Rotational relaxation in ultracold CO+He collisions

P�. M. Florian,M. Hoster,andR. C. Forrey

Penn�

State University, Berks-Lehigh Valley College, Reading, Pennsylvania 19610-6009, USA�Received26 March 2004;published22 September2004�

Cold�

andultracoldcollisions involving rotationallyhot CO moleculesare investigatedusingquantumme-chanical� coupledchannel,coupledstates,andeffective potentialscatteringformulations.Quenchingrateco-efficients� aregiven for initial rotationallevelsnearthe dissociationthreshold.The stability of the CO “superrotors”� againstcollisionaldecayis comparedto previousinvestigationsinvolving homonuclearmolecules.It isfound�

thatquasiresonanttransitionsprovidea significantlystrongercontributionto thetotal relaxationratethanin the comparablecaseof O2. As in the caseof H2, sharpstructuresin the distributionof total quenchingratecoefficients� arefound at rotationallevelswherequasiresonantscatteringis not allowed.

DOI:

10.1103/PhysRevA.70.032709 PACS number� s� : 34.50.Ez

I.

INTRODUCTION

Experimental�

schemesto cool andtrapneutralpolarmol-ecules� usinga time-varyingelectricfield havebeenproposed�1,2� and� recentlyrealized � 3��� . Meijer and co-workershave

applied� the so-calledStark deceleratorto slow down meta-stable� CO molecules� 1� . Schemesto producediatomicmol-ecules� in highly excitedrotationalstateshavealsobeenpro-posed� � 4,5� and� recently realized � 6��� . Theoretical studieshave

suggestedthat the collisional dynamicsof such rota-tionally!

hot moleculeswould be particularly interestingatlow temperatures" 7–1

#1$ . These studies concentratedon

homonucleardiatomic moleculeswhere it was found thatquasiresonant% vibration-rotation& QR

'VR( transitions

!contrib-

uted) significantly in quenchingthe highly rotationally ex-cited* molecules.Another recent investigationshowedthatQR'

VR transitionshavea neglibleeffect on the relaxationofrotationally+ hot oxygenmoleculesandthat the quenchingisvery, efficient for all rotationallevelsanddominatedby purerotationaldeexcitation - 12. . Eachof the theoreticalinvesti-gations/ 0 7–12

# 1assumed� a heliumatomcollision partner. He-

lium2

buffer gascooling 3 134 has

proven to be an effectivetechnique!

for loading moleculesinto a magnetictrap 5 146and� it has recently beenshown that the cooling techniquemay beappliedto a beamof molecules7 158 . Therefore,col-lisions2

involving heliumatomsareof considerableinterestinultracold) molecularphysics.In this work, we performrelax-ation� studiesfor helium collisionswith CO. We computeanextensive� amountof collisional datathat may be usedas apoint� of comparisonto the molecularhydrogenandoxygensystems� studiedpreviouslyand investigatewhetherhetero-nuclearmoleculesintroduceany interestingvariationsto thelow temperaturecollisionaldynamicsseenin rotationallyex-cited* homonuclearmolecules.Becauseheteronuclearpolarmolecules9 like CO possessan electric dipole moment,theradiativedecaysignal would contain information about thecollisions.* With the rapid advancesin experimentaltech-niques: mentionedabove ; 1–6,13–15< it

=is possiblethat CO

+> He may turn out to be an ideal systemfor experimentalinvestigationsof rotational relaxation in ultracold atom-diatom?

collisions.V@

ibrational relaxationin ultracoldCO+He collisionshasbeenA

studied for both isotopesof helium B 16–18C . Strong

resonances+ werepredictedD 16,17E and� therelativeefficiencyofF the processesinvolving exchangeof a double or singlequantum% of vibrationalenergy wasstudied G 18H . Therehavealso� beena numberof theoretical I 16–20J and� experimentalK21,22L M

investigations=

of vibrational relaxationfor this sys-tem!

at higher temperatures.Excellent agreementbetweentheory!

andexperimenthasbeenestablishedfor temperaturesbetweenA

35 and 1500K. The vibrational relaxationstudieshaveconsideredvibrational levels as high as N =2 O 18P and�rotational+ levelsashigh as j

