State-to-state endothermic and nearly thermoneutral reactions in an ultracoldatom-dimer mixture

Jun Rui,∗ Huan Yang,∗ Lan Liu, De-Chao Zhang, Ya-Xiong Liu, Jue Nan, Bo Zhao, and Jian-Wei PanShanghai Branch, National Laboratory for Physical Sciences at Microscale and Department of Modern Physics,

University of Science and Technology of China, Hefei, Anhui 230026, ChinaCAS Center for Excellence and Synergetic Innovation Center in Quantum Information and Quantum Physics,

University of Science and Technology of China, Shanghai 201315, China andCAS-Alibaba Quantum Computing Laboratory, Shanghai 201315, China

Chemical reactions at ultracold temperatureprovide an ideal platform to study chemicalreactivity at the fundamental level, and tounderstand how chemical reactions are governedby quantum mechanics [1–4]. Recent yearshave witnessed the remarkable progress instudying ultracold chemistry with ultracoldmolecules [5–15]. However, these workswere limited to exothermic reactions. Thedirect observation of state-to-state ultracoldendothermic reaction remains elusive. Here wereport on the investigation of endothermic andnearly thermoneutral atom-exchange reactions inan ultracold atom-dimer mixture. By developingan indirect reactant-preparation method basedon a molecular bound-bound transition, we areable to directly observe a universal endothermicreaction with tunable energy threshold andstudy the state-to-state reaction dynamics. Thereaction rate coefficients show a strikinglythreshold phenomenon. The influence of thereverse reaction on the reaction dynamicsis observed for the endothermic and nearlythermoneutral reactions. We carry out zero-range quantum mechanical scattering calculationsto obtain the reaction rate coefficients, andthe three-body parameter is determined bycomparison with the experiments. The observedendothermic and nearly thermoneutral reactionmay be employed to implement collisionalSisyphus cooling of molecules [16], study thechemical reactions in degenerate quantum gases[17, 18] and conduct quantum simulation ofKondo effect with ultracold atoms [19, 20].

Ultracold molecules represent great opportunities toexplore cold controlled chemistry and study state-to-state reaction dynamics at the quantum level [2, 17, 21].Recently, with the development of ultracold moleculetechniques, significant progress has been achieved instudying ultracold bimolecular reactions. Molecule-molecule [9–11] and atom-dimer [12–14] atom-exchangereactions have been observed at temperatures below 1µK. However, these studies are limited to exothermic

∗These authors contributed equally to this work.

reactions, and ultracold reactions in the endothermicregime remain unexplored. Endothermic reactions arequalitatively different from exothermic reactions due tothe presence of energy thresholds. Therefore ultracoldendothermic reactions are expected to show thresholdphenomenon, i.e., the reaction rate coefficients decreaseremarkably when the threshold is changed from zero to asmall positive value. Further, for ultracold endothermicreactions, the kinetic energy converts into internal energyand thus the resultant products are colder than reactants,which means that both the reactants and products shouldbe described by quantum mechanics. This is in contrastto most of ultracold exothermic reactions where onlythe reactants are cold, whereas the products are muchhotter due to the large exothermicity, and consequentlythe products can still be described by classical treatment[22].

Exothermic reactions in ultracold gases always leadto detrimental effects, such as loss of the molecules orheating of the samples, and thus they are expected to besuppressed [4]. In contrast, endothermic reactions do notcause heating and may even result in cold products. Ithas been proposed that endothermic inelastic collisionmay be employed to implement collisional Sisyphuscooling of molecules [16]. Moreover, endothermicreactions may offer a platform to study superchemistrywhere quantum degeneracy may strongly affect thereaction [17, 18], since they will not destroy quantumdegeneracy due to heating. Besides, the thermoneutralreaction (zero energy change, neither exothermic norendothermic) may be employed to implement quantumsimulation of the Kondo effect with ultracold atoms, asproposed in reference [19].

Exploring endothermic reactions at nearly zerotemperature are challenging, because the collisionenergies of the reactants are much smaller than thethreshold of most of endothermic reactions and thusthe reaction should be simply forbidden. Theoreticalschemes have been proposed to use laser to control anisotope-exchange reaction to be endothermic [23]. Thesetechniques remain to be demonstrated. So far, theuniversal atom-exchange reactions with weakly boundFeshbach molecules [12–14] offer a unique possibilityto study the endothermic reaction at nearly zerotemperature. The reaction can be tuned to beexothermic, endothermic, or thermoneutral by simplyvarying the magnetic field.

arX

iv:1

708.

