Enhanced magnetoassociation of 6 Li in the quantum degenerate regime
Vineetha Naniyil, Yijia Zhou, Guy Simmonds, Nathan Cooper, Weibin Li, Lucia Hackermüller
EEnhanced magnetoassociation of Li in the quantum degenerate regime
Vineetha Naniyil, Yijia Zhou,
1, 2
Guy Simmonds, Nathan Cooper, Weibin Li,
1, 2 and Lucia Hackerm¨uller School of Physics and Astronomy, University of Nottingham, Nottingham NG7 2RD, United Kingdom Centre for the Mathematics and Theoretical Physics of Quantum Non-equilibrium Systems,University of Nottingham, Nottingham, NG7 2RD, UK
The association process of Feshbach molecules is well described by a Landau-Zener transition while abovethe Fermi temperature, such that two-body physics dominates the dynamics. However, using Li atoms and theassociated Feshbach resonance at B r = . Introduction.—
Feshbach resonances allow for stronglytunable interactions between ultracold atoms and permit theirassociation into Feshbach molecules [1]. In the past decades,Feshbach molecules formed via magnetoassociation [2–5]have captured much attention in the study of unitary dy-namics [6], collective dynamics [7, 8] and many-body ef-fects [9]. Starting from BCS pairs, deeply bound moleculesare created when the magnetic field is tuned across the Fes-hbach resonance. Molecular formation dynamics can becontrolled by varying the time-profile of the applied mag-netic field. Combined with established cooling and trappingtechniques [10, 11], this opens up opportunities to explorenew fundamental physics [12–16], controlled chemistry [17–21] and the quantum simulation of complex many-body sys-tems [22–25].In the investigation of Feshbach resonances, s-wave scatter-ing lengths play vital roles, characterizing, e.g. the unitary dy-namics. The atom-molecule conversion, on the other hand, isa dynamical, many-body process [1, 26]. A simple model thatcaptures the atom-molecule dynamics is a spin-Boson coupledmodel [27–29], where BCS pairs and molecules are mappedto spin-half and Bosonic particles, respectively. A key param-eter by which the dynamics of this system are characterizedis the atom-molecule coupling coefficient. Many theoreticalworks have shown that the coupling coefficient depends onthe magnetic moment of the atom, s-wave scattering lengthand volume of the gas [2, 29]. It determines the time scale ofthe Landau-Zener transition [30, 31], and many-body dynam-ics [27, 28, 32–36] of the atom-molecule system. At zero-temperature, it was shown that the effective atom-moleculecoupling coefficient becomes collective and depends on N ( N to be the number of total atoms in general), i.e. the as-sociation is enhanced by many-body coherence [28, 34, 37–40]. Although this coefficient is of importance in connectingthe microscopic parameters and macroscopic properties of theatomic gas, a systematic, experimental investigation of thiscoupling parameter has yet to be conducted.In this work, we investigate magnetoassociation of Liatoms below and above the Fermi temperature, where the high Tinter-mediate Tlow T moleculewavefunction (c)
Magnetic field (G) S c a tt e r i ng l eng t h ( a ) B-field ramp50 -700 ms
860 G707 G
600 800 1000 1200-15-10-5051015 (a)
Time (ms)
Magneticfield 860 G 707 GDipole trap ~1 W 0 W Preparationstage Imaging 0 G (b)
Figure 1.
