Table Of ContentVibrational energy transfer in ultracold molecule - molecule collisions
Goulven Qu´em´ener and Naduvalath Balakrishnan
Department of Chemistry, University of Nevada Las Vegas, Las Vegas, NV 89154, USA
Roman V. Krems
Department of Chemistry, University of British Columbia, Vancouver, B.C. V6T 1Z1, Canada
(Dated: February 2, 2008)
We present a rigorous study of vibrational relaxation in p-H2 + p-H2 collisions at cold and ul-
tracold temperatures and identify an efficient mechanism of ro-vibrational energy transfer. If the
colliding molecules are in different rotational and vibrational levels, the internal energy may be
transferred between the molecules through an extremely state-selective process involving simulta-
8 neousconservationofinternalenergyandtotalrotationalangularmomentum. Thesametransition
0 in collisions of distinguishable molecules corresponds to the rotational energy transfer from one
0 vibrational state of thecolliding molecules to another.
2
n
a Thecreationofultracoldmoleculeshasopenedupnew In this Letter, we present rigorous quantum calcula-
J avenues for research in physics and chemistry. Ultracold tions of cross sections for cold and ultracold collisions
5 molecules arefinding exciting applications incondensed- of p-H2 molecules in different levels of vibrational and
1 matter physics [1], quantum information science [2], rotational excitation and identify a new mechanism for
physics of elementary particles [3] and chemical reaction ro-vibrationalenergytransfer,notpresentincollisionsof
]
dynamics [4]. The experimental work on the production moleculeswithatoms. We findthatthe energyexchange
h
p ofdenseensemblesofultracoldmoleculesiscurrentlybe- processincollisionsofindistinguishablemoleculesatcold
- ing pursued by many research groups [3]. The progress and ultracold temperatures may be highly efficient and
t
n of the experiments relies on detailed understanding of state selective when the internal energy and total rota-
a molecular collisionsat ultracoldtemperatures [5, 6]. Re- tional angular momentum of the colliding molecules are
u centtheoreticalstudies [7,8,9,10]haveprovidedimpor- simultaneously conserved. By treating the H2 molecules
q
tant insights into atom - molecule inelastic and reactive as distinguishable particles it is shown that this mecha-
[
collisions at cold and ultracold temperatures. The colli- nism is primarilydrivenby a rotationaltransferprocess.
2 sionenergyofultracoldmoleculesisusuallymuchsmaller We show that the efficiency of the near-resonant energy
v than the energy spacing between ro-vibrational molecu- transferislargelyinsensitivetotheinitialvibrationalex-
1
larlevelsandthedurationofanultracoldcollisionisvery citation of the molecules.
8
0 long, which leads to unusual propensity rules and mech- The H2 + H2 system is the simplest neutral tetra-
3 anisms of inelastic scattering. For example, vibrational atomicsystemanditservesasaprototypefordescribing
. relaxationinultracoldcollisionsofmoleculeswithatoms collisions between diatomic molecules. Using a model
9
0 was found to give rise to quasi-resonant energy trans- rigid rotor potential energy surface (PES) for the H2-H2
7 fer [11] resulting in extremely narrow distributions over system developed by Zarur and Rabitz [17], Forrey [18]
0 finalro-vibrationalstates. Ultracoldmoleculesinexcited investigated rotational transitions in p-H2 + p-H2 col-
: vibrational states are routinely produced in experiments lisions at ultracold temperatures. Boothroyd, Martin,
v
on photoassociation and Feshbach resonance linking of Keogh and Peterson (BMKP) [19] have recently calcu-
i
X ultracoldatoms[12]. Thelifetimeofthemoleculesislim- lated a full-dimensional PES for the H4 system. Pogreb-
r ited by collision-induced vibrational relaxation. Dense nyaandClary[20]reportedratecoefficientsforcollisions
a ensembles of ultracold molecules can also be created in of p-H2 and o-H2 in the first excited vibrational state
experiments on buffer gas cooling [13] or Stark deceler- for collision energies between 10−3 eV (11.604 K) and
ation of molecular beams [14]. Ground-state molecules 1 eV (11604 K). They included all internal degrees of
may then be excited by black-body radiation [15] and freedomandusedthe BMKPPESinaquantumcalcula-
release energy through vibrationally inelastic collisions. tion based on the centrifugal decoupling approximation.
