Atom-dimer and dimer-dimer scattering in fermionic mixtures near a narrow Feshbach resonance

1.5 MB·English

by  J. Levinsen

EPJ manuscript No.
(will be inserted by the editor)
Atom-dimer and dimer-dimer scattering in fermionic mixtures
near a narrow Feshbach resonance
J. Levinsen1,2 and D. S. Petrov2,3
1 T.C.M.Group,UniversityofCambridge,CavendishLaboratory,J.J.ThomsonAve.,CambridgeCB30HE,UnitedKingdom
1 2 Laboratoire de Physique Th´eorique et Mod`eles Statistiques, CNRS and Universit´e Paris Sud, UMR8626, Baˆt. 100, 91405
1
Orsay, France
0 3 Russian Research Center Kurchatov Institute, Kurchatov Square, 123182 Moscow, Russia
2

n Received: date / Revised version: date
a
J
Abstract. Wedevelopadiagrammaticapproachforsolvingfew-bodyproblemsinheteronuclearfermionic
1
mixtures near a narrow interspecies Feshbach resonance. We calculate s-, p-, and d-wave phaseshifts for
3
the scattering of an atom by a weakly-bound dimer. The fermionic statistics of atoms and the composite

natureofthedimerleadtoastrongangularmomentumdependenceoftheatom-dimerinteraction,which
]
s manifests itself in a peculiar interference of the scattered s- and p-waves. This effect strengthens with
a the mass ratio and is remarkably pronounced in 40K-(40K-6Li) atom-dimer collisions. We calculate the
g
scattering length for two dimers formed near a narrow interspecies resonance. Finally, we discuss the
-
t collisional relaxation of the dimers to deeply bound states and evaluate the corresponding rate constant
n
as a function of the detuning and collision energy.
a
u
PACS. XX.XX.XX No PACS code given
q
.
t
a
m 1 Introduction heavyparticlesexperienceafalltothecentreinthe1/R2-
potential[13].Onresonance(interspeciesscatteringlength
-
d a=∞)thisleadstotheEfimoveffect-theexistenceofan
Numerous advances in the field of ultracold Fermi gases
n infinite number of bound heavy-heavy-light trimer states
overthepastdecadehaveenabledtheexplorationofnovel
o
[14–16]. For smaller mass ratios the centrifugal barrier is
strongly interacting regimes in fermionic systems (see [1]
c
dominant. On the one hand this effective three-body re-
[ and [2] for review). The BCS-BEC crossover, extensively
studied in either Potassium (40K) or Lithium (6Li) ho- pulsion excludes the Efimov effect. On the other it sup-
1 presses recombination processes requiring three atoms to
monuclear systems, is now being actively pursued in the
v approach each other to very short distances, which is evi-
new generation of experiments on mixtures of these two
9
dentlyadvantageousforthecollisionalstabilityofthegas.
isotopes [3–10]. The mass ratio is thus a new parameter
7
Thelowerthemassratio,themorestablethisthree-body
9 introducedintothecrossoverphasediagram[11].Itisthen
system is [17].
5 natural to ask whether changing this parameter can lead
. toqualitativelynewcrossoverphysicsand,ifso,howlarge It turns out that even for m /m < 13.6 the ↑↑↓-
1 ↑ ↓
0 shouldthemassratiobeinordertoseenon-trivialeffects? system exhibits non-perturbative effects on the positive
1 For a very large mass ratio (of the order of several (BEC) side of the resonance, where there is a weakly
1 hundreds)acrystallinephasecanemergeontheBECside bound heteronuclear molecular state. One of us found
: of the crossover [12]. The effect is due to a strong long- that the three-body recombination to this state vanishes
v
range repulsion between the heavy fermions originating for m /m ≈ 8.6 [18]. Later, Kartavtsev and Malykh ar-
i ↑ ↓
X from the exchange of their light partners. Another mass gued that this phenomenon is related to the existence of
r ratio dependent change in the behavior of the system oc- a weakly bound not Efimovian trimer state for m↑/m↓ >
a
curs in the problem of two identical heavy fermions of 8.2 [19]. The trimer has unit angular momentum and for
mass m interacting resonantly with a light atom of mass smaller mass ratios turns into a p-wave atom-dimer scat-
↑
m . The exchange of the light atom results in an attrac- teringresonance.Wehaverecentlyshownthatinthecase
↓
tivepotentialbetweentheheavyfermionsproportionalto of a K-Li mixture (m /m = 6.64) the K-(K-Li) atom-
↑ ↓
1/m R2, where R is the distance between them. This ex- dimer scattering should be dominated by this p-wave res-
↓
change attraction competes with the repulsive centrifugal onanceinawiderangeofcollisionenergies[20].Moreover,
barrier∝1/m R2 fortheidenticalfermions.Formassra- by introducing an external quasi-2D confinement, the p-
↑
tiosm /m largerthanthecriticalvalue13.6theexchange waveatom-dimerinteractioncanbetunedfromattractive
↑ ↓
attraction dominates over the centrifugal barrier and the to repulsive, allowing for a trimer formation.
2 J. Levinsen and D. S. Petrov: Scattering in fermionic mixtures near a narrow Feshbach resonance
In this paper we develop a uniform-space diagram- Expanding the denominator of Eq. (2) in powers of kR
e
maticapproachforstudyingfew-bodyprocessesinahete- gives the effective range expansion. In particular, in the
ronuclearfermionicmixturenearaninterspeciesFeshbach s-wave channel ((cid:96)=0) we have
resonance of finite width. We calculate relevant atom-
dimer scattering phaseshifts and partial cross-sections in kcotδ (k)≈−a−1+ 1r k2+..., (3)
0 0
the homonuclear case and in the K-Li case. Passing from 2
m /m = 1 to m /m = 6.64 we observe an increase in
↑ ↓ ↑ ↓ and the corresponding expansion in the p-wave channel
theatom-dimerinteraction,repulsiveinevenangularmo-
((cid:96)=1) reads
mentum channels and attractive in odd ones. The most
dramatic increase is found in the channel with unit angu-
1
larmomentum–thep-wavescatteringvolumechangesby k3cotδ1(k)≈−v−1+ k0k2+..., (4)
2
more than an order of magnitude. Our exact calculations
are complemented by a qualitative explanation of the ob- where v is the p-wave scattering volume, and k is a pa-
0
served effect based on the Born-Oppenheimer approach, rameter analogous to the effective range.