0861

0v1

[co

nd-m

at.q

uant

-gas

] 2

9 A

ug 2

017

2

0.6

0.4

0.2

0.0

35.170 35.180 35.190 35.2000.0

1.0

2.0

RF frequency (MHz)

(10

)AB

N4

NB/(N

B+N

C)

0

2

4

0 50 100 150 200RF duration (µs)

(10

)AB

N

a

b

rfAB

AC

rf

B

C

FIG. 1: Molecular bound-bound rf spectrum and Rabioscillation. a, The measured atomic transition spectrum(red diamonds) and molecular bound-bound spectrum (bluecircles) at 130.29 G. For the atomic spectrum, the fractionof the B atoms is recorded. For the molecular spectrum, thenumber of molecules transferred into the AB state is recordedby imaging the dissociated K atoms in the |9/2,−7/2〉 state.The data points are fitted with Gaussian functions. b, Themeasured bound-bound Rabi oscillation between the AB andAC molecule states at 130.26 G. The bound-bound Rabifrequency is fitted to be 2π × 9.9(1) kHz.

Despite the easy tunability of the energetics, the directobservation of the state-to-state universal endothermicreactions and measurement of the reaction rate remainunsolved questions. In Refs. [12, 13], only the overall lossrates of the molecule reactants are observed. Althoughin Ref. [12] it is claimed that a pronounced thresholdbehaviour is observed in the endothermic regime, theobserved overall loss rate includes the contributionfrom both the endothermic exchange reaction and theexothermic vibrational relaxation into deeply boundstates, and the theoretical model has shown thatdominant loss mechanism is actually the exothermicvibrational relaxation. In Ref. [14], the state-to-statereaction rate can only be measured in the exothermicregime with large released energy.

Here we report on the study of the state-to-stateendothermic and nearly thermoneutral reaction in anultracold atom-dimer mixture. Our experiment startswith approximately 3.0 × 105 23Na atoms and 1.6 × 105

40K atoms at about 600 nK in a crossed-beam opticaldipole trap. The measured trap frequencies for K areh× (250, 237, 79) Hz, with h being the Planck constant.The Na atoms are prepared in the lowest hyperfine|F,mF 〉Na = |1, 1〉 state, and the K atoms can beprepared in any hyperfine ground state |F,mF 〉K =|9/2,−9/2〉...|9/2, 9/2〉.

Magnetic field (G)130.0 130.1 130.2 130.3 130.4

-80

-40

40

0

Exothermic

Endothermic

Thermoneutral

Ener

gy c

hang

e E/

h (k

Hz)

E

AB

C

AC

B

FIG. 2: The reaction energy changes versus magneticfield. The measured energy change ∆E for the reactionAB + C→ AC + B as a function of the magnetic field. Theenergy change is precisely determined by measuring themolecular bound-bound transition frequency and the freeatom-atom transition frequency using rf spectroscopy. Theblue solid line is a polynomial fit to the data points.The vertical dashed line marks the magnetic field Btn =130.249(2) G at which the reaction is thermoneutral with zeroenergy change. For B < Btn the reaction is exothermic, andfor B > Btn, the reaction is endothermic. As a comparison,the grey dashed line shows the energy change obtained fromthe difference between the two binding energy curves. Errorbars inside the circles represent ±1 s.d.

At a magnetic field near 130 G, the interspeciesFeshbach resonance between |1, 1〉 and |9/2,−3/2〉 andthe Feshbach resonance between |1, 1〉 and |9/2,−5/2〉overlap (Supplementary Information). The overlappingFeshbach resonances allow studying the universal atom-exchange reaction between weakly bound Feshbachmolecules and atoms [12–14, 24]. In our experiment,the reaction may be described by AB + C→ AC + B,where A denotes 23Na atom in the |1, 1〉 state, and B andC denote 40K atoms in the |9/2,−5/2〉 and |9/2,−3/2〉state respectively. The AB and AC molecules are thecorresponding NaK Feshbach molecules with bindingenergies Eb

AB and EbAC, respectively. The energy change

in the reaction is given by ∆E = EbAB−Eb

AC. The state-to-state reaction dynamics can be measured by preparingthe reactant mixture and then monitoring the timeevolution of AB reactant and B product, and the reactionrate coefficient is extracted from the time evolution. Fordetection, the AB molecules are dissociated into theA + |9/2,−7/2〉 state and the B atoms are transferredto the |9/2,−7/2〉 state by rf pulses, and the ABmolecule number and B atom number are determinedby measuring the atom number in the |9/2,−7/2〉 state,respectively.

In all previous works studying ultracold reaction withFeshbach molecules, the molecule reactants are directlyformed using magnetic field association [12, 13] or radiofrequency association [14]. However, these methods

3

cannot measure the reaction rate in the endothermicregime, either due to the low formation efficiency or largebackground noise caused by the preparation process.

In our system, the primary experimental challenge is toefficiently prepare the reactants AB + C at the magneticfields where endothermic reactions with small ∆E mayoccur. For the exothermic reactions with large andnegative ∆E, the reactants can be prepared by radiofrequency (rf) association [14, 25] of the AB moleculefrom an A + C mixture. However, this preparationmethod is not applicable for small |∆E|, because in thisregime the scattering lengths aAB and aAC in the twoscattering channels are close to each other. This resultsin a rather low association efficiency since the bound-freeFranck-Condon factor Fbf ∝ (1 − aAC/aAB)2 is largelysuppressed [26].

