Magnetic control of the interaction in ultracold K-Rb mixtures

5 Pages·0.16 MB·English

by  A. Simoni

Magnetic control of the interaction in ultracold K-Rb mixtures
A. Simoni
INFM and Department of Chemistry, University of Perugia, I-06123 Perugia , Italy
F. Ferlaino, G. Roati, G. Modugno and M. Inguscio
INFM and LENS, Universit`a di Firenze, Via Nello Carrara 1, 50019 Sesto Fiorentino, Italy
3
Wepredictthepresenceof severalmagnetic Feshbachresonances inselected Zeeman sublevelsof
0 theisotopic pairs40K-87Rband 41K-87Rbat magneticfieldsupto103 G(1G= 10−4 T).Positions
0
and widths are determined combining a new measurement of the 40K-87Rb inelastic cross section
2
withrecentexperimentalresultsonbothisotopes. Thepossibility ofdrivingaK-Rbmixturesfrom

n theweaktothestronginteractingregimetuningtheappliedfieldshouldallowtoachievetheoptimal
a conditionsforboson-inducedCooperpairinginamulticomponent40K-87Rbatomicgasandforthe
J
production of ultracold polar molecules.

0
1 PACSnumbers: PACSnumbers: 34.50.-s,03.75.-b,34.20.Cf

]
t
f The possibility ofcontrollingthe interactionsin anul- this work we limit ourselves to ℓ = 0 collisions which,
o
tracold atomic gas using magnetically tunable Feshbach at least in the absence of ℓ > 0 low-energy resonances,
s
. resonances [1] has been successfully exploited in recent give the dominant contribution to the interaction in the
t
a remarkable experiments. It has indeed allowed to reach ultracold regime.
m Bose-Einsteincondensation(BEC)forspeciesdifficultto Two key measurements on 40K-87Rb mixtures pro-
- condense in zero field and to study their stability prop- vide the best range of input parameters to our model
d
erties [2, 3], to form brightsolitons [4], to produce ultra- available to date. First, the elastic cross-section for
n
o cold homonuclear molecules [5] and a strongly interact- (9/2,9/2)+(2,2) has been directly measured by inter-
c ing Fermi gas [6]. The presence of Feshbach resonances species thermalization[9] and by damping of coupled K-
[ between atoms of different species could open new pos- Rb dipole oscillations [14]. Combining the systematic

1 sibilities, thereby in this paper we carry out a detailed uncertainties in these two determinations we estimate
9v invWesetigfoactuiosnoinn tthwios dpioretcatsisoiunm. -rubidium mixtures that ascta=linag92o92f,22th=is−va4l1u0e+−88t00oa041(K1-8a70R=b 0a.l0lo5w29s1u7s7tnoms)t.roMngalsys
5 have allowed the production of a two-species BEC (41K- tighten the upper bound on the triplet scattering length
1
01 8tu7Rreb)(4[07K,8-8]7aRnbd)of[9a].strWonegplyreindticetratchtiengpoFseirtmioin-sBoasnedmtihxe- othfeRoefr.ig[1in2a],lleuandcienrgtationtayt(i4n1−th8e7n)u=m1b5e6r+−o15f0tarnipdlteot rbeoduuncde
3 widths of several Feshbach resonances in selected mag- states to Nt(41 87)=31 1.