Laser cooling of molecules, when realized, will result Mat´e et al. [21] reported an experimental study of rota-
in significant populations of vibrationally excited states tionally inelastic collisions at temperatures between 2 K
that may relax through collisions in a gas of sufficient and110Kandcomparedtheirdatatotheoreticalresults
density [16]. Evaporative cooling of molecular ensem- obtained using the PESs calculated by Diep and John-
bles to ultracold temperatures is based on energy dissi- son (DJ) [22] and Schaefer and K¨ohler (SK) [23]. These
pation in molecule - molecule collisions. The spectrum potentialsdescribe the interactionbetweenH2 molecules
ofaccessibleenergylevelsis muchdenserfor collisionsof fixed at the equilibrium internuclear distance. Lee et
moleculeswitheachotherthanforcollisionsofmolecules al.[24]haverecentlypresentedacomparativeanalysisof
with atoms and it is important to understand dominant crosssectionsfor rotationallyinelastic collisionsbetween
energytransfermechanismsinmolecule -moleculeinter- p-H2atlowandultralowenergiescomputedusingtheDJ
actions at ultracold temperatures. andBMKPPESs. Time-dependentapproacheshavealso
2
beenrecentlyusedtodescribethedynamicsbetweentwo
p-H2 [25,26]andbetweenp-H2 ando-H2 [27]atthermal
energies. 2m)
c
The results presented here are obtained using the -160 103
BMKP PES and a full-dimensional quantum dynam- 1
ics calculation without any decoupling approximations. on (
We employed a newly developed quantum scattering ecti
s
code [28] based on the formalism described by Arthurs s
s
and Dalgarno [29], Takayanagi [30], Green [31], and cro 0000
Alexander and DePristo [32]. We consider H2 molecules stic 11020000
initially in different quantum states characterizedby the Ela 1002
vibrational quantum number v and the rotational an-
102
gular momentum j. We refer to a combination of two
ro-vibrational states of H2 as combined molecular state 10-6 10-5 10-4 10-3 10-2 10-1 100 101 102
Collision Energy (K)
(CMS).ACMSrepresentsauniquequantumstateofthe
diatom - diatom system before or after a collision. The 104
sCtMatSesw”ilnlobteadtieonnot[e3d0]asw(hvejrve′j′v).>Wevu′soerdaw“hwenell-vor=derve′d, 2m) 103
tjh≥e qju′.anAtsumthenucmolbliedrisngvjmoorlevc′uj′lecsanarneotinbdeisatisnsiggunieshdatbolea, -1610 c 110021
n (
pabarleticmuolalercmuloelse,cwulee.wTilol udseesctrhibeencootlalitsioionns[vojf;dv′isjt′]i,ngwuhieshre- ectio 100
the quantum numbers vj characterize one molecule, and s s10-1 11020000
s
v′j′ the other. The cross sections for state-resolved ro- cro10-2 1002
c
vibrational transitions are obtained from the solution of asti10-3
coupleddifferentialequations[31]usingthelog-derivative el
method of Johnson [33] and Manolopoulos [34], and ap- In10-4
propriate scattering boundary conditions [29]. 10-5
Fig.1presentsthecrosssectionsforelasticandinelas- 10-6 10-5 10-4 10-3 10-2 10-1 100 101 102
Collision Energy (K)
tic scattering of H2 molecules for collision energies from
10−6 K to 120 K. We consider collisions of H2 molecules FIG.1: Elastic(upperpanel)andinelastic(lowerpanel)cross
in the excited ro-vibrational states (v = 1,j = 0) and sectionsforH2-H2collisionsforcollisionenergiesfrom10−6K
(v = 1,j = 2) with H2 molecules in (v = 0,j = 0) and to 120 K. The curvesare labeled by(vjv′j′) values. See text
collisions of H2(v =1,j =0) with H2(v =0,j =2). The for details.