To circumvent this problem, we develop an indirectreactant-preparation method based on the molecularbound-bound transition. Instead of directly forming theAB molecules, we first associate the AC molecules fromthe A + |9/2,−1/2〉mixture and then transfer the weaklybound molecules from the AC to the AB state by anrf π pulse. A subsequent rf π pulse transferring thefree K atoms from the |9/2,−1/2〉 state to the C statethen prepares the desired reactants. The idea behindthis method is that for a small |∆E|, the suppressionof bound-free Franck-Condon factor is accompanied byan enhancement of the bound-bound Franck-Condonfactor [26] Fbb ∝ 4aACaAB/(aAC + aAB)2. Therefore,although it is difficult to form the AB molecule from theA + C scattering state, it does allow us to create the ABmolecule from the AC bound state.

The Feshbach resonance between the A and C atomstends to be closed-channel dominated, and thus theAC molecules can only be efficiently associated whenclose to the resonance. In our experiment, we canefficiently form the AC molecules in a small magneticfield window between 130.02 G and 130.38 G usingRaman photoassociation (Supplementary Information).The bound-bound rf spectrum and Rabi oscillation aremeasured, as shown in Fig. 1. The difference betweenthe bound-bound transition frequency and the free-free transition frequency precisely determines the energychange ∆E = h(νAC→AB− νC→B) of the reaction, whichis in the range h × (−82, 32) kHz. The measured ∆Eas a function of the magnetic field is shown in Fig. 2.The polynomial fit yields ∆E = 0 at Btn = 130.249(2)G, which agrees well with the value of 130.24(1) Gdetermined from the intersection point of the bindingenergy curves. The bound-bound transition has anefficiency of better than 95%, thanks to the enhancedbound-bound Franck-Condon factor, and approximately2 × 104 AB molecules are formed. After preparingthe reactants, the time evolutions of the AB moleculereactants and the B atom products are recorded.

The reaction dynamics are shown in Fig. 3. Formagnetic fields larger than Btn, the reaction isendothermic with ∆E > 0. With increasing magnetic

field the reaction threshold increases and the maximumnumber of the B products decreases quickly. Using theindirect preparation method, the endothermic reaction isclearly observed up to 130.38 G with a reaction thresholdof 32 kHz.

At 130.25 G, the observed reaction is a nearlythermoneutral reaction with the measured energy change∆E = h × 1.1(2) kHz. The ideal thermoneutralreaction has zero energy change [27], i.e., ∆E = 0.Previously, the thermoneutral reaction usually refersto the “reaction” that the reactants and products areidentical, e.g., H2 + H→ H2 + H, where H2 and H arethe hydrogen molecule and atom in their lowest internalstates, respectively [1]. However, this is not an observablereaction. In our system, by preparing the reactants at themagnetic field where the Feshbach molecules have exactlythe same binding energy, an observable thermoneutralreaction can be realized. In the current experiment,the achievable minimal energy change is limited by theresolution and uncertainty of the magnetic field.

We analyze the data by fitting the time evolution of theAB reactant and B product numbers using exponentials.As shown in Fig. 3a, we find that for the exothermicreactions, the two 1/e time constants τAB and τB agreewith each other within mutual uncertainties, whereasfor the endothermic reactions and nearly thermoneutralreactions, the two time constants are different from eachother. We attribute this discrepancy to the influence ofthe reverse reaction AC + B→ AB + C, i.e., the reactionis reversible.

To demonstrate that this discrepancy is causedby the reverse reaction, we continuously remove theAC molecule products from the reaction mixture bydissociating them into the A + |9/2,−1/2〉 state. To doso, we first remove the A atoms after the AB moleculeshave been prepared. Then during the reaction process,we use the Raman light to continuously dissociatethe AC molecules into the A + |9/2,−1/2〉 state (seeSupplementary Information). This procedure suppressesthe reverse reaction and makes the reaction irreversible.In this way, we find τAB and τB agree with each otherwithin mutual uncertainties also for the endothermic andnearly thermoneutral reactions, as shown in Fig. 3b.

