Momentum-space calculation of four-boson recombination

0.17 MB·English

by  A. Deltuva

Momentum-space calculation of four-boson recombination
A. Deltuva
Centro de F´ısica Nuclear da Universidade de Lisboa, P-1649-003 Lisboa, Portugal
(Received December 12, 2011)
The system of four identical bosons with large two-boson scattering length is described using
momentum-space integral equations for the four-particle transition operators. The creation of
Efimov trimers via ultracold four-boson recombination is studied. The universal behavior of the
recombination rate is demonstrated.
2
PACSnumbers: 34.50.-s,31.15.ac
1
0
2
I. INTRODUCTION been obtained in Ref. [17] for a system of three identical

n bosons plus a distinguishable particle.
a In 1970 V. Efimov predicted that with vanishing two- An extension of the four-body scattering calculations
J
abovethefour-clusterbreakupthresholdisaveryserious
particle binding an infinite number of weakly bound
1 theoreticalchallengebothinatomicandnuclearphysics.
states with total orbital angular momentum = 0 may
1 L In the momentum-space framework the difficulties arise
exist in the three-particle system [1, 2]. Such a situation
due to complicated singularity structure in the kernel of
] takes place if at least two pairs have infinite scattering
h length a [2]. Away from this limit the number the integral equations. Not all but some of these dif-
p → ∞ ficulties are present already at the four-cluster breakup
of trimers is finite but, depending on a, may be large
-
m [1,2]. Thisthree-particleEfimoveffecthasanimpacton threshold. In this work we restrict ourselves to a latter
casewhichis sufficientto calculatethe four-atomrecom-
the behavior of more complex few-particle systems. In
o
particular, for each Efimov trimer the existence of two binationin the ultracoldlimit. Nevertheless,the present
t
a tetramerswaspredictedinRefs.[3,4]. Thesignaturesof work is an important intermediate step towards solving
s. these states were observed recently in experiments with the momentum-space four-body scattering equations at
c positive energies.
ultracoldatomswheretheformationofthetetramersled
si to a resonant enhancement of the recombination or re- We use a system of units where ~=1 and therefore is
y omitted in the equations unless needed for dimensional
laxation processes [5–7].
h
analysis. In Sec. II we recall the AGS equations and de-
These first steps in exploring the four-body Efimov
p
rive the relation between the two- and four-cluster tran-
[ physics experimentally also call for accurate theoretical
studies of the four-body systems with large a. Follow- sition operators. In Sec. III we present results for the
1 four-boson recombination. We summarize in Sec. IV.
ing our recentworksdevotedto the bosonic atom-trimer
v
[8] and dimer-dimer [9] scattering, here we focus on the
6
2 four-atom recombination process in the system of four
II. FOUR-BOSON SCATTERING
3 identical bosons. The description is based on the exact
2 four-particle equations for the transition operators [10]
. derived by Alt, Grassberger, and Sandhas (AGS). For We consider a system of four spinless particles with
1
0 the recombination process we need the transition oper- Hamiltonian
2 ator connecting two- and four-cluster channel states; its
6
1 relation to the two-cluster AGS operators calculated in
H =H + v , (1)
: Refs. [8,9,11,12],willbe giveninthepresentwork. We 0 i
v Xi=1
i solvetheintegralAGSequationsusingmomentum-space
X
framework. In previous works we demonstrated its reli- where H is the kinetic energy operator for the relative
0
r ability for reactions involving many very weakly bound motion and v the short-range potential acting within
a i
trimers as it happens in the universal regime where the the pair i; there are six pairs. For the description of the
accuracy of the coordinate-space methods may be more scatteringprocessinsuchasystemweusetheAGSequa-
limited [4, 13, 14]. For example, in Ref. [15] we pre- tions [10]. They are exact quantum-mechanical equa-
dicted universal intersections of shallow tetramers with tions of the Faddeev-Yakubovsky (FY) type. However,
the corresponding atom-trimer thresholds while the adi- incontrasttotheoriginalFYequations[18]forthewave-
abatichypersphericalcalculationsofRefs.[4,6]werenot function components, the AGS equations are formulated
sufficiently precise to find this remarkable feature that for the transition operators in the integral form and
leadstoaresonantbehaviorintheultracoldatom-trimer thereforearebettersuitedtobesolvedinthemomentum-
collisions. Thus, we expect that also in the case of the space framework preferred by us.
