1  

Coherent multi‐flavour spin dynamics  in a fermionic 

quantum gas  

Jasper  Simon  Krauser1,2,  Jannes  Heinze1,2,  Nick  Fläschner1,  Sören  Götze1,  Christoph Becker1  and Klaus Sengstock1,*  

 

1 Institut für Laserphysik, Universität Hamburg, 22761 Hamburg, Germany 2 These authors contributed equally to this work. 

* e‐mail: klaus.sengstock@physnet.uni‐hamburg.de 

 

 

 

Microscopicspininteractionprocessesarefundamentalforglobalstaticanddynamicalmagneticpropertiesofmany‐body systems.Quantumgasesaspureandwell isolated systemsoffer intriguingpossibilities tostudybasicmagneticprocessesincludingnon‐equilibriumdynamics.Here,wereportontherealizationofawell‐controlledfermionicspinorgasinanopticallatticewithtunableeffectivespinrangingfrom1/2to9/2.Weobservelong‐livedintrinsicspinoscillationsandinvestigatethetransitionfromtwo‐bodytomany‐bodydynamics. The latter results in a spin‐interaction drivenmelting of a band insulator. Via an externalmagneticfieldwecontrolthesystem’sdimensionalityandtunethespinoscillationsinandoutofresonance.Ourresultsopennewroutestostudyquantummagnetismoffermionicparticlesbeyondconventionalspin1/2systems.

Magnetism plays a key role for the fundamentalunderstandingofmaterialsandinmoderntechnologies.A major focus is the understanding of quantumpropertiesofmagnetism,whichhave theirorigin in theunderlying microscopic processes between elementaryspins. However, it remains challenging to derivemacroscopicmagneticphenomenasuchasthefractionalquantum Hall effect or the formation of spin liquidsdirectly from the microscopic level. Here, scalable andcontrollable model systems allow to bridge this gap.Alongside a few tunable magnetic condensed mattersystems1,2, atomic physics experiments came into focusin the last years due to their unrivaled control over allexperimental parameters and nearly perfect isolationfrom environmental influences. Ion chains3,4 as well asbosonic quantumgases, either harmonically trapped orconfined in optical lattices, have produced strikingresultstowardsthesimulationofclassicalandquantummagnetism5‐14. Lacking Pauli blocking, however, theseexperiments did not catch the fermionic character ofelectronicmagnetism.Inthisdirection,thepossibilityofitinerant ferromagnetism15‐18 and spin transport19 inbulk fermionic quantum gases has been recentlydiscussed.Toresembleelectronspinsinrealsolidsevenbetter, it is desirable to have a fully controllablefermionic lattice quantum simulator. In addition, withfermionic quantum gases, completely new systems canbe realized, e.g. high‐spin systems20‐22 1 / 2s , forwhich complex quantum phases are theoreticallypredicted: This includes unconventional BCSsuperfluids23,24, QCD‐like color superfluidity25, SU(N)‐magnetism with alkaline earth atoms26‐29 and furthermulti‐flavoursystems30‐33.

In this article, we demonstrate the first experimentalrealizationofawell‐controlledfermionicspinorgaswithinteraction‐driven spin oscillations. By properlychoosing the initial spin states we can change theeffective lengthoftheatomicspinfrom1/2to9/2.Thecontrol over the magnetic field allows to initialize andstopspindynamicsandtoselectthenumberofinvolvedlevels. We extract the microscopic interactionparameters and find excellent agreement with a two‐particlemodelincludingallspin‐dependentinteractions.Bytuningthedepthoftheopticallattice,weinvestigatedthetransitionfromon‐sitedominatedtoquantummany‐bodyspindynamics,wherespinsdiffusethroughoutthelattice producing highly entangled states. For this case,weobserveanewformofmeltingofabandinsulator.

Principles of fermionic spin‐changing collisions 

We perform our experiments employing a fermionicspinor gas of 40K atoms in the 9 / 2f manifold.Initially, we prepare an equal mixture of two differentspin stateswith typically 54 10 N atoms in anopticallattice, forming a large‐scale band insulator34‐36(diameter about 50 lattice sites) as depicted in Fig. 1a(fordetailsseeMethods).

