Table Of Content

Strong Dependence of Ultracold Chemical Rates on Electric Dipole Moments Goulven Quéméner and John L. Bohn JILA, University of Colorado, Boulder, C0 80309-0440, USA (Dated: January 27, 2010) Weusethequantumthresholdlawscombinedwithaclassicalcapturemodeltoprovideananalyt- icalestimate ofthechemicalquenchingcrosssections andratecoefficientsoftwocolliding particles atultralowtemperatures. Weapplythisquantumthresholdmodel(QTmodel)toindistinguishable fermionic polar molecules in an electric field. Atultracold temperatures andin weak electric fields, thecrosssectionsandratecoefficientsdependonlyweaklyontheelectricdipolemomentdinduced 0 1 by the electric field. In stronger electric fields, the quenching processes scale as d4(L+12) where 0 L>0 is the orbital angular momentum quantum number between the two colliding particles. For 2 p−wavecollisions (L=1) of indistinguishablefermionic polar molecules at ultracold temperatures, the quenching rate thus scales as d6. We also apply this model to pure two dimensional collisions n and find that chemical rates vanish as d−4 for ultracold indistinguishable fermions. This model a provides a quick and intuitive way to estimate chemical rate coefficients of reactions occuring with J high probability. 7 2 I. INTRODUCTION described several cold molecule quantum dynamics cal- ] h culations [10, 11, 13, 16, 17]. p Ultracold samples of bi-alkali polar molecules have - m been createdvery recently in their groundelectronic 1Σ, Within the Bethe–Wigner limit, scattering can be de- vibrational v = 0, and rotational N = 0 states [1–3]. scribed by an elegant Quantum Defect Theory (QDT) o This is a promising step before achieving Bose-Einstein approach[19–22]. Thisapproachmakesexplicitthedom- t a condensatesordegenerateFermigasesofpolarmolecules, inant role of long-range forces in controlling how likely . s providedthatfurtherevaporativecoolingisefficient. For themoleculesaretoapproachclosetooneanother. Con- c this purpose, elastic collision rates must be much faster sequently, quenching rate constants can often be written i s than inelastic quenching rates. This issue is somewhat in an analytic form that contains a small number of pa- y problematic for the bi-alkali molecules recently created, rameters that characterize short-range physics such as h since they are subject to quenching via chemical reac- chemicalreactionprobability. Forprocessesinwhichthe p tions. If a reaction should occur, the products are no quenching probability is close to unity, the QDT theory [ longer trapped. provides remarkably accurate quenching rates [23, 24]. 1 Foralkalidimersthatpossesselectricdipole moments, For dipoles, however,the full QDT theory remains to be v elasticscatteringappearstobequitefavorable,sinceelas- formulated. 2 tic scattering rates are expected to scale with the fourth 6 0 power of the dipole moment [4, 5]. Inelastic collisions of In this article we combine two powerful ideas – sup- 5 polarspeciescanoriginatefromtwodistinctsources. The pressionofcollisionsduetolong-rangephysics,andhigh- . long-range dipole-dipole interaction itself is anisotropic probability quenching inelastic collisions for those that 1 0 and can cause dipole orientations to be lost. This kind are not suppressed – to derive simple estimates for in- 0 of loss generally leads to high inelastic rates, and is re- elastic/reactive scattering rates for ultracold fermionic 1 garded as the reason why electrostatic trapping of polar dipoles. The theory arrives at remarkably simple ex- : molecules is likely not feasible [6]. Moreover,these colli- pressions of collision rates, without the need for the full v i sions also scale as the fourth power of dipole moment in machinery of close-coupling calculations. Strikingly, the X theultracoldlimit[4],meaningthattheratioofelasticto model shows that quenching collisions scale as the sixth r inelasticratesdoesnotingeneralimproveathigherelec- power of the dipole moment for ultracold p wave colli- a − tricfields. Thissortoflosscanbepreventedbytrapping sions. On the one hand, this implies a tremendous de- the molecules in optical dipole traps. greeofcontroloverchemicalreactionsby simplyvarying More serious is the possibility that collisions are anelectricfield,complementingalternativeproposalsfor quenchedbychemicalreactions. Chemicalreactionrates electric field control of molecule-molecule [25] or atom- are known to be potentially quite high even at ultra- molecule [26] chemistry. On the other hand, it also im- cold temperatures [7–17]. Indeed, for collision ener- pliesthatevaporativecoolingofpolarmoleculesmaybe- gies above the Bethe–Wigner threshold regime, it ap- comemoredifficultasthefieldisincreased. InsectionII, pears that many quenching rates, chemical or other- weformulatethe theoreticalmodelforthreedimensional wise, of barrierless systems are well described by apply- collisions. In section III, we apply this model to pure ing Langevin’s classical model [18]. In this model the twodimensionalcollisionsandconclude in sectionIV. In molecules must surmount a centrifugal barrier to pass the following, quantities are expressed in S.I. units, un- closeenoughtoreact,butareassumedtoreactwithunit less explicitly stated otherwise. Atomic units (a.u.) are probability when they do so. This model has adequately obtained by setting ~=4πε =1. 