Stochastic and equilibrium pictures of the ultracold FFR molecular conversion rate
Tomotake Yamakoshi, Shinichi Watanabe, Chen Zhang, Chris H. Greene
aa r X i v : . [ phy s i c s . a t o m - ph ] M a r Stochastic and equilibrium pictures of the ultracold FFRmolecular conversion rate
Tomotake Yamakoshi, Shinichi Watanabe, Chen Zhang, and Chris H. Greene
2, 3 Department of Engineering Science,University of Electro-Communications,1-5-1 Chofugaoka, Chofu-shi, Tokyo 182-8585, Japan Department of Physics and JILA, University of Colorado,Boulder, Colorado 80309-0440, USA Department of Physics, Purdue University,West Lafayette, Indiana 47907, USA (Dated: April 7, 2018)
Abstract
The ultracold molecular conversion rate occurring in an adiabatic ramp through a Fano-Feshbachresonance is studied and compared in two statistical models. One model, the so-called stochasticphase space sampling (SPSS)[E.Hodby et al ., PRL. → K in all the symmetry combinations, namely Fermi-Fermi, Bose-Bose, and Bose-Fermi cases. Toexamine implications of the phase-space criterion for SPSS, the behavior of molecular conversion isanalyzed using four distinct geometrical constraints. Our comparison of the results of SPSS withthose of ChET shows that while they appear similar in most situations, the two models give riseto rather dissimilar behaviors when the presence of a Bose-Einstein condensate (BEC) stronglyaffects the molecule formation. PACS numbers: 31.15.bt,67.85.Pq,82.60.Hc . INTRODUCTION In recent years, the production and application of ultracold molecules have attractedmuch attention. Ultracold polar molecules in particular are being investigated eagerly be-cause of their great promise for numerous applications such as the quantum simulator[1],ultra high precision measurements[2], ultracold chemical reactions[3], etc . However, adap-tation of standard laser cooling techniques to molecules is not readily achieved. Insteadexperimentalists have employed a scheme to convert ultracold atoms into molecules withoutheating, i.e. the combination of Fano-Feshbach resonance(FFR) and Stimulated Raman Adi-abatic Passage(STIRAP)[4]. In this scheme, two atoms get combined to form a quasi-boundmolecule in a higher electronic state via a FFR. The molecules formed get subsequentlytransferred to the ground state by two color laser pulses. In this situation, the total produc-tion rate is limited by the FFR conversion rate, the transfer efficiency of STIRAP being veryhigh. Developing an understanding of the FFR phase of the ultracold molecule formationprocess thus remains important.The first reported experiment[5] swept the external magnetic field to move the unboundlow temperature atoms through the FFR region. According to the experiments, the molec-ular conversion rate saturates at a very slow magnetic sweep, the so-called adiabatic re-gion. The adiabatic magnetic sweep is used for the ground state molecular productionexperiments[4] in order to maximize the number of molecules at a given temperature. Thispaper focuses on molecular production at the adiabatic region. The Landau-Zener(LZ)model has been used for analyzing the experimental results[5–8]. Detailed discussions of theLZ model and of its extensions are given in Ref.[9].At any rate, Hodby et al .[6], developed a semi-classical Monte Carlo simulation method,called the stochastic phase space sampling(SPSS) method to estimate the relation betweenphase-space density and molecular conversion rate by using a phase-space criterion depen-dent on a single parameter fitted to the experimental data at one temperature. No explicitreference to the LZ-type transition mechanism is made in this approach. Nevertheless theSPSS method has been applied to the K- K[6], Rb- Rb[10], and K- Rb[11, 12] cases,and the results show good agreement with the corresponding experimental data down to thelowest temperature realized in the lab. We note that recently an experiment on K- Rbmolecular production has been carried out in a mixed Bose-Fermi system. The result of the2PSS model, however, does not agree well with the experimental formation rate of moleculesin the region of quantum degenerate temperatures[8].On the other hand, Williams et al .[13] developed a theory of the FFR conversion ratebased on a coupled Boltzmann-equation treatment that includes atom-molecule interactions.S. Watabe and T. Nikuni[14] extended the theory of Williams et al . and proposed the so-called (chemical) equilibrium theory (ChET for short in this paper). The ChET has beenapplied to noninteracting ideal atomic gases trapped by a harmonic potential, resulting inagreement with the Rb- Rb experiment down to the lowest temperature realized in thelab. And they commented in Ref.[14] on a possible difference between the SPSS model andthe ChET, at temperatures below the BEC-critical temperature T c , a vital region that hasnot been experimentally examined in detail thus far. Let us note in passing that no explicitreference to the LZ-type transition mechanism is made in ChET either.As regards the SPSS model, no systematic assessment of its basic assumptions has beenmade. There has been no study of the meaning of the phase-space criterion adopted, nor ofpossible alternative criteria. Here we propose a few new phase-space criteria for comparisonand analyze the temperature dependence of molecular conversion rate for each criterion.In doing so, we present an overview of the statistical evaluation of the ultracold molecularconversion rate through comparison between the SPSS and the ChET models. Furthermore,the molecular conversion rate below T c is estimated since its experimental and theoreticalexploration is much desired. To understand the dependence on thermodynamic distributionsof the atoms and on the phase-space criterion adopted, we choose the number-balanced K system in a 3-dimensional harmonic oscillator as a prototype and analyze the features ofSPSS in phase space. The results are compared with the ChET results. Through thesecomparisons, we clarify the distinct temperature dependence of the molecular conversionrate between the SPSS and ChET models qualitatively. Number-imbalanced systems arealso considered for other effects such as the gravitational sag as well as the gap betweentwo different values of T c . Incidentally, the ChET model with interacting atoms has beendeveloped in Ref.