cross sections for elastic collisions of H2 molecules in the
ground state (v = 0,j = 0) are also shown. The cal-
culations are performed for the total angular momenta
J = [0 − 10] and for both parities. While the elastic thatro-vibrationalrelaxationin molecule - molecule col-
scattering crosssections for collisions of molecules in the lisions is determined by the propensity to conserve the
vibrationally excited states are practically the same, the internal energy and the total rotationalangular momen-
probabilityofinelasticscatteringisextremelysensitiveto tum of the colliding molecules. The crosssections for vi-
the initial statesofthe collidingmolecules. The inelastic brationalrelaxationincollisionsofH2(v =1,j =0)with
relaxation of H2(v =1,j =0) is six orders of magnitude H2(v = 0,j =0) are rather small because the minimiza-
more efficient in collisions with H2(v =0,j =2) than in tionof the energy gapbetween the initial andfinal CMS
collisions with H2(v = 0,j = 0). The zero-temperature would require a large change of the total rotational an-
limitingvaluesoftherelaxationrateconstantsfor(1000), gular momentum of the colliding molecules. Conversely,
(1200)and (1002),are respectively, 9×10−18, 4×10−14 the conservation of the total rotational angular momen-
and 6×10−12 cm3 s−1. The inelastic cross section for tum would introduce a large energy gap. Consequently,
collisions between H2(v =1,j =2) and H2(v =0,j =0) there is no single dominant CMS in the final state-to-
issmallerthantheinelastic crosssectionforcollisionbe- state distribution shown in panel a of Fig. 2. The in-
tween H2(v = 1,j = 0) and H2(v = 0,j = 2) at low and elastic relaxation of H2(v = 1,j = 2) in collisions with
ultralow energies. It undergoes a rapid enhancement of H2(v =0,j =0) shown in panel b of Fig. 2 is dominated
two orders of magnitude at the collision energy 25.45 K, by the transition (1200) → (1000). In this case, the en-
at which both inelastic cross sections become similar. ergy gap and the difference in total rotational angular
Toexplainthe resultsofFig.1,wehavecalculatedthe momentum of the two CMSs are simultaneously mini-
corresponding state-to-state cross sections for a collision mized. TheinelasticrelaxationofH2(v =1,j =0)incol-
energy of 10−6 K (see Fig. 2). The final CMSs are pre- lisions with H2(v = 0,j = 2) is however highly efficient.
sented in the order of increasing energy. Figure 2 shows Itis dominatedbythesingletransition(1002)→(1200),
3
E = 10-6 K and the second to purely vibrationalenergy transfer. As
coll
follows from Fig. 2, the rotational energy transfer domi-
103 H (v=1,j=0) + H (v=0,j=0) a natesoverthevibrationalenergytransferbythreeorders
2 2
101 of magnitude. The inelastic transition (1002) → (1200)
10-1 canthusbeviewedmainlyastherotationalenergytrans-
fer. Thesesymmetryconsiderationscanalsoexplainwhy
10-3
Panda et al. [27] found a surprisingly small cross section
10-5 for the resonant distinguishable process p-H2(v = 0,j =
2m) 10-7 0) + o-H2(v = 1,j = 1) → p-H2(v = 1,j = 0) + o-
6 c 103 H2(v=1,j=2) + H2(v=0,j=0) b H2(v = 0,j = 1), despite a small energy gap of 8.5 K.
-10 101 The rotational energy transfer between j = 0 and j = 1
n (1 10-1 is forbidden due to symmetry so this transition is driven
o bythe vibrationalenergytransfer. We concludethatthe
cti 10-3 near-resonant transitions should be generally smaller in
e
s s 10-5 ortho-para collisions than in para-paracollisions.
s
o 10-7
Cr 105 H (v=1,j=0) + H (v=0,j=2) [12;00] c E = 10-6 K
103 2 2 [10;02] 104 coll
[00;12]
1100-11 idnidstisintignugiushisahbalbel em moloelceucluelses 2m) 103 1002 2002 3002 2012 4002 ((vv 02 vv’’ 20))
c (v 0 v’ 0)
1100--53 [02;10] -1610 102 3200 2210 4200
10-7 0) 0) 2) 0) 2) 4) 0) 2) 4) 0) 0) 2) 0) 2) ction ( 101 0200 1200 2200 1210 2010 3010 4010
0 0 0 0 0 0 0 0 0 0 0 0 0 0 e
0 2 2 4 4 4 6 6 6 8 0 8 2 0 s
(0 (0 (0 (0 (0 (0 (0 (0 (0 (0 (1 (0 (1 (1 ss 100 6000
o 1010
FcoImGb.i2n:edStmatoel-etcou-slatartsetactreosfsosretchtieonssamasefsuynsctteimonssaosfitnheFifign.a1l astic cr10-1 3000 4000 2020 5000
and for a collision energy of 10−6 K. nel10-2 2000
I
1000
10-3
Initial combined molecular state
as shownin panel c ofFig. 2. This processconservesthe
FIG.3: Inelasticcrosssectionsasfunctionsoftheinitialcom-
totalrotationalangularmomentumandhasaverysmall bined molecular state for a collision energy of 10−6 K.
