Calculations of collisions between cold alkaline earth atoms in a weak laser field

20 Pages·2012·0.74 MB·English

Calculations of collisions between cold alkaline earth atoms in a weak laser (cid:12)eld
Mette Machholm
Department of Computational Science, The National University of Singapore, Singapore 119260
Paul S. Julienne
National Institute for Standards and Technology, 100 Bureau Drive, Stop 8423, Gaithersburg, MD 20899-8423
Kalle-Antti Suominen
Department of Applied Physics, University of Turku, FIN-20014 Turun yliopisto, Finland
Helsinki Institute of Physics, PL 64, FIN-00014 Helsingin yliopisto, Finland
(cid:31)rsted Laboratory, NBIfAFG, University of Copenhagen, Universitetsparken 5, DK-2100 Copenhagen (cid:31), Denmark
(April 3, 2001)
50 natural linewidths to the red of atomic resonance.
We calculate the light-induced collisional loss of laser-
Studies of small-detuning trap loss, extensively re-
cooled and trapped magnesium atoms for detunings up to
viewedbyWeineret al. [1], havemainlyconcentratedon
50 atomic linewidths to the red of the 1S0-1P1 cooling tran- alkaliatoms[3],forwhichithasbeenverydi(cid:14)culttode-
sition. We evaluate loss rate coe(cid:14)cients due to both radia-
velopquantitativetheoreticalmodelstocomparewithex-
tiveandnonradiativestate-changingmechanismsfortemper-
periment. Thisisbecausealkaliatomshaveextensivehy-
atures at and below the Doppler cooling temperature. We
per(cid:12)ne structure, and thus the number of collisionchan-
solve the Schro(cid:127)dinger equation with a complex potential to
nelsissimplytoolargeforaccuratetheoreticalmodeling.
represent spontaneous decay, but also give analytic models
It has even been di(cid:14)cult to estimate the relative weight
forvariouslimits. Vibrationalstructureduetomolecularpho-
of the di(cid:11)erent possible loss processes. Although one
toassociation is present in the trap loss spectrum. Relatively
broadstructureduetoabsorptiontotheMg21(cid:6)ustateoccurs can develop simpli(cid:12)ed models, these are di(cid:14)cult to test
for detunings larger than about 10 atomic linewidths. Much adequately with complex alkali systems. On the other
sharper structure, especially evident at low temperature, oc- hand,traplossphotoassociationspectrainalkalisystems
cursevenatsmaller detuningsduetoof Mg21(cid:5)g absorption, forlargedetuningcanbemodeledquiteaccurately[4{6].
which is weakly allowed due to relativistic retardation cor- Thisisbecausetheunderlyingmolecularphysicsofalkali
rections to the forbidden dipole transition strength. We also dimermoleculesiswell-known,andthespectraaredeter-
performmodelstudiesfortheotheralkalineearthspeciesCa, minedbyisolatedmolecularvibrational-rotationallevels,
Sr, and Ba and for Yb, and (cid:12)nd similar qualitative behavior for which the photoassociation line shapes can be well-
as for Mg. characterizedeveninthepresenceofhyper(cid:12)nestructure.
Quantitativeanalysisof suchspectra havepermitted the
34.50.Rk, 34.10.+x, 32.80.Pj
determinationofscatteringlengthsforgroundstatecolli-
sions[1]. Thesescatteringlengthsarecriticalparameters
for BEC studies.
Alkalineearthcoolingandtrappinghaverecentlybeen
I. INTRODUCTION
ofconsiderableexperimentalinterest. Traplosscollisions
havebeenstudiedinaSrMOT[7],andintercombination
A. Background
linecoolingofSrhasresultedintemperaturesbelow1(cid:22)K
and raised the prospects of BEC of Sr [8{11]. Ca is of
Laser cooling and trapping of neutral atoms has re- interest for possible applications as an optical frequency
centlyopenedmanynewresearchareasinatomicphysics. standard[12{16],andphotoassociationspectroscopyina
One can cool a gas of neutral atoms in magneto-optical Ca MOT has been reported [17]. Intercombination line
traps (MOT) down to temperatures of 1 mK and be- cooling has also been reported for Yb [18,19], which we
low, and obtain densities up to 1012 atoms/cm3. Evap- have included in our discussion because of its similarity
orative cooling methods have allowed the cooling of al- in structure to alkaline earth atoms.
kali species to much lower temperatures below 1 (cid:22)K so Alkaline earth species provide an excellent testing
that Bose-Einstein condensation (BEC) occurs. Binary ground for cold collision theories, especially given the
atomic collisions play an important role in the physics rapidly developing experimental interest in the subject.
of cold trapped atomic gases, and have been widely in- Since the main isotopes of alkaline earth atoms have no
vestigated [1]. One of the (cid:12)rst cold collisional process hyper(cid:12)ne structure, the number of collision channels be-
to be studied is the heating and loss of trapped atoms comes low enough to allow theoretical calculations even
which result from tuning laser light to near-resonance in the small detuning trap loss regime. Consequently,
withtheatomiccoolingtransition[2]. Herewetakenear- this paper presents theoretical predictions for small de-
resonance,or small-detuning, to mean detuning (cid:1) up to tuningtraplossspectraincoldandtrappedMggasinthe
1
presenceofnear-resonantweaklaserlighttunednearthe state to the loss channel at short range during a single
1S 1P atomic transition, and discusses the nature cycle of oscillation through the short-rangeregion.
0 1
!
of similar processes for Ca, Sr, Ba, and Yb. This work Figure1indicatesthequalitativebehaviorofthetrans-
extends our previous note on Mg trap loss [20] to other ferfunctionJ versusdetuning(cid:1). Atverysmalldetun-
eg
species and lower temperatures, and shows the relation ing of a few atomic linewidths J becomes very small if
eg
between small detuning trap loss and photoassociation there is a high probability of spontaneous emission dur-
theory. It is necessary to include spontaneous emission ingtransitfromtheoutertoinnerregions. Atsu(cid:14)ciently
in modeling trap loss collision dynamics because of the large detuning J exhibits resonance structure due to
eg
long time scale of cold collisions. In addition, relativis- the bound vibrational levels in the excited state poten-
tic retardation e(cid:11)ects play a prominent role at small de- tial, where the vibrational period is much shorter than
tunings by allowing transitions to the dipole forbidden the decay time [22]. This is the domain of photoassocia-
1(cid:5) state, which exhibits resolved vibrational structure, tion to individual vibrational levels. Section IIIC below
g
especially at very low temperature. Although we treat will show simple analytic formulas for J that apply in
eg
the long-rangemolecularinteractionsaccurately,the po- thesetwolimitingcasesofsmalldetuningwithfastdecay
tential energy curves and coupling matrix elements for or isolated photoassociation lines. These formulas show
thedimermoleculesintheshort-rangeregionofchemical how J can in turn be factored as
eg
bonding are not su(cid:14)ciently well-known to determine all
aspects of trap loss. Therefore, we examine the uncer- Jeg =JeePeg; (2)
tainties associated with the unknown phases developed
where P represents the probability of excitation from
in the short-rangeregion of chemical bonding, and show eg
the ground to excited state in the outer zone near the
whichfeaturesarerobustwithrespecttosuchuncertain-
Condonpoint, and J representsanexcited statetrans-
ties and which must be measured or later determined ee
ferfunctionbetweenthelong-rangeouterexcitationzone
from improved theory.
and the short-rangezone. The factorizationin Eq.(2) is
also schematically indicated in Fig. 1.