[15]. This current paper, however, focuses on noninteracting systems onlyto understand the overall behavior.The paper is organized as follows. Sec.II outlines the SPSS and ChET models and givessome additional background. Sec.III A discusses the results of the SPSS model for theprototypical K case with four different phase-space criteria. Sec.III B shows the results for3b and KRb cases, systems for which the temperature dependence of the conversion ratehas been investigated experimentally. Sec.III C compares the results of the SPSS model withthe ChET. Sec.IV concludes the paper. This paper uses T c = 0.94 ~ ¯ ωN / /k B for the unitof temperature throughout, where ~ , ¯ ω , N, and k B correspond to the Planck constant, meantrapping frequency, number of bosonic atoms, and the Boltzmann constant respectively.And we define the “molecular conversion” rate as the number of formed molecules dividedby the initial number of minority atoms, namely χ m = N molecule /N atom , minority . II. THEORETICAL MODELS: SPSS AND CHET
The theory of molecular formation in a cold mixture of two atomic species is based onthe following effective Hamiltonian as a common starting point, namely H = Z d r [ X α =1 , ˆ ψ † α ( r ) H ( α ) A ˆ ψ α ( r ) + ˆ φ † M ( r ) H M ˆ φ M ( r )]+ κ Z d r { ˆ φ † M ( r ) ˆ ψ B ( r ) ˆ ψ A ( r ) + h . c . } + g A Z d r ˆ ψ † A ( r ) ˆ ψ † B ( r ) ˆ ψ B ( r ) ˆ ψ A ( r )where ˆ ψ A and ˆ ψ B are the atomic field operators for the two types of atoms, and ˆ φ M is themolecular field operator[16]; H ( α ) A = − ~ m α ∇ + U α ( r )and H M = − ~ m A + m B ) ∇ + U M ( r ) + ǫ res represent the single-particle Hamiltonian for atoms and molecules, respectively. Here theatomic masses m A and m B differ in general. U α ( r ) and U M ( r ) are the external potentials forthe trap and gravity, respectively. The energy ǫ res of the resonant molecule is tuned close toa collision threshold in experiments by varying an external magnetic field B . The constants κ and g A pertain to the association/dissociation of an atomic pair into/from a molecule,and the atom-atom interaction, respectively. The LZ-type transition leads to κ , but bothSPSS and ChET bypass the use of the interaction term by employing physically-motivatedassumptions. The atom-atom interaction term is dropped in what follows. In the mixtureof two types of atomic gas, the interchannel transition for molecular formation is effective4hen the magnetic field sweeps across the resonance energy. The experimentally observedand reported molecular conversion rate at a given temperature pertains to the saturatedconversion rate that is measured at a reasonably slow magnetic sweep rate. Let us considerfirst the moment when the magnetic field is far away from the Fano-Feshbach resonanceregion so that the relevant classical Hamiltonian is that of an aggregate of independentatoms in the harmonic oscillator trapping potentials. The quantum distribution functionsare given by f α = 1exp[( H α − µ α ) /k B T ] ± . (1)where the Hamiltonian for a single atom of type α is H α = 12 m α ~p + 12 m α ω α ~r , (2)where “+” corresponds to fermions, and “ − ” to thermal bosons.We extend the previously developed SPSS in order to treat the condensate bosons (BEC)using the truncated Wigner approximation to the phase-space distribution. The ChET onthe other hand does not require a specific distribution function for BEC but rather, only thechemical potential. Let us employ the ground state wave function of the harmonic oscillatorto represent a noninteracting BEC and apply the Wigner transformation using φ ( ~r ) ∝ exp (cid:16) − m α ω α ~ ~r (cid:17) . (3)This yields P α ( ~r, ~p ) = ~ π Z V φ ∗ (cid:18) ~r + ~q (cid:19) φ (cid:18) ~r − ~q (cid:19) e i~p · ~q/ ~ d~q ∝ exp (cid:18) − ~p m α ω α ~ − m α ω α ~ ~r (cid:19) . (4)This probability distribution does not depend explicitly on the number of atoms unlike theThomas-Fermi(TF) wave function [10]. On the other hand, the number of BEC atoms inthe harmonic oscillator trap is given analytically by N α = " − (cid:18) TT c (cid:19) N α , (5)where N α , and N α correspond respectively to the number of BEC atoms and the totalnumber of atoms of type α at given temperature T . The probability of finding a condensateatom is then N α P α ( ~r, ~p ). 5efore going farther, note that the BEC transition temperature T c differs in general forthe two types of atoms due to the mass difference. The TF (Thomas-Fermi) wave functionfor a repulsively interacting BEC tends to spread out more and get flatter near the center ofthe trap so that the distribution tends more toward smaller momentum values than in thenoninteracting Eq. (4). To calculate the molecular conversion rate, Papp and Wieman[10]represented the effect of BEC by assuming that if a pair of atoms find themselves within theTF radius, they then form a molecule regardless of their relative momentum. Thus Ref. [10]does not generate a phase-space distribution. Instead, they use only the spatial extent ofthe TF distribution without setting any criterion on the momentum. A. SPSS: Stochastic Phase Space Sampling
The SPSS is a method that samples candidate atoms for molecular formation by semi-classical Monte Carlo simulation. Ref. [6] imposed a certain reasonable-looking phase-spaceconstraint on the initial atom-pair distribution. The conversion rate is calculated under thefollowing two assumptions.[1] Only the atomic pairs satisfying a given phase-space criterion can form a molecule.[2] Once a molecule is formed, it does not dissociate into atoms. Indeed, the lifetime of amolecule is known to be considerably longer than the time scale of an experiment[17].
