Table Of ContentCollisional properties of sympathetically cooled 39K
L. De Sarlo∗, P. Maioli†, G. Barontini, J. Catani1, F. Minardi1,2, and M. Inguscio1,2
LENS - European Laboratory for Non-Linear Spectroscopy and Dipartimento di Fisica,
Universit`a di Firenze, via N. Carrara 1, I-50019 Sesto Fiorentino - Firenze, Italy
1INFN, via G. Sansone 1, I-50019 Sesto Fiorentino - Firenze, Italy
2CNR-INFM, via G. Sansone 1, I-50019 Sesto Fiorentino - Firenze, Italy
(Dated: February 6, 2008)
7
0 Wereport theexperimentalevidenceof thesympatheticcooling of39K with 87Rbdown to1µK,
0 obtained in a novel tight confining magnetic trap. This allowed us to perform the first direct mea-
2 surementoftheelasticcrosssectionof39Kbelow50µK.Theresultobtainedforthetripletscattering
n length,aT =−51(7) Bohrradii, agrees with previousresultsderivedfrom photoassociation spectra
and from Feshbach spectroscopy of 40K.
a
J
PACSnumbers: 34.50.Pi,32.80.Pj,05.30.Jp
3
]
r I. INTRODUCTION tion. Acarefulbalanceof the intensity andthe detuning
e of these two laser frequencies, unsuitable for optimized
h
loading of a Magneto-Optical-Trap (MOT), is the only
t In the field of ultracold and quantum gases Potas-
o waytoobtaintemperaturesattheDopplerlimitbylaser
sium gained a key role since it is the only alkali, be-
at. side Lithium, for which a stable fermionic isotope exists. coofo39liKngat[6z]e.rFoutretmhepremraotruer,etihsemso-wreavtheaenlaosntiecocrrdoesrsosfemctaiogn-
m Furthermore, since the sympathetic cooling of fermionic
nitude smaller than that of 87Rb and, due to the attrac-
40K with 87Rb, the mostwidespreadatomin the labora-
- tivecharacteroftheinteraction,theRamsauer-Townsend
d tories devoted to laser cooling, is particularly favorable,
minimum [7] occurs at a temperature of around 320µK
n it is not surprising that so many experiments with ul-
o wherethecontributionofotherpartialwavesisstillsmall
tracold fermions were carried out or are planned with
c [8]. All these features strongly hamper the efficiency of
this atom. In contrast very little attention was paid to
[ evaporativecooling. Notethat,duetoPauliblocking,an
the two other (bosonic) isotopes of Potassium, 39K and
1 41K. These isotopes offer the possibility of creating dou- even more dramatic decrease in the evaporative cooling
v efficiency occurs for the fermionic 40K.
blespeciesBose-Einsteincondensates,whichdisplayrich
1 For these reasons the most widely employed experi-
and interesting quantum phase diagrams when trapped
5 mental technique to reach ultralow temperatures in a K
in optical lattices [1, 2, 3, 4]. After the seminal work
10 where a double species 41K-87Rb BEC was produced in gas is sympathetic cooling with 87Rb. This amounts to
producing a mixed MOT of the two species and then
0 a magnetic trap [5], essentially no experiments have ad-
7 dressed these topics. evaporativelycool87RballowingKtothermalizewithit.
0 With this technique it has been possible to reach quan-
Along these guidelines, we started an experiment de-
/ tumdegeneracyof40K[9]and41K[10], butno attempts
t voted to the exploration of degenerate Bose-Bose mix-
a have been made to test this method on 39K. For this
m tures in optical lattices. In this work we report that
mixture in fact, the inter-species cross section is again
also 39K can be sympathetically cooled with 87Rb al-
- more than one order of magnitude smaller than the one
d though this is much more difficult due to its unfavorable of 87Rb-40K and 87Rb-41K [11, 12].
n lasercoolingandscatteringproperties,thusdemonstrat-
Wedemonstratehereforthefirsttimethat39Kcanbe
o ingthatallthestableisotopesofPotassiumcanbecooled
c sympathetically cooled to 1µK. Therefore the work pre-
to ultralow temperatures.