0 netic sublevels of the groundelectronic state by combin- Second, bin o−rder to det±ermine a we measure the
/ s
t ing collisional measurements on the two isotopes. These (9/2,9/2)+(2,1)atom-lossrateinanondegeneratesam-
a
m resonances might for instance allow the observation of ple. We prepare a doubly spin-polarized sample in the
boson-induced Cooper pairing [10] or the formation of magnetic trap containing about 104 K atoms and 105
-
d ultracold bosonic or fermionic polar molecules [11]. Rb atoms at temperatures in the range (300 900) nK,
n We use for our calculations the collision model pre- and then transfer about 30% of the Rb sam−ple in the
o sented in Ref. [12], where the Hamiltonian now in- (2,1) state using a fast radio-frequency sweep. After
c
cludestheZeemaninteractionwiththeexternalmagnetic a short rethermalization period we measure the time-
:
v field [13]. Denoting “a” and “b” potassium and rubid- evolution of the number of K atoms, which decays
Xi ium, collisions are labeled as (famfa)+(fbmfb), where through inelastic collisions with Rb according to the
ar ftiao,nfmb faar,emthfeb ahlyopnegrfitnhee qaunagnutlaizratmioonmaexnitsa. Pwliethaseprroejceacl-l Graitneeleq=ua5t.i0o(n2.n5˙)K(1t)0−=12−cGmin3elsn−K91/,2,w9/h2e(rte)ntR2h,eb1(ut)n.ceWrteaifinntdy
thatthemodelisparameterizedbythesingletandtriplet ×
isdominatedby thatonthe atomnumbers. Inthis work
s-wave scattering lengths a and by the long-range dis-
s,t we make a conservative choice and adopt only the order
persion coefficients Cn with n=6,8,10. We adopt a ±2% of magnitude G=(10−12 10−11) cm3 s−1.
uncertainty in the highly accurate van der Waals coeffi- −
Figure 1 showsGinel numericallycalculatedas a func-
cientC =4274a.u. (1a.u.=0.0957345 10−24Jnm6)in
6 × · tion of as varying at within the bounds determined
order to account for the combined uncertainty in all the
above. The horizontal axis is labeled by a scaled “scat-
C ’s. Intheabsenceofweakrelativisticspin-interactions
n tering phase” ξ = (2/π)arctan(a /a ), where a =
s s sc sc
the projection mf = mfa+mfb and the rotational an- 1(2µC )1/4 72a isatypicallengthforapurelyvander
gular momentum of the atoms about the center of mass 2 6 ≈ 0
Waals potential, with µ the K-Rb reduced mass. To be
ℓ are good quantum numbers for the collision event. In
2
cible even in the ultracold regime, see Ref. [15].
TABLE I: Compendium of 40K-87Rb scattering properties.
The zero-field s-wave elastic scattering length a is shown for
different hyperfine sublevels. The uncertainty in the reso-
nancepositions B0 results from theuncertaintyin themodel
parameters quoted in the text. The resonance widths ∆ are
2
10 shown forthenominal valuesas=−185a0,at =−410a0 and
C6 = 4274 a.u. of the scattering lengths and of the van der
Waals coefficient.
) 1
1 10
-s
3m (fa,mfa)+(fb,mfb) a(a0) B0(G) −∆(G)
c
11 100
-stic Rate ( 1010-1 ((99(//922/,,±±2,799///222)))+++(((121,,,1±±)21)) −−−224431550+−+−+−99118822220010 932109..36−+−[phone]..085
ela -2 (9/2,-7/2)+(1,1) −323+−91005 522.9+−4505 0.3
In10 564.8+20 0.1
−24
646.3+35 4.6
−52
658.0+50 0.08
−58
-1 -0.5 0 0.5 1 753.1+50 0.3
ξ −59
s 787.4+−6605 0.9
(9/2,-9/2)+(1,1) −336+89 505.3+31 0.03
−102 −49
FIG. 1: The numerically calculated inelastic rate Ginel 547.1+−2288 0.2
for 40K(9/2,9/2)+87Rb(2,1) collisions at temperature T = 593.9+−3444 4.0
500 nK as a function of the singlet “scattering phase” (see 741.2+60 0.9
−65
Text) for at = -330 a0 (full line) and at = -490 a0 (dashed 921.0+92 10−4
line). The horizontal lines correspond to the bounds deter- −94
(9/2,-9/2)+(1,0) −279+92 477.0+35 0.1
mined experimentally. −114 −47
590.3−31 3.9
+26
notedthestrongsuppressionoftheinelasticratefora
s
∼
a . Comparisonwith the experimental value leads either Amagnetic Feshbachresonanceariseswhen a molecu-
t
toaverylarge(positiveornegative)ortoanegativesin- lar bound-state crosses an atomic threshold [16]. Across
glet scattering length a (40 87) = 185+83 a . Mass- a Feshbach resonance the scattering length a presents a