which we generalize to the case of a narrow interspecies Thescaleofthescatteringlengtha,theeffectiverange
resonance. We predict a very strong interference between r , and other expansion parameters in the higher order
0
s- and p-waves in atom-dimer scattering. Depending on terms in Eq. (3) are set by the length R or its power
e
the collision energy, the scattering is dominant in back- of suitable dimension, and the same holds for higher par-
ward or forward directions, which can be observed exper- tial waves. For k → 0 the partial scattering amplitudes
imentally by colliding an atomic cloud with a cloud of are proportional to (kR )2(cid:96). Thus, in the limit kR (cid:28) 1
e e
molecules. We use our diagrammatic approach to calcu- (ultracoldregime)thes-wavescatteringamplitude,which
late the dimer-dimer scattering length add as a function equals f0(0) = −a, is the most important interaction pa-
of the atomic scattering length a and the width of the rameter in the mixture.
interspeciesresonance.Finally,wediscussthemainmech- Near a scattering resonance the scattering length can
anisms of the collisional relaxation of dimers into deep bemodifiedand,inparticular,cantakeanomalouslylarge
molecular states, and calculate the corresponding atom- values(i.e.|a|(cid:29)R ).AmagneticFeshbachresonanceoc-
e
dimer and dimer-dimer relaxation rate constants as func- curs when the collision energy of the two atoms is close
tions of a, the width of the resonance, and the collision totheenergyofaquasidiscretemolecularstateinanother
energy. hyperfinedomain,whichiscalledclosedchannel.Thetun-
The paper is organized as follows. In Sec. 2 we dis- ing of the scattering amplitude is achieved by shifting the
cuss the two-body problem in the narrow resonance case openandclosedchannelswithrespecttoeachotherinan
and introduce our field-theoretical approach. The main externalmagnetic field(hyperfine statescorrespondingto
part of the paper is structured according to the previous the open and closed channel have different magnetic mo-
paragraph – Secs. 3 and 4 are devoted to the three- and ments). The width of the resonance is determined by the
four-body problems respectively. In Sec. 5 we discuss the strength of the coupling between these two channels. The
inelastic collisional relaxation in atom-dimer and dimer- narrower the resonance, the stronger the collision energy
dimer collisions, and in Sec. 6 we conclude. dependence of the scattering amplitude, and, therefore,
the larger the effective range r . We call a resonance nar-
0
row, if |r | (cid:29) R [21]. In fact, near such a resonance r
0 e 0
2 Two-body problem near a narrow Feshbach
is necessarily negative and it is convenient to use another
resonance length parameter [22]
We assume that all interatomic interactions in the ↑-↓ 1
R∗ =−r /2= , (5)
fermionic mixture are characterized by van der Waals po- 0 2µa µ ∆B
bg rel
tentials.Wealsoassumethattheintraspeciesinteractions
are not resonant, and therefore can be safely neglected where µ = m m /(m +m ) is the reduced mass, a is
↑ ↓ ↑ ↓ bg
in the ultracold regime. Let us denote the van der Waals the background scattering length, µ is the difference in
rel
rangeoftheinterspeciesinteractionbyRe andwritedown the magnetic moments of the closed and open channels,
the partial wave expansion of the on-shell scattering am- and∆B isthemagneticwidthoftheFeshbachresonance.
plitude [13] All 6Li-40K interspecies resonances discussed so far are
∞ characterized by R∗ (cid:38) 100nm [4,7], which is much larger
(cid:88)
f(k,k(cid:48))= (2(cid:96)+1)P(cid:96)(cosθ)f(cid:96)(k). (1) than the van der Waals range Re ≈2.2nm.
Onecanimagineaninteratomicpotentialforwhichthe
(cid:96)=0
higher order terms in Eq. (3) are also anomalously large.
Here k and k(cid:48) are initial and final relative momenta such
For example, we can introduce one or several additional
that |k|=|k(cid:48)|=k, and θ =∠k,k(cid:48) is the scattering angle. closed channels with quasistationary states very close to
We set (cid:126) = 1. The partial wave amplitudes f (k) can be
(cid:96) the open-channel threshold resulting in a rather exotic
written in terms of the phase shifts δ (k) as
(cid:96) scattering amplitude [24]. However, in this paper we as-
1 sumeamorepracticalandsimplecaseinwhichtheterms
f (k)= . (2)
(cid:96) kcotδ(cid:96)(k)−ik denoted by ... in Eq. (3) vanish in the limit kRe → 0.
J. Levinsen and D. S. Petrov: Scattering in fermionic mixtures near a narrow Feshbach resonance 3
Wewillalsoassumethatscatteringwith(cid:96)>0isnotreso-
nantandcanbeneglectedinthislimit.Then,substituting T = + +...
Eq.(3)intoEq.(2)wegetthewell-knownformulaforthe
resonant scattering at a quasidiscrete level [13] (written
as a function of momentum rather than energy)
= + T
1
f(k)=− . (6)
1/a+R∗k2+ik
Fig. 1. Diagrammatic series contributing to the atom-dimer
TheratioR∗/ameasuresthedetuningfromtheresonance T-matrix and a schematic representation of the Skorniakov-
and we distinguish the regime of small detuning, R∗/a(cid:28) Ter-Martirosian integral equation (12). External propagators
1, and the regime of intermediate detuning, R∗/a (cid:29) 1. are included to guide the eye, they do not form part of the
The properties of a few-body system in these two limits T-matrix. Straight and wavy lines denote atomic and dimer
propagators, respectively.
are qualitatively different [22].
In order to describe the ↑↓ mixture near a narrow res-
onance we use the two-channel Hamiltonian [25]
3 Atom-dimer scattering
(cid:88) k2 (cid:88)(cid:18) p2 (cid:19)
Hˆ = 2m aˆ†k,σaˆk,σ+ ω0+ 2M ˆb†pˆbp Knowledgeofatom-dimerinteractionparametersisneces-
k,σ=↑,↓ σ p sary for the correct description of an atom-molecule mix-
+(cid:88)√gV (cid:16)ˆb†paˆp2+k,↑aˆp2−k,↓+ˆbpaˆ†p2−k,↓aˆ†p2+k,↑(cid:17), (7) tmuorme eonntutmhe-spBaEceCfosirdmealoifsmthfeorFetshhebtahchreer-ebsoodnyanpcreo.bTlehme
k,p with short-range interactions was first demonstrated in
wherea† anda arecreationandannihilationoperators the calculation of the neutron-deuteron scattering length
↑,↓ ↑,↓ (total spin S = 3/2 in this case corresponds to our ↑-
of the two fermionic species while b† (b) creates (annihi- ↑↓ scattering problem) [26]. The coordinate formulation
lates) a closed-channel molecule of mass M ≡ m +m .
↑ ↓ canbefoundinRef.[18]wheretheatom-dimerscattering
The atom-molecule interconversion amplitude g is taken
lengthwasobtainedinthemass-imbalancedcase.Herewe
constant up to the momentum cut-off Λ ∝ 1/R , and ω
e 0 extend these results to higher partials waves, finite colli-
isthebaredetuningofthemolecule.Thequantitiesaand
sion energies, and finite Feshbach resonance width.