: sentedinthispaperisacriticalsteptowardstheproduc-
v Laser cooling of 39K has severaldisadvantages if com- tion of a Bose-Einstein Condensate (BEC) of 39K. Such
Xi pared to alkalis such as 87Rb or even 40K. These dis- asystemisapromisingcandidatefor the realizationofa
advantages are related to the hyperfine structure of the
r BECwithinteractiontunablearoundzero,sinceabroad
a P level whose separation is of the same order of mag-
3/2 Feshbach resonance is predicted around 400G [11, 12].
nitude as the natural linewidth of the cooling transition
The achievement of sympathetic cooling allowedus to
|F = 2,m = +2i → |F = 3,m = +3i. The result-
F F performthefirstdirectmeasurementofthes-waveelastic
ingstrongopticalpumpingtowardsthe|F =1imanifold
crosssectionof39K.Assumingfromearlierworkonpho-
mustbecounteredbyarepumpinglightwithanintensity
toassociation spectra [13] and on Feshbach spectroscopy
comparabletothatofthelightdrivingthecoolingtransi-
of 40K [14] the attractive character of the interatomic
interaction (i.e. the negative sign of the s-wave scatter-
ing length), our results agrees with [13, 14] within 1.7
combined standard deviations.
∗Correspondingauthor: [email protected]fi.it
†Present address: Laboratoire de Spectrom´etrie Ionique et The time required for sympathetic cooling depends
Mol´eculaire(LASIM), Universit´eClaudeBernardLyon1,France crucially on the collision rate between the two species.
2
If this time is of the same order of magnitude as the of both species towards the |F =2,m =+2i state and
F
lifetime of the sample, the density will drop during the then trap them in a purely magnetic trap.
evaporation and cooling will eventually stop. Since the
crosssectionfor39K-87Rbcollisionissmallerthaninthe
case of 40K and 41K, one has to increase the lifetime of B. Magnetic trap and evaporation
the sample and/or the confinement before starting sym-
pathetic cooling. In the experiments reported here, the
Our magnetic trap represents a compromise between
confinementofthe 39Ksampleisincreasedbymorethan
the usual magnetic traps generated by coils placed out-
a factor 2 with respect to [5] employing a new kind of
side the vacuum system, typically low confining (ν ∼
magneto-static trap of size intermediate between micro- max
100Hz), high power consuming (∼ 1kW), several cen-
traps, where microscopic current-carrying wires are laid
timetersinsize, whichproduce the largestBEC’s(>106
on a chip, and traditional magnetic traps created with
atoms), and the so called microtraps operating in vac-
multiple winding coils [15].
uum, providing a high confinement (ν ∼ 1 kHz), re-
max
The outline of the paper is the following: in section
quiring low electrical power (∼ 1W), that easily fit on
II we will present a brief description of the experimental
a microchip, but usually produce smaller condensates
apparatus and the results about sympathetic cooling of
(∼104 atoms).
87Rband39K.InsectionIII wewillintroducethe exper-
The potential generated by our trap is of the Ioffe-
imental procedure for studying ultracoldcollision in 39K
Pritchard type and the structure has both current con-
and we will present the theoretical analysis of the data.
ductors laying on a chip and free standing [15]. The free
Finally, we will conclude with the perspectives of 87Rb-
standing conductors are built from a single oxygen-free
39K evaporative cooling and of a 39K BEC.