Q=40 R 20

LTS,U where V and� j

Qare� the

vibrational, androtationalquantumnumbersof the CO mol-ecule.� Here, we considerall boundrotational levels withinthe!

first five vibrational manifolds.At theselow vibrations,the!

moleculecansupportrotationallevelsup to jQ

=230or sobeforeA

dissociating.It would be hopelessto attemptto per-form fully quantummechanicalcoupledchannelW CC

XZYcalcu-*

lationsat suchlargevaluesof jQ. Therefore,we will employa

decoupling?

approximationto obtain the desiredlarge-jQ

scat-�tering!

data.It wasshownin the caseof H2 and� O2 [ 12\ that!

bothA

the coupledstates] CSX_^

and� the effective potential ` EPaapproximations� wereadequatefor obtainingqualitativelyre-liable2

crosssectionsin the limit of ultracold collisions. Byqualitatively% reliable,we meanthat the shapeof a crosssec-tion!

or ratecoefficientcurveasa functionof b orF jQ

is correct,even� thoughthe overall magnitudeof the curvemay be offbyA

a small factor c 12d . This situationis entirely satisfactoryconsidering* that inaccuraciesin the potentialenergy surfacetypically!

introducea comparableamountof error. If morequantitative% accuracyis desired,it is possibleto renormalizethe!

distributionsusing the CC results.In this work, we willagain� test the CS and EP approximationsagainstthe moreaccurate� CC resultsin caseswhereit is possibleto performthe!

computationallyintensiveCC calculations.After deter-mining9 the accuracyof the decouplingapproximations,wecompute* cross sectionsand rate coefficients for ultracoldHe+CO collisions with initial rotational levels ranging allthe!

way to dissociation.For a few specialcasesof initialdiatomic?

states,we extend the investigationsto include arange+ of translationaltemperatures.

II. THEORY

Thee

CC, CS,andEPformulationshavebeengivenprevi-ouslyF f 23–26

L gso� we provideonly a brief review. The atom-

PHYSICAL REVIEW A 70,h 032709 i 2004j

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diatom?

Hamiltonianin the centerof massframeis given by

H = −1

2msut r2v

−1

2wx

R2v

+ y{z r| + VI } r,U R,U�~�� ,U � 1�where� r is the distancebetweenthe carbonand oxygenat-oms,F R is the distancebetweenthe helium atom and thecenter* of massof themolecule, � is

=theanglebetweenr� and�

R�

,U ms is=

the reducedmassof the molecule,and � is=

thereducedmassof the helium atom with respectto the mol-ecule.� The three dimensionalpotential energy surface isseparated� into a diatomic potential ��� r��� and� an interactionpotential� VI � r ,U R ,U���� . The diatomicSchrödingerequation

1

2L

msd� 2

dr� 2 −

jQ��

jQ

+ 1�2L

m rs 2v − ��� r� + ��� j

� ���j��� r  = 0 ¡ 2¢

is solvedby expandingtherovibrationalwavefunction £�¤ j��¥ r¦

in=

a Sturmianbasisset.Eachscatteringformulationthenre-quires% the solutionof a setof coupledequationsof the form

d� 2v

dR� 2 −

l§m¨ª© l§ m¨ + 1«

R� 2 + 2¬ Em¨ Cm¨®­ R = °

n± Cn±³² R�µ³¶ m¨®·UI ¸º¹ n±¼» ,U½3��¾

where� E¿

m¨ is=

the translationalenergy and l§m¨ is=

the orbitalangular� momentumin the ms th

!channel.For theCC formula-

tion,!

the index on the channelfunction À m¨ is assumedtodenote?

somecombinationof the quantumnumbersÁà,U jQ ,U l§ÅÄ ,Uwhereas� for the CS and EP formulations,the index denotesonlyF ÆÃÇ ,U jQ�È . The reducedinteractionpotentialUI is expandedin=

Legendrepolynomials,

UIÉËÊ R� ,U r� ,U�Ì�Í = 2Î VI

ÉËÏ R� ,U r� ,U�Ð�Ñ = ÒÓ=0

ÔUÕ×Ö R� ,U r�ÙØ PÚªÛÝÜ cos* Þ�ß à 4á�â

and� the solutionto Eq. ã 3��ä is=

matchedasymptoticallyto freewaves� to obtainthescatteringmatrix.TheCSandEPformu-lations assumethat the orbital angular momentumof theatom� is decoupledfrom the rotationalangularmomentumofthe!