In order to shed light on the microscopic collisionprocesses, we first consider two fermionic atoms indifferent spin states 1m and 2m in the lowest spatialmode on an isolated lattice site, forming a two‐particlestate 1 2 1 2 2 1, 1 / 2 m m m m m m . For spin 1/2particleswith solely two hyperfine states, analogous to

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Figure  1  |  Principles  of  fermionic  lattice  spin  dynamics.  a,  Sketch  of  atoms  in  an  optical  lattice  with  harmonic confinement. A band  insulator  (with  two particles per site)  is  formed  in  the center within  the  lowest spatial state. b, Microscopic collision processes on an isolated lattice site. Spin states for spin 5/2 particles are exemplarily shown (left). Colored  boxes  represent  possible  two‐particle  states  (right).  These  states  are  coupled  by  spin‐changing  collisions, forming one‐level, two‐level and three‐level systems. Magnetization conservation and Pauli blocking restrict the number of involved two‐particle states. States represented by grey boxes are forbidden due to Pauli blocking. 

 

electrons in solids, there is only one possible two‐particle state. For high‐spin particles 1 / 2s , spin‐changing collisions37‐40 can transfer the atoms intonewstates 1 2, m m at lowmagnetic field.For increasingspinlength, the number of involved two‐particle statestypicallyincreases.Spin‐changingcollisionsoffermionicatoms have to satisfy two physical restrictions:conservation of magnetization 1 2 1 2 M m m m m andPauli blocking.As an example, consider the caseofspin5/2asshowninFig.1b.Dependingonthe initiallychosen spin states, eithera three‐level, a two‐leveloraone‐level systemcanbe realized.This is a consequenceof conservation of magnetization and Pauli blockingwhich restrict the number of allowed final two‐particlestates.Inparticular,allstateswith 1 2 m m areforbiddendue to the Pauli exclusion principle. Employing 40K in

9 / 2f with its ten spin states allows toexperimentallyvarythenumberofinvolvedtwo‐particlestatesbetweenoneandfive.

However, when finite tunneling couples neighboringlattice sites, these restrictionsare significantly lowered.AsdepictedinFig.2,formerlyforbiddenfinalspinstatescan now be generated, leading to much more complexdynamics. For simplicity, we limit the explanation tothree characteristic situations. First, consider twocoupledlatticesitesasshowninFig.2aandb,whichareinitially filledwith the same two‐particle state , ,forming a band insulator (top). Due to Pauli blocking,tunneling is forbiddenandthus thenumberofparticlespersiteisfixed.Spindynamicsoneachindividuallatticesite leads to a time‐dependent occupation of the two‐particle state , and hence a finite probability thatdifferenttwo‐particlestatesarerealizedonneighboringsites (upper middle). In this case, Pauli blocking nolonger prevents tunneling. The dominating processesnowdependonthelatticedepth:ForshallowlatticesasshowninFig.2a,thesystembehaveslikeametal.Single

atoms can tunnel and induce local density fluctuations,e.g. one site contains one and the other site threeparticles (lower middle). This opens additional spinchannels on the triply occupied site and leads to theformationofnewspinstates(bottom).Fordeeplattices,however,when the tunnelingenergy is small comparedto theon‐siteenergy, thesystem is inaMott‐insulatingstate with two particles per site and local densityfluctuations are strongly suppressed. Further two‐particle states can be realized only via super‐exchangeprocesses, as sketched in Fig. 2b (lower middle).Afterwards, theformationofnewspinstates ispossible(bottom). Note, that in contrast to the situation ofmetallic tunneling, only the local magnetizationfluctuateswhile the local density stays constant. In Fig.2c, the scenario at edges or defects of the initial bandinsulator is shown, i.e. when a singly occupied site isadjacent to a doubly occupied site (top). After a spin‐changing collision on the doubly occupied site (uppermiddle), a tunneling process can form another two‐particlestateonthe initiallysinglyoccupiedsite(lowermiddle).Again, this leads to the realizationofnewspinstates(bottom).Inalloftheabovementionedsituations,the local magnetization is no longer conserved, whichallows for the occupation of new spin states, whichwould be forbidden on perfectly isolated lattice sites.The combination of finite tunneling and spin‐changingcollisions thus interestingly leads to a continuousmeltingofthebandinsulator.