We now estimate the reaction rate coefficient βrfrom the measured time evolutions. Taking the reversereaction into account, the time evolution of the numberof B products may be described by

NB = βrnCNAB − kreNACNB, (1)

where the first and second terms on the right side ofEq. (1) account for the forward and reverse reactions,respectively. By continuously removing the AC products,the second term can be safely neglected. The meandensity of the C atoms is approximated by nC = αNC,with α = ((mKω

2)/(4πkBT ))3/2, where ω and T arethe geometric mean of the trap frequencies and thetemperature, respectively. We have assumed nC to bea constant as the number of the AB molecules is about

4

(iii) 130.37 G(ii) 130.25 G(i) 130.04 G

(iii) 130.36 G(ii) 130.25 G

Hold time (ms)

Nearly thermoneutral, Endothermic,Exothermic,

(i) 130.03 G

Atom

s in

(

)-7

/210

4At

oms

in

(

)

-7/2

104

AB=374 47± s

B =400 47± sAB=366 28± s

B =190 22± sAB=715 72± s

B =366 44± s

AB=355 23± s

B =333 42± sAB=292 46± s

B =243 34± sAB=581 37± s

B =485 60± s

0E<0 E>0E

AB reactantB product

a

b

FIG. 3: The reaction dynamics. a, The decay of the AB reactant (blue circles) and the increase of the B product (reddiamonds) as a function of the hold time, for (i) the exothermic reaction at B = 130.03 G with ∆E = −76.7(3) kHz, (ii) thenearly thermoneutral reaction at B = 130.25 G with ∆E = 1.1(2) kHz, and (iii) the endothermic reaction at B = 130.36 G with∆E = 28.0(2) kHz. The blue and red solid lines are exponential fitting curves with 1/e time constants τAB and τB, respectively.For the exothermic reaction, τAB and τB agree with each other within mutual uncertainties, whereas for the endothermic andnearly thermoneutral reactions, τAB and τB are not consistent. b, The time evolution of the AB reactant (blue circles) andthe B product (red diamonds) after continuously removing the AC product in the reaction process. The two time constantsτAB and τB are consistent with each other within the mutual uncertainties also for the endothermic and nearly thermoneutralreactions, and the increased numbers of the B products are larger. Error bars represent ±1 s.d.

one order of magnitude smaller than the number of theC atoms. The reverse reaction complicates the reactiondynamics, but the reaction rate coefficient may still beextracted using the simple equation

βr =NB(0)

nCNAB(0), (2)

where NAB(0) is the initial number of the AB molecules

and NB(0) is the initial derivative of the time evolutionof the B atoms. Eq. (2) is also applicable to thereversible reaction, since we start from the AB + Cmixture and thus at t = 0 the products NAC and NB

are both negligible. We assume that NAB(t) and NB(t)

can be approximated by exponentials, and obtain NB(0)and NAB(0) from exponential fits of the measured datapoints. As shown in Fig. 4, the reaction rate coefficientsobtained in this way are similar to the results in the casewhere the reverse reaction has been suppressed.

For the exothermic reactions, the reaction ratecoefficient increases as the magnetic field increases. Thismay be qualitatively understood from the increase of theoverlap between the wave functions of the two weaklybound molecules, when the difference between the twoscattering lengths decreases [12, 24]. For the endothermicreaction, the reaction rate coefficient shows a strikinglythreshold behaviour, as expected. Besides, the ratio

between the increased number of the B products and theinitial number of the AB reactants is given in the insetof Fig. 4. The data points after removing the A atomsand the AC molecules directly give the branching ratioof the reaction.

To understand the reaction rate, we perform quantummechanical reactive scattering calculations using thezero-range approximation. This is justified sincein our experiments the scattering lengths aAB andaAC are both much larger than the van der Waalslengths, and the binding energies of the AB andAC molecules are well described by the universalmodel (Supplementary Information). In this case,the s-wave atom-dimer scattering may be describedby the generalized Skorniakov–Ter-Martirosian (STM)equations [28–30]. In the STM equations, the onlyinput parameters are the scattering lengths aAB andaAC (Supplementary Information), and the unknownthree-body parameter Λ, which will be determined bycomparing the reaction rates.

By numerically solving the STM equations, wecalculate the reaction rate coefficients with Λ being afitting parameter. The results are compared with theexperiment for the magnetic fields between 129.44 Gand 130.38 G, where the data points for exothermicreactions between 129.44 G and 129.91 G are obtainedfrom our previous work [14]. We find that for a range of

5

Rat

e co

effic

ient

(

10

cm

/s)

3-1

0r

0

5

10

15

Magnetic field (G)129.4 129.6 129.8 130.0 130.2 130.4

E<0 E>0w.o. A and ACwith A and AC

1.0

130.0 130.1 130.2 130.3 130.40.0

0.5N

B/N

AB(0)

FIG. 4: The behaviour of the reaction rate coefficientsas a function of the magnetic field. The blue circlesand red squares represent the data points measured by usingthe indirect reactant-preparation method. The red squaresare the data points obtained by continuously removing theAC molecules (after eliminating the A atoms) when thereverse reactions are significantly suppressed. The grey circlesrepresent the data points obtained from Ref. [14]. Thesolid line is the numerical result with the fitted three-bodyparameter Λ = 22.6/a0 and the normalization coefficientCβ = 1.4. The inset shows the ratios between the net increaseof the B atom product and the initial AB molecule number,where the square points give the reaction branching ratio.Error bars represent ±1 s.d.

values for Λ, the shapes of the theoretical curves and theexperimental data points qualitatively agree with each

other, whereas the measured rate coefficients are largerthan the calculated ones. We attribute this discrepancyto the systematic shifts in our data due to uncertaintiesin the particle numbers, density calibration and so on.Therefore, we include a normalization factor Cβ as thesecond fitting parameter. As shown in Fig. 4, we findgood agreement between the theory and the experimentwith the fitting parameters Λ = 22.6/a0 and Cβ = 1.4,where a0 is the Bohr radius.