coppertubemachinedtoformthefourIoffebarsandtwo
partialringsthatprovidetheendcapsofthepotentialin
theaxialdirection. Theaxialconfinementisincreasedby
II. EXPERIMENTAL APPARATUS
a circular copper trace laid ona direct-bond copper chip
that interfaces the current leads and the free standing
A. Laser system and MOT loading conductors. Electrical and mechanical contact between
the parts is obtainedby vacuumbrazingandthe current
In order to provide the maximum flexibility for the leads are brought outside the Ultra-High Vacuum envi-
loading of the mixed MOT, our experimental apparatus ronment by a custom-made electrical feedthrough. To
featurestwodistincttwo-dimensionalMOT’s(2D-MOT) allow optical access along the horizontal axial direction,
for the separateprecooling of 87Rb and 39K.A complete the feedthroughis hollow and terminates with a vacuum
description of our laser system and a thorough charac- viewport; optical access in the radial direction is easily
terizationofour39K2D-MOTcanbe foundinRef. [16]. obtained through the 2 mm gaps between the Ioffe bars.
The 2D-MOT employed for 87Rb has exactly the same Since these gaps are too small to allow the efficient pro-
structure and similar performances. In the experiments duction of a MOT, the magnetic trap, hereafter called
we used 130(110) mW for the cooling light and 40(6) millimetric-trap or mTrap, is displaced by 26 mm from
mW of repumping light for 39K (87Rb). The atomic flux the center of the MOT. A big coil placed outside vac-
providedbythetwo2D-MOT’siscollectedinsideamixed uum is used to adjust the bias field of the trap thereby
MOTformedinaultrahighvacuumchamber. TheMOT changing the radial confinement. The confinement pro-
is formed by a standard configuration with six indepen- videdbyourmTrapat95A currentand6 Gbiasfieldis
dentlaserbeamswith25mmdiameterandatotalpower ωr =2π×447(7) Hz and ωz =2π×29.20(1)Hz for 39K.
of45(40)mWforthecoolinglightand24(6)mWforthe The displacement of the atoms from the MOT posi-
repumping light of 39K (87Rb). These beams are split tionto the mTrapis accomplishedholding them inside a
from the output of one single mode fiber in which we quadrupole trap which is movedwith a motorizedtrans-
inject the four different frequencies needed for the two lationstage. Thequadrupoletrapisformedrampingthe
species, thereby guaranteeing a perfect geometric over- current of the MOT coils up to 65 A and the motion
lap of the cooling and repumping light and of the two takes less than 500 ms in order to reduce the losses due
MOT’s. The magnetic field gradient of about 16G/cm to Majorana spin-flip.
usedinthe MOT isprovidedbyapairofanti-Helmholtz While the magnetic field of the mTrap adds to that of
coils operating at 4A. the transfer trap along the radial directions, along the
We typically load around 2×109 atoms of 87Rb and axial direction the mTrap generates a field directed in
5×106 atoms of 39K in 10 and 1 s respectively. We can the opposite direction so that an adiabatic transfer of
control very precisely the number of 39K atoms loaded atoms would be only possible decreasing the trap depth
in the MOT adjusting the time of operation of the 2D- during the process and therefore reducing the number
MOT. After the loading of the MOT we reduce the re- of transferred atoms. To overcome this potential pitfall
pumping intensity and increase the magnetic field gra- we employ a mixed approach to load the trap: we ramp
dient to compress the cloud, we apply a 3.5ms stage of up the current of the mTrap in 150ms, a time which is
polarization-gradient cooling, optically pump the atoms fully adiabatic only for the radial direction. The overall
3
III. CROSS-THERMALIZATION
18(2)mK MEASUREMENT
A. Introduction
9(2)mK
We measure the cross section of 39K by analyzing the
relaxation of the atomic cloud towards thermal equilib-
3.1(9)mK rium. The technique can be described as follows. The
dynamics of an ideal gas isolated and confined in an
harmonicpotentialiscompletelyseparablemeaningthat
the different spatial degrees of freedom are decoupled.