Spin dynamics in deep optical lattices 

In a first set of experiments, we studied the timeevolution of exclusively doubly occupied sites (seeMethods) in the initial two‐particle state

1 2, 1 / 2,9 / 2m m inthelow‐tunnelingregime.Asonecentral result of this work, we observe long‐lived

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Figure 2 | Simplified sketch of processes originating from the combination of spin‐changing collisions and tunneling. Shown are two lattice sites coupled by finite tunneling. Three characteristic processes are distinguished, each of them allowing  for  the  generation  of  new  spin  states, which  are  forbidden  on  perfectly  isolated  lattice  sites.  a, Metallic tunneling  occurs  in  the  high‐tunneling  regime.  Starting  from  uniform  filling  of  each  lattice  site with  the  same  two‐particle  state , ,  density  fluctuations  are  induced.  b,  For  large  interaction  strengths,  the  system  is  in  a Mott insulating  state with  suppressed density  fluctuations.  Tunneling  is only possible  via  super‐exchange.  c,  Tunneling  at edges and defects can always occur, independent of the ratio between tunneling and interaction energy. 

 

coherentspinoscillationsoffermionicatomsforthefirsttime. As outlined above, the state 1 / 2,9 / 2 is onlycoupled to 3 / 2,7 / 2 . Since the state 5 / 2,5 / 2 isforbidden due to Pauli blocking, this constitutes a two‐levelsystem.Weobserveoscillationsovermanyperiodsfor more than 250 ms with an amplitude in excess of70% asshowninFig.3a.

Two energy scales are important for the spin‐changingcollisions as depicted in Fig. 3b: the difference ininteractionenergyand thedifference inZeemanenergybetween the individual two‐particle states. At largemagnetic fields, theatomsarepinnedtoa fixedspinbythe Zeeman energy. For smallmagnetic fields, the spininteractionbecomesrelevantandspindynamicsoccurs.The new eigenstates become quantum mechanicalsuperpositionsofthenon‐interactingtwo‐particlestates.AresonantfeatureappearsinthespinoscillationswhenZeeman energy and spin‐dependent interaction energyareequal41‐43.

Fortheexperiments,wefirstpreparedtheatomsintheZeeman‐dominatedregime.Quenchingthemagneticfieldto a lower value expB initializes the observed spinoscillations.We investigated the crossoverbetween theinteraction‐dominated and the polarized regime bystudyingspindynamicsatdifferent expB .Extractingboththe frequency and the amplitude of the oscillation, aclearRabi‐resonancefeatureasexpectedforatwo‐levelsystem is observed anddepicted in Fig. 3c.We identifythe resonance position to be at exp 0.175 0.006 G B ,where the observed amplitude possesses a maximumwhile the frequency is minimal. For higher magneticfields, the system approaches the polarized regime andthe spin oscillations vanish.We compare our data to atwo‐particlemodel and find excellent agreement. Sincethe oscillation frequency at resonance is related to thedifference of the scattering lengths Fa in the respectivescatteringchannelswithtotalspin 8F and 6F ,ourmeasurements provide a high precision test formolecularcalculationsofscatteringlengths.Wecompareour measured value of 8 6 B2.26 a a a to theoretical

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Figure 3 | Coherent fermionic spin dynamics. a, High contrast spin oscillations of a fermionic quantum gas. Plotted are the relative populations  ( )n m  of different spin states  m  as a function of time after initialization. Solid lines are fits to the  data,  from which we  extract  oscillation  amplitude  and  frequency  (for  details  see Methods).  The  lattice  depth  is 35 rE  (with  rE  the  lattice recoil energy) and the magnetic field  is  exp 0.186 GB . b, Sketch of the crossover from the interaction‐dominated  to  the  Zeeman  regime.  The  spin  resonance  appears  at  the  crossing  of  the  two  solid  lines.  c, Frequency and amplitude deduced  from  fits to spin oscillations as shown  in Fig. 3a  for different magnetic  fields. Error bars correspond solely to  fit errors, representing two standard deviations. The red and blue curves result  from a two‐body calculation. The widths of the curves are given by the  lattice depth uncertainty. The  inset shows a zoom  into the spin resonance area. 