R∗ are related to the parameters of the model (7) by [23]
Let us denote the atom-dimer scattering T-matrix by
(see also Appendix A) T(k,k ;p,p ), the arguments of which imply that the in-
0 0
coming four-momenta of the atom and the molecule are
µg2 1 π
a= , R∗ = . (8) (k,k0)and(−k,E−k0),andtheoutgoingonesare(p,p0)
2π g2πµ2Λ −ω0 µ2g2 and (−p,E−p0), respectively. In Fig. 1 we show the di-
agrammatic series for T, the summation of which results
Thebarepropagatorsofatomsandclosed-channelmo-
in the Skorniakov-Ter-Martirosian integral equation [26]
lecules read
(see also Ref. [27])
1
G↑,↓(p,p0)= p −p2/2m +i0, T(k,k0;p,p0)=−g2ZG↓(−k−p,E−k0−p0)
0 ↑,↓
1 (cid:90) d4q
D0(p,p0)= p −p2/2M −ω +i0, (9) −i (2π)4G↑(q,q0)G↓(−p−q,E−p0−q0)
0 0
×D(−q,E−q )T(k,k ;q,q ). (12)
0 0 0
where +i0 slightly shifts the poles of G and D into the
0
lower half of the complex p0-plane. A physical dimer con- Equation(12)isformallyidenticaltotheequal-masswide-
sistsofaclosed-channelmoleculedressedbyopen-channel resonanceone,thedifferencebeinghiddeninthepropaga-
atoms.Thecorrespondingpropagatorisgivenby(seeAp- tors and the factor Z, which serves for correct normaliza-
pendix A) tionofexternalpropagators(seeAppendixA).Theatom-
dimer elastic scattering amplitude is proportional to the
D(p,p )
0 on-shell T-matrix:
2π/µ
= 2µR∗(cid:16)p0− 2pM2 +i0(cid:17)+ a1 −√2µ(cid:113)−p0+ 2pM2 −i0.(10) f(k,k(cid:48))=−2µπ3T(k,k2/2m↑;k(cid:48),k2/2m↑), (13)
where µ ≡ Mm /(M +m ) is the reduced mass of the
ThepoleofD(0,p )determinesthedimerbindingenergy 3 ↑ ↑
0 atom-dimer system, and k = |k| = |k(cid:48)|. Hereafter f, f ,
(cid:96)
(cid:15)0 =−((cid:112)1+4R∗/a−1)2/8µR∗2. (11) δ(cid:96), and σ(cid:96) refer to the atom-dimer scattering parameters,
the two-atom interaction being described by a and R∗.
Equation (11) interpolates between the two limits: for Integration over q in Eq. (12) may be carried out by
0
small detuning we have (cid:15) (cid:39) −1/2µa2 and in the regime closing the complex contour in the lower half plane. The
0
of intermediate detuning (cid:15) (cid:39)−1/2µR∗a. integrationpicksupthecontributionfromthesimplepole
0
4 J. Levinsen and D. S. Petrov: Scattering in fermionic mixtures near a narrow Feshbach resonance
40K-6Li mixture Equal masses 40K-6Li mixture
150
R*=0 75 R*=a/16
(cid:47)/2
100
(cid:98)p 2/a(cid:109)l 50
50
25
(cid:98)
p 0 0
0 0.5 1 0 0.5 1
(cid:98)
d
R*=a/4 R*=a
(cid:98)d (cid:98)s 50 50
2
a
(cid:98) /(cid:109)l25 25
s
-(cid:47)/2
0 0
0 0.5 1 0 0.5 1
0 0.5 1 0 0.5 1 E / |(cid:161) | E / |(cid:161) |
E / |(cid:161) | E / |(cid:161) | coll 0 coll 0
coll 0 coll 0
Fig. 2. (color online). Atom-dimer s, p, and d-wave scatter- Fig.3.(coloronline).PartialK-KLiatom-dimercross-sections
idnogttpedhalisneesshcioftrsrevsps.onEdcotllo/R|(cid:15)0∗|/.aS=oli0d,,1d/a1s6h,e1d/,4,daont-ddRas∗he=d,a,arned- σlissio(nsoelinde)r,gσypfo(dradsihffeedr)e,natnddetσudni(ndgosttRed∗/)ain. units of a2 vs. col-
spectively.Inthehomonuclearcaseweshowδ onlyforR∗ =0.
d
δ ≡ δ , and δ ≡ δ . The results are shown for two
1 p 2 d
of G↑ at q0 =q2/2m↑. The scattering phase shifts are on- mass ratios: m↑/m↓ =6.64 (left) and m↑/m↓ =1 (right).
shell quantities and we let k0 = k2/2m↑, p0 = p2/2m↑, We keep the same vertical scale in both graphs, and one
and the total energy E =k2/2µ3+(cid:15)0. The remaining on- can see that the atom-dimer interaction in the heteronuc-
shell condition, |p| = |k|, should be implemented at the lear case is stronger in every considered channel. Looking
end of the calculations. at the low-energy asymptotes of the phase shifts in the
The kernel of the resulting three-dimensional integral wide resonance case (R∗ = 0) we see that passing from
equationhasasimplepoleat|q|=|k|hiddeninthedimer m /m =1tom /m =6.64theatom-dimers-wavescat-
↑ ↓ ↑ ↓
propagator. We make it explicit by defining functions f˜ tering length increases from a ≈ 1.18a to a ≈ 1.98a,
ad ad
and h: consistentwithRefs.[26]and[18].Atthesametimethep-
wavescatteringvolumeincreasesbymorethananorderof
f˜(k,q) = D(q,E−q2/2m↑)T (cid:18)k, k2 ;q, q2 (cid:19), magnitudefromvad ≈−0.95a3 tovad ≈−10.1a3,whichis
q2−k2−i0 4πg2Z 2m 2m apparentlyduetothevicinityoftheresonanceatthecrit-
↑ ↑
h(k,q)=(k2−q2)D(q,E−q2/2m )/4π. (14) icalmassratiom↑/m↓ ≈8.2[19].Althoughthemassratio
↑ for the K-Li case is quite a bit smaller, our results indi-
Both f˜and h are not singular at |q| = |k|, and f˜is cho- catethatforsufficientlysmalldetuningonehasastrongly
marked p-wave K-KLi scattering resonance. Indeed, for
sen such that f˜(k,k(cid:48)) = f(k,k(cid:48)), for |k| = |k(cid:48)|. Finally, R∗ =0 the p-wave phase shift reaches the unitarity value
we note that Eq. (12) conserves angular momentum and,
π/2 at a relatively small collision energy E ≈0.1|(cid:15) |.