1.0(1)mK
If such a sample is prepared with different energy dis-
tributions along different directions this difference will
not vary with time. On the other hand, due to inter-
FIG. 1: (Color online) Experimental sequence showing sym- actions, the collisions drive the system towards thermal
patheticcooling between87Rb(left)and39K(right). Allthe equilibrium, namely a state in which the total energy is
imagesaretakenafter1msexpansion. Thephasespaceden- conserved and the effective temperatures of the differ-
sity at 1µK is 0.02 and 0.01. Images are taken along the ent degrees of freedom are equal. If the initial state has
vertical radial direction. different effective temperatures, the relaxation towards
thermal equilibrium can provide direct information on
the collisions. In our system this is accomplished in the
following way: after obtaining a cloud of cold 39K as de-
scribed in Section II, 87Rb is blowed away from the trap
with a pulse of resonant light and the radial degrees of
efficiency of the transfer from the MOT to the mTrap freedom of K are excited by parametric excitation, ob-
is around 30%, and we typically trap 7×108 87Rb and tained modulating the value of the trap bias field at the
4×105 39K atoms. frequency of radial confinement for 100 ms. Due to our
elongated trap geometry this excitation is highly selec-
tive and the axial degree of freedom maintains its effec-
Onceatomsareloadedinsidethemagnetictrapweper-
tive temperature. Following this excitation we allow the
form evaporation on the 87Rb cloud by transferring the
cloud to relax for a time t after which we image the
hottestatomstotheantitrapped|F =1,m =+1istate w
F cloud along the vertical radial direction by means of in
witha microwavesweeparound6.8GHz. Inadditionwe
situ absorption imaging. From the measured widths w
applyasecondramptoremoveunwantedatomsfromthe
of the Gaussian density profile we can calculate the av-
|F = 2,m = +1i level. The 39K is unaffected by these
F erage potential energy along the radial (r) and axial (z)
sweeps, owing to the huge difference in frequency with
direction as
the corresponding hyperfine transition (460 MHz), and
onlyminorlossesoccurduringtheevaporation. Provided 1 m
that the number of 87Rb atoms N at a given temper- Ei = kBTi = (ωiwi)2 i=r,z . (1)
Rb 2 2
ature is still largerthan the initial number of 39K atoms
N , the former can act as coolant establishing thermal where m is the atomic mass. Repeating this measure-
K
equilibrium, as it is shown in Fig. 1. For this reason the ment for differenttw allowsus to measure the relaxation
initial number of 39K atoms is reduced in the pictures of the ratio between the axial and radial temperatures.
from 2.7×105 in the first picture to 5.9×104 of the last As shown in Fig. 2 this is well fitted by an exponential
one at a temperature of 1.0(1)µK. decay. Thetimeconstantτ ofthisdecayisrelatedtothe
elastic scattering cross section and to the density of the
sample by the following relation:
We experimentally foundthat a sufficientcondition to
reach thermal equilibrium is NRb/NK > 2.7, but no at- τ−1 = γel = n¯hσvi, (2)
tempthasbeenmadetomeasuretheinter-specieselastic
α α
crosssectionsince itcanbe obtainedwith highprecision
from the results of Feshbach spectroscopy of the 40K- whereαisthenumberofcollisionsrequiredtoanatomto
87Rb mixture [11, 12]. We used instead the known value reachthermalequilibrium,γ istheintraspeciescollision
el
a = +36a to guide the optimization of the evap- rate for 39K, n¯ is the averagedensity, σ is the scattering
K−Rb 0
oration. Indeed we experimentally verified that in pres- cross section, v is the relative velocity of two colliding
ence of 39K atoms, in order to optimize the number of atoms and h·i indicates averaging over the velocity dis-
87Rb atoms below a few µK,the evaporationrampmust tribution in the cloud.