 

predictionsusing coupled‐channel calculations and findverygoodagreement.Beyondthesepureon‐siteeffects,we observe a slow appearance of the spin states

5 / 2mand 1 / 2 ,whichwouldbeblockedatzero

tunneling. In the regime of deep lattices, only super‐exchange tunneling and tunneling at edges and defectscan occur. We attribute the appearance of new spinstates to the latter, as the timescale for super‐exchangetunneling is much too long for our experimentalparameters ( 1 10 s ).We also observe a damping ofthe coherent spin oscillations at a time scale of

1 96.4 4.6 ms ,whichwebelievetobemostlyduetoedge and defect effects, but can also be influenced bymagnetic field noise and gradients, which we haveestimated,however,tobesmall.

In a second set of experiments,we realized a high‐spinsystembypreparingthestate 1 / 2, 1 / 2 (seeMethods)which couples to the states 3 / 2, 3 / 2 , 5 / 2, 5 / 2 ,7 / 2, 7 / 2 , and 9 / 2, 9 / 2 . This constitutes aneffective five‐level system. In this situation,weobservecomplex multi‐flavour quantum dynamics including allfive two‐particle states now governed by up to ten

frequencies.ThisbecomesapparentinFig.4a,wherethecorrespondingbeat‐notesofthesignalareclearlyvisible.

We have investigated the spin dynamics of this multi‐flavoursystematdifferentmagnetic fieldstrengthsandfind that the amount of contributing oscillationfrequencies increases with decreasing magnetic field(Fig.4c).This is ingoodagreementwithournumericalsimulation of the complete multi‐flavour dynamics,which reveals that the externalmagnetic field providesfull control over the effective dimensionality of thesystem in spin space. The number of eigenstatessignificantly overlapping with the initial state dependscruciallyonthemagneticfieldasshowninFig.4b.Eachobservedfrequencycanbeassignedtothesuperpositionof two eigenstates. Typical Fourier spectra for theeffective four‐level system at a magnetic field of

exp 0.372 GB andfortheeffectivetwo‐levelsystematamagnetic field of exp 1.014 GB are exemplarily showninFig.4d.Inthismulti‐flavourspinsystemwithitsrichdynamics, the observed damping is similar to the two‐levelcase.

 

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Figure 4 | Coherent multi‐flavour  spin dynamics of  fermionic atoms. a, Coherent  spin oscillations with  five  two‐particle  states  involved.  Plotted  are  the  relative  populations  ( )n m  of  different  spin  states  m  versus  time.  The lattice  depth  is  25 rE  and  exp 0.372 GB .  At  these  parameters,  four  levels  effectively  participate  in  the  spin evolution, leading to pronounced beat notes in the signal. Solid lines are fits to the data. b, Calculated amplitudes of the overlap  integral 

21 / 2, 1 / 2n  between the  initial state and all five two‐particle eigenstates  1,...,5n  versus 

the magnetic field. At high magnetic fields, the ground state coincides with the initial state. For lower magnetic fields, an  increasing  number  of  eigenstates  overlaps  and  thus  contributes  to  spin  oscillations.  c, Observed  frequencies, obtained  by  discrete  Fourier  analysis  of  spin  oscillation  data  like  in  Fig.  4a  for  different  magnetic  fields  with corresponding errors. Curves show the numerically calculated frequencies. Their widths are given by uncertainties in lattice depth. Shading of the curves is proportional to the expected strength of the transition from Fig. 4b. For lower magnetic field, the number of observed frequencies  increases, fully consistent with calculations. d, Typical discrete Fourier  spectra  for  the  populations  (1 / 2) ( 1 / 2) n n  (black  lines)  and  (3 / 2) ( 3 / 2) n n  (red  lines)  at magnetic fields  exp 0.372 GB  (left) and  exp 1.014 GB  (right). 