Feshbach molecules of Lithium-6 atoms. (a) Broad Fes-hbach resonance of Li at magnetic field B r = . molecules. (b) Experimental timing. An ensem-ble of Li atoms in m f = ± / Li molecules form a Bose-Einstein condensate. magnetic field is linearly ramped across the broad Feshbachresonance at B r = . B > B r ) to Fes-hbach molecules ( B < B r ) [41], as depicted in Fig. 1(a). Thefraction of atoms converted to molecules is measured experi-mentally, as a function of both the temperature of the atomicgas and the sweep rate of the magnetic field. The atom-molecule coupling coefficient is derived from the experimen-tal data through the Landau-Zener formula. We observe thatthe coupling coefficient increases when the temperature of a r X i v : . [ phy s i c s . a t o m - ph ] F e b the atomic gas is lower than the Fermi temperature, due tothe increased spatial coherence of the atoms, as illustrated inFig. 1(c). With the coupling coefficient, we employ an accu-rate spin-Boson model that takes into account of the effects ofmany-body coherence and allows simulation of molecular for-mation dynamics in the regime of quantum degeneracy. Thisprovides insight into the magnetoassociation process at ultra-cold temperatures and will be important for the developmentof quantum technologies based on ultracold molecules. Experiment.—
In order to determine the molecular con-version efficiency, we first prepare a cloud of cold Li atomsin a crossed optical dipole trap. A balanced spin mixture oftwo hyperfine states is loaded from a magneto-optical trapand evaporatively cooled under a static magnetic field of B i = . B ( t ) = B i − α t ,where α is the ramping rate. During this process a certainfraction of the atoms associates into Feshbach molecules. Anabsorption image of the resulting cloud is taken using lightresonant with the D2 line of unassociated atoms of one spinspecies after a time-of-flight of 1.5 ms. Due to the molecularbinding energy, the imaging light is now detuned by manylinewidths (178 MHz binding energy vs. natural linewidthof 6 MHz) from the corresponding transition in magneto-associated atoms. As a result, the absorption imaging processdetects only the unassociated atoms. The molecular conver-sion efficiency can then be determined by comparing the num-ber of unassociated atoms remaining after the magnetic fieldramp to the number present before. For each experimental set-ting a calibration procedure is applied by ramping back overthe resonance thus dissociating molecules back into atoms —see Supplementary Material. A range of different rampingspeeds are used, such that the total magnetic field rampingtime varies from 50 to 700 ms, and by fitting the resulting datato a theoretical prediction [Eq. (3) given later], we obtain thecoupling coefficient corresponding to the experimental condi-tions that have been set. This whole procedure is then repeatedat a range of different atom cloud temperatures (trapping fre-quencies) between 3.2 µ K and 130 nK, i.e. both above andbelow T F . This allows us to explore the behavior of the cou-pling coefficient in a broad range of initial temperatures of thesystem.We first investigate non-equilibrium and equilibriummolecule formation by varying the magnetic field ramp timeover a large range. The experimental results are shown inFig. 2(a). At a given temperature T , the remnant atom frac-tion (as determined via absorption imaging) depends on α − nonlinearly. A general trend is that the remnant atom fractionincreases when the magnetic field is changed faster. The frac-tion of the remnant atoms (molecules) is small (large) when α is small. We find that the remnant atom fraction is non-negligible even in the adiabatic regime. The molecule for-mation efficiency, i.e. the ratio of the molecules formed toinitial atom pairs present, is limited due to, e.g. multiple col- lisions [42], and many-body effects [43]. In the opposite, dia-batic regime when α is large, we find the remnant atom frac-tion increases significantly after sweeping the magnetic field. Figure 2.
Molecule formation at different temperatures andsweeping rates (a) Remnant fraction of non-associated atoms aftermagnetic field ramp. When the inverse ramp rate is low (i.e. fastramp), the atom fraction is large. Decreasing the ramp speed reducesthe remnant atom fraction. In both situations, more atoms are con-verted to Feshbach molecules when the temperature is reduced. Thesolid lines are fitting results according to Eq. (3). The error bars arethe standard error of 5 measurements. We show the temperature de-pendence of the molecule fraction for a fast ramp with α − = α − = T / T F ∼ . B i = . B f =
707 G respectively. The ramping rate is varied by changingthe duration of the ramp.
We find that the molecule conversion rate changes dra-matically at different temperatures. In Fig. 2(b) and (c), themolecule conversion is shown as a function of temperature.When the ramp is fast [Fig. 2(b)], the molecule fraction islow at high temperature and high when the temperature is be-low the Fermi temperature T F = ¯ h / ( mk B )( π n ) / , with k B being the Boltzmann constant. The molecule fractions in-creases monotonically as temperature decreases. Similar de-pendence is found in the case of a slow ramp [Fig. 2(c)]. How-ever one should note that the overall conversion efficiency ishigher in this case. For example, the final molecule fraction (at T =
130 nK) increases from less than 60 % for α − = >
80 % for α − = Atom-molecule coupling coefficient.—
A key parameterto describe the molecule formation dynamics is the atom-molecule coupling coefficient. Many theoretical works [2, 29,34, 44] have shown that the coupling coefficient is given by, g = ¯ h (cid:114) π a bg ∆ ∆ µ m V (1)where V , a bg , and ∆ are the mode volume, background scat-tering length, and resonance width, respectively. The cou-pling coefficient g is a composite parameter and consideredas a constant. It connects the microscopic ( a bg and ∆ µ ) andmacroscopic ( V and ∆ ) properties of the system. Although itis an important parameter when modelling the atom-moleculedynamics [27, 28, 32], the value of g has not been widelydiscussed and a detailed, temperature dependent experimentalmeasurement has not been done so far. Table I. Overview of the calculated and fitted coupling coefficientsand the resulting enhancement factor for the respective temperature.Fermi Temp. Temp. Calculated g Fitted g Enhancement T F ( µ K ) T / T F g c ( ¯ h × kHz ) g f ( ¯ h × kHz ) g f / g c To obtain the coupling coefficient, we note that param-eters a bg and ∆ have been measured in a number of ex-periments [45]. The mode volume of strongly interactingFermions in a harmonic trap is V = π a ho ξ / B √ N , wherethe oscillator length is a ho = ( ¯ h / m ω ) / with ω the geomet-ric mean of the oscillation frequency. The parameter ξ B isthe Bertsch factor [46] accounting for the atomic interactions.In the dilute limit, ξ B ≈ .