FIG. 1: (Color online) Conceptual diagram representing “Stochastic Phase Space Sampling(SPSS)”. See steps [1]-[4] described in the text.
A simulation is implemented in the following four steps, illustrated in Fig. 1:[1] Distribute atoms randomly in phase space in accordance with the thermodynamicequilibrium distribution at the initial temperature T , Eq.(4). This temperature is6xed, meaning that the process is regarded as isothermal. In the presence of a BEC,Eq.(5), is used for the condensed atoms.[2] Search for atomic pairs that satisfy the phase-space criterion. Avoid double-countingby erasing the used atoms from the list of candidates.[3] Count the number of atomic pairs found in the random search. Regard it as thenumber of the formed molecules.[4] Return to Step [1], and repeat the steps until the statistical noise is reduced belowthe target noise level.Since the phase-space volume made available by a geometrical constraint plays a key rolein SPSS, one of our goals is to assess the sensitivity of the method to the criterion adoptedfor pair formation. In what follows we define four independent phase-space criteria, the firstone being the version initially employed by Ref.[6] which yielded an extremely good fit to theexperimental results for the Bose gas and for a two-component Fermi gas. As we will see inthe discussion in Sec. III, all these separate criteria reproduce the temperature dependenceof the experimental molecular conversion rate satisfactorily at thermal temperatures, butshow some departures below the BEC critical temperature.Let us introduce the following four phase-space criteria;Criterion 1: | ∆ r ∆ p | < γ q h, (6)Criterion 2: | ∆ ~r × ∆ ~p | < γ l ~ , (7)Criterion 3: ( | ∆ x ∆ p x || ∆ y ∆ p y || ∆ z ∆ p z | ) / < γ v h (8)Criterion 4: | ∆ x i ∆ p i | < γ s h ( i = x, y, z ) (9)where ∆ r corresponds to the spatial separation of an atom pair and ∆ p = m rm ∆ v , where m rm equals the reduced mass, ∆ v corresponds to the relative velocity, and γ q , γ l , γ s , and γ v are the pairing parameters which are assumed to be independent of temperature and7ensity. Criterion 1 is introduced on the basis of the number of accessible quantum statesin the relative coordinate. Criterion 2 is introduced under the premise that the low partialwaves are important for molecular formation. The use of ~ instead of h for Criterion 2 is thusintentional. Criterion 3 is similar to criterion 1, but represents the full phase-space volumeoccupied by the pair. Criterion 4 is introduced as a counterpart to criterion 3, actually amost familiar form in statistical mechanics. In any case, there is no a priori numerical ortheoretical support for any one of these criteria in the present context of molecular formation.As for the accuracy targeted in the present paper, the simulation program generatesprobability distributions, as given above by Eq.(1), by the rejection sampling, and thensearches for all pairs that meet the phase-space criterion. The period of the random numbergenerator is about 2 . × for this study. This is sufficient for dealing with 10 atoms.The search ends when no remaining atomic pair satisfies the criterion. These searches arerepeated a sufficient number of times, and the results are averaged. The error is of the orderof the inverse of the square root of the number of atoms times the number of iterations.Throughout our calculations this error is reduced to within 1 %. We shall return to thediscussion of these criteria in Sec. III. B. ChET: Equilibrium Theory
The ChET was developed by E.Williams et al .[13], and extended further by S. Watabe etal .[14]. It is based on the result of the coupled atom-molecule Boltzmann equation approach.In the ChET, one solves simultaneous equations consistently with equilibrium conditions toobtain the molecular conversion rate. The first equation is the conservation of the totalnumber of atoms, N tot = N A + N B + 2 N M (10)and the second equation states the constancy of the number of each atomic species, hence α = N B + N M N A + N M (11)where N A , N B , and N M denotes the number of the majority atoms, of the minority atoms,and of the molecules, respectively. And the ratio α is defined by the initial ratio N B /N A ,where we consider N A ≧ N B , where N M = 0 thus 1 ≧ α . There are two equilibrium8onditions, one representing chemical equilibrium, µ A + µ B = µ M + δ (12)where µ denotes the chemical potential of each component, and δ denotes the detuning,namely the energy difference between dissociated atomic state and the molecular boundstate. The other condition represents thermal equilibrium, T A = T B = T M , where T denotesthe temperature of each component. The population of each component is a function of µ and T . ChET uses two assumptions. One is that the magnetic sweep process is adiabatic,thus the total entropy of the system is conserved. And the other one is that molecularproduction halts at δ =0, due to the conservation of momentum and energy. So the methodfirst calculates the total entropy at δ → ∞ at a given T and α , and then traces the adiabaticstate until δ =0. Thus the molecular conversion rate is given by equating the total entropyat δ → ∞ and that at δ =0, S A ( T ∞ , µ ∞ ,A ) + S B ( T ∞ , µ ∞ ,B ) = S A ( T , µ ,A ) + S B ( T , µ ,B ) + S M ( T , µ ,A + µ ,B ) , (13)where S denotes the entropy of each component. III. DISCUSSIONS