four-boson recombination we will be able to achieve the Weaimtodeterminetheamplitudefortherecombina-
universallimitwithhigheraccuracycomparedtothe ex- tionoffourfreeparticlesintoatwo-clusterstate. Dueto
istingcoordinate-spacecalculations[4,16]. Wenotethat time reversal symmetry it is equal to the amplitude for
approximate semi-analytical recombination results have the four-cluster breakup of the initial two-cluster state.
2
The latter is moredirectly relatedto the AGS transition Thefullwavefunctioncorrespondingtotheinitialtwo-
operators calculated in our previous works [8, 9, 11, 12]. cluster channel state Φ is also determined by the
ρ,n
| i
AGS transition operators,i.e.,
A. AGS equations Ψ = Φ + G t G UjkG t G ki φi .
| ρ,ni | ρ,ni 0 j 0 γ 0 k 0Uγρ| ρ,ni
γXjki
InacompactnotationtheAGSequationscanbe writ- (8)
tenas18-componentmatrixequations[10,14]. Thecom- The amplitude for the four-cluster breakup reaction
ponentsaredistinguishedbythechainsofpartitions,i.e., can be obtained as
bythetwo-clusterpartitionandbythethree-clusterpar-
6
tition. Obviously, the two-cluster partitions may be of
Φ T Φ = Φ v Ψ . (9)
3+1 or 2+2 type; we will denote them by Greek sub- h 0| 0ρ| ρ,ni h 0| i| ρ,ni
scripts. All three-cluster partitions are of 2+1+1 type Xi=1
and are specified by the pair of particles that we will de-
Thewavefunction(8)satisfiesalsotheSchr¨odingerequa-
notebyLatinsuperscripts. Forourconsiderationweneed tion, thus, 6 v Ψ = G−1 Ψ . Furthermore,
tohpeereaxtpolrisc,iti.feo.,rmofthe AGS equationsfor the transition G−01|Φ0i = 0Psii=n1ceit|heρ,fnoiur-clus0ter|chρ,anninel state |Φ0i is
an eigenstate of H with eigenvalue E. In fact, Φ is
0 0
| i
ji =(G t G )−1δ¯ δ + δ¯ UjkG t G ki. (2) a product of three plane waves (each is normalized to
Uσρ 0 i 0 σρ ji σγ γ 0 k 0Uγρ the Dirac δ-function) corresponding to the relative mo-
Xγk
tion of four free particles. Thus, the amplitude for the
Here δ¯ =1 δ , four-cluster breakup of the two-cluster initial state is
σρ σρ
−
G =(E+i0 H )−1 (3) Φ T Φ = Φ t G UjkG t G ki φi .
0 − 0 h 0| 0ρ| ρ,ni h 0| j 0 γ 0 k 0Uγρ| ρ,ni
γXjki
is the free resolvent with the available four-particle en- (10)
ergy E, Due to time reversal symmetry it describes also the
four-particle recombination into a two-cluster state, i.e.,
t =v +v G t (4)
i i i 0 i Φ T Φ = Φ T Φ .
ρ,n ρ0 0 0 0ρ ρ,n
h | | i h | | i
If all four particles are identical, there are only two
is the two-particle transition matrix for the pair i, and
distincttwo-clusterpartitions,oneof3+1typeandoneof
Ujk =G−1δ¯ + δ¯ t G Uik (5) 2+2type. Wechoosethosepartitionstobe((12)3)4and
γ 0 jk ji i 0 γ (12)(34)anddenotetheminthefollowingbyα=1and2,
Xi
respectively. TheAGSequations(2)forthesymmetrized
isthesubsystemtransitionoperatorof3+1or2+2type, transition operators become
βα
U
depending on γ. In the summations over the pairs only
the ones internal to the respective two-cluster partition =P (G tG )−1+P U G tG +U G tG ,
11 34 0 0 34 1 0 0 11 2 0 0 21
U U U
contribute.