Crossover to the many‐body regime 

Theexperimentspresentedso farhavebeenperformedin the low‐tunneling limit, in which spin dynamics isstrongly dominated by local interactions on individuallattice sites. It is now particularly interesting toinvestigate the crossover to the tunneling‐dominatedregime, where the system approaches fermionic bulkspindynamics.Inordertoinvestigatethiscrossover,weagain start in the 1 / 2,9 / 2 effective two‐level bandinsulating system and then decrease the lattice depthalong one spatial dimension to a final value LV . As astriking result, high contrast spin oscillations with afrequency consistent with our pure two‐particle modelarealsoobservedinthisregime(seeFig.5a).Note,thattheon‐sitedensityandthusthefrequencychangeswithlattice depth as expected (see Fig. 5b). Besides the

change in frequency,weobserve an enhanceddampingwith decreasing lattice depth up to a completedisappearance of the spin oscillations for very shallowlattices. At the same time, the population of the spinstates 5 / 2 and 1 / 2

increases. Considering the

three tunneling scenarios presented in Fig. 2, wecompare the expected timescales of these processes tothe measured damping constants. For super‐exchangetunneling the expected timescale is proportional to2 2/h J ,with J the single‐particle tunneling energy in

the weak lattice direction and the spin oscillationfrequency.Fortunnelingatedges,theexpecteddampingtimedependson thepropagation timepersitewhich ison the order of /h J 44. A decrease of the global spinoscillation amplitude to1 / e would require a time of( 1) / (4 )hd e Je with d the length of the quasi 1d tube

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Figure 5 | Transition from on‐site to many‐body spin dynamics.  a,  Relative  populations  ( )n m  of  different spin  states  m  as a  function of  time. Solid  lines are fits  to  the  data,  from  which  we  extract  oscillation amplitude  and  frequency.  The  magnetic  field  is 

exp 0.186 GB .  The  lattice  depth  LV  along  one dimension is indicated on the right hand side, for the other two dimensions the  lattice depth  is  35 rE . The transition  from  single‐site  dominated  high  contrast oscillations  to  diffusing  spins  in  the  high‐tunneling regime is clearly visible with decreasing lattice depth. In  the  latter  case,  spin  oscillations  in  the  initial channels  are  strongly  damped,  while  at  the  same time  tunneling  leads  to  the  occupation  of  new  spin states.  b,  Extracted  oscillation  frequencies  from  Fig. 5a compared to a two‐particle calculation (solid line). Errorbars  representing  two  standard  deviations  lie within the datapoints. The width of the curve is given by the lattice depth uncertainty. 

 

alongtheweaklatticedirection.Especiallyatlowlatticedepths,thedampingcannotbeexplainedsolelybytheseprocesses. Since the observed damping rate is on theorderofthetunnelingrate J ,weattributetheenhanceddampingaswell as the increasedpopulationof initiallyforbiddenspinstatestometallictunnelinginthecoreofthe system as described in Fig. 2a. The combination oftunneling and spin‐changing collisions dynamicallyopens up new spin channels and provides a pathwaytowardsamulti‐flavourFermisystem,whichisnolongerlimitedtotwospinsperlatticesite.Inthissituation,weexpectincreasinglocalfluctuationsofthemagnetization

incontrast to the fixed localmagnetization inthe initialband insulator. Regarding the many‐body character ofthesystem,weinterpretthisprocessasanintrinsicspin‐collisionallyinducedmeltingofthebandinsulator.

In conclusion, we have thoroughly studied a high‐spinfermionic spinor gas in an optical lattice. Our resultsrevealthecomplexdynamicsofthesesystemsgovernedby the interplay between spin‐dependent interactions,Zeeman energy, Pauli blocking and tunneling. For thefirst time, we have observed long‐lived coherent spinoscillations of fermionic atoms.We have demonstratedtheability to fully control these spinoscillations,whichallowedustoidentifyaclearspinresonancefeatureandtoextractmicroscopic interactionparameterswithhighaccuracy. Our results show excellent agreement with atwo‐bodycalculationindeepopticallattices.Hence,theyprovideapromisingstartingpointtowardsseveralnewscientific directions: Fermionic spinor gases in opticallattices are ideally suited to studymagneticmany‐bodyproperties on a microscopic level, e.g. spin transport,spin‐charge separation, spin waves or spin‐orbitcoupling. In the strongly‐correlated regime, spindynamics could serve as an intriguing tool to directlyprobe super‐exchange processes. The tunability of thespin length opens up the route to further investigatehigh‐spinmagnetismincomparisontoconventionalspin1/2 electrons and shows the ability to accessfundamentallynewmulti‐flavorspinsystems.Especiallyfor high‐spin systems, our experiments also pave theway for coherent control of entanglement allowingstudyingquantumcoherenceproperties.

