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The nature of Feshbach molecules in Bose-Einstein condensates

a r X i v :c o n d -m a t /0302082v 1 4 F e

b 2003

The nature of Feshbach molecules in Bose-Einstein condensates

T.K¨o hler,1T.Gasenzer,2P.S.Julienne,3and K.Burnett 1

1

Clarendon Laboratory,Department of Physics,University of Oxford,Oxford,OX13PU,United Kingdom

2

Institut f¨u r Theoretische Physik,Philosophenweg 16,69120Heidelberg,Germany 3

Atomic Physics Division,National Institute of Standards and Technology,

100Bureau Drive Stop 8423,Gaithersburg,Maryland 20899-8423

(Dated:February 2,2008)We discuss the long range nature of the molecules produced in recent experiments on molecular Bose-Einstein condensation.The properties of these molecules depend on the full two-body Hamil-tonian and not just on the states of the system in the absence of interchannel couplings.The very long range nature of the state is crucial to the e?ciency of production in the experiments.Our many-body treatment of the gas accounts for the full binary physics and describes properly how these molecular condensates can be directly probed.

PACS numbers:03.75.Kk,34.50.-s,36.90.+f,05.30.-d

Bose-Einstein condensation of molecules is an excit-ing challenge and opportunity in the physics of ultracold gases.As direct laser cooling of molecules is largely pre-vented by their densely lying rovibrational energy lev-els,several approaches now focus on the association of atoms in Bose-Einstein condensates.Present techniques are based on photoassociation [1]or magnetic ?eld tun-able interactions [2,3].The detection of molecular con-densates,however,remains a demanding problem.Re-cent experiments at JILA [4]found evidence for molecu-lar condensation by probing coherence properties of the assembly of atoms plus molecules.The 85Rb conden-sate was exposed to a sequence of two fast magnetic ?eld pulses in the vicinity of a Feshbach resonance.The pulses were separated by a variable period of time with a sta-tionary magnetic ?eld,termed the evolution period.The observed ?nal densities of gas atoms,i.e.the remnant condensate and a “burst”of comparatively hot atoms,in-dicated a coherent coupling between atoms and diatomic molecules in a highly excited vibrational state during the evolution period.Several subsequent theoretical stud-ies [5,6,7]have concluded that the gas contained a molecular condensate at the end of the pulse sequence.The magnitude of the molecular fraction,and the pre-cise mechanism of producing the molecules are a mat-ter of continuing controversy [8,9].To settle this issue the next generation of experiments could determine the coherent superposition of atomic and molecular compo-nents by directly detecting the molecules.The e?ciency of di?erent detection schemes based on laser excitation [10]depends sensitively on the number of molecules and the wave functions of the diatomic bound states produced by a magnetic ?eld pulse.

In this letter we shall give a full description of these wave functions for typical experimental magnetic ?eld strengths and determine the molecular component in the evolution period of the pulse sequence.In particular,we shall show explicitly that the wave functions of the molecules have a spatial extent of the order of the scat-

tering length.At the magnetic ?eld strengths closest to resonance this length scale even becomes comparable with the mean distance between the atoms in the di-lute condensate.Under these conditions a separation of the gas into atoms and diatomic molecules is physically meaningless.The strong binary correlations provided by the long range intermediate molecular states,however,are still crucial for the e?ciency of the association of atoms to molecules.In the evolution period the typi-cal static experimental ?eld strengths are su?ciently far from the resonant ?eld that the highly excited diatomic molecules can exist as a metastable entity of the gas.The microscopic many-body approach we apply to determine the molecular fraction [7,11]accounts for the long range nature of the intermediate molecular states.This ap-proach treats bound and free molecular states formed during the pulse sequence in a uni?ed manner and pro-vides a straightforward understanding of the association of condensate atoms to molecules.

The experimental technique of magnetic ?eld tunable interactions takes advantage of the Zeeman e?ect in the electronic energy levels of the atoms.In the JILA ex-periments [4]the 85Rb condensate atoms were prepared in the (F =2,m F =?2)hyper?ne state.Throughout this letter the open s -wave binary scattering channel of two asymptotically free atoms in the (F =2,m F =?2)state will be denoted as the {?2,?2}open channel with an associated reference potential V bg (r ).When the m F degeneracy of the hyper?ne levels is removed by an ex-ternal magnetic ?eld B the potentials associated with the di?erent asymptotic scattering channels are shifted with respect to each other.Although the {?2,?2}open channel is only very weakly coupled to other open chan-nels,it can be strongly coupled to closed channels.A zero-energy scattering resonance occurs when the ?eld-dependent energy E res (B )of a closed channel vibrational state (a Feshbach resonance level)φres is tuned close to the dissociation threshold energy of V bg (r ).We note that the closed channel state φres is not a stationary state of