The transition operators ji contain full information (11a)
Uσρ
onthescatteringprocess. Theiron-shellmatrixelements =(1+P )(G tG )−1+(1+P )U G tG ,
jihφjσ,n′|Uσjiρ|φiρ,ni are the amplitudes hΦσ,n′|Tσρ|Φρ,ni U21 34 0 0 34 1 0 0U11(11b)
Pfor the two-cluster reactions. Here
=(G tG )−1+P U G tG +U G tG ,
12 0 0 34 1 0 0 12 2 0 0 22
φi =G δ¯ t φj (6) U U U (11c)
| ρ,ni 0 ij j| ρ,ni
Xj =(1+P )U G tG . (11d)
22 34 1 0 0 12
U U
aretheFaddeevcomponentsoftheinitialandfinalchan-
Here the two-particle transition matrix t acts within the
nel states
pair(12)andthesymmetrizedoperatorsforthe3+1and
Φ = φi , (7) 2+2subsystemsareobtainedfromtheintegralequations
| ρ,ni | ρ,ni
Xi U =P G−1+P tG U . (12)
α α 0 α 0 α
and n distinguishes between the different channel states
in the same partition. Φ is given by the nth bound The employed basis states have to be symmetric under
ρ,n
| i
state wave function in partition ρ times the plane wave exchange of two particles in subsystem (12) for 3 + 1
with momentum p between the clusters. and is nor- partition and in (12) and (34) for 2+2 partition. The
ρ,n
malized to Φσ,n′ Φρ,n = δσρδn′nδ(pσ,n′ pρ,n). The correct symmetry of the four-boson system is ensured
h | i −
on-shellcondition for eachchannelstate relatesits bind- by the operators P , P = P P +P P , and P =
34 1 12 23 13 23 2
ing energy, relative two-cluster momentum and reduced P P whereP isthepermutationoperatorofparticles
13 24 ab
mass as ǫ +p2 /2µ =E. a and b. For each two-cluster channel state Φ =
ρ,n ρ,n ρ | α,ni
3
(1 + P )φ there is only one independent Faddeev between all operators in Eqs. (11). Due to rotational
α α,n
| i
amplitude obtained from the integral equation symmetry all operators are diagonal in and indepen-
J
dent of . In addition, U are diagonal in k , J and
|φα,ni=G0tPα|φα,ni. (13) lz, P34 Mis diagonal in kx aαnd lx, t is diagonazl in ky,
The symmetrized amplitudes for the two-cluster reac- k and all ν, and G is diagonal in all quantum num-
z 0
tions are given in Refs. [8, 9, 11, 12]. The four-cluster bers. In this representation the AGS equations for each
breakupamplitudeintermsofthesymmetrizedAGSop- become a system of coupled integral equations in
J
erators (11) becomes three continuous variables k , k , and k . Such three-
x y z
variable equations have been solved in Refs. [11, 12]
Φ T Φ =S Φ (1+P ) [1+P (1+P )]
h 0| 0α| α,ni 0αh 0| 1 { 34 1 for the four-nucleon scattering below the three-cluster
tG0U1G0tG0 1α breakupthreshold. The kernelof integralequations (11)
× U
+(1+P )tG U G tG φ . contains integrable singularities arising from each sub-
2 0 2 0 0 2α α,n
U }| i
(14) system bound state pole of Uα; we isolate these poles
in different subintervals and treat them by the subtrac-
The factors S depend on the symmetry of the final
0α tion technique when integrating over k [11]. However,
channel state Φ . For example, S = √6 and S = 2 z
| 0i 01 02 in the energy regime of the four-cluster breakup, E 0,
for nonsymmetrized Φ , while S =√3 and S =√2 ≥
0 01 02 the AGS equations (11) contain additional singularities
| i
for Φ0 that is symmetrized within the pair (12). arising from G , namely, k′k′k′ν′ G k k k ν =
thrWe|sehodiledn,owtehebrye E|Φ00=i 0thaenfdouarll-cmluosmteerncthaavnanneilshs.taSteincaet δν′νδ(kx′ − kx)δ0(ky′ − ky)δ(αkhz′ x−ykzz)/[|kx2k0|y2kxz2(yEz+iiα0 −
k2/2µ k2/2µ k2/2µ )], where µ and µ are
all particles have equal momenta, the channel state |Φ00i thxe reαspxe−ctivye redαuyc−ed zmassαes. The opeαrxator proαdyucts
is invariant under all permutations, i.e., P Φ0 = Φ0 .