In conclusion, we have experimentally studied state-to-state endothermic and nearly thermoneutral reactionsin an ultracold atom-dimer mixture. The universalcharacter of the reaction allows us to understandthe reaction from a zero-range quantum mechanicalscattering calculation. The study of endothermicreaction opens up the realistic avenue to implementcollisional Sisyphus cooling of molecules [16]. Theobservation of thermoneutral reaction is crucial to thequantum simulation of the Kondo effect with ultracoldmolecules [19, 20]. This scheme requires the Feshbachmolecules to have almost the same binding energy[19], which is exactly the nearly thermoneutral regimedemonstrated in our work.

We would like to thank Peng Zhang, Cheng Chin,and Matthias Weidemuller for helpful discussions, andHui Wang, Xing Ding and Xiao-Tian Xu for theirimportant assistances. We acknowledge Ingo Nosskefor carefully reading the manuscript. This work wassupported by the National Natural Science Foundationof China (under Grant No.11521063, 11274292), theNational Fundamental Research Program (under GrantNo. 2013CB336800), and the Chinese Academy ofSciences.

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Supplementary Information

A. Feshbach resonance

The scattering length near the Feshbach resonancea(B) = abg[1 − ∆B/(B − B0)] has been determined inRef. [S1] by fitting the binding energies of the Feshbachmolecules using the universal model. For the Feshbachresonance between A and B, we obtain abg = −455 a0,B0 = 138.71 G and ∆B = −34.60 G. For the Feshbachresonance between A and C, we have abg = 126 a0,B0 = 130.637 G and ∆B = 4.0 G. The binding energiesof the Feshbach molecules are given by Eb

ij = ~2/2µdaij2,

where aij = aij(B) − a, with a = 51 a0 being themean scattering length. The scattering lengths and

A+B

A+C

-4000

-2000

0

4000

2000

Scat

terin

g le

ngth

(a ) 0

125 130 135 140 145600

400

200

0

Magnetic field (G)

Bind

ing

ener

gy E

/h (k

Hz)

A+B or A+C

AB

AC

b

a

b

FIG. S1: Overlapping Feshbach resonance. a. Thescattering lengths of the collision channels A + B (red) andA+C (blue). Solid lines represent the scattering lengths fromthe universal model. Dashed lines represent the results fromthe coupled-channel calculations. b. The binding energies ofthe weakly bound AB and AC Feshbach molecules obtainedfrom the universal model, which predict the two curves tointersect at 130.24 G. The grey shaded region denotes therange of magnetic fields in which the current experiment isperformed.

binding energies are shown in Fig. S1. Here aAB andaAC are the input parameters in the STM equations.From the binding energy curves, we obtain Eb

AB = EbAC

at B = 130.24(1) G, which is about 10 mG smallerthan Btn = 130.249(2) G which was determined fromthe direct measurement. Therefore, in the numericalcalculations, the magnetic field is shifted by 10 mG tocompare with the experiments.

B. Raman photoassociation and molecularbound-bound transition

In this work, we employ Raman photoassociation toform AC Feshbach molecules [S2]. We use two blue-detuned Raman light fields to couple the |9/2,−1/2〉 andC states with a single-photon detuning of ∆ ≈ 2π × 251GHz, with respect to the D2 line transition of the 40Katom, as shown in Fig. S2. The two Raman beams havea frequency difference of approximately 2π × 37 MHz atthe magnetic fields of around 130 G. Thus, they can besimply created by using two acousto-optic modulatorswith the light field from a single laser. The width ofeach Raman beam is about 0.8 mm at the position ofthe atoms, and the power is 30 mW. The Raman Rabifrequency for the |9/2,−1/2〉 → C transition is about2π × 90 kHz.

7

Na 1,1 +40K 9/2,-1/2

E b

Δ

Na 1,1 +40K 9/2,-3/2

Na 1,1 +40K 423/2P

ω2

ω1

E 0

AC

FIG. S2: Raman photoassociation. The atom mixtureis initially prepared in the A + |9/2,−1/2〉 state. The ACmolecule is formed by the Raman light. Eb denotes thebinding energy of the AC molecules. E0 denotes the freeatomic transition frequency between the two spin states of Katoms.

To associate the AC molecules from theA + |9/2,−1/2〉 mixture, we use a 300-µs Gaussianpulse with a full width at half maximum (FWHM)duration of about 150 µs. Using a Gaussian pulse, thehigh Fourier side lobes can be efficiently suppressed, andthus the transfer of free atoms from the |9/2,−1/2〉 tothe C state can be neglected. The Raman light fieldshave no influence on the active stabilization loop of themagnetic field, and thus it does not cause a shift of themagnetic field.