therefore, can be written as a set of decoupled equations coll 0
In Fig. 2 we also see that the atom-dimer interaction
for each partial wave
decreases with detuning. We attribute this to the fact
(cid:40) (cid:41) that at larger R∗/a the light atom spends more time in
2 (cid:90) ∞ g (p,q)f˜(k,q)
f˜(k,p)=h(k,p) g (k,p)+ q2dq (cid:96) (cid:96) , theclosed-channelmolecularstate,andconsequentlycon-
(cid:96) (cid:96) π 0 q2−k2−i0 tributes less to the atom-dimer exchange interaction. In
(15) a sense, increasing R∗/a is similar to increasing the mass
where we define of the light atom (decreasing the mass ratio): the heavier
the atom, the weaker the exchange interaction.
g (k,p)= 1(cid:90) 1 dxP (x)G (k+p,E−k2/2m −p2/2m ), Althoughthep-waveresonancebecomeslesspronoun-
(cid:96) (cid:96) ↓ ↑ ↑
2 cednearanarrowresonance,inaK-Limixturethep-wave
−1
(16) atom-dimer interaction can be strong, which is demon-
wherexisthecosineoftheanglebetweenkandp.Partial strated in Fig. 3, where we plot the partial wave cross-
atom-dimer scattering amplitudes are related to solutions sections σ (k) = 4π(2(cid:96)+1)k−2sin2δ (k). We clearly see
(cid:96) (cid:96)
of Eq. (15) by the equation f (k)=f˜(k,k), and the cor- that for detunings R∗/a (cid:46) 1 the p-wave partial cross-
(cid:96) (cid:96)
responding phase shifts δ(cid:96) are deduced from Eq. (2). section either exceeds or is comparable to σs in a wide
In Fig. 2 we plot the s-, p-, and d-wave phase shifts as range of collision energies.
functions of the collision energy E = k2/2µ for differ- For comparison, in Fig. 4 we present partial atom-
coll 3
ent detunings R∗/a. We write the phase shifts as δ ≡δ , dimercross-sectionsinthehomonuclearcase.Weseethat
0 s
J. Levinsen and D. S. Petrov: Scattering in fermionic mixtures near a narrow Feshbach resonance 5
Equal masses 40K-6Li mixture
20
4
0
a a
15 a /ad2 -5 r /ad
10 (cid:109)s/a2 0
0 5 10 0 5 10
0 4
5 (cid:109)p/a2 (cid:109)d/a2 3v /aad-20 2 k aad
0
00 0.5 1 -40
E / |(cid:161) |
coll 0 0 5 10 0 5 10
R*/a R*/a
Fig. 4. (color online). Atom-dimer s-wave (black), p-wave
(blue), and d-wave (purple) scattering cross sections for the Fig. 5. (color online). Atom-dimer scattering length aad, s-
homonuclear gas. Solid, dashed, dot-dashed, and dotted lines waveeffectiverangerad,p-wavescatteringvolumevad,andthe
correspond to R∗/a=0,1/16,1/4, and R∗ =a, respectively. p-waveeffectiverangeparameterkad inunitsofcorresponding
powers of a vs. R∗/a for m /m = 6.64. Solid lines are exact
↑ ↓
anddottedlinesareapproximateresults(17)-(20)validinthe
limit R∗ (cid:29)a.
the s-wave contribution is always dominant and the func-
tional form of σ /a2 is fairly insensitive to the detuning.
s Equal masses
We have calculated the scattering parameters for sev-
eral higher partial waves. Their contributions rapidly de- 0
creasewith(cid:96)anditisworthplottingonlythed-wavephase 1
a a
shifts(seeFig.2)andscatteringcross-sections(Figs.3and /d -10 /d
4). The d-wave contribution is comparable to the s- and aa0.5 ra
p-wave ones only in the heteronuclear case and for rela-
-20
tively high collision energies ∼ |(cid:15)0|. In Figs. 2 and 4 the 00 5 10 0 5 10
d-wave contribution for the homonuclear case is plotted
0 20
onlyforR∗ =0asforfinitedetuningsthecurvesareeven
closer to the horizontal axis. 3 15
a a
We have already discussed the atom-dimer scatter- /d 10 ad
ing length and scattering volume for m↑/m↓ ≈ 6.64 and va-5 5 k
m /m = 1 in the case R∗ = 0. In Figs. 5 and 6 we plot
↑ ↓
thesequantitiesandtheeffectiverangeparametersr and 0
ad 0 5 10 0 5 10
kad [atom-dimeranaloguesofr0 andk0 definedinEqs.(3) R*/a R*/a
and (4)] versus the detuning R∗/a. Dotted lines in these
graphsareobtainedbyusingaperturbationtheoryinthe Fig. 6. (coloronline).SameasinFig.5butforthehomonuc-
limit of a very narrow resonance, g → 0, when the atom- lear case.
dimerT-matrixcanbeobtainedbysummingthefirstfew
diagrams in Fig. 1. This gives an expansion in powers of
(cid:112) 3.1 Born-Oppenheimer approximation
a/R∗ (cid:28) 1. The first two terms in the expansion of a
ad
and v and the leading terms for the effective range pa-
ad Itisinstructivetoconsidertheenhancementoftheatom-
rameters r and k read
ad ad dimer scattering and the appearance of trimers for suf-
ficiently large mass ratios in the Born-Oppenheimer ap-
(cid:20) (cid:18) (cid:19)(cid:114) (cid:21)
a ≈aµ3 1+ 1 1− µ3 a , (17) proximation[28].ThismethodwasintroducedinRef.[15]
ad µ 2 µ R∗ to study Efimov physics in the system of one light and
µ (cid:18) µ (cid:19) twoheavyparticles.AlthoughtheBorn-Oppenheimerap-
r ≈−4R∗ 1− , (18) proximation is not exact, it serves well to illustrate the
ad µ 2µ
3 3 essential physics leading to the resonant enhancement of
(cid:20) (cid:18) (cid:19)(cid:114) (cid:21)
v ≈−2a2R∗ µ3 1+ 3 1+ µ3 a , (19) the p-wave scattering. Here we extend it to the case of a
ad 3 m 2 36m R∗ resonance of finite width.
↓ ↓
(cid:18) (cid:19) Themethodtakesadvantageofthelargemassratioby
12m µ
k ≈ ↓ 1− . (20) assumingthatthestateofthelightatomadiabaticallyad-
ad a µ 2µ
3 3 justs itself to the distance R between the heavy fermions.