diatomduring thecollision.At low valuesof jQ

and� l§,U this

assumption� may not be valid, however, the approximationgenerally/ improvesas j

Qis increasedå 26æ . For a basissetof

rovibrationaleigenstatesç�è j� ,U the respectivematrix elements

ofF the reducedinteractionpotentialenergy aregiven by

é�êjlQìë

UIÉîí ï®ð jQòñ l§�óõô = ö÷

=0

ømaxùûú

− 1ü j�+j��ý

−Jþ�ÿ��

2L

jQ

+ 1>���� 2L jQ��

+ 1>�� 2L l§+ 1>��� 2L l

§� + 1>���� 1/2� J l

�j�

jQ��

l§�� j

Q����jQ

0�

0 0�

� l§����

l§

0�

0 0� "!$# j

�"%U&(' )+*(, j�.- / ,U 0 5132

465jQ87:9

UIÉ<; =?> jQ�@BADC = EF

=0

Gmaxù�H

− 1I�JLK�M 2L jQ

+ 1>�N�O 2L jQ�P

+ 1>�Q�R 1/2

S jQ�T�U

jQ

0�

0 0� j

Q�VXWjQY

0 −� Z ["\$] j

��^U_(` a$b(c j�ed f ,Ug6�ih

j6kjQ$l

UIÉnm o?p jQ�qsr = tu

=0

vmaxù w

− 1x j�+j�.y

+ z j� −j�.{}|~

2Li�

+ 1>���� � 2jQ

+ 1��� 2jQ��

+ 1� 1/2

� jQ����

jQ

0�

0 0� �"�+� j

���U�(� �$�(� j�.� � � 7#3�for the CC, CS, and EP formulations.The � � denotes

?a

3-�

jQ

symbol� and � � denotes?

a 6-jQ

symbol.� The CS formu-lation requiresa set of calculationsfor each value of di-atomic� angular momentumprojection quantumnumber �whereas� the EP formulationpreaveragesover the projectionstates� andis independentof � . The respectiveCC, CS, andEPcrosssectionsaregiven by � 23–26� ¢¡

j�¤£¦¥(§

j�.¨ = ©

2ª E« j��¬ 2jQ

+ 1­ ®Jþ =0

¯�°2L

J�

+ 1>�±³²l´= µ Jþ −j

�.¶¶Jþ+j�"·¹¸

lº}»

= ¼ Jþ −j�.½}¾

¾Jþ+j�.¿}ÀÂÁÄÃ

j j�ÆŤÇ

llº+ȤÉ�Ê6Ê(Ë

− SÌiÍ

jl�

;Î(Ï j�.Ð lºÒÑJþ Ó

2v,U Ô 8Õ3Ö

×¢Øj�}Ù¦Ú(Û

j�eÜ = Ý

2L$Þ

E¿àß

j�"á 2L jQ

+ 1>�âäãJþ=0

å�æ2J�

+ 1çéèê=0

ëmaxùíì

2 − î8ï 0ð�ñ�òÄó

j j�Æô¤õ�ö6ö(÷

− Søiù

j�;ú(û j�.üJ

þný þ2,U ÿ 9���

���j�����

j� � = �

2L��

E¿��

j��l�=0

���2l§+ 1����� j j

������ ! �" − S#%$

j�;&�' j�)(l

� *2. + 10,

Thee

quantummechanicalscatteringcalculationswerecarriedoutF for eachof the CC, EP, and CS formulationsusing thenonreactivescattering program MOLSCAT - 27. suitably�adapted� to the presentsystem.The interactionpotentialen-er� gy surfaceof Heijmenet/ al. 0 28

L21was� usedfor all calcula-

tions!

presentedin this work. This potentialallowsfor stretch-ing of the CO bondandis ableto reproducethe boundstateener� giesof the He-CO complex.For the diatomicmolecule,we� usedthe extendedRydberg potential 3 29