B. Trap loss collisions
Loss Transfer Excitation
Light-inducedtraplosstakesplaceasamolecularpro- P J P
pe ee eg
cess. Two colliding cold atoms form a quasimolecule, |eæ
and their motion can be described in terms of the elec-
tronic (Born-Oppenheimer) potentials of the molecular
dimer with light-induced transitions between the molec-
hn
ularstates. Weconsideronlyreddetuning,whichexcites
attractive potentials at long range. Such potentials sup-
port a number of bound vibrational states. |pæ
Figure1schematicallyindicatesthenatureofthe trap
|gæ
lossprocessinaweakradiation(cid:12)eld. Anexcitationlaser
with energy h(cid:23) is tuned a few atomic linewidths below R R R 0 Jeg
the atomic transition energy h(cid:23) . The ground g and p C
0
excited e quasimolecularelectronicstatesarethjusicou- FIG.1. Schematic representation of traploss collisions.
j i
pled resonantly by the laser at some long-range Condon
radius R , where the photon energy matches the di(cid:11)er- Each of the factors in Eqs. (1) and (2) can be af-
C
ence between the excited and ground potential energy fected by unknown phases associated with the short-
curves. The excited state decays to a loss channel p range molecular physics of the dimer molecules. (1) Peg
due to interactions at short range. The fully quantujmi is sensitive to the asymptotic phase of the ground state
mechanicaldescriptioninRef.[21]showsthattheoverall wave function. (2) Vibrational resonance structure in
probabilityPpg of atrap losscollisioncan be factoredas Jeg is sensitive to the position and widths of vibrational
follows: features. (3) Ppe is sensitive to Stu(cid:127)ckelberg oscillations
in short-rangecurve crossingprobabilities. These e(cid:11)ects
P =P J : (1) arediscussedindetailinSecs. IID andIIE. Theoverall
pg pe eg
e(cid:11)ectofsuchsensitivitieswilldependonthetemperature
Here Jeg represents an excitation-transfer probability and the alkaline earth species.
which is proportional to the scattering (cid:13)ux that reaches There are two possible inner zone loss processes char-
the short range region near Rp due to long-range exci- acterizedbyPpe: thestatechange(SC)andradiativees-
tation near the Condon point followed by propagation cape (RE) mechanisms, both represented schematically
on the excited state to the short-range region. Ppe rep- by the loss channel p in Fig. 1. In the SC process the
resents the probability of a transition from the excited excited state couples to another molecular state near
a short-range crossing point R , and population trans-
p
2
fer between them is possible. The products of the col- tion, especially if it has nodes in the region swept by
lision emerge on a state that correlates asymptotically the detuning-dependent Condon point. Analysis of such
with other atomicstatesof lowerenergy,suchas 3P+1S, data would hopefully give a value for the s-wave scat-
therebyreleasingalargeamountof kinetic energytothe tering length for Mg or other alkaline earth species, and
separating atoms. In RE the excited state can decay by consequently determine whether a stable Bose-Einstein
spontaneous emission after the atoms have been acceler- condensate is possible.
ated towards each other on the excited state potential. In this paper we present the calculated estimates for
The ground state atoms then separate with this gain in trap loss rate coe(cid:14)cients in Mg at temperatures around
kinetic energy. If enough kinetic energy has been gained and below the Doppler temperature. The contributions
to exceed the trap depth, this is a lossprocess (early de- from di(cid:11)erent mechanisms and states are identi(cid:12)ed and
cay after insu(cid:14)cient acceleration only leads to radiative compared. We also calculate predictions for other al-
heating). Here R represents the distance at which the kaline earth atoms and Yb by combining the appropri-
p
atomshavereceivedsu(cid:14)cientaccelerationtobelostfrom ate atomic properties with model molecular potentials.
the trap. Any emission with R<R leads to trap loss. Sec. II presents in detail the atomic data, molecular po-
p
In this paper we use the fully quantum complex po- tentials, and lasercouplings used for Mg and other alka-
tentialmethodof Ref.[21]anddonotresorttosemiclas- line earth atoms. Sec. III describes our theoretical ap-
sical methods (with one exception), although semiclas- proach. The results for Mg are given in Sec. IV and for
sical concepts are often useful for interpretation. Our the other atoms (Sr, Ca, Ba, Yb) in Sec. V. Finally we
quantum methods are fully capable of describing the present some conclusions in Sec. VI.
s-wave limit and the quantum threshold properties of
the extremely low collision energies near or below the
critical temperature for BEC. We describe the sponta- II. THE MOLECULAR PHYSICS OF ALKALINE
neous emission processes with a complex potential, and EARTH DIMER MOLECULES
solve the corresponding time-independent multichannel
Schro(cid:127)dinger equation in the molecular electronic state A. Alkaline earth atomic structure
basis. This takesany vibrationalstate structure into ac-
count automatically without the need to calculate wave
Table I gives the basic atomic data for the alkaline
functionsorFranck-Condonfactors,butlimitsourstudy
earth atoms and Yb, which has an electronic structure
toweaklaser(cid:12)eldsonly,whereonlyasinglephotonisex-
similar to the Group II elements. The major isotopes
changed with the (cid:12)eld. Typical cooling lasers are strong
have no hyper(cid:12)ne structure, except for Be. Alkaline-
and detuned only a few linewidths. By alternating with earth atoms have a 1S ground state, and excited 1P
0 1
a weakerprobe laser one can access the particular range and3P statesthatareopticallyconnectedtotheground
1
of detuning and intensity that we study. Future studies
state,thelatterweakly. Figure2sketchestheenergylev-
areneededtoaddressthee(cid:11)ectsofstronglaser(cid:12)eldsand els of the Group II atoms. The 1;3P (cid:12)rst excited states
the consequent revision of our results due to saturation arethe mostimportantforlasercooling. The1;3D states
and power broadening [23,24].
shift downwards as the atomic number increases. For
Wealsoincluderotationalstatesintoourmodel,which Mg the 1;3D states are above the 1;3P states. For Ca
for the molecular ground state correspond to the partial and Sr they are between the 3P and the 1P states, and
wavesofastandardscatteringproblem(angularmomen- for Ba the 1;3D states are below the 1;3P states. In laser
tum quantum number l). The symmetry of spinless al- cooling one uses typically the 1S -1P transition, which
0 1
kaline earth dimers permits only even partial waves. It
is the situation studied in this paper. For Mg this re-
should be pointed out that for near-resonant light the
quires a UV laser source, and for the heavier elements
Condonpointisatrelativelylargedistances. Thismeans
requiresrepumping to recycle atoms that decayto lower
thatalthoughthe collisionenergyislow,oneneedstogo levels. Theweak1S -3P intercombinationtransitionhas
0 1
to relatively large l before the ground state centrifugal
averynarrowlinewidth, and is within the opticalrange.