Acknowledgments 

WeacknowledgestimulatingdiscussionswithE.Demler,A.Eckardt,T.‐L.Ho,M.LewensteinandM.W.Zwierleinand thank T. Hanna and L. Cook for providing uscalculatedvaluesofthescatteringlengths.ThisworkhasbeenfundedbyDFGgrantFOR801.

Methods 

Preparationofafermionicquantumgasinanopticallattice. By sympathetic cooling with 87Rb, about

62 10 N spin‐polarized 40K atoms are cooled toquantumdegeneracy inamagnetic trap.Thepotassiumatoms are transferred adiabatically to a crossed opticaldipole trap operated at 812 nm . After switching off themagnetictrap,aseriesofrf‐pulsesandsweepsisappliedtoprepareanequalmixtureoftwohyperfinestates.Theatoms are evaporatively cooled by exponentiallyramping down the trap laser intensity within 2 s . Theresultingspinmixturenowconsistsofabout 54 10 N atoms at typical temperatures between F0.15T and

F0.25T . For the presented experiments, the magneticfield after the evaporation is 7 G for the mixture1 / 2 1 / 2 and 3G for 3 / 2 7 / 2 . To increasethe chemical potential, we ramp up the depth of the

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optical dipole trap in 50 ms and obtain final trappingfrequencies of 2 125 Hz x in the vertical directionand 2 41Hz y and 2 32 Hz z inthehorizontalplane. Afterwards,we adiabatically rampup a 3d cubicoptical lattice in150 ms . It consists of threeorthogonalretro‐reflected laser beams at 1030 nm with a 21 / eradiusof 200 μm ,detunedwithrespecttoeachotherbyseveral10MHz . The uncertainty of our lattice depthcalibration is 2% . The lattice depth and chemicalpotentialarechosensuchthatabandinsulatorisformedinthecenterofthesystemwhichcontainsabout 40% ofthe atoms which we infer from the measured doubleoccupancy.Notethatonlyatlowenoughtemperaturesasufficiently pure two‐component band insulator isformed in the core of the lattice, which is crucial toobserve high contrast spin oscillations. After thepreparation procedure, we initialize spin dynamics byrapidlydecreasingthemagneticfieldtoavalue expB withanaccuracyofabout 0.003G .

Microwavetransferofsinglyoccupiedsites.Atomsonsingly occupied sites do not participate in the spin‐changing dynamics and only contribute as a constantoffset to the signal. We circumvent this by applyingmicrowavepulsestotransferallsinglyoccupiedsitestothe 7 / 2f hyperfine manifold, where the atoms arenot resonant with our detection light (the 7 / 2f statesareseparatedbyafrequencyof1.3GHz fromthe

9 / 2f states). For this, we prepare the atoms in thestate 3 / 2,7 / 2 andletthesystemevolveonresonancefor one quarter of a spin‐oscillation period. Now, thedoublyoccupiedsitesareinthestate 1 / 2,9 / 2 .Atthismoment we stop the dynamics, by ramping up themagnetic field to 1.69 GB . Then, we apply fourlinearly polarized microwave pulses with a lengthbetween 35μs and 50 μs and transfer all atoms onsingly occupied sites in the remaining spin states (1 / 2 , 3 / 2 , 5 / 2 and 7 / 2 ) into the 7 / 2f

manifold. Subsequently, we restart the spin dynamics,nowfromaverypurestate,whereallatomscontributingto the signal are in the two‐particle state 1 / 2,9 / 2 .Thus, we observe high contrast spin oscillations. Inaddition, theadmixtureofnewspincomponentsdue todefect tunneling is strongly suppressed, since atoms in

7 / 2f do not exchange their spin with atoms in9 / 2f .Note, that thisprocedurewas applied for the

measurements starting from the initial state 1 / 2,9 / 2 inFig.3andFig.5.