L24employed� by

Balakrishnan5

et/ al. 6 167 . A basissetconsistingof 50 Hermitepolynomials� for eachrotationalsymmetrywas usedto rep-resentthe rovibrationalwavefunctions.For the CS calcula-tions,!

we havefound it convenientto restrict the projectionquantum% number 8 to

!be zero. This allows a considerable

speedup� in efficiencywith computationtimes that arecom-parable� to theEPcalculations.As in thecaseof O2

v:9 12; this!

procedure� will typically shift the overall magnitudeof thecross* sectionor rate coefficientcurveswithout altering theshape� or anystructurethat theymaycontain.It alsodoesnotintroduce=

anysignificantlossof accuracybeyondthatwhichis=

already introducedby the decouplingapproximationtobeginA

with. We havealsoadoptedthe l<-labeledvariantof the

CSX

approximation originally proposed by McGuire and

FLORIAN, HOSTER,AND FORREY PHYSICAL REVIEW A 70, 032709 = 2004> ?

032709-2

Khouri @ 24A which� assumesthat the diagonaleigenvalueof

the!

orbital angularmomentumoperatorl< 2v is=

approximatedbyl<CB

l<+1D where� l

<is a conservedquantumnumber. The colli-

sional� ratecoefficientsmay be obtainedfrom the usualther-mal9 averagingprocedure

RE jFHGI�J

jF)K�L TM = N 8Õ k

OBP T/QHRTS�U 1/2 1V

kO

BTWYX 2v

0Z[]\ ^ ,j

FH_`�a,jF)b�c Ek

dfeg

exp�ih − Ekd /Q kO BP Tj Ek

d dE�

kd ,U k 11l

where� TW

is=

thetemperatureandkO

B is=

theBoltzmannconstant.The total quenchingratecoefficientsRm j

Fon Tp are� given by

R��q

jFor TWYs = t u�v

jF)w R��x j

FHyz�{jF)|�} TWY~ . � 12�

Thee

sumin Eq. � 12� includes=

contributionsfrom all possibleexit� channels.The total quenchingratecoefficientis an im-portant� quantity in cooling and trapping experiments.Anytype!

of deexcitationprocesswill leadto unwantedheatingofthe!

gasandlimit furthercoolingefforts. In thecaseof heliumcollisions* with O2,U the total quenchingratecoefficientsweredominated?

for all � and� jQ

byA

pure rotational de-excitationcontributions* j

Q��= jQ

−2. QRVR transitions,which typically oc-cur* at initial j

Q-valueswhererotationalandvibrationalmotion

ofF the diatom are near resonanceand are characterizedbypropensity� rules relating the changein � to

!the changein j

Q�30�2�

,U werefoundto beof inconsequentialimportancefor thissystem� � 12� . By contrast,the energetically allowed QRVR�

jQ

=−2��� transitions!

for He+H2 were� foundto dominatethepure� rotationalcontributionsby about5 ordersof magnitude�9,1�

1� . It is suchcases,wherethe efficiencyof rovibrationaltransitions!

is comparableto or greaterthan pure rotationaldeexcitation,?

that we would expectto seeinterestingstruc-ture!

in the distribution of total quenchingrate coefficientwith� j

Q. This is due not only to the competitivebalancebe-

tween!

the two relaxationpathways,but alsoto theclosingofQR'

VR transitionsthat canoccurfor specific jQ

values, at lowtemperatures.!

III. RESULTS

Figure�

1 comparesthetotal quenchingratecoefficientsforzero-temperature� collisions calculatedwith the CC, EP, andCSX

scatteringformulations.The moleculeis initially in the� =0 level for eachcalculationwith thebasissetrestrictedtojQ

−10 � jQY���

jQ

+> 2 and � max� =1. Theanisotropyof thepotentialener� gy surfacerequires� max� =20 with 40 integrationnodesfor theanglebetweenthediatomandthe line connectingtheatom� to the centerof massof the diatom.Theseparametersare� twice those used for He+O2

v�� 12� which� allowed CCcalculations* for j

Q �40á

before becomingintractable.In thepresent� case, the CC calculationsbecomeintractable forjQ ¡

20 and a decouplingapproximationsuch as EP or CSmust9 be used.As in the caseof He+O2

v ,U it appearsthat suchapproximations� will provide estimatesfor large j

Qthat!

arereliableto within a factor of 2.