energy gap of 25.45 K. The energy gap arises from the
slightly different centrifugal distortion of the vibrational
states v = 0 and v = 1. Therefore, the conservation of Fig. 3 demonstrates the effects of vibrational excita-
internal energy and total rotational angular momentum tionof the colliding molecules onthe inelastic relaxation
is largely satisfied and the process becomes highly effi- cross section at a collisionenergy of 10−6 K. The results
cient and selective. This near-resonant transition is 150 are presented as functions of the initial CMSs (v0v′0)
timesmoreefficientthanthetransition(1200)→(1000). (dashed line), (v2v′0) (dashed bold line) and (v0v′2)
The reverse transition (1200) → (1002) is responsible (solid line). We use the full basis set for the calcula-
for the rapid enhancement of the inelastic cross section tion involving the initial CMSs (0j0j′), (1j0j′), (2j0j′)
for H2(v = 1,j = 2) + H2(v = 0,j = 0) collisions at and (1j1j′) and truncated the basis set, as described
25.45 K shown in Fig. 1 (the state (1002) becomes ener- in Ref. [37], for the calculations focusing on higher vi-
getically accessible at this collision energy). The mech- brational states. The truncation eliminates all channels
anism is thus a highly specific process, reminiscent of corresponding to ∆v and/or ∆v′ > 1 transitions. The
quasi-resonant energy transfer in collisions of rotation- magnitude of the inelastic cross sections for collisions of
ally excited diatomic molecules with atoms [11, 35, 36], H2(v,j = 0) with H2(v′,j = 0) depends strongly on the
but with a purely quantum origin. initial vibrationalstates of the colliding molecules which
In order to elucidate the mechanism for the (1002) is consistentwith the previous analysisof vibrationalre-
→ (1200) transition, we have repeated the calculation laxationinH-H2 [7],He-H2 [8],andLi-Li2 [10]collisions.
in Fig. 2 for the collision process H2(v = 1,j = 0) + For v = [1−6] and v′ = 0, the cross sections show a
H2(v = 0,j = 2) assuming that the colliding molecules monotonousincreasewiththevibrationalquantumnum-
are distinguishable. The inelastic transition (1002) → ber v as observed for He-H2 collisions [8] at ultralow en-
(1200) in collisions of indistinguishable molecules corre- ergies. When both molecules are initially vibrationally
sponds to two transitions: [10;02] → [12;00] and [10;02] excited as in v = [2−4] and v′ = 1, the inelastic cross
→ [00;12] in collisions of distinguishable molecules. The sections become significantly larger and less sensitive to
firstonecorrespondstopurelyrotationalenergytransfer, the initial vibrational levels of the two molecules. For
4
collisions of H2(v,j =2) with H2(v′,j =0), the inelastic cientlyexchangedbetweenthemolecules,leadingtonear-
cross sections are quite large and show a slight increase resonantenergytransfer. Ifthemoleculesaredistinguish-
with the vibrational quantum numbers. Finally, for col- able,thisprocesscorrespondstorotationalenergytrans-
lisions of H2(v,j = 0) with H2(v′,j = 2), the inelastic fer from one vibrational state of the colliding molecules
cross sections are very large due to the near-resonant toanother. Thisinelasticprocessthatdominatesoverall
energytransferanddo notdisplay anyparticulardepen- otherinelastictransitionsislargelyinsensitivetotheini-
dence on the vibrational levels. We have found that the tial vibrational excitation of the molecules. The process
elasticcrosssectionsgenerallydonotdependstronglyon may be an important mechanism for collisional energy
the vibrational excitation of the molecules. They rather transfer in ultracold molecules formed by photoassocia-
depend on quantum statistics effect, whether or not the tion of ultracold atoms and for chemical reactions pro-
vibrational and rotational quantum numbers of the col- ducing identical molecules. We hope that these results
lidingmoleculesarethesame[38]. Thisdifferencecanbe willstimulatefurtherstudiesofvibrationalenergytrans-
seen in the elastic cross sections of (0000) and (1000) in fer in molecular collisions at ultracold temperatures.
Fig. 1.
In summary, we have presented the first study of vi- This work was supported by NSF grant # PHY-
brational relaxation in cold and ultracold molecule - 0555565 (N.B), Natural Sciences and Engineering
molecule collisions and found a new mechanism of state Council (NSERC) of Canada (R.V.K.), NASA Grant
selective ro-vibrational energy transfer. If the colliding NNG06GJ11G from the Astrophysics Theory Program
molecules are in different levels of vibrational and rota- and NASA grant # NNG06GC94G to David Huestis of
tional excitation, the ro-vibrational energy may be effi- SRI International.
[1] H.P.Bu¨chleret al., Phys.Rev.Lett. 98,060404 (2007); 523 (2002).