barrier stops the quasimolecule from reaching R . We
C Thus alkaline earth atoms are good candidates for opti-
correctheresomemistakesthatwediscoveredinthesum
cal atomic clocks when cooled to low temperatures. For
over partial wavesin Ref. [20].
clockapplicationsweneedtounderstandtheirlasercool-
The probe laser can be tuned over a wide range from
ing properties, including the magnitude and nature of
a few to many atomic linewidths. For su(cid:14)ciently large
laser-induced collisional trap loss.
detuning the rotational structure becomes sharp, and it
Beryllium is not likely to be a serious candidate for
should be possible to resolve the vibrational and rota-
lasercooling. Itistoxic,hasaveryshortwavelengthcool-
tional states. However, even at 27 GHz detuning, rota-
ing transition, and the intercombination \clock" transi-
tional features in the Ca 1(cid:6) 1(cid:6) photoassociation
2 g u tion is so weak as to be e(cid:11)ectively forbidden. Therefore,
!
spectrum are only partially resolved [17]. Photoassocia-
we will not consider Be in the rest of this paper.
tionstudiescanyieldpreciseinformationonthe molecu-
larpotentials. Also,the photoassociationlineshapesare
sensitive to the near-threshold ground state wave func-
3
TABLE I. Atomic data for major isotopes of Group II elements and Yb without hyper(cid:12)ne structure (natural abundance
shown). Most data is derivedfrom [25]. The lifetime (cid:28) for 1P1 is takento be the inverse of the 1P1-1S0 spontaneous emission
rate(cid:0)at=h(cid:22),thusneglectingweaktransitionstootherstatesforBaandYb. The 3Plifetimes arefromseveral sources[26]. The
linewidthinfrequencyunitsis (cid:0)at=h=(2(cid:25)(cid:28))(cid:0)1. Thewavelengths(cid:21)and(cid:12)nestructuresplittings forSr andBa aretakenfrom
[27]. TheDopplertemperatureisde(cid:12)nedasTD =(cid:0)at=(2kB). WetaketherecoiltemperaturetobeTR =(h(cid:22)2)=(m(cid:21)2kB),where
(cid:21)=(cid:21)=2(cid:25) and m is the atomic mass. The dipole moment is d0 = 3(cid:0)at(cid:21)3=4 (in a.u.). The atomic units for dipole moment,
length, andenergy are ea0 =8:4783 10(cid:0)30 Cm, a0 =0:0529177 nm, ande2=(4(cid:25)(cid:15)0a0)=4:3597482 10(cid:0)18 J respectively.
(cid:2) p (cid:2)
Be Mg Ca Sr Ba Yb
Major isotopes without
hyper(cid:12)nestructure 9Be(100%) 24Mg(78.99%) 40Ca(96.94%) 88Sr(82.58%) 138Ba(71.70%) 174Yb(31.8%)
(abundance) 26Mg(11.01%) 86Sr(9.86%) 172Yb(21.9%)
(cid:28)
1P1 (ns) 1.80 2.02 4.59 4.98 8.40 5.68
3P1 (ms) 5.1 0.48 0.021 0.0014 0.00088
(cid:0)at=h
1P1 (MHz) 88.5 78.8 34.7 32.0 18.9 28.0
3P1 (kHz) 0.031 0.33 7.5 120 181
Doppler-cooling limit
1P1 (mK) 2.1 1.9 0.83 0.77 0.45 0.67
3P1 (nK) 0.75 8.0 179 2.8 103 4.4 103
Recoil limit
1P1 ((cid:22)K) 39 9.8 2.7 1.0 0.45 0.69
3P1 ((cid:22)K) 3.8 1.1 0.46 0.22 0.36
d0 (a.u.)
1S0 1P1 1.89 2.38 2.85 3.11 3.16 2.35
(cid:0)
(cid:21)(nm)
1S0 1P1 234.861 285.21261 422.6728 460.733 553.548 398.799
(cid:0)
(cid:21)=(cid:21)=(2(cid:25)) (a0)
1S0 1P1 706.4 857.8 1271.2 1385.7 1664.9 1199.4
(cid:0)
FSsplitting
3P2 3P0 (a.u.) 1.36 10(cid:0)5 2.77 10(cid:0)4 7.20 10(cid:0)4 2.65 10(cid:0)3 5.69 10(cid:0)3 1.10 10(cid:0)2
(cid:0)
Because of the lack of hyper(cid:12)ne structure the ba-
sic laser cooling mechanism for alkaline earth atoms is
Dopplercooling,forwhichthetemperaturelimitT ,de-
D
(cid:12)ned in the caption of Table I, is set by the linewidth
(cid:0)at ofthecoolingtransition(widthsinthispaperareex- 0.2 1P 1S
pressedin energy units, so that the decay rateis (cid:0) =h(cid:22)).
at 3S
The lifetime of the alkaline earth 1P state is between
1
1.8 and 8.4 ns, giving a Doppler-cooling limit between 1P
2.1 and 0.45 mK for the elements in Table I. On the 1S
otherhandthe3P statehasalonglifetimewithDoppler- 3S 1S
1
cthoeolpinhgotliomnitrsecionilthliemnitKTra,ndgee(cid:12).nTehdisincatnhebTeacbolmepcaarpetdiotno; a.u.] 1P 3S 1P
Treacboliel IlismhiotwisstahbaotvTeRthisebRienttweerecnom0b.2inaantdio4n(cid:22)liKn.eTDhoupsptlheer nergy [ 0.1 3P 3P 13DD 1P13DD 1P 3P
coolinglimitforMgandCa,nearlycoincidentwithitfor E 3P 3P
Sr, and below it for Ba and Yb. 3P
Although Sisyphus cooling and magnetic trapping is 1D
not available for 1S atoms as it is for alkali atoms, in- 3D
0
tercombinationlinecoolingispossibleforsomeGroupII
species. This has been used to cool Sr to 400 nK with 0 1S0 1S0 1S0 1S0 1S0 1S0
(cid:24)
relatively high phase space density > 0:1 [8,9]. If com- Be Mg Ca Sr Ba Yb
bined with faro(cid:11)-resonantopticaltraps andevaporative
FIG.2. Energy levels of Group IIatoms andYb.
cooling it may become possible to obtain Bose-Einstein
condensation with optical methods alone.
