Table Of ContentSpontaneous dissociation of long-range Feshbach molecules
Thorsten K¨ohler,1 Eite Tiesinga,2 and Paul S. Julienne2
1Clarendon Laboratory, Department of Physics, University of Oxford,
Parks Road, Oxford, OX1 3PU, United Kingdom
2Atomic Physics Division, National Institute of Standards and Technology,
100 Bureau Drive Stop 8423, Gaithersburg, Maryland 20899-8423
5 (Dated: February 2, 2008)
0
0 We study the spontaneous dissociation of diatomic molecules produced in cold atomic gases via
2 magnetically tunableFeshbachresonances. Weprovidea universalformula for thelifetime ofthese
molecules that relates their decay to the scattering length and the loss rate constant for inelastic
n spinrelaxation. Ouruniversaltreatmentaswellasourexactcoupledchannelscalculationsfor85Rb
a
dimers predict a suppression of the decay over several orders of magnitude when the scattering
J
lengthisincreased. Ourpredictionsareingoodagreementwithrecentmeasurementsofthelifetime
9 of 85Rb2.
1
PACSnumbers: 34.50.-s,03.75.Nt,03.75.Ss
]
r
e
h ThediscoveryofFeshbachresonancesincoldgasespro- statewavefunction thatcanprovidethe crucialstability
t
o vides the unique opportunity to widely tune the inter- of the Feshbach molecules with respect to their sponta-
. atomicinteractionsviamagneticfields nearasingularity neous decay.
t
a of the scattering length. This singularity is due to the In the following, we shall consider pairs of atoms in
m
near degeneracy of the very small collision energy of the a cold gas that interact via s waves. We denote the bi-
- coldatomswiththebindingenergyofanextremelyloose nary scattering channel of a pair of asymptotically free
d
n diatomicmolecularstateformagneticfieldsatwhichthe atomsasthe entrancechannelandchoosethe zeroofen-
o scattering length is positive. A number of recent exper- ergy at its dissociation threshold. This threshold is de-
c iments, as summarised in the accompanying experimen- termined by the Zeeman energy of the separated atoms.
[ tal paper of Thompson et al. [1], have taken advantage The Hamiltonian H of the relative motion can then be
2 of this virtual energy match to populate this molecular divided into the contribution H of the closed channels
cl
v state. The interacting pairs of atoms in the first stud- with dissociation thresholds at or above zero energy (in-
7
ies of these cold Feshbach molecules [2, 3] were subject cluding the entrance channel), the contribution H of
8 op
todeeplyinelasticspinrelaxationcollisionsinvolvingthe the open channels with negative threshold energies, as
3
8 deexcitation of at least one of the colliding atoms. Spin well as the weak coupling Hint between the closed and
0 relaxationcanaffectboth free [4]andbound atompairs. open channels:
4
0 Inthisletterwepredictthespontaneousdissociationof H H
op int
t/ Feshbach molecules due to spin relaxation. We provide H = H† H . (1)
a a universal treatment of the molecular decay by which (cid:18) int cl (cid:19)
m
werelatethemolecularlifetime τ tothespatialextentof
We shall consider magnetic field strengths on the side
- the boundstatewavefunctionandthe lossrateconstant
d of positive scattering lengths of a zero energy resonance
for inelastic spin relaxation collisions. Both the size of
n (i.e.asingularityofthescatteringlength). Thelargepos-
o the molecule and the loss rate constant exhibit a pro- itivevaluesofthescatteringlengthimplytheexistenceof
c nouncedresonanceenhancementatthesingularityofthe aweaklybound(metastable)molecularstateφcl ,which
: −1
v scattering length. General considerations on the nature corresponds to the highest excited vibrational (v = 1)
Xi of this enhancement allow us to predict the functional bound state of Hcl. The wave function φc−l1 thus fu−lfils
form of the magnetic field dependence of τ. We show the stationary Schr¨odinger equation H φcl = Ecl φcl ,
r cl −1 −1 −1
a that the near resonant molecular lifetime increases with whereEcl isthenegativebindingenergy. Fermi’sgolden
−1
increasing scattering lengths and cantherefore be varied
rule determines the exponential decay of the Feshbach
over several orders of magnitude in related experiments.