The wavefunction of the light atom can be written in the
6 J. Levinsen and D. S. Petrov: Scattering in fermionic mixtures near a narrow Feshbach resonance
form
0.5
e−κ±(R)|r−R/2|/R e−κ±(R)|r+R/2|/R
ψ (r)∝ ± . (21)
R,± |r−R/2| |r+R/2|
It satisfies the free-particle Schr¨odinger equation with the )
R
energy (
V1
(cid:15)±(R)=−κ2±(R)/2m↓. (22) 2a0
(cid:63)
m
The singularities of ψR,±(r) at vanishing r˜ = r ±R/2 2
satisfy the Bethe-Peierls boundary condition [29]
[r˜ψ](cid:48)/r˜ψ| =iκ (R)cotδ [iκ (R)]
r˜ r˜→0 ± 0 ±
=−1/a+R∗κ2(R), (23)
± -0.5
0 2 4
R/a
where the light-heavy s-wave phase shift, again denoted
by δ , is calculated at the light-heavy collision energy
0 Fig.7.TheBorn-Oppenheimeratom-dimereffectivepotential
(cid:15)±(R). The equation for κ±(R) is obtained by applying V1(R) [in units of 1/2m↓a2] in the wide resonance case (R∗ =
the boundary condition (23) to the wavefunction (21): 0) for mass ratios m /m =5 (dashed), m /m (solid), 8.2
↑ ↓ K Li
(dash-dotted), and 13.6 (dotted).
κ (R)∓exp[−κ (R)R]/R=1/a−R∗κ2(R). (24)
± ± ±
ThesecondstepoftheBorn-Oppenheimermethodcon- 0.5
sists of solving the Schr¨odinger equation for the heavy
fermions by using (cid:15) (R) as the potential energy surface.
±
Letusdenotethecorrespondingheavy-fermionwavefunc-
tion by φ(R). Since the total three-body wavefunction,
)
R
proportional to the product φ(R)ψR,±(r), should be an- (
V1
tisymmetricwithrespecttothepermutationoftheheavy
2
a
fermions,thesymmetryofφdependsonthechoiceofsign (cid:63)
m
in Eq. (21). As ψ (r) is symmetric with respect to the 2
R,+
permutation R ↔ −R, the heavy-atom wavefunction φ
is antisymmetric and describes odd atom-dimer scatter-
0
ing channels. Accordingly, the lower sign in Eqs. (21-24)
corresponds to even channels. We see how the compos-
ite nature of the dimer leads to the (cid:96)-dependent effective 0 2 4
R/a
atom-dimer potentials: by solving Eq. (24) one arrives at
a purely attractive (cid:15) (R) for odd channels and purely re-
+
Fig.8.TheBorn-Oppenheimeratom-dimereffectivepotential
pulsive (cid:15) (R) for even ones.
− V (R)[inunitsof1/2m a2]intheK-LicaseforR∗ =0(solid),
FromtheviewpointoftheradialSchr¨odingerequation 1 ↓
R∗ = a/16 (dashed), R∗ = a/4 (dash-dotted), and R∗ = a
it is convenient to introduce the total effective potential
(double-dotdashed).Thedottedlineisthecentrifugalbarrier.
for each φ (R):
(cid:96)
V (R)=(cid:15) (R)−(cid:15)(∞)+(cid:96)((cid:96)+1)/m R2, (25)
(cid:96) (−1)(cid:96)+1 ↑ mass ratios the presence of the well leads to the resonant
enhancement of the p-wave interaction.
whichincludesthecentrifugalbarrierandshiftsthethresh-
old to zero by subtracting the dimer binding energy TheeffectoffiniteR∗ istodecreasethestrengthofthe
exchangepotentials(cid:15) .InFig.8weshowV (R)intheK-
± 1
(cid:15)(∞)=−((cid:112)1+4R∗/a−1)2/8m R∗2. (26) Li case for different values of the detuning R∗/a. One can
↓
see that the p-wave attraction becomes less pronounced
In the limit m (cid:29)m Eq. (26) reduces to Eq. (11). and the well eventually disappears with increasing R∗/a.
↑ ↓
For the p-wave atom-dimer interaction the central is- It is important to distinguish the p-wave trimer for
sue is the competition between the attractive exchange m /m (cid:38) 8.2 from Efimov trimers. The former exists
↑ ↓
potential(cid:15) ∝1/m andtherepulsivecentrifugalbarrier, only for a > 0 and is a result of the peculiar competi-
+ ↓
which is inversely proportional to m . In Fig. 7 we show tion between the exchange potential and the centrifugal
↑
V (R)inthelimitofvanishingdetuningfordifferentmass force at distances of the order of a, which determines its
1
ratios. Remarkably, for m /m ∼ m /m this potential, size. In contrast, the Efimov effect occurs at larger mass
↑ ↓ K Li
beingrepulsiveinbothlimitsR(cid:28)aandR(cid:29)a,develops ratios, m /m >13.6, when the effective potential at dis-
↑ ↓
a well at distances R ∼ a. For m /m > 8.2 the depth tances R (cid:28) a is no longer repulsive. Then, in the Born-
↑ ↓
of this well is enough to accomodate a trimer state with Oppenheimer description the heavy atoms fall to the cen-
unit angular momentum [19] and for somewhat smaller ter in an attractive 1/R2-potential. This is accompanied
J. Levinsen and D. S. Petrov: Scattering in fermionic mixtures near a narrow Feshbach resonance 7
3.5
14
3
2.5 0 1.28
10
2
8
1.5 0 6.6
1
4
0.5
0 N)-0 02.4
+
+0.2
N
xˆ )/( 0
N-
zˆ -+-0.2
N
Fig. 9. (color online). The integrated column density for K- (
-0.4
KLiatom-dimerscatteringinarbitraryunits.Thecollisionen-
ergies are Ecoll =0.05|(cid:15)0| (left) and Ecoll =0.25|(cid:15)0| (right). In -0.6
both cases R∗ =a/4. Backward direction corresponds to neg-
0.001 0.01 0.1 1
ative z. For presentation purposes we imitate a small thermal E / !(cid:161) !
smear. coll "
Fig. 10. The contrast vs. collision energy for detuning R∗/a
equal to 0 (solid), 1/16 (dashed), 1/4 (dot-dashed), and 1
by the appearance of an infinite set of Efimov states, irre-
(double-dotdashed).Dottedlineisthehomonuclearcaseresult
spective of the sign of a.
for R∗ =0.