In this particular case the relation (14) simabp|lifi0eis to| 0i likeP34P1G0 renderthesingularitystructureofEqs.(11)
very complicated. It is considerably simpler at E = 0
Φ0 T Φ =S Φ0 (12tG U G tG where there is only one singular point k =k =k =0.
h 0| 0α| α,ni 0αh 0| 0 1 0 0U1α (15) x y z
+6tG0U2G0tG0 2α)φα,n . In this case the integrals involving G0 over any of the
U | i Jacobi momenta k (except for the trivial integrals in-
j
We use the momentum-space partial-wave framework
volving momentum δ-functions) are of type
to solve the AGS equations (11). Two different types of
basis states |kxkykz[(lxly)Jlz]JMiα = |kxkykzνiα with ∞ f(kx,ky,kz)kj2dkj
α = 1 and 2 are employed. Here k , k , and k denote (19)
x y z Z i0 k2/2µ k2/2µ k2/2µ
magnitudesoftheJacobimomenta. Forα=1theJacobi 0 − x αx− y αy− z α
momenta describe the relative motion in the 1+1, 2+1,
where the function f(k ,k ,k ) is regular at k = 0.
and3+1subsystemsandareexpressedintermsofsingle x y z j
Thus, the the singularity of G is cancelled by k2 mak-
particle momenta k as 0 j
a ing the integral numerically harmless. Nevertheless, G
0
1 varies very rapidly near k = k = k = 0 and there-
k = (k k ), (16a) x y z
x 2 2− 1 fore one should avoid interpolation of G ; this will be
0
1 explained in Sec. IIB.
k = [2k (k +k )], (16b)
y 3 3− 1 2 The four free boson channel state |Φ00i has kx = ky =
1 k =0andl =l =l =J = =0. Duetothethresh-
k = [3k (k +k +k )], (16c) z x y z J
z 4 4− 1 2 3 oldlawtheonlynonvanishingpartial-wavecomponentof
the breakup amplitude (15) is the one with =0, i.e.,
while for α = 2 they describe the relative motion in the J
1+1, 1+1, and 2+2 subsystems, i.e.,
Φ0 T Φ = Φ0 TJ=0 Φ /(4π)2. (20)
1 h 0| 0α| α,ni h 0| 0α | α,ni
k = (k k ), (17a)
x 2 2− 1 Although Φ0 has only the l = J = = 0 compo-
| 0i j J
1 nent, we emphasize that angular momenta l and J are
k = (k k ), (17b) j
y 2 4− 3 not conserved. Nonzero lj and J basis states must be
1 included when solving the AGS equations (11).