Figure 2 showsthe energy gapsbetweenthe initial andfinal¢

rovibrational statesof CO£¥¤ =1,jQ§¦

as� a function of jQ.

Thee

boxed region on the right shows where ¨ jQ

=−2©�ªQR'

VR transitionsarelikely to takeplace.Unlike thecaseofO«

2,U thereexistsa positivecrossingpoint for the upwardanddownward?

curves.Therealsoexist QRVR ¬ jQ

=−3­�® transi-!

tions! ¯

leftmost2

boxedregion° that!

wereabsentin the homo-nuclear: O2 and� H2 cases* studied previously. The positivecrossing* pointsin theboxedregionsrepresentj

Qvalues, where

classical* trajectorycalculationswould generallypredict thestrongest� correlationbetween±² and� ³ j

Qand� thereforethe

FIG. 1. Total quenchingratecoefficentsfor He+CO ´¶µ =0,j· ¸

inthe limit of zerotemperature.The curveswerecomputedusingtheCC ¹ solid lineº , Eh P » short¼ dashedline½ , andCS ¾ long dashedline¿scatteringformulations.For j

·ÁÀ20, the CC calculationsbecomein-

tractableanda decouplingapproximationsuchasEPor CSmustbeused.Fromthefigure, it appearsthatsuchapproximationswill pro-vide estimatesfor large-j

·that arereliableto within a factor of 2.

FIG. 2. Energy gapsbetweeninitial andfinal rovibrationalstatesfor COÂÄÃ =1,j

·)Å. The boxedregionsshowwhereQRVR transitions

typically occur. In both cases,the energy gap is positive at thecrossingpoint betweenthe two curves.Interestingthresholdbehav-ior may be expectedfor cold collisions near these j

·values.The

respectiveenergy gapsfor the crossingpoints are 12.8cm−1 andÆ4.2 cm−1 for

�the Ç j

·=−3È�É and Ê j

·=−2Ë�Ì curves.

ROTATIONAL RELAXATION IN ULTRACOLD CO+HeÍ PHYSICAL REVIEW A 70, 032709 Î 2004Ï

032709-3

largest and most specific transition efficiency Ð 7,8,30# Ñ

. AtordinaryF temperatures,thesmall positiveenergy gapsdo notsuppress� the efficiencyof the QRVR transitionsbecausethemoleculecan borrow energy from the translationalmotion.At ultracoldtemperatures,however, thesetransitionsareen-er� getically closed.Becauseneighboringj

Qvalues, havesmall

negative: energy gaps that allow QRVR transitionsto takeplace,� we shouldexpectto seestructurein thedistributionofratecoefficientsfor the j

Qvalues, neara crossingpoint.

Thee

crosssectionshave beenobtainedfor Ò =0−5 andjQ Ó

100usingtheCSformulation.Thecalculationswerecon-ver, ged to within a few percentfor ÔTÕ 0

�using basis sets

restricted to Ö −1 ×ÙØ Ú�ÛÝÜ +1 and jQ

−10 Þ jQYß�à

jQ

+10. Thezero-temperature� ratecoefficientsareshownin Figs.3–5. InFig.�

3, the state-to-staterate coefficientsare plotted as afunction of j

Qfor He+COá¥â =0,j

Q�ãin the limit of zero tem-

perature.� The pure rotational deexcitation curves aresmoothly� varying with an upturn at j

Qvalues, neardissocia-

tion.!

The curves representingrovibrational transitionsarepeaked� at the j

Qvalues, where vibrationally upward QRVR

transitions!

first becomeenergetically allowed. Becausetheefficiencies� of rovibrational and pure rotational transitionsare� comparablein magnitude,thesepeakswill appearas“steps” in the total quenchingrate coefficient distribution.Similarä

behaviorwas found in the state-to-staterate coeffi-cients* for H2

v and� O2v . In the caseof O2

v ,U however, the effi-ciency* of pure rotational relaxation was too large for thesteps� to be observedin the total quenchingrate coefficientdistribution? å