Atom-dimer scattering in even channels is described
by the potential (cid:15) (R), which is defined at distances R>
+
a. It has a form of a purely repulsive soft-core potential, favors either backward or forward scattering. In Fig. 9 we
whichincreaseswiththemassratioanddecreaseswithR∗, simulate an absorption image (column density) of scat-
consistent with the exact results above on s- and d-wave teredparticlesthatinitiallymovedinthepositivez direc-
atom-dimer scattering. tion. In the K-Li case backward scattering dominates at
smallcollisionenergieswhileforwardscatteringisfavored
at higher energies, when δ −δ >π/2.
p s
3.2 Interference of s- and p-waves In Fig. 10 we plot the contrast, defined as the normal-
ized difference between the numbers of particles scattered
Now we would like to discuss one of the implications of forward,N ,andbackward,N ,asafunctionofcollision
+ −
the channel dependent atom-dimer interaction, a pecu- energy for different detunings R∗/a. For comparison we
liarity, which is strongly pronounced in the K-Li mixture. alsopresentthehomonuclearwide-resonancecase(dotted
Inthiscasethep-ands-wavephaseshiftsarecomparable line). We see that in this case backward scattering always
in magnitude and can be large (see Fig. 3). It is thus not dominates,thehighestcontrastachievedforE ≈|(cid:15) |/3.
coll 0
necessary to go to very high collision energies for observ- The collision experiment described above requires the
ing the quantum interference between these partial waves ability of manipulating atoms and molecules individually,
[30]. which points to an advantage of heteronuclear mixtures
Let us consider a gedankenexperiment in which a cold – in the heteronuclear case different atomic species feel
thermal cloud of KLi dimers collides with a cloud of K opticalpotentialsinadifferentmanner,andthisobviously
atomsatcollisionenergiesbelowthedimerbreak-upthresh- holds for the two components of the corresponding atom-
old.Themeasurablequantityisthentheangulardistribu- molecule mixture.
tion of scattered dimers (or atoms), which is proportional As far as the energy scale is concerned, in the K-Li
to the differential cross-section. We can write it in terms case with R∗ = a = 100 nm the dimer binding energy
of the phase shifts by using Eqs. (1) and (2): givenbyEq.(11)equals|(cid:15) |≈1.8µK,anditdecreasesto
0
200nKwhena=400nm.Wealsomention,forreference,
dσ = 1 (cid:2)sin2δ +6cos(δ −δ )sinδ sinδ cosθ that the relative atom-dimer velocities corresponding to
dΩ k2 s p s s p the collision energy E = |(cid:15) | in these two cases equal
+9sin2δ cos2θ(cid:3)+... (27) 3.7 cm/s and 1.2 cm/scorlelspect0ively.
p
Here k is the relative atom-dimer momentum and the an-
gle θ is measured with respect to the collision axis, which 4 Dimer-dimer scattering
wedenotebyzˆ.Thedotssignifythecontributionofhigher
partialwaves.Wehavecheckedthattheycanbesafelyig- Dimer-dimerinteractionparametersarecrucialforthede-
nored. scriptionoftheBCS-BECcrossoverintheBEC-limit,i.e.
The first term on the right hand side in Eq. (27) gives when the gas of molecules is dilute. In the lowest order
thewell-knownsphericallysymmetricscatteringhalo.The the chemical potential, condensate depletion, and speed
last term corresponds to the pure p-wave scattering. It of sound in the BEC of dimers are determined from the
contributes equally to the forward (0 < θ < π/2) and densityanddimer-dimerscatteringlengtha inthesame
dd
backward(π/2<θ <π)directions,butvanishesinthedi- manner [31] as in the usual Bogoliubov theory of dilute
rectionperpendiculartozˆ.Thesecond(interference)term Bose gases.
8 J. Levinsen and D. S. Petrov: Scattering in fermionic mixtures near a narrow Feshbach resonance
In the case of a homonuclear mixture near a wide res-
onance the dimer-dimer scattering length equals a ≈ T = ! + T !
dd
0.6a. This number was obtained in Ref. [32] by solving
an integral equation derived directly from the four-body Fig.11.Diagrammaticrepresentationoftheintegralequation
Schr¨odingerequationincoordinatespace.Latertheresult (28), which relates the dimer-dimer T-matrix with the sum of
wasconfirmedbydiagrammaticapproaches[31,33]andby all the two-dimer irreducible diagrams, Γ.
Monte-Carloandvariationaltechniques[34,35].Inthehe-
teronuclear case the molecule-molecule scattering length
wasalsocalculatedinthecaseofawideinterspeciesreso- ! = + ! +( )
nance[36,35,37].VonStecheretal.[35]alsocomputedthe
dimer-dimer effective range. The inelastic scattering and Fig. 12. The integral equation satisfied by χ, the sum of the
the formation of Efimov trimers in dimer-dimer collisions two-dimer irreducible diagrams, in which one of the outgoing
for m /m >13.6 is discussed in Ref. [38]. dimers is split into its constituent parts.
↑ ↓
A qualitative summary of the results cited above is
that the dimer-dimer interaction can be thought of as
The corresponding s-wave averaged sum is denoted by
a soft-core repulsion. It strengthens with the mass ratio,
Γ(q,q ;p,p ), where the four-momenta of the incoming
0 0
but is always weaker than the s-wave repulsion between a [outgoing] dimers equal (±q,(cid:15) ±q ) [(±p,(cid:15) ±p )]. The
0 0 0 0
dimer and a heavy atom. This picture can be understood
equation for the T-matrix then reads
from the Born-Oppenheimer analysis, when one assumes
(cid:90)
thatthewavefunctionofthetwolightfermionsisgivenby i
T(p,p )=g4Z2Γ(0,0;p,p )+ q2dqdq T(q,q )
the antisymmetrized product of ψR,+(r1) and ψR,−(r2) 0 0 4π3 0 0
[seeEq.(21)],whichisantisymmetricunderthepermuta- ×Γ(q,q ;p,p )D(q,(cid:15) +q )D(q,(cid:15) −q ) (28)
0 0 0 0 0 0
tion of the heavy fermions. Accordingly, the heavy-atom
part of the wavefunction should be symmetric, consistent andisillustratedinFig.11.Asinthethree-bodycase,the
with the fact that only even scattering channels are al- prefactor of the first term on the right hand side serves
lowed between identical bosons. The Born-Oppenheimer for the correct normalization of external propagators. In
potential energy surface is given, independent of the an- order to avoid poles and branch cuts we solve Eq. (28) by
gular momentum, by the sum (cid:15) (R)+(cid:15) (R). It increases rotatingthecontouroftheq -integrationtotheimaginary
+ − 0
with decreasing the mass of the light atom and is repul- axis [31].
sive,butnotasstrongastheatom-dimers-wavepotential The sum of the two-dimer irreducible diagrams is cal-
(cid:15) (R). culated as follows. We first sum the two-dimer irreducible
−
As far as we know, higher partial waves in scatter- diagrams which end in a dimer and two fermionic atoms.