k = [(k +k ) (k +k )]. (17c)
z 4 3 1 2
2 −
Therespectiveorbitalangularmomental ,l ,andl are
x y z
B. Separable potential
coupledto the totalangularmomentum with the pro-
J
jection ;alldiscretequantumnumbersareabbreviated
M
by ν. An explicit form of integral equations is obtained Theuniversalpropertiesofthefour-bosonsystemmust
by inserting the respective completeness relations beindependentoftheshort-rangeinteractiondetails. We
∞ choosethetwo-bosonpotentialofaseparableformacting
1= k k k ν k2dk k2dk k2dk k k k ν in the S-wave only, i.e.,
Z | x y z iα x x y y z zαh x y z |
Xν 0
(18) v = g λ g δ . (21)
| i h | lx0
4
The form factor ThediscretizationofintegralsinEqs.(25)usingGaus-
sianquadraturerulesleadstoasystemoflinearalgebraic
kx g =[1+c2(kx/Λ)2]e−(kx/Λ)2 (22) equations. We reduce its dimension even further by ap-
h | i
plying the so-called -matrix technique [14] converting
is taken over from Ref. [8]. Two very different choices K
the complex equations into the real ones, and by elimi-
of c , namely, c = 0 and c = 9.17, are used in order
2 2 2 − nating the α=2 component, i.e., by inserting Eq. (25b)
to confirm the universality of the obtained results. The
into (25a) and Eq. (25d) into (25c). To avoid difficulties
strength
due to possibly slow convergence of iterative methods,
−1 the resulting linear system is solvedusing the direct ma-
2 1 c 3c Λ
λ= 1+ 2 1+ 2 (23) trix inversion. Further details of the calculations can be
πm(cid:26)a −(cid:20) 2 (cid:18) 8 (cid:19)(cid:21)√2π(cid:27) found in Refs. [11, 14].
isconstrainedtoreproducethegivenvalueofthescatter-
ing length a for two particles of mass m. The potential
III. FOUR-BOSON RECOMBINATION
(21)supportsoneshallowtwo-bosonboundstateata>0
andno one at a<0. There are no deeply bound dimers.
The resulting two-boson transition matrix The number of the four-atom recombination events in
a non-degenerated atomic gas per volume and time is
t= g τ g (24) K ρ4 where ρ is the density of atoms and K the four-
| i h | 4 4
atom recombination rate. In the ultracold limit the ki-
is separable as well with τ = (λ−1 g G g )−1. This
−h | 0| i netic energies of initial atoms are much smaller than the
allows to reduce the AGS equations (11) to a system of
final two-cluster kinetic or binding energies, and the ini-
two-variable (k and k ) integral equations and thereby
y z tial momenta are much smaller than the momenta of
simplifies considerably the numerical calculations. The
the two resulting clusters, kx,ky,kz << pα,n′. Under
AGS equations with separable interactions are solved in
these conditions K can be approximated very well by
4
the form
the zero-temperature limit K0 calculated at E = 0 with
4
k =k =k =0 in the initial state, i.e.,
g G φ =P g P φ x y z
0 11 1,n 34 1 1,n
h | U | i h | | i
+P g G U G g τ g G φ
+ g34Gh |U 0G1g0τ| giGh | 0Uφ11| 1,,ni K40 =16π7 µαpα,n′|hΦ00|T0Jα=0|Φα,n′i|2. (26)
h | 0 2 0| i h | 0U21| 1,ni αX,n′
(25a)
g G φ =(1+P ) g P φ Ithascontributionsfromalltwo-clusterchannels(α,n′).
0 21 1,n 34 1 1,n
h | U | i h | | i
Concerningthe four-atomrecombination,the mostin-
+(1+P ) g G U G g τ g G φ ,
34 0 1 0 0 11 1,n
h | | i h | U | i teresting regime lies at large negative two-bosonscatter-
(25b)
inglengthawherenoshallowdimersexistbuttheEfimov
g G φ = g P φ
0 12 2,n 2 2,n trimers and tetramers cross the zero-energy threshold.
h | U | i h | | i
+P g G U G g τ g G φ In our nomenclature we characterize the trimers by one
34 0 1 0 0 12 2,n
h | | i h | U | i
+ g G U G g τ g G φ , integer number n, starting with n = 0 for the ground
0 2 0 0 22 2,n
h | | i h | U | i (25c) state, and the tetramers by two integers (n,k), where n
refersto the associatedtrimerandk =1(2)foradeeper
g G φ =(1+P ) g G U G g τ g G φ .
