the atoms originally in the |↓〉 state (nowmapped to the remaining |1,−1〉 state) with single-siteresolution [8].

We analysed the extracted atom positions in terms of

a joint probability Pi,j to simultaneously detect atoms onlattice sites i and j along the tubes. Only data sets withexactly two atoms per tube were included. Approxim-ately 50% of the data are discarded through this process,mainly due to the finite spin flip fidelity. In Fig. 3a, weshow the resulting probability distributions. The boundstate population is directly reflected in the strong signalalong the diagonals j = i± 1. The spread along this dir-ection increases with evolution time, which is a signaturefor the correlated motion of the spin pair forming thebound state. To subtract uncorrelated detection eventscaused by finite temperature effects and finite prepara-tion fidelity (see Supplementary Information), we calcu-late the correlation function Ci,j = Pi,j − PiPj , wherePi =

∑j Pi,j is the probability to find one atom on site i

(see Fig. 3b). Due to the hard-core constraint, trivialanti-correlations are present for i = j, which we dis-card in the analysis. Next to the strong signal of boundmagnons, a second feature is visible along the orthogonaldiagonal. It corresponds to those free magnon states withwhich the prepared initial state has finite overlap (seeFig. 2). As we show below, these are spins detected atlargest distance from each other given by the maximalfree magnon velocity Jexalat/~. Their anti-bunching be-haviour of propagating in opposite directions can be un-derstood intuitively from the aforementioned mappingof the Heisenberg model to a fermionic Hamiltonian.Numerical results based on exact diagonalization of the

4

Heisenberg chain (assuming zero temperature), shown inFig. 3c, are in remarkable agreement with the experi-mental data. Analog to the free magnon wavefront, theone for bound magnons spreads also at its maximum ve-locity Jexalat/(2~∆) due to a singularity in the probabil-ity density of propagation velocities (see SupplementaryInformation and ref. [6]).

To investigate the dynamics of the magnon boundstates in more detail, we concentrate on the diagonalsj = i±1 in Fig. 3 and analyse the evolution of the normal-ized distribution P ↑↑i = Pi,i+1/

∑j Pj,j+1, that is we use

only data where two atoms on adjacent sites have beendetected. We expect the magnon bound states to spreadas compound objects almost freely across the lattice. Wetherefore extract the width w of the distributions P ↑↑iby fitting the data with Bessel functions of the first kind[Ji (w)]2 (ref. [28]). To measure the propagation velocityof the free magnon excitations, we analyse the correla-tions Cd =

∑i Ci,i+d as a function of distance, shown in

Fig. 4b. Here, the correlation signal at d = 1 is due tothe magnon bound states, while the second positive cor-relation signal, at a distance increasing with evolutiontime, is the free magnon contribution. We determine theposition of the free magnons via Gaussian fits and definethe wavefront as the center plus one Gaussian sigma totake the dispersion into account (see Supplementary In-formation). Figure 4c shows the measured wavefront po-sitions of both the free and bound magnon states versustime. A linear fit yields the velocities vf = 60(3) sites/s

for the free and vb = 26(+2+6−2 ) sites/s for the bound

magnons, where the first uncertainty of vb is due to thefit and the second one takes a systematic underestima-tion of the bound state velocity into account (see Sup-plementary Information). The ratio of the two velocitiesis vf/vb = 2.3(+0.2

−0.7), consistent with the predicted valuevf/vb = 2∆ = 2 for the isotropic case [6].

Above we analysed the data in the context of the iso-tropic Heisenberg chain and found good agreement withthe theoretical predictions. However, the experiment wasnot carried out at zero temperature, resulting in a fi-nite density of particle or hole excitations (approxim-ately 10%) in the atomic chain. We expect that coup-ling of these thermal excitations to the magnon boundstates leads to a finite lifetime. To extract this lifetime,we plot the probability to find two atoms on adjacentsites (

∑i Pi,i+1) versus time in Fig. 5. For comparison,

we show the zero temperature prediction of the Heisen-berg chain for the isotropic experimental case and forthe case ∆ = 0, for which no bound states exist. Here,we take the finite preparation fidelity (87(1)%) for flip-ping the spin of two atoms at adjacent sites into ac-count (see Supplementary Information). For long evol-ution times (inset), the probability approaches zero forthe non-interacting case (∆ = 0), while it reaches a fi-nite value of 38% in the isotropic model. This value issmaller than the overlap between our initial state and themagnon bound states because of the finite extension ofthe bound states beyond neighbouring sites (see Fig. 2c).