G. K. Brennen et al., New J. Phys. 9, 138 (2007). [21] B. Mat´e, F. Thibault, G. Tejeda, J. M. Fern´andez, and
[2] A. Micheli, G.K. Brennen, and P. Zoller, Nature Phys. S. Montero, J. Chem. Phys. 122, 064313 (2005).
2, 341 (2006). [22] P. Diep and J. K. Johnson, J. Chem. Phys. 112, 4465
[3] J. Doyle, B. Friedrich, R. V. Krems, and F. Masnou- (2000).
Seeuws, Eur. Phys.J. D 31, 149 (2004). [23] J.SchaeferandW.E.K¨ohler,Z.Phys.D:At.,Mol.Clus-
[4] R. V. Krems, Int. Rev. Phys. Chem. 24, 99 (2005); Na- ters 13, 217 (1989).
turePhysics 3, 77 (2007). [24] T.-G. Lee, N. Balakrishnan, R. C. Forrey, P.C. Stancil,
[5] P. Staanum, S. D. Kraft, J. Lange, R. Wester, and M. D.R. Schultz, and G. J. Ferland, J. Chem. Phys. 125,
Weidemu¨ller, Phys.Rev. Lett. 96, 023201 (2006). 114302 (2006).
[6] N. Zahzam, T. Vogt, M. Mudrich, D. Comparat, and P. [25] S.Y.LinandH.Guo,J.Chem.Phys.117,5183(2002).
Pillet, Phys.Rev. Lett. 96, 023202 (2006). [26] F. Gatti, F. Otto, S. Sukiasyan, and H.-D. Meyer, J.
[7] N. Balakrishnan, R. C. Forrey, and A. Dalgarno, Chem. Chem. Phys.123, 174311 (2005).
Phys.Lett. 280, 1 (1997). [27] A. N. Panda, F. Otto, F. Gatti, and H.-D. Meyer, J.
[8] N. Balakrishnan, R. C. Forrey, and A. Dalgarno, Phys. Chem. Phys.127, 114310 (2007).
Rev.Lett. 80, 3224 (1998). [28] R. V. Krems, TwoBC - quantum scattering program,
[9] J. M. Hutson and P. Sold´an, Int. Rev. Phys. Chem. 26, University of British Columbia, Vancouver, Canada
1 (2007). (2006).
[10] G.Qu´em´ener,J.-M.Launay,andP.Honvault,Phys.Rev. [29] A. M. Arthurs and A. Dalgarno, Proc. R. Soc. A 256,
A 75, 050701(R) (2007). 540 (1960).
[11] R.C.Forrey,N.Balakrishnan, A.Dalgarno, M.R.Hag- [30] K. Takayanagi, Adv.At. Mol. Phys. 1, 149 (1965).
gerty,andE.J.Heller,Phys.Rev.Lett.82,2657(1999). [31] S. Green, J. Chem. Phys. 62, 2271 (1975).
[12] K.M. Jones et al., Rev.Mod. Phys. 78, 1041 (2006); T. [32] M.H.AlexanderandA.E.DePristo, J.Chem.Phys.66,
K¨ohler et al., Rev.Mod. Phys. 78, 1311 (2006). 2166 (1977).
[13] J.D.Weinstein,R.deCarvalho, T. Guillet, B.Friedrich, [33] B. R.Johnson, J. Comp. Phys. 13, 445 (1973).
and J. M. Doyle, Nature395, 148 (1998). [34] D. E. Manolopoulos, J. Chem. Phys. 85, 6425 (1986).
[14] J. J. Gilijamse et al., Science 313, 1617 (2006). [35] B. Stewart, P. D. Magill, T. P. Scott, J. Derouard, and
[15] S.Hoekstra et al., Phys.Rev. Lett. 98, 133001 (2007). D. E. Pritchard, Phys. Rev.Lett. 60, 282 (1988).
[16] M. D. DiRosa, Eur. Phys. J. D 31, 395 (2004). [36] A. Ruizand E. J. Heller, Mol. Phys. 104, 127 (2006).
[17] G.ZarurandH.Rabitz,J.Chem.Phys.60,2057(1974). [37] R.V.Krems,J.Chem.Phys.116,4517(2002);J.Chem.
[18] R.C. Forrey,Phys.Rev. A 63, 051403(R) (2001). Phys. 116, 4525 (2002).
[19] A. I. Boothroyd, P. G. Martin, W. J. Keogh, and M. J. [38] L. Biolsi, J. Chem. Phys. 58, 5331 (1973).
Peterson, J. Chem. Phys. 116, 666 (2002).
[20] S.K.PogrebnyaandD.C.Clary,Chem.Phys.Lett.363,