s − − −225 0
scalingwithNs(41 87)=98 2givesasingletscattering typical dispersive dependence on the strength of the ap-
lengthforthebboso−nspaira (±41 87)=5.0+23a and,by plied field B, a(B)=a˜(1 ∆ ), where B is the field
s − −34 0 − B−B0 0
compatibility with the a (41 87) of Ref. [12], rules out strengthonresonance,∆isthe resonancewidthanda˜ is
s
−
therangeofverylarge a (40 87). Withthesea ,a we the off-resonancebackgroundscatteringlength. The res-
s s t
| | −
canpredictthe zero-fieldscatteringlengthandFeshbach onance parameters B and ∆ have been extracted from
0
resonances of different magnetic polarization states. Let anaccurateclose-coupledcalculationonafineB-gridand
us first consider the boson-fermion pair 40K-87Rb. are shown in Tab. I.
The inverted hyperfine structure of the fermionic Homonuclearcollisionsbetweenalkaliatomsinamag-
potassium, with the f = 9/2 lying below the f = 7/2 netic field have been studied in great detail. Only a rel-
level, leads to a peculiar and very favorable feature of atively small amount of Feshbach spectroscopy is often
this pair, namely the stability under spin-exchange col- necessary as input to theoretical models in order to gain
lisions of any combination where either atom is in its an impressive power in determining the interaction po-
lowest Zeeman sublevel. We therefore find it important tential and in predicting the position of additional reso-
to summarize in Tab. I the scattering properites of the nances [17, 18]. The parameters a ,a and C ’s of our
s t n
(9/2, 9/2)+(1,m )statesandofatleastthemosteas- collision model can in principle be determined following
fb
−
ily experimentally realizable (9/2,m )+(1,1) magnetic a similar procedure. At least three resonances between
fa
states. Noteallofthesehyperfinestateshavearelatively the ones shown in Tab. I would be needed as input to
large ( i.e. a > a ) and negative zero-field scattering the model. Inclusion of the high-field (9/2,7/2)+(1,1)
sc
| |
length. Thisensuresthatthetwocloudswillremainmis- resonance would improve the procedure since this latter
3
resonance is nearly purely triplet in character with av-
erage electron spin S2 1.95 and very sensitive to C ,
6
h i ≈
δB /B = -1.2 δC /C .
0 0 6 6
The relatively strong and tunable interaction be-
tweendifferentmagneticpolarizationstatesopensupthe
possibility of realizing Bose-Fermi mixtures where the
fermions undergo a Cooper-pairing phase transition to
a superfluid state via exchange of density fluctuations of
the background BEC [10, 19]. Due to identical particle
symmetrization,directaswellasboson-mediateds-wave 1.0
collisionsbetweenapairoffermionicKatomsareallowed
onlyinaFermigaswithatleasttwointernalcomponents,
)
whichwedenoteas“1”and“2”inthefollowing. Wealso a 0
letaRb betheintra-speciesscatteringlengthforcollisions 30 0.0
1
of Rb atoms, aK the intra-species scattering length for (
collisionsofKinstate“1”andKinstate“2”anda , a
KRb,i
with i=1,2, the inter-species scattering lengths for col-
lisions of Rb and K in state “i”. -1.0
The critical temperature T for boson induced s-wave
c
Cooper pairingcanbe givenananalytic expressionvalid
in the limit of low boson-densities and of a BEC larger
than the fermionic cloud, see Eq. (13) of Ref. [19]. As 722 724 726
long as aKRb,1aKRb,2 >> aRbaK, the critical tempera- B ( G )
ture is independent of a , and increases exponentially
K
with a /(k a a ), where k is the Fermi vec-
Rb F KRb,2 KRb,1 F
tor. Itmustbenotedhoweverthatassoonask a2
F KRb,i ∼ FIG. 2: Field dependence of the 40K(9/2,−7/2)+ 87Rb(1,1)
a , the i-th component of K and Rb will become dy-
Rb (fullline)and40K(9/2,−9/2)+87Rb(1,1)(dashedline)scat-
namically unstable and collapse [14, 20]. As a con- tering lengths for as = −105a0 and at = −380a0 and the
sequence the optimal condition for Cooper pairing are nominal C6. The arrow shows the best working point for
aKRb,1 ∼aKRb,2 ≡aKRb and a2KRbkF ∼aRb. inducingthe Cooper pairing instability (see Text).