12æ . Figure4 showsthestate-to-statecurvesforç =1. The rovibrationalcurvesappearto have“holes” at thejQ

values, wherethe classicallyallowedQRVR transitionsareener� getically closedè i.e.,

=at the crossingpoints in the boxed

regionsof Fig. 2é . The magnitudeof the rovibrational ratecoefficient* on eithersideof a holeis comparableto or greaterthan!

that of the pure rotationaldeexcitationratecoefficient.Therefore,e

when all of the possibledeexcitationrate coeffi-cients* areaddedtogether, the distributionof total quenchingratecoefficientswill alsoappearto containtheholes.Figure51

showsthe total quenchingratecoefficientsasa functionofê and� jQ. Thestepsin the ë =0 curveandtheholesin the ìTí 0

�curves* are labeledaccordingto their QRVR propensityruleî30�2ï

. Thefigureshowsthat theholestendto increasein sizeand� shift downwardin j

Qas�ñð is

=increased.

Atò

jQ

values, very closeto dissociation,we find anupturninthe!

zero-temperatureratecoefficientswith jQôó

see� Fig. 6õ . ThisbehaviorA

differs from the casesof H2v and� O2

v studied� previ-ouslyF . In orderto makesenseof this result,we speculatethatthere!

maybea strongercompetitionbetweenenergy gapandangular� momentumgapminimizationfor thepresentsystem.In previouswork ö 12÷ ,U it wasshownthat theratecoefficientsfollowedø

an exponentialenergy gapfit. Here,we attempttofit¢

the numericalCS resultsusing

FIG. 3. State-to-stateratecoefficientsfor He+COùÄú =0,j·)û

in thelimit of zero temperature.“Steps” occur at j

·valuesü wherethe vi-

brationallyupwardQRVR transitionsbecomeenergetically open.

FIG. 4. State-to-stateratecoefficientsfor He+COýÄþ =1,j·)ÿ

in thelimit of zero temperature.“Holes” occur at j

·valuesü whereQRVR

transitionsareenergetically closed.

FIG. 5. Total quenchingratecoefficientsfor He+CO��� , j·��

in thelimit of zero temperature.The curvesare given for � =0,1,2,3,4,5andareincreasingin orderon the left sideof the figure.The struc-tures are labeled according to the nearest available QRVRtransitions.

FLORIAN, HOSTER,AND FORREY PHYSICAL REVIEW A 70, 032709 � 2004> �

032709-4

�E −

jQI and� � � jQ ����� − 1 −

1

8Õ

jQ 2v

⇒� lim2T� 0Z R��� TW���� c� 1 exp��� − � j

Q��+> c� 2

v exp� − ! /QjQ 2" # 13$

with� “best-fit” parametersgiven in TableI. The solid curvesin=

Fig. 6 showthat this type of balancebetweenthe energyand� angularmomentumgapscangive qualitativefits to thedata.?

This situationis not entirely satisfactory, however, andwe� look for otherwaysto analyzetheneardissociationdata.One«

possibilityis to applytheBornapproximation.TheBornapproximation� may be usedwhen both the initial and finaltranslational!

energies tend to zero.This situationcan occurforø

transitionsbetweenstatesthat havea very small energygap./ Figure2 showsthat theenergy gapsfor % j

Q=−2&(' tran-

)sitions* arenearzerowhen + =1 and j

,.-230.In this case,the

leading order behaviorof the zero-temperaturerate coeffi-cient/ is given by the Born approximationto be

limT0 0Z R 1 T2 = O

3 4�5E6 5/27

. 8 149

Figure7 showstheratecoefficientsfor : j,

=−2;(< transitions)

for He+CO=?> =1,j,�@

in the limit of zero temperature.The

solid* curvesattemptto fit the datato Eq. A 14B . Apart fromoneC anomolousdatapoint, it appearsthat the Born approxi-mation is able to fit the dataas the energy gap approacheszero.D The anomolousdatapoint at j

,=230 is most likely the

resultE of a boundor virtual stateof the three-bodysystemthat)

is too closeto zero F 31GIH

.The resultspresentedabovehave beenrestrictedto the

limitJ

of zerotemperature.It would be interestingto seehowthe)

holesin theratecoefficientdistributionsbecomefilled asthe)

temperatureis increased.Tabulationof temperaturede-pendentK rate coefficientsfor all values of L andM j