h | 0U22| 2,ni 34 h | 0 1 0| i h | 0U12| 2,ni (shallower) tetramer. The a-dependence of the nth fam-
(25d)
ily trimer (b ) and tetramer (B ) binding energies is
n n,k
shown in Fig. 1 for a < 0. We denote by a0 the specific
We emphasize that with respect to angular momentum n
negative value of a where the nth trimer binding energy
quantum numbers, the full complexity of the four-body
vanishes, i.e., b =0. In the universal limit, i.e., for suf-
problem is still present in Eqs. (25). Basis states up n
ficiently large n, the ratio a0 /a0 = eπ/s0 22.6944
to ly,lz,J < 3 are included to achieve the partial-wave where s 1.00624 [2]. Trimne+r1binnding energy≈bu taken
convergencefor the solutions of the AGS equations (25). 0 ≈ n
in the unitary limit 1/a = 0, is used to build dimen-
Regarding the continuous variables k and k , the treat-
y z sionless ratios b /bu and B /bu. These ratios become
ment is taken over from Ref. [11]. It requires inter- n n n,k n
independent of the short-range details of the interaction
polation of g G U G g in initial and final momenta
0 α 0
h | | i provided that its range is small enough compared to the
k . To take care of the G singularity at k = 0
y 0 j
size of the few-boson states. This condition is fulfilled
we factorize gk′k′ν′ G U G gk k ν = (k′2/2µ +
h y z | 0 α 0| y z i y αy for high excited states, i.e., for sufficiently large n. We
kkz′)2//k22µαs)u−ch1U¯thανa′νt(kU¯y′ν,′kνy(,kk′z,)k(k,yk2/2)µisαya+vkerz2y/2sµmαo)o−t1hδ(fukzn′c−- cfohremckinedg ctahlicsuilnadtieopnesnwdeitnhceno=f t3haenrdes4ulftosritnwoFicgh.oi1cepseor-f
z z α y y z
tion and is interpolated very reliably using spline func- the potential form factor (22).
tions as in Ref. [11], while the factors of the type We remind that only two lowest n = 0 tetramers are
(k2/2µ +k2/2µ )−1 arecalculatedexplicitly wherever true bound states. All higher tetramers lie above the
y αy z α
needed. lowest atom-trimer threshold and therefore are unstable
5
12
10
100
10-1 B = Bn,1 109
B = B
n,2
un
B/b 10-2 B = bn n 6
κ 10
10-3 n = 1
3 n = 2
10
10-4 n = 3
n = 4
-2 -1 0 100
0
|a |/a
n 0.2 0.4 0.6 0.8 1.0
0
FIG. 1. (Color online) Trimer and tetramer binding energies a/a
n
as functions of thetwo-boson scattering length.
FIG. 2. (Color online) Dimensionless four-boson recombina-
bound states with finite width and lifetime. Their posi- tion rate κn as a function of two-boson scattering length.
tions B shown in Fig. 1 are extracted from the atom-
n,k
trimer scattering calculations at E < 0 as described in n a0 /a0 a0 /a0 a0 /a0
Refs. [8, 15]. The specific negative values of a = a0 n,1 n n,2 n n+1 n
n,k 0 0.4435 0.8841 17.752
where the respective tetramers emerge at the four free
1 0.4412 0.9162 21.935
particle threshold can be obtained by extrapolating the
results in Fig. 1 to B =0. This procedure yields uni- 2 0.4267 0.9128 22.639
n,k
versal ratios 3 0.4254 0.9124 22.691
4 0.4254 0.9125 22.694
a0 /a0 =0.4254(2), (27a)
n,1 n 3 0.4253 0.9125 22.695
a0 /a0 =0.9125(2). (27b) 4 0.4254 0.9125 22.694
n,2 n
The uncertainties, apart from the numerical accuracy,
TABLE I. Ratios of special two-boson scattering length val-
arise due to the residual dependence on n 3 and c in
2
≥ ues corresponding to the peaks and cusps of the four-boson
Eq. (22).
recombinationrate(n≥1)orvanishingtetramerbindingen-
We will show our results for the four-boson recombi-
ergy(n=0). Resultsareobtainedusingpotentialformfactor
nation rate as functions of the dimensionless ratio a/a0.
Thenintheinterval(e−π/s0,1)thereareexactlyntrimenr with c2 =0 (top) and c2 =−9.17 (bottom).
states with 0 n′ n 1 that contribute to Eq. (26).