2

FIG.1:Scheme of a typical magnetic ?eld pulse shape in the low density (n 0=3.9×1012cm ?3)experiments in Ref.[4].The minimum magnetic ?eld strength of the ?rst and second pulse is B min =15.55mT.In the evolution period the ?eld strength is chosen as B evolve =16.0mT.In the course of the experiments the evolution time t evolve as well as B evolve were varied.The dashed line indicates the position of the resonance at B 0=15.49mT.

the full two-body Hamiltonian,which has a coupling be-tween the channels.If E res (B )approaches the threshold from below,however,the overall potential matrix sup-ports a shallow multi-channel bound state φb with energy E b .This proper stationary molecular state ceases to ex-ist at the position of the resonance (B 0=15.49mT [4]),for which the s wave scattering length a of two asymp-totically free atoms in the {?2,?2}open channel has a singularity.When E b is su?ciently close to thresh-old,the component of φb in the {?2,?2}open channel then exhibits the universal form [12]exp(?r/a )/r at large relative distances of the two atomic constituents of the molecular state.

We shall now describe more fully the long range nature of the molecular states for typical experimental magnetic ?eld strengths in the fast sequence of pulses shown in Fig.1.To this end we have reduced the complete multi-channel Hamiltonian of the relative motion of two atoms to a two-channel model:

H 2B = ?

2m ?2

+V cl (B,r )

.(1)Here m is the atomic mass of 85Rb.For the potential in

the {?2,?2}open channel we use a Lennard-Jones form V bg (r )=4ε (σ/r )12?(σ/r )6 with σ=37.3292a Bohr (a Bohr =0.052918nm)and 4εσ6=C 6=4660a.u.[13](1a.u.=0.095734yJ nm 6).V bg (r )then reproduces the background scattering length of a bg =?450a Bohr [4].In accordance with Ref.[14]we model the closed channel potential as V cl (B,r )=V bg (r )+E cl (B ),where E cl (B )follows the dependence of the energy di?erence of the corresponding Zeeman hyper?ne levels on the magnetic

We use h ?1?E cl /?B =?34.6MHz/mT.In this model V cl (B,r )supports only two vibrational We assume the excited one to be the resonance φres which thus satis?es [? 2?2/m +V cl ]φres =φres .We have chosen the o?diagonal potential as W (r )=βexp(?r/α)with β/k B =38.5mK and 5a Bohr ,which gives a width of the resonance of =1.1mT [4].With this choice of parameters the also produces the correct shift between the reso-position B 0and the magnetic ?eld strength B res ,the resonance state crosses the dissociation thresh-of V bg (r )(E res (B res )=0).For the 85Rb resonance 0=15.49mT this shift is negative and of the or-der of B 0?B res =?0.9mT,so that the multi-channel bound state φb persists when the resonance state has crossed the threshold.The corresponding wave functions and negative binding energies are obtained from Eq.(1)by H 2B φb =E b φb and the scattering length is approxi-mated by

a (B )=a bg

1?

?B

n 0a 3is comparable

to 1,and one can no longer identify a particular pair of atoms in the gas as a molecule in the state φb because its molecular wave function would overlap with other gas atoms.One and the same atom could thus contribute to several diatomic molecules.After the pulse sequence and during the evolution period,however,the gas is weakly interacting (n 0a 3?1)and diatomic molecular bound states are su?ciently con?ned in space for the number of molecules to be a meaningful quantity.

We shall illustrate the resulting remarkable evolution of the molecular fraction in the JILA experiments [4]with the quantum mechanical many-body description devel-oped in Ref.[7]for a homogeneous gas with the exper-

3

FIG.2:Coupled-channel bound states at the magnetic?eld

strengths of B evolve=16.0mT and B min=15.55mT.

The radial coordinate is given on a logarithmic scale.The

dotted(dashed)curves indicate the closed({?2,?2}open)

channel components.The solid curves are the correspond-

ing bound state wave functions of the separable potential in

Ref.[7].The separable potential wave functions agree with

the{?2,?2}open channel components of the coupled chan-

nels bound states at distances large compared to the van der

Waals length of1

?t Ψ(t)=?Ψ?(t) ∞t0dτΨ2(τ)?

2 t t0dτΨ2(τ)?