After creating the AC molecules, we determine themolecular bound-bound transition frequency νAC→AB bymeasuring the molecule radio-frequency (rf) spectrum.This is achieved by first transferring the AC moleculeto the AB molecule via a rf pulse, and then the ABmolecules are detected by dissociating them into theA + |9/2,−7/2〉 state for imaging. Besides, we alsomeasure the free atomic transition frequency νC→B bymeasuring the rf spectrum between the C and B states.From these measurements, the reaction energy change∆E = νAC→AB − νC→B can be accurately determined,and the results are shown in Fig. 1 in the main text.After determining the bound-bound transition frequency,the Rabi oscillation between the AB and AC moleculestates is measured, and thus the duration of the π pulseis determined.

C. Remove the remaining A atoms and the ACmolecule products

To demonstrate that the discrepancy between thetime constants τAB and τB for the endothermic reactionand nearly thermoneutral reaction is caused by thereverse reaction, we remove the AC molecule productscontinuously during the reaction process by dissociating

Na 2,2

E b

Δ

Na 1,1

Na 323/2P

ωbωa

cyclingtranstion

ReactionAssociation

AB transfer

Na Blast

C transfer300 53 12 5 50 12 5 50 50

rf π

Raman

1,1 2,2

2,2 3 ,3 wait wait

Na Remove

μsrf π

π πNa Raman

a

b

FIG. S3: Removal of the remaining A atoms. a. Theenergy levels of the two-photon Raman transition used fortransferring the Na atoms from the A to the |2, 2〉 state.After the Raman transfer, the Na atoms in the |2, 2〉 stateare resonantly blasted out of the optical trap by a lightpulse resonant with the cycling transition. b. The timesequence of the preparation process including the removal ofthe remaining A atoms.

them into the A + |9/2,−1/2〉 state. However, theremaining A atoms and the residual |9/2,−1/2〉 atomsdue to the imperfection of the π pulse may cause acompetition between the dissociation and associationprocess. Therefore, we first remove the A atoms afterthe AB molecules have been prepared. This is achievedas follows. We first transfer Na atoms from the A to the|2, 2〉 state using a blue-detuned Raman π pulse with asingle-photon detuning of ∆ ≈ 2π×215 GHz with respectto the D2 line of Na atom, and then blast them outof the optical trap by applying a light pulse resonantlycoupling the |2, 2〉 → |3′, 3′〉 cycling transition, as shownin Fig. S3a. The Raman fields for Na atoms have awidth of about 1.1 mm at the position of the atoms, andeach beam has a power of about 50 mW, which yields aRaman Rabi frequency of 2π× 46 kHz for the A→ |2, 2〉transition. The Raman-Blast sequences are applied twiceto completely remove the remaining A atoms. The timesequence including the removal of the A atoms is shownin Fig. S3b. The Raman-Blast process also causes smallloss of the AB molecules. Typically, we still keep 90% ofthe AB molecules after removing the remaining A atoms.

Then we can remove the AC molecule products duringthe reaction process by continuously dissociating theminto the A + |9/2,−1/2〉 state. This is achieved by usingthe Raman light coupling the |9/2,−1/2〉 and the Cstates, which is also used to associate the AC molecules.For the dissociation, the two-photon detuning is tuned tobe 220−320 kHz above the bound-free transition, and thepowers of both beams are reduced to obtain half the peak

8

0.0 0.5 1.0

1.5

2.0

2.5

1.5

3.0AB + -1/2

with Raman removing

0 200.0

1.0

2.0

40

AB + -1/2

0.0 0.1 0.20.0

1.0

2.0

0.3

3.0 AC + -1/2with Raman removing

0 100.0

1.0

2.0

20

AC + C

Hold time (ms)

Dis

soci

ated

NAB

( 1

0 )4

(12.1±1.4) ms (2.7±0.1) ms

(78±4) μs(8.6 ±0.8) ms

a b

c d

Ramanremoving

-3/2

-1/2Δω

E b

ACD

isso

ciat

ed N

AC(

10

)4

Hold time (ms)

FIG. S4: Lifetime of the molecules under different conditions. The measurements are taken after removing the remainingA atoms at 130.26 G. a and b. The lifetime of the AC molecules without and with the Raman removing fields, respectively.The two-photon frequency is tuned to be about 220 kHz above the bound-free transition frequency. The AC molecules aredetected by first transferring them to the AB state and then dissociating them into the A + |9/2,−7/2〉 state. c and d. Thelifetime of the AB molecules without and with the Raman removing fields, respectively. Error bars represent ±1 s.d.

intensity as was used in the association process. ThisRaman dissociation light reduces the lifetime of the ACmolecules to around 80 µs, as is shown in Fig. S4b, andthus the AC molecules can be quickly eliminated. Wechecked that the Raman removing pulse has negligibleeffects on the C atoms. The Raman removing pulse alsoaffects the lifetime of the AB molecules. However, theAB molecule still has a lifetime of about 2.7 ms under theRaman removing fields, as is shown in Fig. S4d, whichis long enough for the study of the reaction dynamics.