0

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0 50 1000

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Time t (ms)

Wav

efro

nt p

ositi

on (l

attic

e si

tes)

a

c

0 msb

40 ms

60 ms

80 ms

120 ms

~

Figure 4. Spreading wavefront velocity of bound andfree magnons. a, Bound state probability distributions P ↑↑ifor different evolution times. The green bars show the ex-perimental data. We extract the widths via Bessel functionfits to the data (solid green lines). b, Propagation of the freemagnons. The extracted correlation functions Cd versus dis-tance d are plotted for the same evolution times as used in a(blue circles). The signal at d = 1 is due to the bound stateswhile the outwards moving peak stems from free magnons.The position and width of this peak are captured by Gaussianfits (dark blue lines). c, Comparing the propagation velocit-ies. Linear regression of the extracted widths for the boundstates (green) yields a velocity of 26(+2+6

−2 ) sites/s comparedto 60(3) sites/s for the wavefront of the free magnons. Errorbars represent one s.e. of the mean.

5

0 50 1000

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Time t (ms)

Pro

bab

ility

0 100 200 3000

0.5

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Time t (ms)

Pro

bab

ility

Figure 5. Stability of the bound state. Probability tofind two spins at neighbouring sites as a function of the evol-ution time. The green circles are the experimental data andstatistical error bars are smaller than the circles. We show nu-merical calculations (exact diagonalization) for the isotropic∆ = 1 case (green shaded area) and for ∆ = 0 (blue shadedarea), taking the preparation fidelity of 87% and the resultinguncertainty into account. The darker green line is a fit basedon the isotropic numerical result multiplied by an exponen-tial decay. Inset: Numerical prediction for longer evolutiontime and without correcting for the preparation fidelity. Thenearest-neighbour probability approaches zero in the ∆ = 0case (blue line) while it converges to a finite value of 38% for∆ = 1 (green line).

We find the experimental data to lie in between the twoscenarios (see Fig. 5). We fit the data with a heuristicmodel, which assumes the numerical prediction of theisotropic Heisenberg chain multiplied by an exponentialdecay. The extracted decay time of the bound magnonstate is τ = 210(20)ms, where the uncertainty includesboth the fitting error and the uncertainty in the numer-ical prediction. We believe this decay time to be de-

termined by both thermal density fluctuations that arepresent already initially and technical heating during theevolution dynamics. It remains an interesting challengefor future theory work to explain the lifetime due to theinteraction of bound magnons with density fluctuationson the spin chain.

In conclusion, we deterministically realized a localquantum quench in a Heisenberg spin chain. We micro-scopically tracked the resulting dynamics and directly ob-served distinctive magnon bound state correlations andtheir evolution with time. This is the first realization ofan interacting quantum walk in a magnetic spin chain.From the quantum simulation perspective, our resultsalso constitute the first observation of interacting spinsin optical lattices. Future studies might address the ques-tion of the stability of magnon bound states in an envir-onment containing thermal as well as stronger quantumfluctuations or even the binding of two impurities in a su-perfluid environment, where one expects a “bi-polaron” toform. Other interesting extensions would be the study ofuniversal Efimov physics using three magnons [31]. Thereported results also pave the way towards the determin-istic microscopic engineering of complex magnetic many-body states and the study of magnetic correlations innon-equilibrium situations.

ACKNOWLEDGEMENTS

We thank H. G. Evertz, M. Haque, J.-S. Caux andW. Zwerger for discussions. We thank J. Zeiher forproofreading the manuscript. This work was suppor-ted by MPG, DFG, EU (NAMEQUAM, AQUTE, MarieCurie Fellowship to M.C.), and JSPS (Postdoctoral Fel-lowship for Research Abroad to T.F.).