4

FIG.3:Evolution of the densities of the atomic conden-sate n c and the molecular fraction n b for di?erent evolu-tion times of the magnetic?eld pulse sequence in Fig.1 (t evolve=10,11,12,13μs).The initially pure homogeneous 85Rb condensate has a density of n0=3.9×1012cm?3.

of n b/n0=6%.The small oscillations in n c and n b during the evolution period in Fig.3indicate the small (n0[a(B evolve)]3=8×10?5)but?nite overlap between the atomic condensate and the molecular fraction.At the lowest value of the?eld pulses in Fig.1one obtains n0[a(B min)]3=0.27and the observable that determines the number of bound pairs in a weakly interacting gas even yields n b>n0/2(not shown explicitly in Fig.3). This clearly indicates the signi?cant overlap between the molecular wave function of a pair of atoms with its sur-rounding gas atoms.The gas is then described by a strongly correlated non-stationary many-body state. The entire dynamics of the di?erent components of the gas can be understood in an intuitive way when the lin-ear ramps of the two?eld pulses in Fig.1are idealized as sudden switches of the magnetic?eld strength.The ?rst?eld pulse then shifts the virtually uncorrelated ini-tial condensate into a many-body state with the crucial binary correlations.The overlap of this many-body state with the multi-channel bound state of the pairs of atoms at the?eld strength B evolve determines the molecular fraction in the evolution period.The overlap with the

excited binary scattering states provides a burst fraction[4,7]of correlated pairs of compar-hot atoms.In the evolution period the atomic and the molecular component are virtually

and evolve coherently.At the end of the evo-period the di?erence in phase between the ampli-of the atomic condensate and the molecular fraction ?=E b(B evolve)t evolve/ .The second?eld pulse gets

components to overlap and,thereby,probes their

di?erence.The second burst fraction as well as ?nal atomic condensate and molecular component exhibit an interference depending on??.As the pulses are chosen as mirror images in Fig.1,at con-interference,n b(t?n)can be expected to result

twice the molecular density in the evolution pe-have shown in this letter that highly excited di-

bound states produced in recent experiments[4]

by a large spatial extent that by far ex-the size of all known ground state molecules[17].

corresponding85Rb2wave functions are strongly by their{?2,?2}open channel component.

have analyzed the non-adiabatic association mecha-

of Ref.[4]on the basis of a microscopic quantum many-body description of the gas.Our pre-molecular fraction of6%in the evolution period

pulse sequence[4]exceeds the results of Ref.[5] by more than an order of magnitude.This provides a signi?cantly better perspective for proposed experimen-tal schemes to detect the molecules.Corrections imposed by the atom trap[7]do not a?ect the orders of magnitude reported here.Their systematic study in connection with possible improvements of molecular production schemes will be subject to future work.

We thank Eleanor Hodby,Neil Claussen,Simon Gar-diner and Bill Phillips for inspiring discussions.This work was supported by the U.K.EPSRC(T.K.).K.B.is

a Royal Society Wolfson Merit Award holder.

[1]R.H.Wynar et al.,Science287,1016(2000).

[2]J.Stenger et al.,Phys.Rev.Lett.82,2422(1999).

[3]S.L.Cornish et al.,Phys.Rev.Lett.85,1795(2000).

[4]E.A.Donley et al.,Nature(London)417,529(2002).

[5]S.J.J.M.F.Kokkelmans and M.J.Holland,

Phys.Rev.Lett.89,180401(2002).

[6]M.Mackie,K.-A.Suominen,and J.Javanainen,

Phys.Rev.Lett.89,180403(2002).

[7]T.K¨o hler,T.Gasenzer,and K.Burnett,Phys.Rev.A

67,013601(2003).

[8]References[5,6]separate out physical observables as-

sociated with diatomic bound states in terms of a phe-nomenological quantum?eld that is included in the Hamiltonian.The molecular fraction reported in[5]is less than0.5%throughout the evolution period of the pulse sequence and rises steeply at the?nal ramp.The

5

results in this letter will exceed these predictions by more than an order of magnitude.

[9]E.Braaten,H.-W.Hammer,and M.Kusunoki,e-print

arXive cond-mat/0301489.

[10]Private communication from Eleanor Hodby and Neil

Claussen.

[11]T.K¨o hler and K.Burnett,Phys.Rev.A65,033601

(2002).

[12]This is the long range asymptotic form of the free Green’s

function evaluated at the near resonant binding energy

E b=? 2/ma2(see,e.g.,[7]).[13]J.L.Roberts et al.,Phys.Rev.A64,024702(2001).

[14]F.H.Mies,E.Tiesinga,and P.S.Julienne,Phys.Rev.A

61,022721(2000).

[15]J.Fricke,Ann.Phys.(N.Y.)252,479(1996).

[16]For a general description of the detection of composite

particles see,e.g.,J.D.Dollard,J.Math.Phys.14,708 (1973).

[17]The bond length of the largest known diatomic ground

state molecule4He2is of the order of100a Bohr[see R.E.Grisenti et al.,Phys.Rev.Lett.85,2284(2000)].

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