We also compare the time evolutions when onlyremoving the remaining A atoms and the time evolutionswhen both the remaining A atoms and the AC moleculesare removed. After removing the A atoms, the dominantatoms in the mixture are the C atoms. The AC moleculehas a long lifetime of about 8 ms when coexisting with theC atoms, as shown in Fig. S4a. In this case, the effects ofthe reverse reaction may also be observed on a long timescale. As shown in Fig.S5, when only the A atoms areremoved, we observe a long-term decay for the B atomproducts. However, this long-term decay disappearswhen the AC products are continuously removed duringthe reaction process. Therefore, we attribute this long-term decay to the influence of the reverse reaction.

Besides, we also measure the influence of the Ramanremoving light on the molecular and atomic transitionfrequencies. Our measurements show that the atomictransition frequency νC→B is shifted by about +0.3 kHz,

whereas the molecular bound-bound transition frequencyνAC→AB is shifted by about −3.4 kHz. The energychange ∆E = νAC→AB − νC→B under the Ramanremoving fields is plotted in Fig. S6. The polynomialfitting shows the magnetic field of zero energy change isshifted to 130.258(2) G under the Raman removing fields.

D. Reaction dynamics

Taking into account the reverse reaction, the timeevolution of the AB reactant and B product may bedescribed by

NAB = −γABNAB + kreNACNB, (3)

NB = βrnCNAB − kreNACNB, (4)

where the first and second term of the right side accountfor the forward and reverse reaction respectively. Thetime evolution of the AC molecule may be written ina similar form as the AB molecule. Since the timeevolution of the AC molecule is difficult to detect, itsequation is not shown. Here the overall loss rate of NAB

is γAB = βAnA + βCnC + βrnC, with nA and nC themean densities of the A and C atoms, respectively. Thefirst term accounts for the loss due to inelastic collisionswith the remaining A atoms, with βA the loss ratecoefficient. When the A atoms are removed, this term can

9

1.5

1.0

0.5

0.00 5 10 15

0.0 1.0

1.0

0.02.0

Hold duration (ms)

Atom

incr

ease

(

)N

B10

41.5

1.0

0.5

0.00.0 0.5 1.0 1.5

Hold duration (ms)

Dis

soci

ated

(

)N

AB10

4a

b

FIG. S5: Time evolution after removing the remainingA atoms. The red (blue) data points represent the timeevolution without (with) applying the Raman removing lightat 130.26 G. a. The time evolution of the AB reactants. Thetime constants are measured to be τAB = 0.319(21) ms (red)and τAB = 0.262(28) ms (blue), respectively. b, The timeevolutions of the B products, which are fitted with the model,NB(t) = N0

B(e−t/τ2−e−t/τ1). The fitting gives τ1 = 0.184(24)ms and τ2 = 21(4) ms (red), and τ1 = 0.220(18) ms andτ2 = +∞ (blue). (Inset). The short term evolution of the Bproducts. The simple exponential fitting gives a time constantof τB = 0.166(22) ms (red), and τB = 0.220(18) ms (blue).Error bars represent ±1 s.d.

be neglected. The other two terms describe the losses dueto collisions with the atom reactant C, which include thedesired reactive collision with a reaction rate coefficient ofβr, and the losses due to reactions in other channels andvibrational relaxations with an overall loss rate of βC.The collisions between the AB molecules are neglectedsince the Feshbach molecules are fermionic molecules. Wehave assumed nA and nC are constants since the numberof molecules is about one order of magnitude smaller thanthat of the atoms. The reverse reaction is caused by thereactive collisions between the AC molecule and B atomproducts with kre the reverse reaction rate.

In our experiment, the time evolutions of NAB(t)and NB(t) are measured. To extract the reaction rate

Ener

gy d

iffer

ence

ΔE

(kH

z)

Magnetic field (G)130.0 130.1 130.2 130.3 130.4

-80

-40

40

0

FIG. S6: Influence on the energy change of the Ramanremoving fields. The red points represent the measuredenergy change under the Raman removing fields. The bluepoints represent the energy change in the main text. The solidlines are polynomial fitting to the measured values, whichshows the zero-energy-change magnetic field is shifted by 9mG under the Raman fields. The grey dashed line is theenergy change calculated from the universal binding energycurves. Error bars inside the circles represent ±1 s.d.

coefficient, we use

βr =NB(0)

nCNAB(0). (5)

This is reasonable as we start from the AB + C mixtureand thus at t = 0 the number of products NAC and NB

can be neglected. The exact form of NAB(t) and NB(t)are difficult to know. Therefore, we assume they maybe approximated by exponentials and use exponentialsto fit the data points. The reaction rate coefficientsare extracted with NAB(0) and NB(0) obtained from thefitting curves. Note that when the reverse reaction issuppressed or can be neglected, the analytic solutions ofNAB(t) and NB(t) are exponentials, and thus reactionrate coefficients obtained from Eq. (5) are exact.