4
B. Alkaline earth dimer molecular structure interatomicdistanceR with amagnitudeonthe orderof
twicetheatomiclinewidth. Thusthevibrationallevelsof
Figure 3 shows the lowest electronic potentials for the 1(cid:6)u state near the 1S+1P dissociation limit overlap
Mg [28]. Thereareonlytwostateswith attractivelong- strongly, and one does not expect to resolve them. One
2
range potentials correlating with 1P1+1S0 that can be interesting point is that the dipole-forbidden 1(cid:5)g (cid:0)1(cid:6)g
resonantly coupled to the ground state by laser light, transitionbecomesallowedatlargeRduetoretardation
namely 1(cid:6)+u and 1(cid:5)g. Both states o(cid:11)er the possibil- corrections. This means that the 1(cid:5)g state can be ex-
ity for the SC and RE mechanisms. When comparing cited at RC, but the spontaneous emission probability
Figs. 2 and 3 one can see that Mg is special. For other goes down quickly as R decreases. Consequently, 1(cid:5)g
alkaline earth atoms the molecular state picture is fur- vibrational states near the dissociation limit have nar-
ther complicated by the atomic D states below the 1P row emission linewidths. Thus the assumption that the
1
state. Thisincreasesthe numberof molecularstatesand vibrational states overlap strongly and can not be re-
thus the number of energetically available exit channels. solvedatdetunings of a few linewidths is not necessarily
The small number of molecular states is one reason we valid. One must determine if resolvable features persist
have chosen Mg as the basis for our studies. Theoretical whenonesumsoverallinvolvedrotationalstates/partial
calculationsrequirepreciseinformationaboutthemolec- waves, and takes the energy average over a thermal dis-
ular potentials and couplings over a wide range of inter- tribution. We show that one can indeed see vibrational
atomic distance. Ab initio calculations of ground and structure, especially if the temperature is well below the
excited molecularpotentials arealso available forSr [29] 1S0-1P1 Doppler limit. As mentioned above, this is by
and Ba [30]. no means impossible, if one does the cooling using the
While working on the manuscript we received new ab 1S0-3P1 transition.
initio data on Mg from E. Czuchaj, including both im-
2
provedpotentialcurvesandspin-orbitcouplings[31]. Al-
C. Long-range properties of states correlating with
though the new data di(cid:11)er to some extent from the re-
sultsofStevensandKrauss[28]becauseofimprovedcal- 1P1+1S0 atoms
culationsofelectroncorrelatione(cid:11)ects,wedonotexpect
that use of the new data would lead to any strong mod- There are four molecular states correlating with
i(cid:12)cation of our basic results, which should be viewed as 1P1+1S0 atoms: two long-range attractive states, 1(cid:6)+u
qualitative model calculations for the reasons to be dis- and 1(cid:5)g, and two long-range repulsive states, 1(cid:6)+g and
cussed in the following sections. 1(cid:5) . Herelong-rangemeansthatR islargecomparedto
u
the short-range region of chemical bonding and van der
0.18 Waals interactions shown in Fig. 3 so that the potential
1P is determined by the (cid:12)rst-orderdipole-dipole-interaction
u
with C = 2d2 and d2 for the 1(cid:6) and 1(cid:5) states, re-
0.16 1S+1P spective3ly. (cid:7)Her0e d i(cid:7)s th0e z-component of the atomic
0
21S transition dipole matrix element, which is related to the
g
atomic linewidth (cid:0) and the reduced wavelength of the
u.] 0.14 1S u 1S0(cid:0)1P1 transitionat((cid:21) = (cid:21)=2(cid:25)) by d20 = 3(cid:21)3(cid:0)at=4. The
[a. 1P exact long-range (lr) potentials including relativistic re-
R) 0.12 g tardation corrections are [32]
(
V 3P 3(cid:0)
u 3S V (u;1(cid:6)+)= at[cos(u)+usin(u)]; (3)
0.1 g 1S+3P lr u (cid:0)2u3
3(cid:0)
V (u;1(cid:5) )= at[cos(u)+usin(u) u2cos(u)];
3S lr g (cid:0)4u3 (cid:0)
0.08 3P u
g where u = R=(cid:21) is the scaled distance. The molec-
ular linewidths with relativistic retardation corrections
4 6 8 10 12 14
R / a are [32]
0
FIG. 3. The molecular states of Mg2 in atomic units cor-
responding to the asymptotic atomic states 1S0+1P1 and (cid:0)(u;1(cid:6)+)=(cid:0) 1 3 [ucos(u) sin(u)] ; (4)
1S0+3P0;1;2 [28]; the zero of energy is at the ground state u at(cid:26) (cid:0) u3 (cid:0) (cid:27)
1S0+1S0 asymptote. There are four states correlating with 3
(cid:0)(u;1(cid:5) )=(cid:0) 1 ucos(u) (1 u2)sin(u) :
each asymptote, of which two are attractive and two are re- g at (cid:0) 2u3 (cid:0) (cid:0)
pulsiveatlargeR,wherethesystemisexpectedtoberesonant (cid:26) (cid:27)
(cid:2) (cid:3)
with the laser (cid:12)eld. In the region with R < (cid:21), the potentials vary as 1=u3,
and (cid:0)(1(cid:6)+) and (cid:0)(1(cid:5) ) respectively vary as 2(cid:0) and
Thelinewidth of the excited 1(cid:6) statedepends onthe u g at
u (cid:0) u2=5.
at
5
The very long-range excited state potentials result and show that this is not a serious limitation. There are
in large Condon points for excitation of the attractive tworeasonsforthis. Oneisthatthe groundstatepoten-
states. If the laser detuning relative to the atomic tran- tial is(cid:13)at in the long-rangeregion,and the amplitude of
sitionisexpressedinunitsof(cid:0)atandthedistanceinunits (cid:9)(R;l;k1) has its asymptotic value in Eq. (7) indepen-
of(cid:21),thescaledCondonpoint(u )fortheground-excited dent of R as long as R > x = 1(2(cid:22)C =h(cid:22)2)1=4 [34,35];
C 0 2 6
state transition becomes independent of atomic species. theuseof x (oracloselyrelatedlength)asanappropri-
0
Table II shows u forseveraldetunings (cid:1), wherewe de- atelengthscaleforvanderWaalspotentialsisdescribed
C
(cid:12)ne red detuning to be positive. For Mg at (cid:1)=(cid:0) we in the Appendix of Ref. [6]. The condition R x is
at C 0
(cid:29)
have R (1(cid:6)+)=1132a and R (1(cid:5) )=728 a . easily satis(cid:12)ed in our case. Second, at the Doppler limit
C u 0 C g 0
T for 1S 1P cooling, a number of partial waves l
D 0 1
!
TABLE II. Condon points in scaled distance for selected contribute to trap loss in our detuning range (1-50 (cid:0) ).
at
detunings We demonstrate in Sec. IVA that a sum over l removes
Detuning(cid:1) uC(1(cid:6)+u) uC(1(cid:5)g) the dependence on the short-rangepotential.
Our trap loss spectra for s-wave scattering at the low
(cid:0)at 1.32 0.849
5(cid:0)at 0.716 0.511 temperatures available via intercombination line cooling
10(cid:0)at 0.555 0.411 willbesensitivetothe actualscatteringlengthA0 of the
30(cid:0)at 0.376 0.288 ground 1(cid:6)+g potential. However, we demonstrate a sim-
ple scaling relationship that will allow our low T s-wave
resultsto bescaledtoanyvalue ofthe scatteringlength.