37 is obtained from Monte Carlosimulations [47]. As some atoms do not participate in the for-mation of molecules, the volume becomes V = √ . × V ,where 0 .
687 is the mean molecule fraction. Using the exper-imentally obtained atom number N (and hence V ), we obtainthe coupling coefficient g c . The related parameters and thecoupling coefficient are summarized in Table I. The data ta-ble shows that the coefficient g c varies only marginally as wechange temperature.Next, we obtain the coupling coefficient by fitting the ex-perimental data. To this end, the molecule formation is de-scribed by a simple, two-level model [45], where two atomsform a molecule through a Landau-Zener (LZ) transition. Inthis two-state description, the dynamics is governed by theHamiltonian H = (cid:18) gg δ ( t ) (cid:19) , (2) Figure 3.
Temperature dependence of the coupling coefficient .We show the ratio R = g f / g c obtained from the experimental data infigure 2 (see Table I). The orange line represents the geometric ratiofactor K as defined in equation (4). The theory estimation is boundedby 1 (blue horizontal line) in the thermal case, and the orange line isdashed where it exceeds this bound. where δ ( t ) = ∆ µ B ( t ) , with ∆ µ = µ B being the difference ofmagnetic moment of the two states. Based on this model, thetime-dependent Schr¨odinger equation can be solved analyti-cally. The molecule occupation probability P is given in thelimit t → + ∞ by P = exp (cid:2) − π g / ( ∆ µα ) (cid:3) . To take into ac-count the remnant atom fraction even in the adiabatic limit,we fit the atom fraction n a based on the LZ result with, n a = n + ( − n ) exp (cid:32) − π g f ∆ µα (cid:33) , (3)where g f is the fitted coupling coefficient and n the remain-ing atom fraction in the adiabatic limit ( α → g f , also shown in Table I. Thefitted coefficient g f , however, depends on the temperature. Wefind that g f is small at higher temperatures and for T > T F , g f is nearly identical to g c . At lower temperatures g f grows grad-ually and is almost twice g c when T / T F = .