From Tab. I a mixture composed of K(9/2, 9/2),
−
K(9/2, 7/2) and Rb(1,1) seems to be the most appro-
priate i−n order to fulfill this condition. However, for for as,t larger than the nominal values (which however
| |
the nominal values of the zero-field scattering lengths implies larger zero-field a’s) such a field may not exist.
between these components, we estimate that at least a We now consider the boson-boson pair 41K-87Rb. We
factor of four increase in the average trapping frequency summarizeinTab.IIthecollisionpropertiesofthestates
would be necessary with respect to our current exper- (2,2)+(2,2)and(1,1)+(1,1),whicharecollisionallysta-
imental setup. If the actual a turned out to be ble for any field, and of the state (1, 1)+(1, 1) which
KRb,i − −
smallerthanthenominalvalue,thisfactorwouldincrease is only stable up to a threshold value Bt 143.45 G,
≈
evenfurther. Fortunately,atleastforsomepotentialpa- where the (2, 2)+(1,0) channel opens up and becomes
−
rameters, a pair of resonances exist that allow simulta- the stable one. The a11,11 and a1−1,1−1, which are equal
neous tuning of a and a to a large value at at zero field, are relatively large.
KRb,1 KRb,2
the same time, making such a large trapping frequency The (1,1)+(1,1) collision always presents at least two
unnecessary. broadandonenarrowresonancebelow103 G,seeTab.II
A very favorable situation is shown in Fig. 2, where and Fig. 3. The analytical expression for the scattering
for B 725.89 G we get a 687a . In order to length of a pair of overlappingresonances with positions
KRb 0
have t≈his value stabilized to wit≈hin−10 a0 the magnetic B0,± and widths ∆±
field must be stabilized to within 0.1 G, which is a strin-
a(B) (B B0,+)∆−+(B B0,−)∆+
gentyetpossibletobemetrequirement[18]. Thefieldat =1 − − ,
which aKRb,1 =aKRb,2 and the common value aKRb are a˜ − (B−B0,+)(B−B0,−)
extremely sensitive to the details of the potential, and a has been used to fit both low-field resonances in the ta-
conclusive word on what is the largest achievable com- ble. The lowest-field resonance exists only if a > 0,
11,11
mon aKRb is at this stage left to forthcoming precision for a11,11 < 0 it turns into a real bound-state below the
experiments on K-Rb Feshbach resonances. As a trend, (1,1)+(1,1)atomic thresholdand cannotbe observedin
as,t smallerthanthenominalvaluestendtohaveagood a standard scattering experiment.
| |
workingfield for inducing the Cooper pairing instability, The (1, 1)+(1, 1) collision presents a feature near
− −
4
TABLEII:SamethanTab.Ibutfor41K-87Rb. Theresonance
widths ∆ are shown for the nominal values as =5.0a0, at =
156a0 and C6 = 4274 a.u. of the scattering lengths and of
the van der Waals coefficient. In cases where the sign of a
isundeterminedwehaveshownthenominalvalue,theupper
bound for a<0 and thelower boundfor a>0.