Theattractivepotentialssupportaseriesofvibrational We may expect that the 1(cid:6)+ scattering length can be
g
levels leading up to the dissociation limit. Assuming a determinedfromoneortwo-colorphotoassociationspec-
potentialwithalong-rangeform C3=R3 givesthebind- tra,ashasbeendoneforalkalispecies[1]. However,such
(cid:0)
ing energy of vibrational level v [33]: analysis will require an accurate C coe(cid:14)cient and for
6
optimumresultsneedsareasonablyaccurateshort-range
(cid:25) 2 h(cid:22)2 potential as well.
"1=3 = (v v )2; (5)
v (cid:18)2a3(cid:19) 2(cid:22)C32=3 (cid:0) D
where a =(p(cid:25)=2)(cid:0)(5=6)=(cid:0)(4=3)= 1:120251, v is the E. Excited state short-range potentials
3 D
vibrationalquantumnumberatthedissociationlimit(v
D
isgenerallynonintegral)and(cid:22) is the reducedmass. The Traplossspectradepend stronglyon the excitedstate
vibrational spacing function, which we need later, is short-rangepotential structure in three ways:
(1) The curve crossings leading to SC trap loss occur
@" 3(cid:25) h(cid:22)2 1=2 at short range, and determine Ppe. In a Landau-Zener
@vv =h(cid:23)v = a3 2(cid:22)C2=3! "5v=6; (6) interpretation,
3
Ppe(J0)=4e(cid:0)2(cid:25)(cid:3) 1 e(cid:0)2(cid:25)(cid:3) sin2((cid:12)J0); (8)
where h(cid:23) is the vibrational frequency. (cid:0)
v
where(cid:3)= V (R )2=(h(cid:22)v(cid:0) D )and(cid:1)J0 designatesexcited
pe p p p
j j
state total angular momentum (See Section IIF). Here
D. Ground state
V (R ), v , D are the respective coupling matrix ele-
pe p p p
ment,speed,andslopedi(cid:11)erenceattheR crossing,and
p
The ground-state van der Waals potential varies at (cid:12)J0 is a semiclassical phase angle [36]:
long-range as R(cid:0)6 and is essentially (cid:13)at for the range
of Condon points we consider. The energy-normalized Rp Rp (cid:25)
0 0
ground state scattering wave function for collisional mo- (cid:12)J0 = ke(R;J )dR(cid:0) kp(R;J )dR+ 4; (9)
mentum h(cid:22)k1 andpartialwavel hasthe long-rangeform ZR0e ZR0p
0
where R and h(cid:22)k (R;J ) are the respective inner clas-
1 0i i
2(cid:22) 2 (cid:25) sical turning point at zero energy and local momentum
(cid:9)(R;l;k1)= (cid:25)h(cid:22)2k1 sin k1R(cid:0) 2l+(cid:17)l(k1) : for state i = e or p. The attractive singlet potentials
(cid:18) (cid:19)
h i mayhaveoneormorecrossingswith repulsivepotentials
(7)
from lower-lyingstates, e.g., those correlatingto 3P+1S.
We de(cid:12)ne the collisional energy as "=h(cid:22)2k12 =(2(cid:22)). The Thusthenumber,positionsofcrossingsandthecoupling
betweenstateswithcrossingpotentialsareimportantfor
short-rangepotentialsarenotsu(cid:14)ciently well-knownfor
themagnitudeofP . Althoughwecanmakereasonable
alkalineearthdimerstodetermineaccuratelythescatter- pe
ing phase shifts (cid:17)l(k1). Therefore, our calculations will oesfttihmeapteostefonrti(cid:3)al,stahnedpchaanseoannlyglbee(cid:12)cJa0licsusleantesditiavcectuoradteetlayilisf
havetobemodelcalculations. However,wetestthe sen-
very accurate potentials are available [37].
sitivity of our trap loss spectra to the unknown phases,
6
(2) The vibrational structure in the trap loss spectra based on the Mg-model, and explore the e(cid:11)ects of mass,
depends on both short and long-range potentials. The radiative properties, and coupling strength (size of spin-
spacing between the vibrational levels given by Eq. (6) orbit splitting) in these model calculations. Thus we do
depends onlyonthe long-rangepotentials,but the exact not use the ab initio potentials of Refs. [29,30] discussed
positions of the levels in Eq. (5) are determined by the in the paragraph above, because even if we were to in-
short-range potentials in the region of chemical bonding clude all the curves, there would still be uncertainties
(throughthev parameter). Thus,themagnitudeofthe associated with unknown phases and unknown coupling
D
vibrationalspacingsinourmodelcalculationswillbecor- matrix elements. Rather, our goal is to indicate qualita-
rect,whereastheactualpositionscanonlybedetermined tive di(cid:11)erences in magnitude and spectral shapesamong
by measurement. the various species. We trust that these will be helpful
(3)Theshort-rangeSCprocessalsocontributestothe in providing guidance for future experimental and the-
widthofvibrationalfeaturesinthetraplossspectra[38]. oretical studies of these systems. As we discuss in Sec-
Depending on species, temperature, and detuning, the tionIIIB,ourcalculationof the excitation-transfercoef-
widths may be primarily determined by natural or ther- (cid:12)cient(cid:20)willbetoalargeextentindependentofdetailsof
mal broadening or by the predissociation decay rate re- thecomplicatedshort-rangemolecularphysicsandcurve
lated to P . However, we are able to place approximate crossings.
pe
bounds on the magnitude of P . Sections IIIC, IV, and
pe
V discuss the contributions to feature widths and show
that natural and thermal broadening tends to be domi- F. Molecular rotational structure and coupling to
nant at small detuning, whereasSC broadeningmay be- the laser (cid:12)eld
come dominant at large detuning.
The available data on short-range potentials varies The full three-dimensional treatment of the collision
among the Group II elements. Ab initio potentials are of a 1S atom coupled to a 1P atom by a light (cid:12)eld is
available for Mg [28,31]. The structure of the molecular worked out in Ref. [40]. We adapt this treatment to our
potentials is fairly simple because only states correlat- simpli(cid:12)ed model with molecular Hund’s case (a) transi-
ing to 1P+1S (4 states) and 3P+1S atoms (4 states) are tions between two molecular states with the usual rota-
present (see Fig. 3) and the potentials provide quali- tional branch structure. The angular momentum J00 in
tative data for possible SC mechanisms. On the other the groundstatecanonlybethatofmolecularaxisrota-
hand, Ca, Sr and Ba have a very complex short-range tion, J00 = l00 with projection m00 on a space-(cid:12)xed axis.
structure,becauseoflarge(cid:12)nestructuresplittingsinthe The excited rotationallevels in a Hund’s case(a) molec-
triplet states, and the states correlating to 1D+1S and ularbasisfora1(cid:6)+ or1(cid:5) statedonothavemechanical
u g
3D+1S coming into play. For example, Boutassetta et rotation l0 as a good quantum number but instead the
al.[29]andAlloucheetal.[30]calculatedtheshort-range total angular momentum J0 with space projection M0.
potentialsforSr andBarespectively,but the largenum- The three possibletransition brancheshaveJ0 =l00+B,
ber of states (e.g., 38 states for Ba) makes it excessively where the branch labels P, Q, and R respectively desig-
complicatedtotreattheshort-rangeSCmechanismeven nate the cases B = -1, 0, and 1. The quasi-molecule
qualitatively. ground state can couple to the 1(cid:6)+ state only by P
u
The coupling between the singlet and triplet states at and R branches, but to the 1(cid:5) state by P, Q, and R
g
short-range is due to spin-orbit couplings. The exact branches. All potentials have a centrifugal term added,
magnitude of the couplings is unknown for all Group n(n+1)=(2(cid:22)R2), depending on ground state n = l00 or
II elements, but can be estimated using Table A1 of excited states n=J0 =l00+B.