11. The changein the fitted coefficient is seen clearly in Fig. 3, where the ra-tio R = g f / g c is shown. We note that at T / T F = . g f islarger than its neighboring values. It is unclear what causesthis discrepancy. Coherence enhanced molecule conversion.—
We inter-pret the enhancement of the coupling coefficient due tomany-body coherence at ultra-low temperatures, where manymolecules are condensed [48]. As a result, the coupling co-efficient is amplified to be g T = √ N T g , where N t = ρ V T isthe number of molecules in a thermal volume V T at density ρ . Assuming that the molecules have the same temperatureas the atoms, their de Broglie wavelength at temperature T isgiven by λ T = ¯ h (cid:112) π / Mk B T , with M being the mass of theLi molecule. Thermal volumes of molecules at temperature T are hereafter V T = λ T . We then compare g T with the cou-pling coefficient at the Fermi temperature g F = √ N F g , where N F = ρ V F , with V F = π ¯ h (cid:112) ( mk B T F ) − to be a Fermi vol-ume at the temperature T F . We characterize the temperaturedependence with a geometric ratio factor, K = g T g F = V T V F = √ √ π (cid:18) T F T (cid:19) / , (4)which is proportional to T − / .This temperature dependence of the geometric ratio factorshows qualitative agreement with the experimentally fitted g f when the temperature T < T F , as shown in Fig. 3. We thus caninterpret the experimental result as follows. When the temper-ature is high T > T F , the thermal volume is smaller than theFermi volume, V T < V F . The atom-molecule coupling takesplace at the two-body level in this case, for high temperatures.When T < T F , however, the thermal volume is larger than theFermi volume, which leads to many-body enhanced collectiveatom-molecule coupling. This means that the molecule con-version efficiency will be high at lower temperatures, which isconsistent with the experimental result. Quantum dynamics of finite systems.—
With the cou-pling coefficient at hand, we study the atom-molecule cou-pling dynamics. We consider a low temperature regime, T < T F , using a typical value of the coupling coefficient. TheHamiltonian [27, 28, 48] describing the dynamics of moleculeformation is given by H = ∑ j H j , where the j -th pair Hamil-tonian H j reads H j = δ ( t ) ˆ b † j ˆ b j + ε j (cid:16) ˆ c † j ↑ ˆ c j ↑ + ˆ c † j ↓ ˆ c j ↓ (cid:17) + g (cid:16) ˆ b † j ˆ c j ↓ ˆ c j ↑ + H.c. (cid:17) . Here ˆ b j (ˆ b † j ) is the Bosonic annihilation (creation) operatorof a molecule in the j -th energy level of the harmonic trap,while ˆ c j σ ( ˆ c † j σ ) denotes the annihilation (creation) operatorof a Fermionic atom with spin σ ( σ = ↑ , ↓ ) . The parameter δ ( t ) = ∆ µ ( α t + B ) gives the molecular energy, where B isthe initial magnetic field, and ε j is the kinetic and trap energyof the atom pair. It is a good approximation to neglect thisterm when the temperature is low [28]. The atom-moleculecoupling coefficient, g , is given by equation (1). At low tem-peratures, molecules condense into the ground state, such thatwe can neglect their index, i.e. ˆ b j → ˆ b (ˆ b † j → ˆ b † ).We have solved the Hamiltonian H for different numbersof atoms numerically, as shown in Fig. 4. For this, we haveset the magnetic field B > B r ( B < B r ) when t < t >
0) tomimic the Feshbach molecule formation dynamics. With onlyone pair of atoms, the molecule fraction is negligible when themagnetic field is larger than B r (i.e. t < t →
0) the molecule fraction increases rapidly. Abovethe resonance ( t > -0.1 0 0.1 0.2 0.3 time (ms) f r a c . m o l . averageN=1N=4N=7 Figure 4.
Quantum dynamics of Feshbach molecule formation .When the total number of atom pairs is small, the molecule fractionis low. For a fixed number of atom pairs, the molecule fraction os-cillates rapidly. Due to strong dephasing, the average molecule frac-tion quickly reaches a steady value. In the simulation, we have used g =
50 kHz, which is in the range of the experimental data. Otherparameters are α = / ms and B is tuned from B r − .
158 G to B r + .
316 G.
However, the molecule fraction when t > ∼√ Ng , see Fig. 4. We have carried out calculations for up to tenpairs of atoms, where fast oscillations are seen in the moleculefraction when t >
0. We then take the average value of theseindividual realizations, assuming they are equally weighted.This procedure averages out the large amplitude oscillations,where the molecule fraction quickly reaches a steady valuewhen t >
0. We note that such dephasing can also be obtainedwhen a thermal average is performed [48]. The simulationshows also that molecules form during a short period ∝ / g where the magnetic field changes about g / ∆ µ in the vicinityof the resonance. This time scale is different from what isobserved in the experiment and worth careful investigation inthe future. Conclusion.—