be slowed down in the last part to be sure that the two If one indicates the geometric averageof the tempera-
species are always in thermal equilibrium. ture along the different direction as T¯ = (T2T )1/3 and
r z
4
introduces an average,temperature-dependent crosssec-
tion defined as:
1.1
hσvi
σ˜(T¯)= , (3)
hvi Er1.0
since the average density of a harmonic trap is: E / z
o 0.9
m 3/2 ati
n¯ = 4πk T¯ NKωr2ωz (4) gy r 0.8
(cid:18) B (cid:19) er
n
and the average relative velocity is hvi = E 0.7
4(k T¯)1/2(πm)−1/2, the relaxation rate can be ex-
B
pressed as:
0.6
τ−1 = γel = σ˜(T¯)N mωr2ωz . (5) 0 5 10 15 20 25
α α K 2π2k T¯
B Wait time [s]
We will come back to the relation between σ˜ and the FIG. 2: (Color online) Plot of the relaxation dynamics of
actual energy-dependent cross-sectionin Sec. IIIC. an ultracold sample of 39K after parametric heating in the
Eq.(5)holdsonlyifoneassumesthattheaverageden- radial direction. Data are taken after expansion (see text).
sity of the sample does not change during the relaxation Each point is an average of several experimental realization
process. Actuallythelossrateinducedbycollisionswith andthelineisanexponentialdecayfitwithequilibriumvalue
atomsofthebackgroundgasisabout0.03s−1 fortheex- fixedto1,initial valueandtimeconstantas freeparameters.
Average initial number of atoms is 379(10)×103 and initial
perimentsreportedinthiswork. Ifthisrateisnotnegligi-
average temperature is 16.2(7)µK. From the fit we obtain
bleascomparedtothemeasuredrelaxationrates,itmust
τ =3.66(46)s.
be takeninto account: one expects that the reductionof
the density slows down the relaxation so that after an
initial decay with a rate corresponding to Eq. (5), equi-
oftheinitialnumberofatomsN . Figure3(a)showsthe
K
librium apparently sets for a value of E /E lower than
z r measurementstakenafterexpansionwithanaverageini-
1. Thispictureisfurthercomplicatedbytheunavoidable tial temperature T¯ =16(1)µK, while Fig. 3(b) presents
heatingratepresentintheexperimentthatincreasesthe measurements taken in situ with T¯ =28.9(1.3)µK. One
temperature throughout the measurements: in our sys-
can see that the relaxation rate has the expected linear
tem we measure a different heating rate in the axial and
dependence on the number of atoms and the extrapola-
radial direction so that it is possible to achieve equilib-
tion towards zero is consistent with zero.
rium also for a value of E /E greater than 1.
z r InaIoffe-Pritchardtrap,asufficientlybigatomiccloud
In principle, one could take these effects into account
can experience a potential that is not strictly separable
havingthe equilibriumvalue ofthe energyratioasa free
so that different degrees of freedom are coupled and re-
parameter: for our measurements however the equilib-
laxation can occur even in the absence of collisions. We
riumvalue is almostalwaysconsistentwith 1. We there- refertothisprocessasergodic mixing. Apriori,forequal
fore choose to fix it to 1 in order to improve the esti-
harmonic frequencies, ergodic mixing could play a more
mation of the time constant, as it is shown in Fig. 2.