The initial molecule number has to be corrected bytaking into account the finite rf dissociation rate [S1],since the reaction is fast. The finite π-pulse transferefficiency of the K atoms from the B to |9/2,−7/2〉 statesalso needs to be included, which is about 90% with theNa atoms, and 95% without the Na atoms. Finally, alluncertainties of the parameters are included to give theuncertainty in the reaction rate coefficients.

10

E. STM equations

The scattering amplitudes satisfy the STM equations

tii(k, p, E) =mA

πµd

√aAC

aAB

∫ Λ

dqq

2kK(k, q, E)

×DAC(q, E)tfi(q, p, E)

tfi(k, p, E) =2π~4

µ2d

√aABaAC

mA

2pkK(k, p, E)

+mA

πµd

√aAB

aAC

∫ Λ

dqq

2kK(k, q, E)

×DAB(q, E)tii(q, p, E)

(6)

where

K(k, p, E) = log2µdE − p2 − k2 + 2µdpk/mA

2µdE − p2 − k2 − 2µdpk/mA

Djl(q, E) = [− ~ajl

+

√−2µd(E −

q2

2µad)]−1.

We denote the input channel AB + C by channel iand the reaction channel AC + B by channel f , andthus tii(k, p, E) is the elastic scattering amplitude, or inother words, the back-reflection amplitude of scatteringinto the input channel, and tfi(k, p, E) is the reactivescattering amplitude, with p and k being the relativemomentum of the incoming and outgoing atom-dimerpairs and E the total energy, µd = mAmB/(mA +mB) isthe reduced mass, and µad = mA(mA+mB)/(2mA+mB)is the reduced of mass of atom and dimer. In the integralequations, the only input parameters are the scatteringlengths aAB and aAC, which have been given in Sec.A in the Supplementary Information, and the unknownthree-body parameter Λ, which will be determined bycomparing the reaction rates. Since the magnetic fieldwindow in the experiment is narrow, we assume Λ is aconstant.

We numerically solve the STM integral equations tocalculate the on-shell reaction amplitudes tfi(kf , pi, E)with E = ~2k2

f/2µd − EbAC = ~2p2

i /2µd − EbAB

using the method introduced in Ref. [S3, S4]. Theincident relative momentum is calculated for pmin ≤pi ≤ pmax, where pmin = 0 for the exothermicreactions and pmin =

√2µd|∆E| for the endothermic

reactions, and pmax =√

2µdEbAB, i.e., we only

consider the collisional kinetic energy smaller thanthe binding energy of the AB molecule. This is agood approximation since in our experiment we haveEb

AB > 4kBT . The total reaction cross section isgiven by σr(pi) = (8π3/h4)µ2

ad(kf/pi)|tfi(kf , pi, E)|2.The reaction rate coefficient is thus calculated byβr =

∫vσr(v)f(v)dv where we have v = p/µad and

f(v) = 4πv2[µad/(2πkBT )]3/2 exp[−(µadv2)/(2kBT )] is

the Maxwell-Boltzmann distribution. In the calculation,the temperature is assumed to be 700 nK, and themagnetic field has been shifted 10 mG so that theoverlapping magnetic field obtained from the universalbinding energy curves is consistent with the directmeasurement.

The calculated rate coefficients are compared withthe experiment for the magnetic fields between 129.44G to 130.38 G, where the data points between 129.44G and 129.91 G are obtained from our previous work.The reaction rate at 130.03 G is also measured in ourprevious work [S1]. However, due to the suppression ofthe bound-free Franck-Condon coefficients, the numberof the AB molecule that can be formed from the A+Cmixture is only about 5000. In comparison, using theindirect method in this work, 1.5 × 104 AB moleculescan be formed. Therefore, at 130.03 G, the reaction ratemeasured in this work has much better signal-to-noiseratio and thus is used in the fitting. For the magneticfields larger than 130.03 G, the reaction rates can onlybe measured using the current method.

[S1] Rui, J. et al. Controlled state-to-state atom-exchangereaction in an ultracold atom-dimer mixture. Nat. Phys.13, 699 (2017).

[S2] Fu, Z. et al. Production of feshbach molecules inducedby spin–orbit coupling in fermi gases. Nat. phys. 10, 110(2014).

[S3] Braaten, E., Hammer, H.-W., Kang, D. & Platter,

L. Three-body recombination of 6Li atoms with largenegative scattering lengths. Phys. Rev. Lett. 103, 073202(2009).

[S4] Helfrich, K., Hammer, H.-W. & Petrov, D. S. Three-body problem in heteronuclear mixtures with resonantinterspecies interaction. Phys. Rev. A 81, 042715 (2010).