Our experiment shows that the fraction ofatoms associated into molecules increases when both thetemperature of the atomic gas and the sweeping rate of themagnetic field are decreased. We have measured the atom-molecule coupling coefficient, which increases at lower tem-peratures and in the adiabatic regime, as a result of many-body coherence. The qualitative trends predicted by our the-ory agree with our experimental findings, and quantitativeagreement appears strong at temperatures only slightly belowthe Fermi temperature. The quantitative differences at evenlower temperature indicates that a more sophisticated model istherefore needed to fully describe the experiment. Our studyprovides an accurate experimental measurement of the atom-molecule coupling coefficient. Exploitation of these enhancedcoupling coefficient might lead to a path for more efficientmolecule creation.Daniele Baldolini is thanked for historic contributionsto construction of experimental apparatus. This workwas supported by the EPSRC grants EP/R024111/1 andEP/M013294/1 and by the European Comission grantErBeStA (no. 800942). W. L. acknowledges support fromthe UKIERI-UGC Thematic Partnership (IND/CONT/G/16-17/73), the Royal Society through the International ExchangesCost Share award No. IEC \ NSFC \ [1] E. Hodby, S. T. Thompson, C. A. Regal, M. Greiner, A. C.Wilson, D. S. Jin, E. A. Cornell, and C. E. Wieman, Produc-tion efficiency of ultracold Feshbach molecules in Bosonic andFermionic systems, Phys. Rev. Lett. , 120402 (2005).[2] E. Timmermans, P. Tommasini, M. Hussein, and A. Ker-man, Feshbach resonances in atomic Bose–Einstein condensates,Phys. Rep. , 199 (1999).[3] F. A. van Abeelen and B. J. Verhaar, Time-dependent Feshbachresonance scattering and anomalous decay of a Na Bose-Einsteincondensate, Phys. Rev. Lett. , 1550 (1999).[4] F. H. Mies, E. Tiesinga, and P. S. 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Vineetha Naniyil, Yijia Zhou, , Guy Simmonds, Nathan Cooper, Weibin Li, , and Lucia Hackerm¨uller School of Physics and Astronomy, University of Nottingham, Nottingham NG7 2RD, United Kingdom Centre for the Mathematics and Theoretical Physics of Quantum Non-equilibrium Systems,University of Nottingham, Nottingham, NG7 2RD, UK
Herein we give additional details of the experimental procedure, in particular with reference to howabsorption images of the atom cloud are used to determine the fraction of Li atoms that have associatedinto Feshbach molecules. We also provide more details on analysis of Landau-Zener transition and thequantum spin-Boson model.
TECHNICAL DETAILS OF EXPERIMENTAL PROCEDURE
Generation of the cold atom cloud prior to magnetoassociation begins with a magneto-optical trap (MOT) [S1]. The MOT isloaded via a Zeeman slower [S2], which slows an atomic beam that is transmitted through a differential pumping stage from asource chamber. Over a 10 second loading cycle, the MOT captures ∼ × Li atoms. An additional cooling step, in whichthe trapping lasers are tuned to half a natural linewidth below resonance (for optimal Doppler cooling), brings the temperatureof this atom cloud down to ∼ µ K.An optical dipole trap is loaded from this cold cloud. A 100 W fibre laser, operating at 1070 nm, is used to produce a crossed-beam dipole trap, in which each beam is focused to a waist of 80 µ m. The crossing angle is 14 degrees. This captures up to2 × atoms.These atoms are then evaporatively cooled to a regime close to quantum degeneracy to temperatures between 0 . − . F with total atom numbers between 100.000 - 200.000 atoms. After the loading, the dipole trap is first held at constant powerfor 600 ms, following which the power in the optical dipole trap is ramped down to the range of tens to hundreds of mW. Theend point depends on the final trap depth desired and is reached in a series of linear ramps that collectively approximate anexponential decay of the trapping power. The power is lowered through a combination of reducing the laser current and theuse of an acousto-optic modulator. A photodiode is used to measure the optical power that passes through the dipole trap, withservo-controlled feedback to the acousto-optic modulator enabling active stabilisation of the dipole trap’s depth to its set value.This is necessary to reduce unwanted heating effects arising from small variations in trap depth.At the end of this evaporative cooling cycle, which lasts ∼
10 s, on the order of 10 atoms typically remain, at temperaturesranging from tens of nK to several µ K. The cloud is then held at constant trap depth corresponding to trapping frequenciesbetween 622 - 750 Hz (radially) and 74 - 90 Hz (longitudinally).The magnetic field is then ramped linearly from 860.6 G to the BEC side of the Feshbach resonance (707 G). The linearmagnetic field ramp is applied through a change in the current in the Feshbach coils as shown exemplary in the Fig.S1. -0.10 -0.05 0.00 0.05 0.10 0.15
Time (s) I n p u t a n d O u t p u t s i g n a l ( V ) Input signalCurrent transducer signal
Figure S1.
Magnetic field ramp.
Current transducer signal for a 50 ms ramp.