Features of the molecular conversion rate are now explored for three statistically distincttypes of atom-atom pair, namely Fermi-Fermi, Bose-Bose, and Bose-Fermi. To this end,we employ K as a prototype for our analysis, considering K for the boson, and K forthe fermion. The K system simplifies the situation on two accounts. Firstly, there isnegligible gravitational sag thanks to the nearly equal masses of the isotopes. Secondly, wemay presume that there are no more than two characteristic temperatures in the K system,namely the Fermi temperature T F and the Bose-Einstein condensation point T c [18]. It is alsoworthwhile pointing out that in a theoretical proposal as well as in a recent experiment with K- Rb[19], the authors of the experimental study employed an optical dipole trap withthe so-called “magic frequency”, which eliminates the gravitational sag. For this reason, themolecules K- K and K- Rb can be considered equivalent under an appropriate scalingof parameters[19]. At any rate, the issues gravitational sag and the gap in T c are exploredas supplementary items later in Subsection B.9et us now compare the results of SPSS models and that of the ChET. To begin with,observe that the ChET is based on entropy conservation during an adiabatic magnetic sweepand on the condition that the statistical distribution of atomic pairs and that of moleculesare always in equilibrium. The SPSS model presumes the conservation of the two-bodylocal phase-space volume but no equilibrium. However, when atomic pairs are chosen theSPSS appears to be adiabatic since conservation of the phase-space volume amounts toadiabatic invariance and thus to conservation of entropy. But consistency with thermalequilibrium is not met. In the presence of an atomic BEC in the Bose-Bose and Bose-Fermicases, ChET assumes the chemical potential of the bosonic atom component is 0, so thatthe conversion rate gets flat-lined. However, in the presence of two atomic BECs in theBose-Bose case, ChET reveals certain temperature dependence of the conversion rate. Forthe sake of comparison, we determine the molecular conversion rate using the proceduredeveloped in Ref. [14] and review it in Sec. III C. A. Features with Prototypical K Molecules
The following specific systems are considered. • Fermi-Fermi: K( f = , m f = − )- K( f = , m f = − ) • Bose-Bose: K- K • Bose-Fermi: K- KNote that experimental data on K formation is known to us only for the Fermi-Fermisystem[6]. TABLE I: Experimental parameters from [6] used in the present simulation. N K is the number ofK atoms, ω trap the trap frequency, and γ ’s pertain to the criteria. N K ω trap γ Q γ L γ V γ S × π × (470, 470, 6.7) Hz 0.19 0.26 0.0085 0.050 For the isotropic trap considered in this paper, we rescale the length and momentumaccording to the trap frequencies. The values of γ q , γ l , γ v , and γ s are thus determined bythe fitting to the experimentally available K( f = , m f = − ) K( f = , m f = − ) data,10nd are used for the other Bose-Bose and Bose-Fermi cases as well. Table I shows the valuesof various parameters employed in this section for K . The accessible phase-space volumefor the atomic pairs increases as γ increases. The fitting parameters thus become smaller inthe order of γ Q > γ S > γ L / π > γ V .
1. Temperature dependence of the conversion rate
Stereotypical behavior of the conversion rate is summarily displayed as a function of tem-perature in Fig 2 for the four criteria altogether. The thermal limits look similar for the
FIG. 2: (Color online) SPSS molecular conversion rates for four criteria shown altogether. Thered line corresponds to the result of Criterion 1, green to Criterion 2, blue to Criterion 3, andpink to Criterion 4. From left to right the panels correspond to K- K, K- K, and K- K,respectively. three types of systems. It is probable that this is due to the fact that the Hamiltonian andthe geometrical constraints are all quadratic in r and p . And also the chemical potentialplays little role in the thermal limits. Then the relevant quantities can all be expressed interms of temperature-scaled distance and momentum, namely ˜ r = r/ √ T and ˜ p = p/ √ T .The magnitude of the formation rate depends then on the value of γ / T which controls thephase-space volume. Specific temperature dependences become apparent when the chemicalpotential µ ( T ) plays a role, namely at T ≤ T c for boson and at T ≤ T F for fermion wherethe chemical potential changes sign. See Fig.2. In the Bose-Bose case the conversion rategrows close to 100% as T →
0, suggesting complete overlap of the two BECs in this limitwhich makes it easy to satisfy the imposed criterion. However, the slope of the molecularconversion rate is discontinuous at T c because of the BEC phase transition across this tem-perature. In the Fermi-Fermi case, the conversion rate rises monotonically as T lowers. The T =0 limit is finite, however. No discontinuity in the slope occurs at T F since there is no11hase transition. (Note that the Fermi temperature T F =1.93 T c is out of the range displayedin this figure.) The Bose-Fermi case shows a noticeable discontinuity in slope because of theBEC phase transition and of the subsequent diminishing overlap in phase space.