molecule by a rate
We demonstrate the predictive power of our universal
treatment in a comparison with exact coupled channels 2
γ = Im φcl H† G (Ecl +i0)H φcl . (2)
calculations of the lifetime of the 85Rb dimers produced −¯h h −1| int op −1 int| −1i
in the experiments of Refs. [1, 2, 5]. The exact calcula- h i
tionsshowthatτ variesintheremarkablerangebetween HereG (Ecl +i0)= Ecl +i0 H −1 istheGreen’s
op −1 −1 − op
only100µsandroughly30msintheexperimentalrange function of the open binary scattering channels, whose
(cid:0) (cid:1)
of magnetic field strengths. Our studies indicate that energy argument Ecl + i0 indicates that Ecl is ap-
−1 −1
it is the large spatial extent of the near resonant bound proached from the upper half of the complex plane. As
2
Ecl is muchsmallerthantypicaltransitionenergies,it factor 1/ assures that φcl is unit normalised. The as-
−1 N −1
canbeneglectedintheargumentoftheGreen’sfunction. sociated binding energy Ecl is determined by [7] Ecl =
(cid:12) (cid:12) −1 −1
(cid:12) Fer(cid:12)mi’s golden rule also determines the loss rate con- E (B)+ φ WG (Ecl )W φ . Thestrongcoupling
res h res| bg −1 | resi
stant for inelastic spin relaxation collisions to be betweentheentrancechannelandthebareFeshbachres-
onancestateφ (r)alsodeterminesthezeroenergyscat-
4 res
K2 =−¯h(2π¯h)3 Im hφ(0+,c)l|Hi†ntGop(i0)Hint|φ(0+,c)li . (3) tering state φ(0+,c)l of Hcl by the general formula [7]:
h i
aHteerde wφi(0t+,hc)l tihsethHeamzeilrtooneinanergHycls,cawt(h+tei)crhingfuslfitalstethaesssotcai-- φ(0+,c)l = φ(0+,b)g+Gφbrges(0A)W(Bφ)res A(B) !. (6)
tionary Schr¨odinger equation H φ = 0. The pref-
cl 0,cl
actors in Eq. (3) correspond to a cold thermal gas of Here φ(+) denotes the bare zero energy state associ-
identical Bosons. The associated rate equation reads [4] 0,bg
aN˙to(tm)s=an−dKn2(htn)(t)isiNth(eti)r, awvherearegeNde(tn)siitsy.the number of aVtbegd(rw)]φit(0h+,b)gth(re)b=ac0k,garonudntdhescaamttperliitnugd,ei.Ae.(B[−)¯hi2n∇E2/qm. (6+)
At magnhetic fiield strengths in the vicinity of a zero is given by [7]:
energy resonance, φ(+) and φcl are determined by their
0,cl −1 φ W φ(+)
wave functions in the entrance channel and in a closed A(B)= h res| | 0,bgi . (7)
channel strongly coupled to it [6, 7]. This strong inter- −Eres(B)+ φres WGbg(0)W φres
h | | i
channel coupling is due to the near degeneracy of the
The scattering length a(B) can then be obtained from
energy E (B) ofa closedchannel vibrationalstate (the
res the asymptotic behaviour (2π¯h)−3/2[1 a(B)/r] of the
bareFeshbachresonancelevel)φres withthe dissociation −
entrance channel component of Eq. (6) at large inter-
threshold of the entrance channel. The restricted two-
atomic distances r. This yields
body Hamiltonian H of the entrance channel and the
cl
closed channel can be effectively described by: ∆B
a(B)=a 1 , (8)
bg
Hcl = −h¯m2∇W2+(r)Vbg(r) −h¯m2∇2W+(rV)cl(B,r) !. (4) where ∆B = (2π¯h)3|hφres(cid:18)|W|−φ(0+B,b)g−i|2Bm0/(cid:19)(4π¯h2abgµres) is
theresonancewidthanda isthebackgroundscattering
Here r denotes the relative distance between the atoms bg
length. ThesingularityofA(B)thusdeterminesthemea-
and m is twice their reduced mass, B is the magnetic
surable position B of the zero energy resonance, which
field strength, V (r) is the background scattering po- 0
bg
is distinct from the zero B of the bare energy E (B)
tential of the entrance channel, and W(r) provides the res res
by the shift B B = φ WG (0)W φ /µ of
inter-channel coupling. The closed channel potential 0 res res bg res res
− −h | | i
V (B,r) supports the resonance state, i.e. [ ¯h2 2/m+ the denominator on the right hand side of Eq. (7).