ing of bosonic dimers have not been studied, but it has Wedenotethesumofsuchtwo-dimerirreduciblediagrams
been shown that the ground state of four ↑ − ↑ − ↓ − ↓ by χ(q,q0;p1,p2), where the two incoming dimers have
fermions in an anisotropic harmonic potential has zero four-momenta (±q,(cid:15)0±q0), and the outgoing ↑ [↓] atom
angular momentum, independent of a and the mass ratio may be put on-shell with four-momentum (p1,p21/2m↑)
[35,39]. Besides, the qualitative Born-Oppenheimer anal- [(p2,p22/2m↓)].Byenergy-momentumconservationtheout-
ysisdoesnotprovideargumentsforanyresonantenhance- going dimer then has four-momentum (−p1 −p2,2(cid:15)0 −
mentofhigherpartialwaves.Wethusconjecturethatthe p21/2m↑ −p22/2m↓). In Fig. 12 we illustrate the integral
s-wave channel should dominate the dimer-dimer interac- equation satisfied by χ. The equation itself reads
tion, at least for sufficiently small collision energies. (cid:90) dΩ (cid:26) (cid:18) p2 (cid:19)
The aim of this section is to compute the dimer-dimer χ(q,q0;p1,p2)=− q G↓ q−p1,(cid:15)0+q0− 1
4π 2m
s-wavescatteringlengtha forthehomonuclearandhete- ↑
ronuclear cases (having inddmind the Li-K mixture) taking ×G (cid:18)−q−p ,(cid:15) −q − p22 (cid:19)+[(q,q )↔−(q,q )](cid:27)
↑ 2 0 0 0 0
into account the finite width of the Feshbach resonance. 2m
↓
Our derivation follows Refs. [31,37] where the problem (cid:90) d3Q(cid:26) (cid:18) Q2 p2 p2 (cid:19)
was studied in the regime of small detuning. − (2π)3 G↑ Q+p1+p2,2(cid:15)0− 2m − 2m1 − 2m2
↓ ↑ ↓
We consider the scattering of two dimers with four-
(cid:18) Q2 p2 (cid:19)
momenta(0,(cid:15)0)intodimerswith(±p,(cid:15)0±p0)andproject ×D Q+p ,2(cid:15) − − 1 χ(q,q ;p ,Q)
1 0 0 1
onto the s-wave (average over directions of p). The four- 2m↓ 2m↑
body T-matrix with these kinematics is denoted T(p,p0). (cid:18) Q2 p2 p2 (cid:19)
Similarly to the three-body case the four-body T-matrix +G↓ Q+p1+p2,2(cid:15)0− − 1 − 2
2m 2m 2m
consists of an infinite sum of diagrams, which may again ↑ ↑ ↓
be reduced to integral equations. In order to perform this ×D(cid:18)Q+p ,2(cid:15) − Q2 − p22 (cid:19)χ(q,q ;Q,p )(cid:27),
2 0 0 2
summation, we first construct the sum of two-dimer ir- 2m 2m
↑ ↓
reducible diagrams beginning and ending in two dimer (29)
propagators. These are the diagrams that cannot be di-
vided in two by cutting only one pair of dimer propaga- where the frequency integration in the closed loop of the
tors (external lines are excluded from the summation). iterated term is already performed. The configurational
J. Levinsen and D. S. Petrov: Scattering in fermionic mixtures near a narrow Feshbach resonance 9
! = + "
0.8
Fig. 13. Γ expressed in terms of χ (see text).
0.6 Li-K mixture
a

/

space of Eq. (29) is in fact three-dimensional. It consists dd
a0.4
of the moduli of the vectors p and p , and the angle
1 2
between them. The pair (q,q ) enters parametrically. In
0
order to express Γ in terms of χ, it is advantageous to 0.2 Equal masses
separate out the simplest diagram, in which the dimers
exchange identical atoms. Then, the remaining diagrams
in Γ are obtained by closing the fermionic loop in χ (see 0
0 5 10
Fig. 13).
R*/a
The relation between Γ and χ is
Fig.14.Thedimer-dimerscatteringlengthvs.R∗/aforequal
Γ(q,q ;p,p )=Γ(0)(q,q ;p,p )
0 0 0 0 masses (solid line) and for the Li-K mixture (dashed). The
1(cid:90) d3p d3p (cid:26) (cid:18) p2 (cid:19) dottedlinescorrespondtotheasymptote(34)validinthelimit
−2 (2π)13(2π)23 G↓ p−p1,(cid:15)0+p0− 2m1 R∗ (cid:29)a.
↑
(cid:18) p2 (cid:19) (cid:27)
×G p+p ,(cid:15) −p − 2 +[(p,p )↔−(p,p )]
↑ 2 0 0 0 0
2m↓ undergo relaxation into deep bound states in collisions
(cid:18) p2 p2 (cid:19) with each other or with unbound atoms. The process is
×D p1+p2,2(cid:15)0− 1 − 2 χ(q,q0;p1,p2), (30) local as it requires at least three atoms to approach each
2m 2m
↑ ↓
other to a distance comparable to the size of the future
where the factor 1 is needed for correct counting of dia- molecular state, i.e. ∼Re. The released binding energy is
2 of the order of 1/m R2 and is much larger than all other
grams. The quantity Γ(0) is the first diagram on the right ↓ e
energy scales in the problem including the height of the
hand side of Fig. 13 and is given in Appendix B.
trappingpotential.Thus,therelaxationproductsarelost.
The dimer-dimer scattering length is related to the T-
Althoughtherelaxationisashort-rangephenomenon,
matrix by
it can be treated in the zero-range approximation. For
M
a = T(0,0). (31) wide resonances in the Efimov case, i.e. for bosons or for
dd 4π
fermions with m /m > 13.6, the Efimov physics is well
↑ ↓
Fig. 14 shows our results for the dimer-dimer scattering described by the motion of three atoms in an effective at-
length in the equal mass case and for the Li-K mixture. tractive1/R2 potential[41,42].Thethree-bodywavefunc-
In the limit of small detuning we recover the results [32, tion can be separated in an incoming wave and an outgo-
36,35,37] ing one, and the relaxation process is taken into account
byaddinganimaginaryparttothethree-bodyparameter
a =0.60, m /m =1, (32)
dd ↑ ↓ [43]. It fixes the ratio of the corresponding incoming and
add =0.89, m↑/m↓ =mK/mLi. (33) outgoing fluxes. The physical range of the potential Re
does not enter the resulting relaxation rate constant.