Counting of spin components. After a variableevolution time,we stop the spin oscillations anddetectthe relative occupations in the different states m . Tostop the oscillations we increase the magnetic fieldrapidly,whichpinstheatomstotheirmomentarystate.Afterwards, we ramp down the lattice potential in500 μs andhencemapthequasimomentaof theatomsonto real momenta. After all optical potentials areswitched off, we perform a Stern‐Gerlach separation ofthe different spin states within a time‐of‐flight of18.5 ms . The atoms are finally detected via absorptionimaging with a short pulse of resonant laser light. Wecountthenumberofatomsinallspinstates,accounting

for saturationeffects due to the spatial variationof thepulseintensity.

Fitting procedure of the spin oscillations. For eachexperiment,wemeasurethetime‐dependentoccupation( ) ( )mn t oftendifferentsingle‐particlespinstates m .To

each of the spin states showing an oscillatory behaviorwefit:

( )( ) ( ) ( ) ( )( ) cos m

jtttm m m m

j j jj

n t A B e C e t .

Here, ( )mA is a general offset. The slowoverall increaseor decrease of the population of the individual spinstates is expressed in the second term, employing anexponential function with amplitude ( )mB and timeconstant ( ) m

t .Thespinoscillationsaredescribedbythesum in the third term: ( )m

jC is the amplitude, j thefrequency,and j theinitialphaseoftheoscillation.Weassume a damping of the oscillations described by anexponentialfunctionwiththetimeconstant j .Thesumistakenoverthenumberofallcontributingfrequenciesj asexpectedfromtheory.Note, thatthevalues j , j and j foreachcontributing frequency j areextractedin conformity with all participating single‐particle spinstates.

Forthetwo‐levelsystem(seeFig.3),foursingle‐particlespinstatesareinvolvedfortheon‐sitedynamics( 1 / 2 ,3 / 2 , 7 / 2 , 9 / 2 ). Here, the switching time of themagnetic field is small compared to the observedoscillation period, which allows us to set the initialphases 0j . Further spin states, which are slowlypopulated due to tunneling processes, do not showcoherentoscillations,butaslowoverallincreaseoftheirpopulation. We fit the occupation of these spin stateswith an exponential function

( )( ) ( )( ) (1 ) mt tm mn t A e .

Again, ( )mA istheamplitudeand ( ) mt thetimeconstant.

For the investigated five‐level system (see Fig. 4), fivetwo‐particle states ( 1 / 2, 1 / 2 , 3 / 2, 3 / 2 ,5 / 2, 5 / 2 , 7 / 2, 7 / 2 , 9 / 2, 9 / 2 ) participate inthespinevolutioninvolvingall tenspinstates( 9 / 2 ,…, 9 / 2 ). In this case, the oscillations are slightlyinfluenced by the finite switching time of themagneticfield ( 500 μs ), which we account for byincludingindividual phases 0j for each oscillation. The fitresults coincidewith the values of the discrete Fouriertransformgiveninthepaper.Note,thatallfittingerrorsaregivenbythe 95% confidence‐interval.

Fitting procedure to extract the scattering lengthdifference. In Fig. 3c, the observed frequencies andamplitudes of the two‐level system are depicted. Thefrequencydepends on themagnetic field aswell as thescattering energy difference of the participatingchannels. Since the lattice depth and themagnetic fieldare known precisely, the largest uncertainty originatesfrom the scattering length difference 8 6a a .We fit thedependenceofthefrequencyonthemagneticfield ( )B to the data for expB up to 0.6 G , where the spin‐dependent interaction plays a crucial role.We obtain ascatteringlength 8 6 B2.26 a a a withastatisticalerrorof B0.02 a and systematicerrorsdue to latticedepth (

B0.04 a ) and magnetic field uncertainties ( B0.01 a ).

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This is in good agreement with the theoreticallypredicted value 8 6 B B B168.53 166.00 2.53 a a a a a .To adapt for a total spin oscillation amplitude smallerthan unity, we rescale ( )A B by a fitted factor

0.69 0.02 inFig.3c.

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