0 2 II. COLLISIONS IN THREE DIMENSIONS A. Cross sections and collision rates short range In quantum mechanics, the quenching cross section of chemistry apairofcollidingmolecules(oranyparticles)ofreduced R) massµ foragivencollisionenergyEc andapartialwave V( L,ML is given by V long range E b physics c ~2π σqu = Tqu 2 ∆ (1) L,ML 2µE | L,ML| × 0 c R b + whereTquisthetransitionmatrixelementofthequench- ing process, Tqu 2 represents the quenching proba- | L,ML| R bility, and the factor ∆ represents symmetrization re- quirements for identical particles [27]. If the two col- FIG. 1: (Color online) Effective potential barrier V(R) as a liding molecules are in different internal quantum states functionoftheintermolecularseparationR. VbandRbdenote (distinguishable molecules), ∆ = 1 and if the two theheight and theposition of the centrifugal barrier. colliding molecules are in the same internal quantum state (indistinguishable molecules), ∆ = 2. The total quenching cross section of a pair of molecules is σqu = σqu . The quenching ratecoefficientofa We consider now the case of collisions between two in- L,ML L,ML pair of molecules for a given temperature T (collisional distinguishable molecules (∆ = 2 in Eq. (1)). During P event rate) is given by the time dt = τ the number of molecules N lost in each ∞ collision is two. Then we get dN/dt = 2/τ. The vol- Kqu =<σqu v >= σqu vf(v)dv (2) ume per colliding pairs of molecules is V−/(N(N 1)/2) L,ML L,ML × L,ML − Z0 where we have taken into account the indistinguisha- where bility of the molecules. For the same reason explained 3/2 above, the volume associated with the collisional event µ f(v)=4π v2 exp[ (µv2)/(2k T)] (3) during the time τ should be equal to the volume occu- B 2πk T − (cid:18) B (cid:19) pied by just one colliding pair of molecules. And then is the Maxwell–Boltzmann distribution for the relative Kqu τ = V/(N(N 1)/2). The rate equation for the × − velocitiesforagiventemperatureandk istheMaxwell– number of molecule N is then given by B Boltzmann constant. The total quenching rate coefficient of a pair of molecules is Kqu = Kqu . To avoid confusion, we will also write theLc,MorLrespLo,MndL- dN = 2Kqu N(N −1)/2. (6) P dt − × V ing rate equation for collisions between distinguishable and indistinguishable molecules. First, we consider collisions between two distinguishable molecules in quantum If n=N/V and N(N 1) N2, then − ≈ states a and b (∆=1 in Eq. (1)). During a time dt=τ, where τ is the time of a quenching collisional event, the dn = Kqu n2. (7) number of molecules N lost in each collision is one and a dt − × the number of molecules N lost in each collision is one. b Then dN /dt = 1/τ and dN /dt = 1/τ. The volume a b − − per colliding pairs of molecules is V/(N N ), where V a b stands for the volume of the gas. During the time τ, the B. Quantum threshold model quenching collisional event is associated with a volume <σqu v > τ =Kqu τ. Bydefinitionofτ,thisvolume × × × shouldbe equaltothe oneoccupiedbyjust onecolliding We consider the case of two identical ultracold pair of molecules. Then we get Kqu τ = V/(N N ). fermionic polar molecules, as has been achieved very re- a b The rate equation for the number of m×olecule N or N centlyforKRbdimers[2,28]intheirro-vibronic(1Σ,v = a b is then given by 0,N = 0) ground state. Under these circumstances, be- cause of Fermi exchange symmetry, the relative orbital dN N N a,b = Kqu a b. (4) angularmomentumquantumnumberLbetweenthe two dt − × V molecules must take odd values L = 1,3,5,7.... These If na = Na/V and nb = Nb/V are the densities of molecules are polar molecules and can be controlled by molecule a and b in the gas, then anelectric field . In the usualbasis set of partialwaves E dna,b = Kqu nanb. (5) |mLoMleLcui,letshienlaonpgre-rsaenncgeeobfeahnaveiloerctroifctfiweoldcioslgliodvinergnepdolbayr dt − × 3 an interaction potential matrix whose elements are laws, the QT quenching tunneling probability below the barrier can be written as LM V(R)L′M′ = h L| | