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> ?
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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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032709-7�
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