Ref. [39], which relates the coupling matrix elements to The radiative coupling terms are
the 3P 3P (cid:12)ne structure splitting. This approxima-
2 0
(cid:0)
tion ignores any R-dependence of the spin-orbit matrix 2(cid:25)I 1=2
elements due to chemical bonding. The (cid:12)ne structure VA;B;l00;m00;q(u)= A;J0M0 e^q d~l00m00 ; (10)
c h j (cid:1) j i
splitting of the 3P state increases with atomic number (cid:18) (cid:19)
(see Table I). whereA=1(cid:6)+ or1(cid:5) labels the excitedstate,I islight
u g
We have opted for Mg as a model system, because intensity, d~ is the lab frame dipole operator, e^ is the
short-range potentials are available. Each singlet state q
polarization vector of light with polarization q = 0; 1,
withattractivepotentialcouplestoonlyonetripletstate 0 00 (cid:6)
andM ,m arelabframeangularmomentumprojection
in the inner zone. Thus the SC trap loss problem for
quantum numbers. In our weak-(cid:12)eld case, where transi-
Mg decouples into two three-state calculations, one for
the 1(cid:6)+u and one for the 1(cid:5)g excited state. For Mg we tdieo(cid:12)nnperaonbamb0i0l-iatvieesraagreedlirnaedairatiinvejVcAou;Bp;lli00n;mg0m0(ua)t;qrijx2,ewleemceannt
provide qualitatively correct SC trap loss spectra and 00
(the sum over m removesthe dependence on q):
even give some quantitative estimates.
In the case of the other Group II elements and Yb, 00
V (u;l ;B;I)
eg;A
we treat their complicated inner zone physics as one ef-
fective crossing, that is, we use three-state calculations
7
l00 1=2 modeled the ab initio potentials with Lennard-Jones3-6
1
=02l00+1m00=(cid:0)l00jVA;B;l00;m00;q(u)j21 pCo3tevnatluiaelsgkiveeenpiinngEthqe. (lo3n)g-rangeform(cid:12)xedtoitsknown
X
@2(cid:25)I 1=2 A (cid:27) 6 (cid:27) 3
= c (cid:11)A;B;l00dA(u) Vsr(u)=4(cid:15) (cid:21)u (cid:0) (cid:21)u : (14)
(cid:18) (cid:19) (cid:20)(cid:16) (cid:17) (cid:16) (cid:17) (cid:21)
=2:669 10(cid:0)9(cid:11)A;B;l00 I(W=cm2)dA(u)(a:u:): We have two (cid:12)tting parameters (cid:15) and (cid:27), and three
(cid:2) given values: well depth of the ab initio potential (the
q (11)
depth of the model potential is (cid:15)), position of minimum
u and C (1(cid:6)+) = 2d2 or C (1(cid:5) ) = d2. Because
The molecularelectronic transitiondipole moment dA is min 3 u (cid:0) 0 3 g (cid:0) 0
we want to (cid:12)x the long-range potential C we can not
3
(cid:12)t u and the well depth at the same time and have
3(cid:21)(a:u:)3 min
dA(u)(a:u:)= (cid:0)A(u)(a:u:): (12) chosen the latter:
4
r
(cid:27)((cid:6)) 6
wsthaetereas(cid:0)iAn(Euq).i(s4)t.hTehmeofalecctuolrasrpli2nle00w+id1th(cid:11)Ao;Bf ;tl0h0eareexgciivteedn Ve(u;1(cid:6)+u)=4(cid:15)((cid:6))(cid:18) (cid:21)u (cid:19) (15)
in Table III. 2d2
0 [cos(u)+usin(u)];
(cid:0)(cid:21)3u3
TABLE III. The rotational line strength factors
p2l00+1 (cid:11)A;B;l00. V (u;1(cid:5) )=4(cid:15)((cid:5)) (cid:27)((cid:5)) 6 (16)
e g
00 00 (cid:21)u
StateA BranchB l =0 (s-wave) l =0 (cid:18) (cid:19)
(cid:6) P 0 l600=3 d20 [cos(u)+usin(u) u2cos(u)];
(cid:6) R 2=3 (l00+1)=3 (cid:0)(cid:21)3u3 (cid:0)
p
(cid:5) P 0 (l00 1)=3
p p (cid:0) where
(cid:5) Q 0 (2l00+1)=3 (cid:15)((cid:6))=0:0347a.u., (cid:27)((cid:6))=4:339 a ,
p 0
(cid:5) R 2=p3 (l00+2)=3 (cid:15)((cid:5))=0:0681a.u., (cid:27)((cid:5))=2:751 a .
p 0
The well minima are (cid:21)u = 5:5 a and 3.5 a for 1(cid:6)+
p min 0 0 u
and 1(cid:5) , respectively, compared to the ab initio values
g
(cid:21)u =6:1 a and 5.4 a .
min 0 0
BothexcitedstateshaveaSCmechanismintheshort-
G. Model potentials for Mg rangeregionwith coupling to a triplet state. The triplet
states are purely repulsive, and modeled with
Ourresultsarenotsensitivetothedetailedformofthe
C
gTrhouusn,dwsetamtoedpeolttehnetigarlo,ufonrdtshteatreeapsootnesntgiiavlebnyinaSLeecn.nIaIrDd-. Vp;A(u)= (cid:21)6u66 +V1; (17)
Jones 6-12 form where A = 1(cid:6) or 1(cid:5) labels the molecular state, and
u g
(cid:27) 12 (cid:27) 6 V1 = 0:0601a.u.