2. Relevant single- and two-particle phase-space regions
Let us begin this subsection with the following question: Should the four alternativecriteria pick out similar regions of the phase space when γ Q , γ L , γ V , and γ S are each madeto fit to the observed molecular conversion rate? Let us look into the initial phase-spacedistribution of atoms that lead to molecular formation by comparing the results of the fourcriteria. FIG. 3: (Color online) Master diagram for Q(Criterion1) showing phase-space distribution ofmolecule-forming pairs, ρ (∆ r, ∆ p ) Eq. 14. Each consists of three panels corresponding to T > T c , T ∼ T c , and T < T c . The triplet of distributions are arranged so that the left one is the Bose-Bosecase, the middle one the Fermi-Fermi case, and the right one the Bose-Fermi case. The horizontalaxis shows the relative spatial separation ∆ r , while the vertical axis shows the relative momentum m red × ∆ v . The brighter the color, the higher the density. γ q =0.19 as in Table. I. To this end, we define the “two-particle” distribution function represented by the numberof candidate pairs averaged over the implemented iterations, N iter , within a suitably cho-sen phase-space domain D (∆ r, ∆ p ) of a moderately small volume in the neighborhood of12 IG. 4: (Color online) Master diagram for L(Criterion2) showing phase-space distribution ofmolecule-forming pairs, ρ (∆ r, ∆ p ) same as Fig. 3. γ q =0.26 as in Table. I.FIG. 5: (Color online) Master diagram for V(Criterion3) showing phase-space distribution ofmolecule-forming pairs, ρ (∆ r, ∆ p ) same as Fig. 3. γ q =0.0085 as in Table. I. (∆ r, ∆ p ), ρ (∆ r, ∆ p ) = 1 N iter X j N [(∆ r j , ∆ p j ) ∈ D (∆ r, ∆ p )] (14)Figs. 3, 4, 5, and 6 are our master diagrams showing ρ (∆ r, ∆ p ) comprehensively. Eachpanel consists of the three symmetry cases. Down each column, the temperature varies from13 IG. 6: (Color online) Master diagram for S(Criterion4) showing phase-space distribution ofmolecule-forming pairs, ρ (∆ r, ∆ p ) same as Fig. 3. γ q =0.050 as in Table. I. T > T c to T < T c . And a represents the Bohr radius.Next consider the physical implications of the way the candidate pairs are distributed.First, we consider how the candidate pair distributions differ, depending on the combinationof atomic species. In the Fermi-Fermi case, the shape of the candidate pair distributionsremains unchanged throughout these temperature regions. In the Bose-Bose case, we cansee a change in the pair distribution below T c , namely a reduction in size toward the origin.This change comes from the condensation of the bose gases where their phase space volumesbecome very small. In the Bose-Fermi case, there are special features near zero temperature.And these features depend critically on the phase-space criterion. Above T c on the otherhand, there are no marked differences in the candidate pair distributions for the threesymmetry cases because they are all representable by the Maxwell-Boltzmann distribution.One major difference throughout the diagram is that Criteria 2, 3, and 4 permit more widelyspread distributions than does Criterion 1. Stated somewhat differently, the distributionssimulated by Criteria 2, 3, and 4 are generally sparse with their maxima roughly in theregions cut away sharply by Criterion 1. The sharp cut-aways with respect to Criteria 2,3, and 4 do not introduce sharp edges in this ∆ r -∆ p representation unlike Criterion 1.Incidentally, the converse is not necessarily true. Let us note that in the phase space ofangular momentum and relative kinetic energy, both Criterion 1 and 2 show sharp edges.14he maximum value of the angular momentum corresponding to the edge is about 1.2 ~ so that the s-wave contribution is dominant, but with nonnegligible p-wave contribution.In all the panels, the high concentration of most likely candidates appear along similar-looking hyperbolic-shaped arcs near the origin for the Bose-Bose and Fermi-Fermi cases ifon different scales. FIG. 7: (Color online) The upper two figures represent phase-space density distributions of indi-vidual components; (a) the fermion to the left and (b) the boson to the right at T =0.1 T c . (c)shows the density of the molecule-forming pairs under Criterion 1. (d) shows the density of thefermion cut away by Criterion 1 when the boson is assumed to be concentrated entirely at theorigin. These two resemble each other. On the other hand, the Bose-Fermi case exhibits a considerably different appearance. Toaid in understanding, plotted in the two upper panels of Fig. 7 are the phase-space densitydistributions of the individual atoms in single-particle phase space. The one on the leftrepresents the fermion, and the right one the boson which is condensed near the origin. Asa result, the molecular formation is restricted to the narrow overlapping regions of thesesingle-particle distributions. The lower two panels of the same figure mark the fermionicatoms that contribute to the molecular formation simulated by Criterion 1. The one on theleft shows the distribution of the fermion atoms in the formed molecules which is extractedin the course of numerical simulations. The one on the right shows the same distributionbut constructed by applying Criterion 1 to the single-particle distributions of the upper15wo panels. We note that they look rather similar. The boson (BEC) distribution beingconcentrated near the origin, the criterion is naturally met either at short distances or atsmall momentum as in Fig. 3. (Bose-Fermi case at the lower temperature column.)There is another feature worth noting for SPSS. The likely distance between the atoms inthe initial pair turns out to be much larger than the experimentally known final size of themolecule whose diameter is comparable to the scattering length at the final magnetic field.To demonstrate this point, Fig. 8 shows the reduced distribution obtained by summing overthe momentum at T =0.1 T c . The likely distance is considerably greater for Criteria 2, 3, and4 than for Criterion 1. This observation reflects certain traits of the SPSS approximation.The molecular formation takes place within the finite duration of the adiabatic sweep whichChET presumes is sufficiently long for thermalization. The SPSS prespecifies the phase-space volume that would be involved in the isothermal transport (which happens