cl
− ∇ At near resonant magnetic field strengths the binding
V (B,r)]φ (r) = E (B)φ (r). The dissociation
cl res res res energy is determined by Ecl = ¯h2/(ma2) (cf., e.g.,
threshold of Vcl(B,r) is determined by the energy of a −1 −
Refs. [2, 5, 7]). The energy argument of the Green’s
pair of non-interacting atoms in the closedchannel. The
function in Eq. (5), therefore, vanishes at large positive
relative Zeeman energy shift between the two channels
scattering lengths. In this limit the background scatter-
aswellasthe bareenergyE (B) canbe tunedby vary-
res
ing the magnetic field strength B. We denote by B ing contribution φ(+) to the right hand side of Eq. (6)
res 0,bg
the magnetic fieldstrength,atwhichE (B) crossesthe also becomes negligible for inter-atomic distances on the
res
dissociation threshold energy of the entrance channel, lengthscalesetbytherangeoftheinter-channelcoupling
i.e. Eres(Bres) = 0. The bare energy Eres(B) varies vir- Hint in Eqs. (2) and (3). Equations (5) and (6) thus co-
tually linearly in B, and an expansion about B yields incide up to a factor A(B), and Eqs. (2) and (3) yield
res N
Emraegs(nBet)ic=mµormese(nBt−beBtwreese)n. tHheereFeµsrhebsaicshthreesodniffaenrceenscteaitne nKa2n(tBa)m/γp(lBit)ud=e A2((2Bπ)¯h)is3|rAe(aBdi)l|y2Nob2t(aBin).edTfrhoemnethare rreessoo--
and a pair of atoms in the entrance channel. nancewidthandshiftofEq.(8),whilethenormalisation
Under the assumption that the spatial configuration constant 2(B) =(µres∆B)mabga(B)/(2h¯2) can be de-
N
of a pair of atoms in the closed channel is restricted to terminedfromhφc−l1|φc−l1i=1 [7]. The molecularlifetime
the resonance state φ (r), the dressed highest excited τ(B)=1/γ(B) is then given by the universal formula
res
vibrational bound state φcl of H is given by [7]:
−1 cl τ(B)=4πa3(B)/K (B)=[4/K (B)]4π r3 /3. (9)
2 2
h i
1 G (Ecl )Wφ
φc−l1 = bg φ−1 res . (5) Here r3 istheexpectationvalue ofr3 forthenearreso-
res h i
N (cid:18) (cid:19) nant wave function φcl (r)=exp( r/a)/(r√2πa) [7, 8].
HereG (Ecl )=(Ecl +h¯2 2/m V )−1istheGreen’s Generalconsiderati−o1ns[9]show−thatinthepresenceof
bg −1 −1 ∇ − bg
function associated with the entrance channel, and the opendecaychannelsthe resonanceenhancedzeroenergy
3
binaryelasticandinelasticcollisioncrosssectionscanbe 10000
describedintermsofacomplexscatteringlengtha(B)
−
ib(B), whose imaginary part is related to the loss rate
constant by ]a0
h [1000 a(B)
K2(B)=16π¯hb(B)/m. (10) ngt
e 1000xb(B)
ItsrealpartiswellapproximatedbyEq.(8),exceptthat g l
n
a2(B) approaches a large but finite value at B = B0, eri
which is determined by the decay width. We shall pre- att 100
c r
suppose in the following that only the bare resonance S vdW
state φ decays to a set of final channels with a total
res
rate γ = 1/τ , where τ is the associated lifetime.
res res res
The imaginary part of the complex scattering length is 10
15.6 15.8 16 16.2 16.4
then given by a Lorentzian function B [mT]
¯hγ /(4µ )
b(B)=a ∆B res res . (11) FIG. 1: Magnetic field dependence of the real (solid curve)
bg (B B0)2+[h¯γres/(4µres)]2 andimaginary(dashedcurve)partsofthecomplexscattering
−
lengtha(B)−ib(B). Thedottedlineindicatesthelengthscale
Inserting Eqs. (8), (10) and (11) into Eq. (9) then yields
[10]rvdW =(mC6/¯h2)1/4/2set bythelong rangeasymptotic
τ(B)=τresma2bg4µh¯r2es∆Bx2 x1 −1 3 =τresf(B), (12) −teCnt6i/arl.6WvaenndoetreWthaaatlsb(tBai)l oisfmthueltbipalcikedgrbouynadfsaccattotreroifng10p0o0-.