prominentrolein ourmTrapthanin the usualtrapsdue
to its small size. For this reason we take ergodic mixing
into account by separating the component of the relax-
ationrate,whichislinearinN,fromtheextrapolationin
B. Measurements of the elastic cross section
the limit ofzero density where relaxationcanonly occur
through ergodic mixing. One can therefore assume that
In order to reduce systematic errors on the determi-
nation of σ˜, we performed several measurements at dif- γ =α−1 dγel N +γ =AN +γ , (6)
K mix K mix
ferent densities and for two different temperatures of dN
K
29 and 16µK. Furthermore, to check that the inhomo-
where γ is the time constant of the ergodic mixing
geneous magnetic field present in our in situ imaging mix
process. The two parameters A and γ are obtained
was not a source of error, the dataset at 16µK is taken mix
from a linear fit of the experimental data, as shown in
with a slightly different procedure. First, we adiabati-
Fig. 3. By using Eq. (5) one has
cally decompress the trap to ω = 2π ×290(1)Hz and
r
ωpazr=am2eπtr×ic2h1e.a2t4i(n1g)Hatzωanr,dwthaeitnfborlowa vaawraiayb8le7Rtbim, eapapnldy A= ασ˜ 2mπω2kr2ωTz¯ . (7)
image the cloud after a 2ms expansion. B
The results of our measurements are shown in Fig. 3 As pointed out above, since the values for γ are con-
mix
wherewereporttherelaxationrateγ =τ−1asafunction sistent with zero in both the datasets, we can conclude
5
tion on the interatomic potential, one has to make some
0.40 (a) assumptiononthebehaviorofthecrosssectionasafunc-
0.35 tion of temperature. A simple partial wave expansion of
1] the scattering amplitude shows that, for the experimen-
-s 0.30
[ tal condition reached in this work, collisions can only
-1t 0.25 occuratl =0angularmomentum,oddvaluesbeingsup-
ate 0.20 pressed on symmetry ground and even values due to the
r low temperature [27]. In a very general form the s-wave
n 0.15
o cross section for identical boson is:
ati 0.10
x
a σ(k) = 8π|f (k)|2 (8)
el 0.05 0
R
0.00 where k = mv/(2h¯) is the relative wavevector and f (k)
0
0 100 200 300 400 500 600 700 800x 103 is the s-wavescattering amplitude that can be expressed
in terms of the S-matrix phase shift δ (k): f (k) =
0 0
0.40 (b) eiδ0(k)sinδ (k)/k.
0
In the limit of vanishing energy, given that the poten-
0.35
-1s] 0.30 tciraolssdesceacytisonmaosrearfaupnidctlyionthoafna1/sirn3g,lewepacraanmeexteprr,estshethse-
[
-1t 0.25 wave scattering length which, for two alkali atom in the
ate 0.20 stretched |F =2,mF =+2i state, is the triplet one aT :
r
on 0.15 |f (k)|2 ≃ a2 (9)
axati 0.10 0σ(k) ≃ 8πTa2T (10)
el 0.05
R For higher temperature one has to calculate the scat-
0.00 tering amplitude up to order k2 : f (k) = (−1/a +
0 T
0 100 200 300 400 500 600 700 800x 103 ik+1k2r +...)−1,wherer denotes the effectiverange,
2 e e
Number of atoms which is a function of the scattering length a and the
T
C coefficient of the van der Waals potential [17]
6
FIG. 3: (Color online) Plot of the measured relaxation rate
as a function of the numberof atoms in the sample after ex- a2
σ(k)=8π T . (11)
pansion (a) and in situ (b). The measured initial average (1− 1k2r a )2+k2a2
temperature is 16(1)µK and 28.9(1.3)µK respectively. The 2 e T T
slope of the linear fit to the data is related to the elastic Since the C coefficients are well known for all alkali
6
cross-section,whiletheinterceptisameasureofergodicmix-
dimers [18, 19], we numerically invert Eq. (3) after in-
ing as described in the text. The fit results are: γmix =
−0.003(0.065)s−1 and A= 5.5(1.6)×10−7s−1 for figure (a) serting Eq. (11), averaging over the Boltzmann thermal
and γmix = −0.0005(0.034)s−1 and A = 4.0(1.0)×10−7s−1 distribution and assuming aT < 0, to obtain the scat-
tering length: a = −67(11)a from data at 16µK and
for figure (b). Error bars on the vertical direction are statis- T 0
ticalerroron theexponentialdecayfit,whileonthehorizon- aT =−48(5)a0 fromdataat29µK[28]. Forcomparison,
tal direction they are the statistical fluctuation of the initial we notice that, if we neglect the effective range, these
numberof atoms not including thecalibration uncertainty. values would be −57(11)a and −36(5)a , respectively.