to be also“adiabatic” in SPSS because there is no change in the phase-space volume). The thermaldistribution for each component of the final state can be evaluated at the temperature of theinitial state, but then the system cannot be in chemical equilibrium. This is where the SPSSand ChET differ, yet the molecular conversion rate comes out surprisingly similar above T c . Let us note in passing that as the magnetic field is swept, the scattering length growsrapidly near the resonance. Across the unitarity limit, the scattering length grows withoutlimit, causing even the widely separated pairs to interact. However, without dynamicalcalculations, it is not possible to assess the contribution of this resonant region. FIG. 8: (Color online) Pair density distributions as functions of spatial separation ∆ r at T =0 . T c . From left to right, Fermi-Fermi, Bose-Bose, and Bose-Fermi. Each solid line pertains to theindicated Criterion. All the peaks representing the initial separation are significantly larger thanthe experimentally known final size of the molecule. Now consider very low temperatures below T c , as we explore the case T =0.1 T c in what16ollows. Figs. 3-6 summarize the phase-space distributions under consideration. Fig. 8 plotspair density distributions as functions of the spatial separation ρ s (∆ r ) = R ρ (∆ r, ∆ p ) d ∆ p .In the Fermi-Fermi case, the maximum appears near the origin, and then the number ofcandidate pairs decreases rapidly as the spatial and relative momentum separations increaseaway from the origin. This trend is independent of the applied criterion, but the height of thepeak and the decline in distribution depend on the criterion. Because of the sparsely spreaddistribution as seen in Figs. 4-6 away from the origin, the initial distance distributionsof criteria 2, 3, and 4 are seen to decrease more gradually compared to criterion 1. Inthe Bose-Bose case, pair density distributions ρ s (∆ r ) show similar tendencies because two-particle distributions have the same trend, a consequence of the small size of the BEC for allthe four criteria. In the Bose-Fermi case, a structure appears in addition to the peak nearthe origin. In Criterion 1, there are two peaks, one coming from atomic pairs with smallspatial separation and high relative velocity, and the other one with small relative velocityand large spatial separation as noted earlier in relation to Fig. 7. In the other criteria 2,3, and 4, they also have double-peaked structure but the profile of these peaks stronglydepends on the criterion. For example, the outer peak, somewhat broad and round, reachesthe highest value for Criterion 3. This profile comes from the elliptic-shaped structure seenin the low right panel of Fig. 5. In any combinations of thermodynamic functions, all thepeaks yield ∆ r much larger than the experimentally estimated final size of the molecule.SPSS merely enumerates the candidate pairs no subsequent dynamics is implemented.To summarize, we have explored the one- and two-particle distribution functions andhave observed the following. • The two-particle distribution function behaves more or less the same under all the fourcriteria examined except for the Bose-Fermi combination at temperatures below T c . • For the Bose-Fermi system, this strong dependence on the imposed criterion is reflectedin the molecular conversion rate. • Compared to the molecular size, the peak(s) in the two-particle distribution functioncorresponds to an atomic pair separation which exceeds the known size of the molecule.17 . Other Features
Two aspects of the molecular conversion rate are now analyzed. One is the case wherethere exist two distinct BEC critical temperatures, and the other is the case of a differentialgravitational sag that partially separates the two species. We begin with the former. Asthe temperature is lowered, the condensation of the component with higher T c introducesa decline in the molecular production since the other component still remains in a thermaldistribution. In a real experiment, as was discussed in Ref. [10] with Rb- Rb, the secondcomponent fails to attain lower temperature due to technical difficulty and remains thermal.In such a case, the decline continues until T =0. See the left panel in Fig. 9. The Fermi-Fermi case is not treated since no conspicuous features arise due to the absence of any phasetransition across T F . FIG. 9: (Color online) SPSS conversion rates for Rb- Rb using Criteria 1, 2, 3, and 4. Theleft panel assumes the same temperature for the two components. The right one simulates thesituation where Rb fails to get cooler at T =2.2 T c [10]. To understand the implications of gravitational sag, it is instructive to see the phase-spacedensity as a function of temperature for K- Rb. The BEC phase transition marks the pointwhere the growing spatial gap between the two atomic species leads to almost nil conversionrate at lower temperatures. See Fig. 10 for the phase-space density representation of howthe overlap evolves as T lowers. The gravitational sag can be made insignificant, however,by choosing an appropriate optical trap frequency referred to as the “magic” frequency inRef. [19]. 18 IG. 10: (Color online) The left panel shows the SPSS conversion rate for the K- Rb case. Theright column of panels shows the evolution of the phase-space density distributions for K and Rb as functions of temperature. The upper one is for
T > T c , and the lower one for T < T c .The rather large mass difference in K- Rb results in almost nil conversion rate toward T → C. Comparison with ChET
The ChET model is now applied to Fermi-Fermi, Bose-Bose, and Bose-Fermi cases withnumber ratio α set to either 1 or 2/15, and compared with the results of the SPSS model forcriterion 1. The α = 1 case corresponds to the Fermi-Fermi experiment[6], and the α = 2 / FIG. 11: (Color online) Temperature dependence of the molecular conversion rate for the Fermi-Fermi case. The red line corresponds to the result of ChET, and the green corresponds to the SPSSresult. (a) the α =1 result, (b) the α =2/15 result. In the figure for α =2/15, the characteristictemperature T F corresponds to the Fermi temperature for the majority atom. The results for the Fermi-Fermi case are shown in Fig. 11. The ChET produces a con-version rate that increases with decreasing temperature. And at T →