(cid:18) (cid:19)
where we have introduced x = (B B )/∆B. This for-
0
−
mulaprovidesthegeneralfunctionalformofτ(B)aswell
as its order of magnitude in terms of the constant factor and (3, 2;3, 2). A fit to Eq. (8) predicts the param-
− −
τres, which can be obtained either from a measurement eters B0 = 15.520 mT, ∆B = 1.065 mT, and abg =
of K2(B), at a single magnetic field strength, or from -484.1 a0 (a0 = 0.0529177 nm) for the experimentally
coupled channels calculations. well characterised 15.5 mT zero energy resonance of
We shall demonstrate the predictive power of Eqs. (9) 85Rb [5]. The decay of the Feshbach molecules is due
and (12) for the example of the Feshbach molecules as- to spin-dipole interactions, which couple s waves to d
sociated recently from pairs of 85Rb atoms in a Bose- waves. Amongthe23dwavechannelscoupledtotheen-
Einstein condensate [1, 2, 5] at magnetic field strengths trance channel only three are open. These d wave decay
of about 15.5 mT. The nuclear spin I = 5/2 of 85Rb exit channels are characterised by the internal quantum
gives rise to two energetically relevant hyperfine levels numbers (2, 2;2, 1), (2, 2;2,0) and (2, 1;2, 1) in
− − − − −
with total spin quantum numbers f = 2 and 3. A ho- order of increasing energy release. The transition en-
mogeneous magnetic field B splits the Zeeman sublevels ergies are all on the order of several mK (in units of
into a non-degenerate set of levels (f,mf), labelled by kB = 1.3806505× 10−23 J/K), which implies that the
the projection quantum number m along the axis of decay products are lost from an atom trap. Including
f
the magnetic field and the f value with which it cor- spin-dipole interactions in the coupled channels calcula-
relates adiabatically at zero field. In the experiments tions shifts the predicted position B0 by less than 1 µT
[2, 5] the atoms were prepared in the (f = 2,m = 2) butitprovidesthesmallimaginarypartb(B)ofthescat-
f
−
internal state. The binary asymptotic scattering chan- tering length (see Fig. 1), whichdetermines the loss rate
nels are then characterisedby pairs of internal quantum constant K2(B) by Eq. (10).
numbers (f,m ;f′,m′), as well as the orbital angular Given the magnetic moment difference µ /h=34.66
f f res
momentum quantum number ℓ associated with the rel- MHz/mT, a fit of Eq. (11) to the curve of b(B) in Fig. 1
ative motion of the atom pair and its projection quan- determines the lifetime of the bare resonance level to be
tum number m . The s wave (ℓ = 0,m = 0) entrance τ = 1/γ = 32 µs. The solid curve in Fig. 2 shows
ℓ ℓ res res
channel is thus characterised just by the internal quan- theassociateduniversalestimatesofthelifetimesτ(B)of
tum numbers (2, 2;2, 2). The cylindrical symmetry the dressed 85Rb Feshbach molecules [2, 5] determined
2
− −
of the setup implies that the projection quantum num- from Eq. (9), using the exact loss rate constant K (B)
2
ber m +m′ +m = 4of the totalangularmomentum [cf. Eq. (10) and Fig. 1]. The dashed curve indicates the
f f ℓ −
is conserved in a binary collision. predictions of Eq. (12). We expect these estimates to
We have applied the coupled channels approach of be accurate between B and about 16.0 mT, where the
0
Ref. [6] to determine the scattering length a(B) (see scattering length a(B) by far exceeds the length scale
Fig. 1), including spin exchange interactions,which cou- r =84a setbythelongrangeasymptoticbehaviour
vdW 0
pletheentrancechanneltothefourclosedswavescatter- ofthevanderWaalsinteraction(seeFig.1). Thistrendis
ing channels (3, 3;2, 1), (3, 2;2, 2), (3, 1;3, 3) confirmedbyourdirectcalculationsofthemolecularlife-
− − − − − −
4
100 is the event rate constant for molecular decay (which is
experiment the same as that for the inelastic collision of a pair of
4πa3(B)/K (B) identical Bose condensed atoms [11]). The large extent
10 2 of the molecular wave function at near resonant mag-
exact
s] τ f(B) neticfieldstrengthsthusprotectstheFeshbachmolecules
m res
against their spontaneous decay. We caution, however,
[
me 1 that purely two-body theory may eventually become in-
ti adequatesufficientlycloseto resonance;forexample,the
e
f mean distance between atoms at the experimental [1]
Li peakdensityof6.6 1011cm−3isonly2 104a ,whichis
0.1 × × 0
τ comparabletotheestimatedbondlengthofthemolecules
res [8] of about 3 103 a at 15.6 mT.