0 0
In the first case the two values differ by approximately
1.57 combined standard deviations, while in the second
that ergodic mixing plays a negligible role in our mea- case the difference is 1.75. The quoted uncertainties de-
surements. rive mainly from the fits of Fig. 3 and the atom number
The resulting values for the temperature-dependent calibration (±20%), which was done independently for
average elastic cross section, computed from Eq. (7) each data set. Finally, we take a weighted average of
with α = 2.7, as obtained from numerical simulation the two scatteringlength values calculatedwith effective
described in Sec. IIID, are 2.2(0.8) × 10−12cm2 and rangeandmultiplytheassociateduncertaintybyafactor
0.91(0.22) × 10−12cm2 for 16 and 29µK respectively. χ2 =1.65, to set the confidence level to 68% [20]: the
These values are a very good approximation of the ac- result is a =−51(7)a .
T 0
tual cross section as we show in the next section. pPrior to this work, the 39K triplet s-wave scatter-
ing length has been measured by photoassociation spec-
troscopy, a = −33(5)a [13], and by mass-scaling from
T 0
C. Extracting the value of the scattering length rethermalizationof 40K, a =−37(6)a [14]. Indeed our
T 0
measurementisthefirstdirectdeterminationoftheelas-
As pointed out above,the measured cross sections de- tic cross section for this isotope. To different extents, all
pend on the temperature. In order to extract informa- reported values of the scattering length depend on the
6
C coefficient of the van der Waals long range potential:
6
we take C6 = 3927a.u. [19], while in Ref. [13, 14] C6 is 4.5
1.00
assumed equal to 3897a.u. [18]. We checked that our
4.0
v[(ana.oluu.se],uccahhpapnrragotexesimiwsaiatthvivaCeilla6ybw5le0itithnimaReresaf.tsem[1δa4|l]al)etr|a=tnhda0n.t0h0ien1reaRf0oe×rfe.δ[t1Ch36e] E / Ezr0.95 33..05 particle
rdiivffeedresnccaettienritnhgelCen6gdtho.esAnsootusrigrnesifiucltandtifflyercshafrnogme tthheedave-- gy ratio 00..8950 22..05 ons per
edreavgiaetoiofnt,hweepcuobnlicslhueddetvhaalutetshoefa1g.r7e0emcoemntbiisnseadtisstfaancdtoarryd. Ener 0.80 11..05 Collisi
0.5
0.75
0.0
D. Numerical simulations
0 2 4 6 8 10 12 14 16 18
Time [s]
As shownin Eq.(7), to compute the value ofthe scat-
tering length from the fit to the data one needs to know FIG. 4: (Color online) Plot of the simulated energy ratio
thevalueoftheparameterα,whichistheaveragenumber (black dots, left scale) and collisions per particle (blue tri-
of collisions per particle needed for thermalization. This angles, right scale) as a function of time for 20×103 atoms
parameterdepends onthe temperature andthe trapfre- in a harmonic trap whose frequencies are 440 and 29.4Hz in
the radial and axial direction respectively. The red curve is
quenciesinanontrivialwayandcanvaryfrom2.4to3.4
anexponentialdecayfit. Thevalueα=2.72canbeobtained
[21]. Following [22, 23, 24] we estimate this parameter
directly from the graph identifying the time constant of the
from a numerical simulation of the system. Taking ad-
decay (7.74 s).
vantageofthelowdensityandsmallatomnumberofthe
sample, we make a direct simulation of the gas in which
weconsider3Dpositionandvelocityofeverysingleatom.
where σ is the elastic scattering cross section and |v | is
Choosingatimestepδtmuchsmallerthanboththeaver- r
themodulusoftherelativevelocity;collisionisprocessed
age collision time γ−1 and the faster timescale of single
coll if y <℘. In order to reproduce a correct scattering rate,
particle dynamics, namely ω−1, one may assume that
r onemustchooseδxsothattheaveragenumberofatoms
during the time interval δt interactions between atoms
in a volume δx3 is much smaller than 1 and M so that
are weak enough that they decouple with the center of
thedistancetraveledbyanatomduringthetimeMδtis
massmotionandthattheforceexperiencedbytheatoms
of the order of δx.