0, it always reaches1900% and is independent of α . SPSS shows a similar trend, but indicates a saturation effectat near zero temperature as seen in Fig. 11 (a). Whether the saturation effect appears ornot in the SPSS depends on α .S. Watabe et. al. [20] also applied ChET with resonant interaction within the limita-tions of the mean field approximation. Their result indicates that the saturation effect issmall. They concluded that the resonant interaction introduces a suppression of moleculeconversion, which reduces the conversion rate somewhat from its maximal rate. FIG. 12: (Color online) The same as Fig. 11, except here for the Bose-Bose case. Upper panels cor-respond to δ =0 − , lower to δ =0 + . For the α =2/15 case, T c corresponds to the critical temperaturefor the major bosonic component. In ChET, T c gets shifted and the plateau regions emerge. In the Bose-Bose case, both SPSS and ChET show similar behavior at thermal tem-peratures except for a subtle difference stemming from the relationship between the initialtemperature T and the BEC transition temperature T c in the latter theory. See the encir-cled features in Fig. 12. ChET assumes that the system follows the isentropic curve as themagnetic field is swept so that the final temperature differs from the initial one. This causesthe difference in the number of atoms in the BEC as reflected in the shift of T c , in contrastto SPSS. The ratio r c = T c ( t final )/ T c ( t initial ) can be simply evaluated near the two limitingcases. Setting T equal to the condensation temperature of the major atomic component at δ =0, and µ B, = µ M, and applying an approximation to the chemical potential, we obtainfrom Eq.13 at α ≃ ζ (4) αζ (3) r c = 2(2 − α ) ζ (4) αζ (3) + 4 + ln (cid:18) − ααζ (3) (cid:19) − αζ (3)8(2 − α ) , (15)and at α ≃ ζ (4) αζ (3) r c + ln (cid:18) r c αζ (3) (cid:19) − αζ (3)8 r c = 2(2 − α ) ζ (4) αζ (3) + ln (cid:18) − ααζ (3) (cid:19) − αζ (3)8(2 − α ) . (16)20hese equations could allow a simple estimation of the shift in T c . Thus at α =1, r c =0.91 andat α =2/15, r c =0.99. Fig. 13 compares these approximate analytical results with numericalones. This predicted region of a shift in T c , where the trend of the conversion rate suddenlychanges, could provide an experimental test for the validity of ChET. FIG. 13: (Color online) Comparison between approximations and a numerical result for the ratio r c . At α ≃
1, both initial atomic clouds have a BEC component. However at α ≃
0, only themajority component has a BEC component. This causes differences in the ratio r c to appear as afunction of α . Moreover, in ChET a plateau appears when the temperature lies between the two valuesof T c for α = 2 /
15 as displayed in Fig. 12. In ChET, the chemical potential of the majorityatoms goes to 0, and thus the chemical potentials of the rest, i.e. that of the minorityatoms and of molecules must become equal(Eq. 12). So in this situation, the conversion ratebecomes independent of temperature while the rate attained depends on the trap frequencyof the minority atoms and that of the molecules. Below the lower critical temperature,there are two possibilities; one is when the atomic BEC is more stable than the molecularBEC, and the other is the opposite situation. In the former case, the SPSS conversion ratedecreases with decreasing temperature, and it goes to zero at T →
0, and in the latter caseit goes to 100% as T=0. And the results of ChET behave similarly.A marked difference between the two methods appears in the Bose-Fermi case as inFig. 14. Consider a system with α =2/15 for which there are more bosons than fermions.The SPSS conversion rate begins to decrease at T c for both α = 1 and 2 /
15. However therates of descent differ in two cases. For α = 1 the decline is steady, but for α = 2 /
15, theSPSS conversion rate is almost flat, decreasing very slowly with decreasing temperature;but it begins to drop suddenly at the temperature where the number of fermions exceeds21
IG. 14: (Color online) Temperature dependence of molecular conversion rate for the Bose-Fermicase, the same as in Fig. 11. In the ChET, shift of the T c and the plateau regions appear as inFig. 12. the number of thermal bosons. But in ChET the conversion rate gets flat-lined below T c .We emphasize that the temperature dependence of the molecular conversion rate comes outmore or less the same for SPSS and ChET, but a noteworthy difference manifests itself in theBose-Fermi case below T c , and it would be especially desirable to have more experimentaltests available in this regime. IV. CONCLUSIONS
This work has extended the SPSS analysis of the molecular conversion rate in a quadratictrap, including temperature ranges where no experimental explorations have been made todate. In particular, the temperature dependence of the conversion rate below T c has beenconsidered at great length. This extension has introduced four distinctive geometrical con-straints for the pairing criterion in phase space that controls which atomic pairs contributesignificantly to the molecular formation. Each constraint contains, apart from the geomet-rical information, a single parameter γ which serves to fix the overall magnitude of thephase-space volume for molecular formation. Our study has examined the sensitivity of themolecular conversion rate to the geometry and to the magnitude of the parameter γ . Auseful tool for analysis is the quantity we have denoted the “two-particle distribution func-tion”, which is the probability distribution of the molecule-forming pairs in their relativecoordinate | ∆ ~r | = | ~r A − ~r B | and relative momentum | ∆ ~p | defined similarly. Our investigationshows that once the parameter γ is fitted to the experiment at T higher than T c , then theSPSS method reproduces the experimental results well in the known temperature rangesirrespective of the constraints (or criteria) used (Sec.III A).22he benchmark system considered is K+K → K in all the three symmetry combinationsFermi-Fermi, Bose-Bose, and Bose-Fermi, and the results of the calculations and analysisare the following. • The two-particle distribution function behaves more or less the same under all the fourcriteria examined except for the combination of Bose-Fermi at temperatures below T c . • For the Bose-Fermi system, this strong dependence on the imposed criterion manifestsitself in the molecular conversion rate. • For all the three cases, the peak(s) in the two-particle distribution function corre-sponding to the separation of a pair of atoms exceeds the known size of the molecule.Thus, the Bose-Fermi case proved different because the single-particle distribution for thebosonic atoms and that for the fermionic atoms behave totally differently as they undergocondensation (Sec.III A 2). Temperatures below T c remain a challenge for SPSS.The results of SPSS have been compared with those of ChET. This comparison exhibitsa very similar temperature dependence of the conversion rate in Fermi-Fermi and Bose-Bosecases. One marginal difference is that in the Fermi-Fermi case the conversion rate of ChETgoes to 100% at T →