DETERMINATION OF MOLECULE FRACTION VIA ABSORPTION IMAGING
To reduce the impact of technical noise sources on the absorption imaging, the atom cloud was released from the dipole trapand allowed to expand for a period of 1 to 2 ms (depending on exact experimental parameters) prior to imaging. The size ofthe atom cloud after this period was typically some hundreds of micrometers, which greatly exceeds our imaging resolution of3 µ m. Each absorption image is background-subtracted and then normalised to an equivalent image taken 50 ms after the atomshave been dispersed, which greatly reduces the influence of technical noise sources on our data.We also carry out additional control experiments to account for the effect of loss of unassociated atoms from the dipole trapduring the magnetic field ramp. If not properly accounted for, this could cause overestimation of the molecular fraction afterthe ramp, since we assume that atoms not seen in the absorption image are associated into molecules. We therefore conduct,for each set of experimental conditions under which we take data, a control experiment in which the magnetic field is rampedacross the Feshbach resonance and then back again, thus dissociating any molecules that were previously formed. This processis time-symmetric, taking twice as long as the unidirectional ramp, and we therefore assume that the fraction of the atomsremaining after this process is equal to the square of the total fraction remaining (in both associated and unassociated forms)after a unidirectional ramp. This allows us to estimate the reduction in apparent atom number that results from atom loss duringthe magnetic field ramp under each set of experimental conditions employed. By dividing the apparent unassociated atomfraction that we measure using absorption imaging by this value, we can thus eliminate the systematic bias resulting from atomloss during the magnetic field ramp. Landau-Zener Transition of a two-level system
The molecule formation via sweeping magnetic field through the Feshbach resonance can be modeled to be a Landau-Zener(LZ) transition. Using a two-state process pictrue, LZ describes the transition under the Hamiltonian H = (cid:18) ε gg δ ( t ) (cid:19) , (S1)where δ ( t ) slowly increases from − ∞ to + ∞ at a constant speed ˙ δ . Analytical solution reveals that the flip, or transition,probability is P = exp (cid:16) − π g / ˙ δ (cid:17) . (S2)Near Feshbach resonance, δ ( t ) = ∆ µ ( B ( t ) − B r ) , where ∆ µ is the difference of magnetic moment, B r is the resonant magneticfield strength. g is the atom-molecule coupling strength. For fermions, it is equal to g = ¯ h (cid:112) π a bg ∆ ∆ µ / m / √ V [S3]. For Li at B r = . ∆ µ = µ B , the resonance width ∆ = −
300 G and background scattering length a bg = − . V is the modevolume. To obtain the volume, we note that typically two-body interactions will change the shape and density of atoms in thetrap. Papenbrock and Bertsch [S4] introduced a parameter ξ B such that the chemical potential is scaled by the Fermi energyof the non-interacting case µ = ξ B E F . The trapping frequency is then scaled by (cid:112) ξ B ω i accounting for the change of effectivetrapping frequency. Then the radii of the atomic cloud read R i = ξ / B a ho ω ho ω i ( N ) / , (S3)yielding the volume of a spherical gas, V = π a ho ξ / B √ N . (S4)In the BCS regime, ξ B is calculated by the Monte Carlo method [S5]. Though depending on trapping profile and particle density, ξ B converges to ≈ .
37 in dilute limit, which is used in the calculation.