Instead of K0≤we bu≤ild a−dimensionless quantity
4 interactionrange[13]. Wenotethatournon-universalre-
sults at n 1 depend not only on c but also on m and
κn =K40m/(~|a0n|7). (28) Λ;ourcalc≤ulationsareperformedwi2thmbeingthemass
of 4He atom and Λ=0.4 ˚A−1.
With1 n 4inFig.2weexploreabroadrangeofthe
≤ ≤ Thefour-bosonrecombinationratehasresonantpeaks
two-boson scattering length a and recombination chan-
nels n′ whereas α = 1. The four-boson recombination ata=a0n,k wheretherespectivetetramersareatthefour
rate varies over many orders of magnitude, but all κn as free atom threshold. The ratios a0n,k/a0n extracted from
functions of the respective a/a0n show qualitatively the the peak positions (from Bn,k =0 in the case n=0) are
same behavior. For higher n, i.e., n 3, there is also a collected in Table I. For n 3 they approach the uni-
≥ ≥
verygoodquantitativeagreement,indicating the univer- versal limit with high accuracy and are fully consistent
sality of our κ results. Indeed, while κ in Fig. 2 are with the ones in Eqs. (27). Furthermore, the four-boson
n n
obtained with c =0 in Eq. (22), additional calculations recombinationratehassharpcuspsata=a0,or,equiva-
2 n
with c = 9.17 for n = 3 and 4 agree very well with lently,ata=e−π/s0a0,whereanewatom-trimerchannel
2 n
the correspo−ndingpredictions inFig.2, therebyconfirm- opens. WelistinTableIalsotheratiosa0 /a0 toprove
n+1 n
ing our conclusion on κ universality. In contrast, κ that our numerical predictions converge towards analyt-
n 1
shows significant quantitative deviations from the uni- ical result eπ/s0 22.6944[1, 2]. Furthermore, for n 3
≈ ≥
versal behavior due to finite-range effects: in the regime we obtain a0 mbu/~= 1.5077(1).
n n −
a/a0 <1 the only available recombination channel leads At a/a0 >p1 not only the ground state but also ex-
1 1
tothegroundstatetrimerwhosesizeiscomparabletothe cited trimers are produced via four-atom recombination
6
100 Refs. [4, 16] for the peak positions, i.e., a0 /a0 0.43
n,1 n ≈
10-1 n = 4 n’ = 0 and a0n,2/a0n ≈ 0.90, are consistent with our results. On
the other hand, the available experimental data [6] are
n’ = 1
-2 obtained in the system of ultracold 133Cs atoms with
10 n’ = 2
n’ deeply bound non-Efimov-like few-atom states and refer
νn,10-3 n’ = 3 to n = 0. Therefore it is not surprising that the exper-
imental results a0 /a0 0.47 and a0 /a0 0.84 devi-
-4 n,1 n ≈ n,2 n ≈
10 ate from the universal values even more. Nevertheless,
they remain roughly consistent with the theoretical pre-
-5
10 dictions. Thus, our calculations present no improvement
0.2 0.4 0.6 0.8 1.0 over the ones of Refs. [4, 16] when comparing with the
existing data, but serve as a theoretical benchmark and
0
a/a
n maybe valuableforthe future experiments performedin
a truly universal regime.
FIG. 3. (Color online) Relative weights νn,n′ for theproduc-
′
tion of the nth state trimers via four-boson recombination.
IV. SUMMARY
n=4 atom-trimer channels contributein theshown regime.
We studied ultracold four-boson recombination. Ex-
act four-particle scattering equations in the AGS ver-
and contribute to the total rate (26). In Fig. 3 we com- sionhavebeenused. Werelatedthefour-clusterbreakup
pare the relative weights νn,n′ for the production of the andrecombinationoperatorsto the standardtwo-cluster
trimers in the individual states n′ when n atom-trimer AGS operators. The scattering equations at the four-
channels are available in total. The weights are normal- cluster breakup threshold have been precisely solved in
ized to nn′−=10νn,n′ = 1. The results in Fig. 3 clearly the momentum-space framework. The zero-temperature
demonstPrate that more weakly bound trimers are pro- limitofthefour-bosonrecombinationratewascalculated
duced more efficiently and the four-atom recombination for negative values of the two-boson scattering length.