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SUPPLEMENTARY INFORMATION

I. EXPERIMENTAL PROCEDURE

The general experimental procedure closely followedthe one published in [28]. Additionally, for the long evol-ution time (120ms) experiments presented here, we useda vertically propagating, blue-detuned (∼ 667 nm) beamto reduce the harmonic confinement in the horizontalplane. This enabled us to create larger Mott insulatingplateaus with unity filling and thereby avoid reflectionsof the magnons from the boundaries of the atomic spinchains. The typical length of the chains was 20 sites withthe deconfinement and 13 without. We generated thisdeconfinement beam by a broadband superluminescentdiode in order to avoid possible interference of the beamwith reflections from the vacuum window. The beam wassuccessively amplified by two tapered amplifiers.

II. EXTRACTION OF THE WAVEFRONTVELOCITY FROM THE FITS

The validity of our method to extract the wavefrontvelocities was checked by analysing the results obtainedfrom simulated data. To extract the velocity of the boundmagnons, we used fits with Bessel functions. The Besselfunction is not the exact distribution to describe the evol-

0 100 200 300 4000

5

10

Time t (ms)

Wid

th (l

attic

e si

tes)0

0.540 ms

0

0.2

Prob

abilit

y 80 ms

−5 0 50

0.2 120 ms

Position (lattice sites)

a b

Figure S1. Propagation of bound magnons. a Calculatedprobability distribution P ↑↑i (black lines) together with theBessel function fit (green lines) for different evolution times(40, 80, and 120ms). The red vertical lines show the widthextracted from the fit. b Determination of the velocity. Thered line is the extracted width from the Bessel function fitversus the evolution time. The black line corresponds to theexpected maximum velocity (Jexalat/2~).

ution of the bound magnons that we prepare, but, as weshow below, it is suitable to capture the position of thewavefront in the distributions (see Fig. S1a). The po-sitions extracted from the simulated data for differenttimes are shown in Fig. S1b. In the long-time average,the resulting velocity agrees with the theoretical predic-tion (Jexalat/2~) for the bound magnons with ∆ = 1.For shorter, experimentally accessible times, the fits canunderestimate the velocity up to 20%. This is the reasonfor the systematic error on vb given in the main text.

The free magnon velocity is extracted from the timeevolution of the position and width of the outwardmoving peak in the correlation functions Cd by usingGaussian fits: A exp

[− (d− c)2 /s2

]. To focus on the

propagating peak, we exclude from the fit both the pointsat d = 1, which show the strong positive signal of thebound state, and the points with negative values. InFig. S2, the peak position c and the wavefront positionc+ s are plotted. The wavefront velocity yields twice thevelocity of a single free magnon since two free magnonspropagate separately in opposite directions. The devi-ation of the extracted velocity from the maximal velocityJexalat/~ of the single free magnon is found to be only3%.

III. PREPARATION FIDELITY

The preparation fidelity for flipping the spin of twoatoms on neighbouring sites is estimated to be 87%. Thisvalue is limited by two factors. First, the spin-flippingprocess might have addressed two spins initially separ-ated by a larger distance. Second, the flipping processmight have succeeded for one atom only, while the secondatom observed is one from the majority component thatwas not removed during the push-out process because of

7

0 100 200 300 4000

20

40

Time t (ms)

Dist

ance

(lat

tice

site

s)

0 5 10 15−0.05

0

0.05

0.1

Distance d (lattice sites)C

d~

Figure S2. Propagation of free magnons. The blue andred lines show the Gaussian center c and the center plus thewidth c+s. Note that the center moves slower than the max-imum wavefront velocity. The black line, almost overlappingwith the red line, corresponds to twice the expected singlemagnon maximum velocity (2Jexalat/~). The inset shows anexample of the Gaussian fit (green line). The gray circles rep-resent the calculated correlation function Cd for the evolutiontime of 80ms. The blue shade highlights the region used forthe fit.

its finite efficiency (98–99%). These two effects have dif-ferent contributions on the probability to find two atomson neighbouring sites after the evolution time. In the firstcase, the effect can be calculated by solving the dynam-ics with the measured initial distributions (Pi,j (t = 0)).For the second case, we can assume that falsely measuredatoms are uniformly distributed over the chain (they weregenerated after the dynamics). The calculated probabil-ities shown in Fig. 5 take both these effects into account.The width of the shaded region displayed in Fig. 5 is dueto the uncertainty of the ratio between the two effects.