0
×
0.01 Our general theory applies to a variety of Feshbach
15.6 15.8 16 16.2 16.4
molecules with open decay exit channels [12]. Among
B [mT]
these species are the 40K [3] and 6Li [13] molecules
2 2
FIG.2: Two-bodydecaylifetimeτ(B)ofthe85Rb2 Feshbach producedin Fermi gasesin the s waveentrance channels
associated with the pairs of atomic quantum numbers
molecules [2, 5] versus magnetic field strength B near 15.5
mT.ThesquaresindicatetheexperimentaldataofThompson (9/2, 9/2;9/2, 5/2)and (1/2,1/2;3/2, 3/2), respec-
− − −
et al. [1]. The circles indicate exact predictions of coupled tively. Measurements of two-body decay lifetimes could
channels calculations, whereas the solid and dotted curves allow the characterisation of the spin-dipole interaction
show the results of the universal formulae (9) and (12), re- between various atomic species with an unprecedented
spectively. The magnitude of the bare state lifetime τres is precision. Our universal estimate determines the con-
indicated by the dotted line. For the purpose of compari-
ditions for the stability of Feshbach molecules, which is
son, we have slightly corrected the axis of the magnetic field
crucialtoallfuturestudiesofcoldmoleculargasesinthe
strengthinsuchawaythatthetheoreticalresonanceposition
recovers themeasured value of B0 =15.5041 mT [5]. presence of open decay exit channels.
We are grateful to Eleanor Hodby, Sarah Thompson
and Carl Wieman for providing their data prior to pub-
time showninFig.2. Theexactlifetime canbe obtained
lication. This research has been supported by the Royal
from the energy dependence of the inelastic scattering
Society(T.K.)andtheUSOfficeofNavalResearch(E.T.
cross section between any pair of open decay channels.
and P.S.J.).
We have chosen transitions between the open d wave
channels associated with the internal atomic quantum
numbers(2, 2;2, 1)and(2, 2;2,0). Fittingthestan-
− − −
dard Breit-Wigner formula to the resonance peak at the
binding energy E (B) [2, 5] of the metastable Feshbach
b [1] S.T. Thompson, E. Hodby, and C.E. Wieman,
moleculesthendeterminesthedecayrateγ(B)=1/τ(B)
cond-mat/0408144.
through its width. [2] E.A. Donley et al.,Nature (London) 417, 529 (2002).
The squares in Fig. 2 indicate lifetime measurements [3] C.A. Regal et al., Nature(London) 424, 47 (2003).
by Thompson et al. [1] in which the dimer molecules [4] J.L. Roberts et al.,Phys. Rev.Lett. 85, 728 (2000).
wereproducedinadilutevapourof85Rbatomsatatem- [5] N.R.Claussenetal.,Phys.Rev.A67,060701(R)(2003).
[6] F.H. Mies, E. Tiesinga, and P.S. Julienne, Phys. Rev. A
perature of 30 nK using a Feshbach resonance crossing
61, 022721 (2000).
technique (see, e.g., Refs. [6, 7]). Our theoretical predic- [7] K. Goral et al., J. Phys. B 37, 3457 (2004).
tions agree with the experimental lifetimes in both their [8] T. K¨ohler et al.,Phys. Rev.Lett. 91, 230401 (2003).
systematic trends and magnitudes. Figure 2 shows the [9] J.L. Bohn and P.S. Julienne, Phys. Rev. A 56, 1486
large differences in the order of magnitude of τ(B) from (1997); ibid.60, 414 (1999).
roughly 30 ms to only 100 µs in the experimentally ac- [10] G.F. Gribakin and V.V. Flambaum, Phys. Rev. A 48,
cessible [1] range of magnetic field strengths from about 546 (1993).
[11] H.T.C. Stoof et al., Phys.Rev.A 39, 3157 (1989).
15.6 to 16.2 mT. The lifetime correlateswith the admix-
[12] InthecaseofmoleculescomposedofFermionsindifferent
ture of the bare resonance level to the dressed molecular
spinstatestheeventrateconstantK2/4inEq.(9)needs
state, which, in turn, is related to the spatial extent of
to be replaced by K2/2.
themolecules[8]. Equation(9)canindeedbeinterpreted [13] Based on the results of this letter, the lifetimes of 6Li2
in the intuitive way that 4π r3 /3 sets the scale of the FeshbachmoleculeswerecalculatedbyM.Bartensteinet
volume confining the boundhatoim pair, while K (B)/4 al., cond-mat/0408673.
2