Vg(u)=4(cid:15) (cid:21)u (cid:0) (cid:21)u : (13) The(cid:0)SCfrom1(cid:6)+u to 3(cid:5)u takesplacearoundthe inner
(cid:20)(cid:16) (cid:17) (cid:16) (cid:17) (cid:21) turning point of the 1(cid:6)+u potential well. We have chosen
For Mg we model the ground state potential from [28] C6 =392 a.u. for the model of the 3(cid:5)u state.
with awelldepth of(cid:15)=0:002825a.u.andaninnerturn- TheSC from 1(cid:5)g to 3(cid:6)g takesplaceabout1.5a0 out-
ing point of (cid:27) = 6:23 a0. This potential has a scat- side and 0.019 a.u. above the minimum of the 1(cid:5)g state
tering length of 95 a0 for the 24Mg reduced mass of potentialwell. WithC6 =81a.u.wehaveamodelofthe
23.985042/2 atom(cid:0)ic mass units. The scattering length crossing where the corresponding values are: 1.0 a0 and
A is determined from the k 0 behavior of the s-wave 0.019 a.u. The di(cid:11)erence in slope of the crossing poten-
0
phase shift: (cid:17)0 = k1A0. ! tials is 0.030 a.u. for the ab initio potentials and 0.037
Sincetheexcite(cid:0)dabinitiopotentialsofRef.[28]donot a.u. for the model.
permit a quantitative calculation of spectroscopic accu- The coupling between the crossing states are approx-
racy,forthe reasonsgivenin Sec.IIE,the speci(cid:12)c forms imated using Table A1 of Ref. [39]. The 1(cid:6)+u-3(cid:5)u and
of the short-range (sr) excited-state potentials are not 1(cid:5)g-3(cid:6)+g matrixelementsare(cid:16)=p2and(cid:16)=2,respectively,
important for our purposes of modeling the qualitative where (cid:16) = 1:84 10(cid:0)4 a.u. is 2/3 of the atomic 3P -
2
structure and magnitude of the collisional loss. How- 3P splitting. For(cid:2)example, weestimate anupper bound
0
ever,itisimportanttoretainthecorrectlong-rangeform. (sin2(cid:12)J0 = 1) to the Landau-Zener version of Ppe in
Therefore for simplicity in the calculations and because Eq. (8) for the 1(cid:5) -3(cid:6) crossing to be 2.6 10(cid:0)3 for the
g g
wewanttomodelthe other alkalineearthsystemstoex- model potentials and 2.8 10(cid:0)3 estimated(cid:2)from the ab
(cid:2)
plore the e(cid:11)ect of di(cid:11)erent C , (cid:0) , (cid:21) and mass, we have initio potentials. This upper bound is consistent with
3 at
8
0
the calculated P as a function of J from our complex Thespin-orbitcouplingconstant(cid:16) scaleswiththespin-
pe
potential calculation described below. orbit splitting of the 3P atomic states. We use the same
Since we will use a semiclassical method to determine de(cid:12)nitionofthematrixelementsasforMg. TheLandau-
the P factor for the RE process via the 1(cid:6)+ state (see Zener adiabaticity parameter 2(cid:25)(cid:3) in Eq. (8) for Ba and
pe u
Sec.IIIDbelow), wedonotneedanexplicitprobechan- Ybislargerthanunity,leadingtoamodi(cid:12)edshapeofthe
nel for RE. However, we introduce a probe channel to short-range adiabatic potentials and very small P .
pe;SC
simulate RE in order to show that the same J factor Thus for Ba the 1(cid:6)+-probe state coupling and for Yb
eg u
in Eq. (1) applies for both SC and RE processes, irre- the 1(cid:5) and the 1(cid:6)+-probe state couplings have been
g u
spective of the choice of the short-range R . Since we reduced by about a factor 5 to obtain values of P
p pe;SC
may take any form we like for a RE probe state, we use close to unity, in order to test the limit of very strong
a probe state potential which crosses the excited state broadening of the vibrational structure. We believe this
at a distance u , where the kinetic energy gained by the limitisphysicallymorerealistic. Thevarietyofcrossings
p
collisionpairis1K,correspondingtoatrapdepth of0.5 in these systems might lead to a strong SC process.
K. The RE probe potential has a repulsive inner wall
C
VRE;probe(u)= ((cid:21)u1)212 (cid:0)Vkin;RE;1; (18) III. COMPLEXCPAOLTCEUNLTAIATILOCNLSOSE COUPLED
whereVkin;RE;1 =3:17 10(cid:0)6a.u. andC12 =5 106a.u.
(cid:2) (cid:2) A. Description of method
Norotationaltermisincludedinthisprobechannel. The
same probe state potential is used for all collision ener-
gies,whicharesmall(mKrangeandbelow)comparedto The weak (cid:12)eld approximation assumed in this study
the 1 K kinetic energy at u . The coupling between the allows us to apply a complex potential method [21,41],
p
excited state and the probe state is chosen to be weak: since re-excitation of any decayed quasimolecular popu-
10(cid:0)9 a.u. lation can be ignored. Furthermore, the weak (cid:12)eld only
coupleseachground-statepartialwavetoatmost3rota-
tionalstatesofthe1(cid:6) or1(cid:5) statethroughtheP,Q,or
u g
H. Model potentials for Ca, Sr, Ba and Yb R branches. However, in the weak(cid:12)eld the excited-state
rotational states do not couple further to other ground-
Sincethe di(cid:11)erentgroundstatevaluesof C andinner state partial waves. Therefore we can ignore any partial
6
potentialshape makeno di(cid:11)erenceforthese modelstud- wave ladder climbing. Thus, for each trap loss mecha-
ies, for the reasons given in Sec. IIG, we take the same nism we have three dressed states: a ground state g, an
ground state Lennard-Jones 6-12 potential, Eq. (13), as excitedstatee,andaprobestate,p. Wesolvethethree-
in the Mg case to model the other alkaline earth ground channel, time-independent radial Schro(cid:127)dinger equation
00
states, and only change the reduced mass. This proce- for ground state collision energy ", partial wave l = l ,
dure yields respective model scattering lengths of 67 a , for each transition branch B and for a given intensity I
0
-65a ,-41a ,and97a for40Ca,88Sr,138Ba,and174Yb.
0 0 0 d2 2(cid:22)
Thus, A0 RC in all model cases. (cid:30)(";R)+ ["1 V(R;l;B;(cid:1);I)](cid:30)(";R)=0;
Thejlonjg(cid:28)-range of the excited state potentials for Ca, dR2 h(cid:22)2 (cid:0)
Sr, Ba and Yb is still exact, using the data from Table (20)
I with the form in Eq. (3). Due to lack of accurate ex-
cited state short-rangemolecularpotentials and because V is the 3 3 potential matrix
(cid:2)
of their more complicated structure, we model the trap
lossforCa,Sr,BaandYbbyscalingthe potentialsfrom V(u;l;B;(cid:1);I)= (21)
the Mg model Eqs. (15) and (16). The well depth (cid:15) of (cid:1)+V (u;l;B) i(cid:0)(u) V (u;l;B;I) V (u)
sthineg1le(cid:6)t-+utraipnldet1s(cid:5)tagtpesotsepnltitiatilnsgarceomscpaalerdedbtyotthheastizseploitfttinhge 0 Veeg(u;l;B;(cid:0)I) 2 egVg(u;l) pe0 1:
V (u) 0 V (u;l;B)
pe p
in Mg, e.g.:
@ A
The elements of V are described in Sec. II above. A
E(1P;Ca) E(3P;Ca)
(cid:15) =(cid:15) (cid:0) : (19) complexterm i(cid:0)(u)=2isaddedtothe excited-statepo-
Ca MgE(1P;Mg) E(3P;Mg) (cid:0)
tential to simulate the e(cid:11)ect of excited-state decay. The
(cid:0)
full retarded form of the molecular linewidth, Eq. (4), is
The short-range structure is treated as one e(cid:11)ective
used.