Intheoppositelimitthediagrammaticexpansionbecomes
The suppression of relaxation in the non-Efimovian
perturbativeasisthecasefortheatom-dimerproblemdis-
cases (i.e. the ↑↑↓ system of fermions with m /m <
↑ ↓
cussedinSec.3.Thedominantcontributiontothedimer-
13.6) originates from the centrifugal barrier for identical
dimer T-matrix is provided by Γ(0). Including also the
fermions, which, in turn, leads to the repulsive effective
next order, we find [40] three-body 1/R2 potential [14,44]. In order to recombine,
the atoms have to tunnel under this barrier to distances
(cid:114) (cid:26)
add = M a + a × 0.13, m↑ =m↓ , R∗ (cid:29)a. ∼ Re. The zero-range approach in this case is perturba-
a 8µ R∗ R∗ 0.23,m↑/m↓ =6.64 tive. It uses the unperturbed few-body wavefunction to
(34) predict the probability of finding three atoms at small
ThefirsttermontherighthandsideofEq.(34)hasbeen distances and gives the functional dependence of the re-
derived in the equal-mass case in Ref. [23]. laxationrateconstantonthescatteringlengthforagiven
massratio[17,36].Iftherelaxationrateconstantisknown
for a certain a, one can predict its value for any other
5 Relaxation rates a(cid:29)R .
e
To be more specific, let us demonstrate how one can
The weakly bound dimers that we are considering are in estimate the atom-dimer relaxation rate in the case of a
factmoleculesinthehighestrovibrationalstate.Theycan wide resonance, for example in s-wave collisions. For an
10 J. Levinsen and D. S. Petrov: Scattering in fermionic mixtures near a narrow Feshbach resonance
atom and a molecule in a unit volume, the probability of Using Eq. (39) it is straightforward to show that the
findingthemwithinthedistanceafromeachotherequals wavefunctionoftheweaklyboundmolecularstateisgiven
a3 (we assume that there is no s-wave atom-dimer reso- by
√ (cid:113)
nance).Atdistancessmallerthana,thethree-bodywave- (cid:112)
φ = Z = 1−1/ 1+4R∗/a (40)
function(inthecenter-of-massreferenceframe)factorizes 0,b
iρn=to(cid:112)Ψ((RR11,−R2R,2r))2∝+ρmνs↓−/1(Φ2m(Ωˆ↓)+, wmh↑e)r(e2trh−e Rhy1p−errRad2i)u2s,Ωiˆs and ψb(R)=√1−Z(cid:112)κ/2πexp(κR)/R, (41)
√
is a five-dimensional set of all the remaining coordinates
whereκ= 2µ(cid:15) [seeEq.(11)]andZisdefinedinEq.(65).
0
(hyperangles), and Φ is a normalized hyperangular wave-
We see that the probability of finding the atoms in the
function. The power νs for the ↓↓↑-system is given by the open channel equals 1−Z and is small in the regime of
root of the transcendental equation [14] intermediate detuning. Therefore, as R∗/a → ∞ the re-
laxation rate constant tends to a constant value [45]
πν sin[φ(ν +1)]
(ν +1)tan s −2 s =0 (35)
s 2 sin(2φ)cos(πνs/2) αsa,dbare ∼Re/m↓, (42)
in the interval −1 < Reνs < 3. In Eq. (35) φ is defined which corresponds to the relaxation in collisions of atoms
as φ=arcsin[m↑/(m↑+m↓)]. For m↑/m↓ =1 we obtain and bare molecules.
νs ≈1.166 and for m↑/m↓ =6.64 we get νs ≈2.02. Equation (39) can be used in a more general situa-
The probability of finding the three atoms at hyper- tion as it applies to a pair of atoms when they are very
radii smaller than ρ scales with ρ as P(ρ) ∝ |ρνs−1|2ρ6. close to each other. Even in a system of more than two
The last term is the volume factor of the six-dimensional atoms and/or in an external potential we can look at a
configurational space of the three-body problem (in the particular pair of atoms and observe that the probability
center-of-mass reference frame). Thus, the total probabil- offindingthemintheopenchannelatseparationssmaller
ity of finding the three atoms in the relaxation region is than R equals (cid:82)R|ψ(R(cid:48))|24πR(cid:48)2dR(cid:48) = |φ |2R/R∗, i.e. in
∼ a3P(Re)/P(a) ≈ a3(Re/a)2νs+4. The relaxation rate thecaseR(cid:28)R∗0itismuchsmallerthanth0eprobabilityof
constantisobtainedbymultiplyingthisprobabilitybythe finding them in the closed channel. In particular, we can
frequency with which the relaxation process takes place conclude that locally, when three atoms are at the hyper-
once the atoms are within the range of the potential. It radius ρ (cid:28) R∗, they can be considered as an atom and
can be estimated as ∼1/m↓Re2. Finally, for the rate con- a closed-channel molecule. The bare interaction between
stant we obtain themisneglectedinEq.(7)asitisassumednon-resonant.
Their induced interaction (via the exchange of the open-
R (cid:18)R (cid:19)2νs+1
αad ∼ e e . (36) channel atoms) has a Coulomb form [22,46] and can also
s m↓ a be neglected at very small distances.
In order to estimate αad for narrow resonances in the
s
In the case of a narrow resonance the three atoms can regimeofsmalldetuning,R∗ (cid:28)a,weshouldslightlymod-
approachtherecombinationregioneitherasfreeatomsor ify the speculations that lead us to Eq. (36). At distances
as a closed-channel molecule and an atom. One can show, ρ (cid:29) R∗ the three-body wavefunction behaves practically
however, that in this case the probability of the former is inthesamemannerasinthewideresonancecase.Thede-
much smaller than the probability of the latter. Indeed, viation is important at distances smaller than R∗, where,
let us consider two atoms, ↑ and ↓, in the center-of-mass as we have just mentioned, the three-body wavefunction
reference frame. The state of the system is given by describesanon-interactingatomandabaremolecule.The
rate constant reads
(cid:32) (cid:33)
(cid:88)
|Ψ(cid:105)= ψkaˆ†k,↑aˆ†−k,↓+φ0ˆb†0 |0(cid:105). (37) (cid:34) (cid:18) 1 (cid:19)3(cid:35) (cid:34)(cid:18)R∗(cid:19)2νs+4 (cid:35)
αad ∼ αad × a3 , (43)
k s s,bare R∗ a
DemandingthatEq.(37)beaneigenstateoftheHamilto-
nian (7) with energy E we get two coupled equations for where the first factor is the relaxation rate for an atom
ψ and φ , one of which in coordinate space reads and a bare molecule confined to a volume of size R∗3 and
k 0
the second factor is the probability to find three atoms
−∇2Rψ(R)+gφ δ(R)=Eψ(R). (38) in this volume. We see that the ratio ηs = αsad/αsa,dbare
2µ 0 interpolates between ηs ∼ (R∗/a)2νs+1 for small R∗/a to
η =1 for large R∗/a.
s
From Eq. (38) one can see that the singularity of ψ(R) at Theatom-moleculerelaxationrateinthep-wavechan-
the origin is related to φ by nel can be estimated in the same fashion. The difference
0
√ fromthes-wavecaseistheadditionalfactor(ka)2,wherek
ψ(R→0)=φ / 4πR∗R, (39) is the relative atom-molecule momentum. It enters due to
0
theunitangularmomentumwhenwecalculatetheproba-
where we have used the second of Eqs. (8) to express g in bility to find the atom and the molecule at distances ∼a.
terms of R∗. For the same reason the relaxation rate constant in the