IV. PARAMETERS OF THE HEISENBERGMODEL

The superexchange coupling Jex and the anisotropy ∆are given by [25, 26, 32, 33]

Jex =4J↑J↓U↑↓

, (3)

Jex∆ =

(4J2↑

U↑↑+

4J2↓

U↓↓− 2

J2↑ + J2

↓

U↑↓

). (4)

Here Jσ are tunnelling matrix elements for a boson withspin σ =↑, ↓ and Uσσ′ are the on-site interaction energiesbetween bosons with spin σ and σ′. In our case, thetunnelling matrix elements are spin independent (J↑ =J↓ = J), and the interaction energies are almost the same(U↑↑ ≈ U↓↓ ≈ U↑↓ = U). The anisotropy is ∆ = 0.986for our ratios of the interaction energies (U↑↑ : U↓↓ :U↑↓ = 100.4 : 99.0 : 99.0) that follow from the respectivescattering lengths [34, 35].

V. NUMERICAL CALCULATIONS USINGEXACT DIAGONALIZATION

We calculated the dynamics of the effective Heisenbergchain by directly diagonalizing the Hamiltonian. Sincethe number of magnons, or the total magnetization, isconserved, we considered only the Hilbert space contain-ing two magnons. The numerical calculation was donefor a superexchange coupling of Jex/~ = 54Hz, whichhas been estimated from Jex = 4J2/U . Here, the tun-nelling matrix element J was obtained from the observa-tion of the quantum walk of a single free atom, as shownin [27]. The on-site interaction energy U was obtainedfrom an ab initio band-structure calculation using lat-tice depths, that were calibrated from amplitude mod-ulation spectroscopy. For the simulations we used openboundary conditions and lattice sizes of 61 or 81 sites,depending on the evolution time, making sure that themagnons remain sufficiently far away from the edges toavoid spurious reflections.

VI. DENSITY OF STATES AND INITIALDISTRIBUTION OF GROUP VELOCITIES

Our initially prepared state |Ψi〉 can be decomposedinto two parts:

|Ψi〉 = |Ψb〉+ |Ψf 〉 ,

that describe the overlap with magnon bound states(|Ψb〉) and free magnon scattering states (|Ψf 〉). Thebound magnon part is expanded in bound magnon ei-genstates |ψk〉 with center-of-mass wave-vector k as

|Ψb〉 ∝∫k

dk〈ψk|Ψb〉 |ψk〉 . (5)

The probability density P (v) to find our initial state inmagnon bound states that have a group velocity v canbe written as:

P (v) ∝ 〈Ψb|∫k

dk δ(v − vg(k)) |ψk〉 〈ψk| |Ψb〉

=

∫k

dk δ(v − vg(k))|〈ψk|Ψb〉|2

=1√

v2max − v2∑kv

|〈ψkv |Ψb〉|2. (6)

The group velocity is vg(k) = dεdk = J

2 sin(k) with thebound state dispersion ε = J

2 (1 − cos(k)) [9, 19]. Themaximum group velocity is vmax = J/2. The sum in thelast line runs over all wave-vectors kv that yield a partic-ular group velocity vg(kv) = v. The quantity |〈ψk|Ψb〉|2describes the probability to find the initial state in amagnon bound state with k and is plotted in Fig. 2a.It is nonzero for the values k = ±π/2 that yield the max-imum group velocity vg(k) = ±vmax. Therefore, P (v)shows a divergence for v = ±vmax.

8

The quantity∫kdk δ(k−vg(k)) ∝ 1√

v2max−v2essentially

describes the density of states for a particular group ve-

locity v [6]. The singularity in this density of states isthe origin of the singularity in P (v).