crossing. The probe states are qualitatively like those in
Application of standard asymptotic scattering bound-
the Mg model. The position (in energy) of the crossing
between the 1(cid:5) and probe state potentials is scaled as ary conditions to the three-component state vector (cid:30)
g
above. The 1(cid:6)+ and probe state potentials come very gives the S-matrix elements Sij(";l;B;(cid:1);I). If " > (cid:1),
u
close at the inner wall of the 1(cid:6)+ potential around the all three channels are open: i;j = g, p, or e. When
u
" < (cid:1), as is normally the case in our model, channel e
classical turning point V (u)=".
e
9
is closed, and S is only de(cid:12)ned for i;j = g or p. We ground-statepartialwavesfromcontributingto the loss,
ij
choose the light intensity I low enough that the results becausetheexcited-statepopulationneverreachesthein-
arein the weak(cid:12)eldlimit wherethe P = S 2 matrix nerzonewhereREdecayandSCtakeplace. Inthiscase,
pg pg
j j 00
element scales linearly in I. Our results are normalized l may decrease from the value de(cid:12)ned by the above
max
to a standard intensity of I =1 mW=cm2. inequality. We (cid:12)nd that the ground state cuto(cid:11) applies
We (cid:12)nd that we can make a change in the asymptotic except for the case of high energy and small detuning.
shapeof the arti(cid:12)cialprobepotentialtomakethe model Ineither case S 2 decreasesmanyordersof magnitude
pg
00 j j00 00
much more manageable computationally. The deep po- as l varies from l over the next few l -values. The
max
tential well of the excited state and the large kinetic en- upperlimitforthesuminEqs.(22)and(23)issettothe
ergy in the probe channel require a small stepsize in u l00-valuewhere(2l00+1)S (";l00;B;(cid:1);I)2 is10(cid:0)6 ofthe
pg
((cid:21)(cid:1)u 0:005 a ). However, with the large range of u maximum previous (2l0j0+1)S (";l00;B;j(cid:1);I)2 value.
0 pg
(cid:25) j j
((cid:21)u 1500-3000 a ) a small (cid:1)u increases the com-
max 0
(cid:25)
putation time and may compromise the numerical sta-
bility. Therefore, we modify the probe state potential to B. Factorization of trap loss probability
bringV (u)toasmallnegativevalueatintermediateand
p
asymptoticu. Thisresultsinasmallasymptoticmomen- The factorization in Eq. (1) allows us to separate the
tum h(cid:22)k in the probe channel, and allows us to gradually physics of the long-range excitation and the short-range
increasethestepsizeto(cid:21)(cid:1)u 0:5a0 asuincreases. The decaytothe traplosschannel. Reference[21]showshow
(cid:25)
coupling between the excited and probe states is turned to determine the short-range probability P from a dif-
pe
o(cid:11) exponentially before the change in Vp(u). The prob- ferent coupled channels calculation where the complex
athbeiliatsieysmjpStpogtji2c panrodpePrpteiesaroef tchoempprloetbeelypointednetpiaenldifetnhteroef ddeectearymtienremd i(cid:0)ni(cid:0)a=r2eginionEqn.ea(2r1R) iswohmeritetetdh.e lSoicnaclekPinpeetiics
p
are no asymptotic barriers. Since we have no centrifu- energy is very high in relation to ", this probability is
gal potential in the asymptotic probe channel, there are nearlyindependentof"overawiderange. Therefore,we
no asymptotic centrifugal barriers. The presence of such calculate [21]
barriers in our previously published model [20] resulted
in some errors at larger l00 which we have now corrected P (J0)= S (">(cid:1);l00;B =0;(cid:1);I =0)2: (24)
pe pe
j j
in the present model.
Thethermallyaveragedlossratecoe(cid:14)cientviastatee Here " is taken above the threshold energy (cid:1) where the
is: e channel becomes open and Spe is de(cid:12)ned.
k T 1 d" Ournumericalcalculationsshow,asexpected,thatPpe
K((cid:1);T)= B e(cid:0)"=kBT (22) is independent of " over a wide range, typically of "=k
B
hQT Z0 kBT from 0.3 mK to 300 mK at low J0 and 3 mK to 300
(2l00+1)S (";l00;B;(cid:1);I)2; mK at high J0, and also independent of (cid:1) in our small
pg
(cid:2) j j
l00 ;B range of detuning. The Landau-Zener interpretation of
evXen
P in Eq. (8) leads us to expect that P will vary with
pe pe
where QT = (2(cid:25)(cid:22)kBT=h2)3=2 is the translational parti- J0. This variation should be stronger for the outer 1(cid:5)g-
tion function. Identical particle exchange symmetry en- 3(cid:6) crossing than for the inner 1(cid:6) -3(cid:5) crossing. For
g u u
sures that only even partial waves exist for the ground the latter crossing, our calculations do give P values
pe
state. Wealsode(cid:12)neanon-averagedratecoe(cid:14)cientfora which vary slowly with J0. We calculate P (J0 = 1)
pe
(cid:12)xedcollisionenergy",wherewede(cid:12)neT" "=kB. Only to be 0.024, 0.44, 0.31, 0.34 and 0.64 for Mg, Ca, Sr,
(cid:17)
the sum over partial waves and branches is performed: Ba,andYbrespectively. Theseprobabilitiesarealllarge
(orderunity) except for the case of Mg. This qualitative
k T
K((cid:1);")= B " (2l00+1)Spg(";l00;B;(cid:1);I)2: conclusion is likely to be robust, even though our model
hQ j j
T" l00 ;B calculations are only quite approximate.
evXen
In contrast to our results for the 1(cid:6) -3(cid:5) cross-
(23) u0 u
ing, Fig. 4 shows that the calculated P (J ) values for
pe
Therearetwopossiblecuto(cid:11)slm00ax tothe partialwave the outer 1(cid:5)g0-3(cid:6)g crossing indeed depend much more
sum provided by the ground and excited-state centrifu- strongly on J . A test of the Landau-Zener formula for
gal potentials, respectively. For the ground state we can Mg shows that the result of Eq. (8) is indistinguishable
takel00 tobethelargestintegerforwhichh(cid:22)2l00 (l00 + from the calculated line on the (cid:12)gure. The qualitative
1)=2(cid:22)mRaC2x < " at the Condon point RC. Thusm, athxemCaoxn- feature of a dip in Ppe as it goes near zero for some J0
don point is classically accessible for l00 l00 and clas- is associated with the phase factor in the LZ formula,
sically forbidden for l00 > l00 . For th(cid:20)e emxacxited state, Eq.(9). Sincethespeci(cid:12)c J0-rangewherethisdipoccurs
max
the centrifugal potential may create a barrier inside the is sensitive to the potentials used [37], our model calcu-
Condonpointfortheg eexcitation. Thepositionand lationscan only be aqualitative guide evenfor Mg. The
the height of the barrie!r depend on J0. For collision en- relativevaluesfortheotherspeciesarealsoonlyqualita-
ergiesaround"=k T thisbarriermaypreventallowed tive guides, since other curvecrossingsare also involved.
B D
10