varies little. Therefore the dynamics of the gas can be
In Fig. 4 we report a typical result of the simulation
directly simulated with the following procedure:
showing the radial to axial energy ratio and the number
1. Positionand velocity are updated according to ex- of collisions per atom as a function of time. The simula-
ternal forces with a Verlet integrator. tion is performedin a cylindricalharmonic trap with ra-
dial and axial frequency of 440 and 29.4Hz respectively.
2. Every M steps, the positions of the atoms in real The number of atoms is 2×104 and the initial energy
space are discretizedon a lattice of spacing δx and distribution has a width of 15(19.5)µK along the axial
iftwoatomsarelocatedatthesamesiteacollision (radial) direction. The cross section is 8.6×10−16m2 so
test is made. that the scattering rate in the simulation is close to the
experimentalone. The valueofα=2.7canbe extracted
3. If the collision test is positive, the collision is re-
directly from the graph as explained in the caption of
solved using simple classical mechanics and s-wave
Fig. 4.
scattering:
v′ =v + |vr|eˆ
1 CM 2 R IV. CONCLUSIONS
( v2′ =vCM − |v2r|eˆR
wherev′ indicatesthevelocityofparticleiafterthe In summary we have demonstrated sympathetic cool-
collisioni, v and v are the center of mass and ing of 39K with 87Rb, in spite of the low interspecies
CM r crosssection,andwehavemeasuredthecrosssectionfor
relative velocity respectively, before the collision,
and eˆ is a random direction on the unit sphere. elastic collision between two 39K atoms in a pure triplet
R state. Wehavehenceobtainedthetriplets-wavescatter-
The collision test consists in comparing a random real inglengthaT,assumingitsnegativesignandaneffective
number y, uniformly distributed in [0,1), with the colli- rangeapproximationforthe s-wavescatteringamplitude
sion probability calculated according to kinetic theory: at low but finite temperature.
The possibility of sympathetic cooling and the knowl-
℘=σ|v |Mδtδx−3 (12) edge of the collisional properties of bosonic potassium
r
7
open the way to its use as a quantum degenerate gas. Whileadoublecondensateof41Kand87Rbwasproduced
Bosonic potassium isotopes, 39K and 41K, are predicted in a magnetic trap [5], 39K has not been brought to de-
to feature Feshbach resonances, several Gauss wide, at generacy so far. This work represents a decisive step in
moderate fields (< 1 kG). As a consequence, optically thisdirectionaswereachedaphase-spacedensityof0.01,
trapped bosonic potassium appears a suitable candidate with a gain of several orders of magnitude with respect
to realize a condensate with a scattering length tunable to the state-of-the-art [6].
around zero. Such a condensate would be interesting for
interferometricpurposes,asthe interactionenergylimits
the accuracy of interferometers employing Bose-Einstein
Acknowledgments
condensates. In addition, wide Feshbach resonances
makepotassiumcondensatesparticularlyattractivetore-
alize double condensates in optical lattices, for quantum We thank all the degenerate gas group at LENS for
simulation purposes. For 39K, in particular, Bohn and many stimulating discussions and fruitful advice and A.
coworkers have predicted a Feshbach resonance around Simoni for useful data on interatomic potential. This
40 G for two atoms in the |F =1,m =−1i state, with work was supported by MIUR, by EU under Con-
F
a width of several tens of Gauss [25], while another 60 tracts HPRICT1999-00111 and MEIF-CT-2004-009939,
G broad resonance has been predicted around 400 G for by INFN through the project SQUAT and by Ente CR
atomsintheabsolutegroundstate|F =1,m =1i[26]. Firenze.
F
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