0, however in the SPSS model the saturation appears below 100%.In the Bose-Bose case, there are two limits as T →
0, depending on the value of the BECchemical potential, i.e. exothermic δ < δ >
0. The molecular conversionrate tends either to 100% or to 0%. As a result, there is no substantial difference in thetemperature dependence of the conversion rate between either theory. And in the Bose-Fermi case the conversion rate as a function of temperature has a different structure below T c . ChET yielded a flat-lined conversion rate whereas SPSS gives a slowly temperature-dependent conversion rate whose behavior is further dictated by the value of α . This paperdoes not consider effects of inelastic collision which may play an important role below T c .Further theoretical investigation is needed for deeper understanding of the dynamical behav-ior of the molecular conversion and an assessment between the theories and the experiments.This work has revealed what aspects of the atom distributions in phase space affect theconversion rate and how. The gained information may serve to give an insight when rigorousquantum transport theory is effected to visualize the phase-space evolution of the systemduring the magnetic sweep. After all, experimental effort for going below T c to examine thetemperature dependence of the conversion rate appears seriously needed.23his work was supported by Grants-in-Aid for Scientific Research (NO.22340116 andNO.23540465) from the Ministry of Education, Culture, Sports, Science and Technology ofJapan. S. W. acknowledges support from a JILA Visiting Fellowship in the summer of 2010.T. Y. acknowledges support from the JSPS Institutional Program for Young ResearcherOverseas Visits. The group of C. H. G. and C. Z. has been supported in part by the U.S.National Science Foundation. [1] A. Micheli, G. K. Brennen, and P. Zoller, Nature Phys. , 341(2006)[2] D. DeMille, S. B. Cahn, D. Murphree, D. A. Rahmlow, and M. G. Kozlov,Phys. Rev. Lett. , 023003(2008)[3] K.-K. Ni, S. Ospelkaus, D. Wang, G. Quemener, B. Neyenhuis, M. H. G. de Miranda,J. L. Bohn, J. Ye, and D. S. Jin, Nature , 1324(2010)[4] K.-K. Ni, S. Ospelkaus, D. J. Nesbitt, J. Ye, D. S. Jin, Physical Chemistry Chemi-cal Physics , 9626(2009)[5] C. A. Regal, C. Ticknor, J. L. Bohn, and D. S. Jin, Nature , 47(2003);K. E. Strecker,G. B. Partridge, and R. G. Hulet, Phys. Rev. Lett. , 080406(2003)[6] E. Hodby, S. T. Thompson, C. A. Regal, M. Greiner, A. C. Wilson, D. S. Jin, E. A. Cornell,and C. E. Wieman, Phys. Rev. Lett. , 120402(2005).[7] A.-C. Voigt, M. Taglieber, L. Costa, T. Aoki, W. Wieser, T. W. H¨ansch, and K. Dieckmann,Phys. Rev. Lett. , 020405(2009).[8] T. D. Cumby, R. A. Shewmon, M.-G. Hu, J. D. Perreault, and D. S. Jin, Phys. Rev.A. ,012703(2013). In the paper, the authors mention that the failure of the SPSS model in thequantum degenerate region is may be due to heating by inelastic collision.[9] The Landau-Zener (LZ) crossing theory to each independent atomic pair is believed to beapplicable to the inelastic transition to the Fano-Feshbach molecular bound state. The two-body LZ model predicts that the molecular conversion rate goes to 100% at the adiabaticregion, but the actual experimental results do not agree. For one thing, a theory showsthat many-body effects alter the two-body picture, causing non-LZ behavior. See for ex-ample A. P. Itin and S. Watanabe, Phys. Rev. E. , 026218(2007). In addition, the LZmodel in the presence of atom-atom and atom-molecule interactions suggests an obstruction f 100% molecular conversion at the adiabatic region when temperature goes to 0. See J. Liu,L.-B. Fu, B. Liu, and B. Wu, New. J. Phys. , 123018(2008). However, in this paper weconsider non-interacting systems for brevity. See also B. Borca, D. Blume and C. H. Greene,New J. Phys. ,111(2003); J. von Stecher and C. H. Greene, Phys. Rev. Lett , 090402(2007);C. Zhang, J. von Stecher and C. H. Greene, Phys. Rev. A , 043615(2012); E. Altman andA. Vishwanath, Phys. Rev. Lett , 110404(2005)[10] S. B. Papp, and C. E. Wieman , Phys. Rev. Lett , 180404(2006)[11] J. J. Zirbel, K.-K. Ni, S. Ospelkaus, T. L. Nicholson, M. L. Olsen, P. S. Julienne, C. E. Wie-man, J. Ye, and D. S. Jin, Phys. Rev. A. , 013416(2008).[12] Michele Lynn Olsen, PhD thesis, University of Colorado(2009)[13] J. E. Williams, N. Nygaard, and C. W. Clark, New J. Phys. , 123(2004); J. E. Williams,N. Nygaard, and C. W. Clark, New J. Phys. , 150(2006)[14] S. Watabe, and T. Nikuni, Phys. Rev. A. , 013616(2008)[15] T. Nishimura, A. Matsumoto, and H. Yabu. Phys. Rev. A. , 063612(2008).[16] Y. Ohashi and A. Griffin, Phys. Rev. Lett. , 130402 (2002); J. E. Williams, T. Nikuni,N. Nygaard, and C. W. Clark, J. Phys. B: At. Mol. Opt. Phys. , L351(2004)[17] C. A. Regal, M. Greiner, and D. S. Jin, Phys. Rev. Lett. , 083201(2004)[18] In the case of two bosonic isotopes such as K and K used in actual experiments, thereare two different condensation temperatures. Likewise two fermionic isotopes give rise to twodifferent Fermi temperatures. Of course, the differences are negligible for our purpose.[19] Silke Ospelkaus-Schwarzer, PhD thesis, Universtat Hamburg(2006); C. Ospelkaus, S. Os-pelkaus, L. Humbert, P. Ernst, K. Sengstock, and K. Bongs, Phys. Rev.Lett. , 120402(2006).When the trap frequencies of the two atomic components are made identical by setting thelaser frequency to the “magic frequency”, the overlap in phase space can be scaled by themass ratio.[20] S. Watabe, T. Nikuni, N. Nygaard, J. E. Williams, and C. W. Clark, J. Phys. Soc. Jpn. ,064003(2007),064003(2007)