MANY-BODY MODEL OF THE ATOM-MOLECULE COUPLING
The formation of bosonic molecules from pairs of fermionic atoms is modelled by a spin-boson coupled system [S6–S8]. TheHamiltonian consists of different molecular states, such that H = ∑ i H i , where Hamiltonian H j reads H j = δ ( t ) ˆ b † j ˆ b j + ε j (cid:16) ˆ c † j ↑ ˆ c j ↑ + ˆ c † j ↓ ˆ c j ↓ (cid:17) + g (cid:16) ˆ b † j ˆ c j ↓ ˆ c j ↑ + H.c. (cid:17) . Here δ ( t ) = ∆ µ ( α t + B ) gives the molecular energy, where α and B are the ramping rate and the initial value of the magneticfield. ε j is the kinetic and trap energy of the atom pair. Here ε j denotes the energy of a pair of . Typically, it can be the harmoniclevels ε j = ¯ h ω ( j + / ) , or free space by replacing j by k , ε k = k / m . δ ( t ) is the molecular energy. If all molecules are in theground state, i.e., forming a molecular BEC, then we can neglect the index j , as all molecules have identical one.When sweeping the magnetic field from 860 G to 707 G, the molecule energy changes from δ ( t ) / ¯ h = −
457 MHz to δ ( t f ) / ¯ h = ε j (Fermi level) is roughly ε F = ¯ h / ( m )( π n ) / ≈ ¯ h × . n = cm − . If we take the full range of magnetic field, the numerical cost in the simulation will be very expensive. Tosimplify the calculation, we have chosen initial value of magnetic field relatively close to the resonance, which captures the LZtransition dynamics. ANDERSON PSEUDOSPIN REPRESENTATION
When the molecules are condensed to the ground, this allows us to further simplify the model using the Anderson pseudospin.The Andreson pseudospin connects the spin operator viaˆ S + i = ˆ c † i ↓ ˆ c † i ↑ ˆ S − i = ˆ c i ↑ ˆ c i ↓ ˆ S zi = ˆ c † i ↑ ˆ c i ↑ + ˆ c † i ↓ ˆ c i ↓ − S xj = ( S + j + S − j ) / S yj = − i ( S + j − S − j ) /
2. The commutationrelations are [ S + j , S − j ] = S zj , [ S + j , S zj ] = − S + j and [ S − j , S zj ] = S − j . The total number 2 N of particles are conserved, 2 N b + ∑ Nj ( S zj + ) = N , which can be seen from the commutation relation [ H , N ] = H = δ ( t ) ˆ b † ˆ b + ∑ i ε i ˆ S zi + g ∑ i (cid:0) ˆ b † ˆ S − i + ˆ S + i ˆ b (cid:1) . (S5)A c -number part ∑ i ε i is discarded, i.e. neglecting the kinetic energy of atom pairs. This is a good approximation when thetemperature is low. The Heisenberg equation of the operator can be obtained, i ∂∂ t ˆ b = δ ( t ) ˆ b + g ∑ i ˆ S − i (S6a) i ∂∂ t ˆ S − i = ε i ˆ S − i − g ˆ S zi ˆ b (S6b) i ∂∂ t ˆ S + i = − ε i ˆ S + i + g ˆ b † ˆ S zi (S6c) i ∂∂ t ˆ S zi = g (cid:0) − ˆ b † ˆ S − i + ˆ S + i ˆ b (cid:1) (S6d)As the total number of atoms is conserved, one can solve the dynamics numerically when N in the order of a few hundred tothousand with a normal desktop PC. For numbers close to the experimental situations (hundreds of thousands of atoms), onemany use the mean field theory, i.e. decoupling mean value of operator products as (cid:104) AB (cid:105) ≈ (cid:104) A (cid:105)(cid:104) B (cid:105) to solve the coupled equation. [S1] E. Raab, M. Prentiss, A. Cable, S. Chu, and D. Pritchard, Trapping of neutral sodium atoms with radiation pressure, Phys. Rev. Lett. ,2631 (1987).[S2] A. Paris-Mandoki, M. Jones, J. Nute, J. Wu, S. Warriar, and L. Hackerm¨uller, Versatile cold atom source for multi-species experiments,Rev. Sci. Instrum. , 113103 (2014).[S3] R. A. Duine and H. T. C. Stoof, Atom–molecule coherence in Bose gases, Phys. Rep. , 115 (2004).[S4] T. Papenbrock and G. F. Bertsch, Pairing in low-density Fermi gases, Phys. Rev. C , 2052 (1999).[S5] J. Carlson, S. Gandolfi, K. E. Schmidt, and S. Zhang, Auxiliary-field quantum Monte Carlo method for strongly paired fermions, Phys.Rev. A , 061602(R) (2011).[S6] J. Javanainen, M. Ko˘strun, Y. Zheng, A. Carmichael, U. Shrestha, P. J. Meinel, M. Mackie, O. Dannenberg, and K.-A. Suominen,Collective Molecule Formation in a Degenerate Fermi Gas via a Feshbach Resonance, Phys. Rev. Lett. , 200402 (2004). [S7] E. Pazy, I. Tikhonenkov, Y. B. Band, M. Fleischhauer, and A. Vardi, Nonlinear Adiabatic Passage from Fermion Atoms to BosonMolecules, Phys. Rev. Lett. , 170403 (2005).[S8] E. Pazy, A. Vardi, and Y. B. Band, Conversion Efficiency of Ultracold Fermionic Atoms to Bosonic Molecules via Feshbach ResonanceSweep Experiments, Phys. Rev. Lett.93