,wouldN

requireE extensivecomputationaleffort and is not the pathpursuedK here. Instead,we focus our attention on the O j

,=−3P(Q hole thatexistsin theratecoefficientdistributionforR =1 and j

,betweenS

170and174 T see* Fig. 5U . In orderto seehowV

this hole in the distribution becomesfilled as the tem-peratureK is increased,it is necessaryto computecrosssec-tions)

for a largerangeof translationalenergies.Figures8–10show* the dominantenergy-dependentcrosssectionsmulti-pliedK by collision velocity in orderto providethedimensionsofC a ratecoefficient.The curvesin Figs. 8 and9 arefor theupwardW and downwardQRVR transitions,whereasthoseinFig. 10 are for the pure rotational transitions.The resonantstructure* in the crosssectionsthat is seenin eachcurveoc-curs/ at energy valuesthat are similar to thosefound at lowrotational levels by BalakrishnanetX al. Y 16Z usingW the CCformulation.Thetranslationalenergy rangefor resonancesinweaklyN interactingatom-diatomsystems,such as thosein-volving[ helium, is not strongly affectedby the internal di-atomicM state,so it is typical for thesesystemsto possessalargenumberof resonancesthatareeachassociatedwith thesame* orbital angularmomentum \ 32

GI]. Figure 8 showsthat

the)

j,

=170 curve extendsall the way to ultracold energies.Likewise, the j

,=174 curve in Fig. 9 is nonzerofor the low

ener^ gy limit. The approximateenergieswherethe remainingcurves/ becomenonzeroareindicatedin thefigures.ThermalrateE coefficientsmay be obtainedby convolutingthe results

FIG._

6. Ratecoefficientsfor `ba =0, c j·

=−1 transitionsin thelimit of zero temperature.The solid lines attempt to fit the datausing a balancebetweenthe energy and angularmomentumgapsdseeTableI e .f

TABLE I. Fitting parametersfor gbh =0, i j·

=−1 ratecoefficientsin the limit of zerotemperature:R=c1 expjlk − m j

·�n+c2o expjlp −q /

rj· 2s .f

t cu 1 v cmw 3x

s−1y cu 2 z cmw 3x

s{ −1| } ~0 9.80336� 10−12 6.67083� 10−12 0.026638 176777

1 9.15125� 10−12 3.08254� 10−11 0.025021 235517

2 8.32866� 10−12 2.54698� 10−10 0.023285 322256

3 8.40136� 10−12 5.24745� 10−10 0.022427 340188

4 1.19161� 10−11 2.54660� 10−11 0.024494 187721

5 2.16808� 10−11 2.51188� 10−12 0.028880 80936

FIG. 7. Rate coefficients for � j·

=−2�b� transitions�

for He+CO��� =1,j

·��in�

the limit of zero temperature.The solid curvesattemptto fit thedatato the leadingorderbehaviorpredictedby theBorn approximation.

ROTATIONAL RELAXATION IN ULTRACOLD CO+He� PHYSICAL REVIEW A 70, 032709 � 2004�

032709-5

ofC Figs.8–10with a Maxwell-Boltzmanndistributionof col-lisionJ

velocitiesas in Eq. � 11� . The resultsaregiven in Fig.11 for severaltemperatures.The T

W=0.1 K curve nearly ap-

proachesK the ultracoldT � 0�

limit shownin Fig. 5. As tem-peratureK increasestoward 1 K, the hole in the distributionbeginsS

to shrink.At TW

=30 K, theholehascompletelydisap-pearedK andthedistributionis smoothlyvaryingwith j

,. These

results suggest that a buffer gas with a temperatureof300G

–500 mK would be sufficient to experimentallytest theexistence^ of suchholesin thedistributionof ratecoefficients.

The�

presentresults also suggestthat apart from odd-integerQRVR transitionsthat are allowed in heteronuclearsystems,* theredoesnot appearto beany fundamentaldiffer-ences^ betweenhomonuclearand heteronucleardiatomic re-laxationJ

at ultracold temperatures.Each system studiedshows* sharpstructurein the state-to-statecrosssectionsat j

,

values[ where quasiresonantscattering is not allowed.Whether�

or not this structureappearsin the total relaxationrateE dependson the relative efficiency of the rovibrationalandM pure rotational deexcitation.For ultracold He+O2,� thepureK rotationaltransitionsarealwaysdominant,whereasthetotal)

relaxationratesfor ultracoldhelium collisionswith H2�

andM CO are strongly influencedby the rovibrational transi-tions.)