process is strongly dominated by the n′ = (n 1)th We demonstrated its universal behavior in the regime
−
channel with the shallowest available trimer. We note with high excited Efimov trimers where the finite-range
that a similar conclusion was drawn in Ref. [9] about effectsarenegligible,anddeterminedthepositionsofres-
the trimer production in dimer-dimer collisions. Except onantrecombinationpeakscausedbythetetramerstates.
for the vicinity of a/a0n = e−π/s0 where the shallowest WealsohaveshownthatmostoftheEfimovtrimerspro-
trimerdisappears,therelativeweightsνn,n′ dependquite duced via the four-boson recombination are in the state
weakly on a. Thus, the tetramer states around a =a0n,k withweakestbinding. Ourresultsareconsistentwithex-
enhance the production of all trimers by equal factors. isting experimental data [6] and theoretical predictions
Asmentionedintheintroduction,therecombinationof [4, 16], but exceed the latter ones with respect to the
four identical bosons has been calculated in Refs. [4, 16] accuracy in the universal limit. In addition, the present
usingtheadiabatichypersphericalframework. Thereisa work constitutes a step towards solving the momentum-
goodqualitativeagreementbetweenourresultsandthose space four-body scattering equations above the breakup
of Refs. [4, 16]: the shape of the recombination rate, in- threshold.
cluding its resonant peaks and cusps, is reproduced in
both cases. However,the calculations of Refs. [4, 16] are
limited to n = 1 where the finite-range effects are not ACKNOWLEDGMENTS
entirely negligible and deviations from our universal re-
sults can be expected much like in the case of our n=1 TheauthorthanksP.U.Sauerfordiscussions. onfew-
results in Fig.2. Within this accuracythe predictions of body recombination problem.
[1] V.Efimov, Phys.Lett. B 33, 563 (1970). 101, 023201 (2008).
[2] E. Braaten and H.-W. Hammer, Phys. Rep. 428, 259 [6] F. Ferlaino, S. Knoop, M. Berninger, W. Harm, J. P.
(2006). D’Incao, H.-C. N¨agerl, and R. Grimm, Phys. Rev. Lett.
[3] H. W. Hammer and L. Platter, Eur. Phys. J. A 32, 113 102, 140401 (2009).
(2007). [7] S. E. Pollack, D. Dries, and R. G. Hulet, Science 326,
[4] J. von Stecher, J. P. D’Incao, and C. H. Greene, Nature 1683 (2009).
Phys.5, 417 (2009). [8] A. Deltuva,Phys. Rev.A 82, 040701(R) (2010).
[5] F. Ferlaino, S. Knoop, M. Mark, M. Berninger, H. [9] A. Deltuva,Phys. Rev.A 84, 022703 (2011).
Sch¨obel, H.-C. N¨agerl, and R. Grimm, Phys. Rev. Lett.
7
[10] P. Grassberger and W. Sandhas, Nucl. Phys. B2, 181 [15] A. Deltuva,EPL 95, 43002 (2011).
(1967);E.O.Alt,P.Grassberger,andW.Sandhas,JINR [16] N. P. Mehta, S. T. Rittenhouse, J. P. D’Incao, J. von
report No.E4-6688 (1972). Stecher,andC.H.Greene,Phys.Rev.Lett.103,153201
[11] A.Deltuvaand A.C. Fonseca, Phys.Rev.C 75, 014005 (2009).
(2007). [17] Y. Wang and B. D. Esry, Phys. Rev. Lett. 102, 133201
[12] A. Deltuva and A. C. Fonseca, Phys. Rev. C 76, (2009).
021001(R) (2007). [18] O.A.Yakubovsky,Yad.Fiz.5,1312(1967)[Sov.J.Nucl.
[13] R.LazauskasandJ.Carbonell,Phys.Rev.A73,062717 Phys. 5, 937 (1967)].
(2006).
[14] A. Deltuva, R. Lazauskas, and L. Platter, Few-Body
Syst.51, 235 (2011).