3.0
(fa,mfa)+(fb,mfb) a(a0) B0(G) −∆(G) f=2
(2,2)+(2,2) 156+10 2.0
−5
(2,2)+(2,1) 49.5+5 6.5
−7 )
(1,1)+(1,1) 431,<−144,>249 53.0+−4533 30 a 01.0 (1,1) + (1,1)
(1,-1)+(1,-1) 431,<−144,>249 75848701...550+−+−−+3511117635466466 017..0733 3e ( a ) ( 10 0.0 (1,-1) + (1,-1) f=3
R
-1.0
B
-2.0 t
the threshold value Bt. In this situation virtual excita-
25 50 75 100 125 150
tions to the (2, 2)+(1,0) channel can strongly modify
− B ( G )
therealpartofthescatteringlength(seeFig.3),defined
as a1−1,1−1 = limk→0S1−1,1−1/(2ik), with S the
R −R
scattering matrix and k the wave vector of the incom-
ing atoms. Such fluctuations-induced feature is always FthIeGr.ea3l:pFarietldRad1e−p1e,n1−d1enfocer4o1fKt-h87eRpbucreollylisiroenasl aan11d,1t1heanndomo-f
present in our range of parameters if a1−1,1−1 > 0 but inal as, at, C6. Approximate zero-field quantum numbers f
turns out to be important for a1−1,1−1 <0 only for very , with f~=f~a+f~b, are assigned to the resonances. Near the
large |a1−1,1−1|. To be noted the cusp at B = Bt that (2,−2)+(1,0) channelthreshold Bt (vertical line) Ra1−1,1−1
follows from the analytical properties of the scattering changes dramatically.
matrix near the opening of a new threshold [21].
Finally, a field induced resonance in the (2,2)+(2,1)
inelastic loss-rate has been included in Tab. II since it ing K-K and Rb-Rb pairs. For instance, the condition
may be important in order to constrain the collision aKRb(11,11) =0.5√aK(11,11)aRb(11,11)requiredinRef.[11]
modelfromFeshbach-resonancespectroscopycarriedout in order to load a single K and a single Rb per cell can
on the boson-boson isotope. The width ∆ shown in the easily be reached. A Raman photoassociation pulse or
table is in this case the inelastic one, defined through a direct microwave pulse can then be used to associate
the expression of the total scattering eigenphase across atoms into molecules at each lattice cell. In alternative,
resonance, δ(B)=δ arctan ∆inel , with δ the back- a time-dependent magnetic field has also been proposed
0− (B−B0) 0 for near-unity conversion of atoms into molecules, see
ground scattering phase. Moreover, near this resonance
Ref. [22]. This technique can of course be used for both
wefindadramaticreductionofthe(2,2)+(2,1)inelastic
decay rate down to G 10−15 cm3 s−1, a value that en- the isotopic pairs investigated in this work.
≈ We are grateful to P. S. Julienne, E. Tiesinga, and C.
suresthecollisionalstabilityofatwo-species(2,2)+(2,1)
J. Williams for stimulating discussions and for making
BEC.
available the ground-state collisions code. We aknowl-
An interesting application of our results is to the pro-
edge useful discussions with F. Riboli and L. Viverit.
duction of polar molecules via photoassociation, as pro-
This work was supported by MIUR under a PRIN
posed in Ref. [11]. First step in this scheme is a Mott-
project, by ECC under the Contract HPRICT1999-
insulator transition of a double K-Rb BEC into a state
00111,and by INFM, PRA ”Photonmatter”.
where only pairs of molecules are loadedinto the cells of
an optical potential. Whether K-K and Rb-Rb or K-Rb
pairs are loaded together depends upon the ratio of the
interaction strength between all pairs. Using the low-
field resonance shown in Fig. 3 one can clearly vary the
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