One important differencebetweenhomonuclearandheteronuclearsystemsis that heteronuclearpolar moleculespossessK an electric dipole moment.The emitted radiationfrom�

such moleculeswould contain information about thecollisions./ Therefore,it shouldbepossibleto seeevidenceof

FIG. 8. Cross section times collision velocity for He+CO�� =1,j

·¢¡as a function of translationalenergy. The crosssectionsare

for £¥¤ =−1, ¦ j·

=3 transitions.

FIG. 9. Cross section times collision velocity for He+CO§�¨=1,j

ᢩas a function of translationalenergy. The crosssectionsare

for ª¥« =1, ¬ j·

=−3 transitions.

FIG. 10. Crosssectiontimes collision velocity for He+CO ­¯®=1,j

·¢°as a function of translationalenergy. The crosssectionsare

for ±¥² =0, ³ j·

=−1 transitions.

FIG. 11. Thermal rate coefficients for He+CO ´¯µ =1,j·¢¶

as afunctionof temperatureandthefive rotationallevelsshownin Figs.8–10.The ratecoefficientsincludedeexcitationto all possiblefinalstatesof CO. WhenT · 10 K, the ¸ j

·=−3¹¥º transitions

�aredomi-

nant for all five rotational levels. As the temperatureis reducedbelow1 K, theseQRVR transitionsbecomeenergeticallyclosedforj·

=171–173 and the distribution of rate coefficientsbeginsto ap-proachthe limiting ultracoldresult.

FLORIAN, HOSTER,AND FORREY PHYSICAL REVIEW A 70, 032709 » 2004¼ ½

032709-6

QR¾

VR transitionsandtheassociatedj,-dependentstructurein

the)

emissionspectrum¿ 33GIÀ

for thesesystems.

IV. CONCLUSIONS

W�

e haveinvestigatedcold andultracoldcollisionsinvolv-ingÁ

rotationallyexcitedCO moleculesusingCC, CS,andEPscattering* formulations.Like in thecasesof H2 andM O2 stud-*ied previously  12à ,� the decouplingapproximationsyield re-sults* that appearto be qualitatively correctas a function ofthe)

initial j,

levelJ

when j,

isÁ

larger than 20 or so. It is theselarge j

,values[ wherethe numericallyexactCC methodbe-

comes/ too slow to be of any practicaluse.In caseswherecomparisons/ can be made, the decouplingapproximationswereN found to bequantitativelyaccurateto within anoverallfactorof 2, which is acceptableconsideringthe lossof accu-racy introducedby a typical potentialenergy surfacewhenappliedM to ultracold collisions.More importantly, the calcu-

lationsrevealthepresenceof significantrovibrationaltransi-tions)

arising from QRVR energy transfer. ThesetransitionsgiveÄ rise to sharpstructurein the total quenchingratecoef-ficientsÅ

asa function of j,. WhenQRVR transitionsareener-

geticallyÄ closed,the structureappearsasa “hole” in the dis-tribution)

of rate coefficients. The holes in the zero-temperature)

distributionswerefoundto persistall theway upto)

temperaturesthat are typical of buffer gascooling Æ e.g.,^300G

–500 mKÇ . Therefore,it shouldbe possibleto performexperimental^ testsof QRVR energy transferusing a buffergasÄ setup È 13É withoutN the need for magnetictrapping.ApolarK moleculesuchas CO would be an ideal candidatetoperformK suchmeasurementsdueto the strongsignalarisingfrom�

the radiatingelectricdipole moment.

ACKNOWLEDGMENTSÊ

This work was fundedby the National ScienceFounda-tion,)

GrantNos.PHY-0331073andPHY-0244066.

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ROTATIONAL RELAXATION IN ULTRACOLD CO+He} PHYSICAL REVIEW A 70,Ù 032709 ~ 2004�

032709-7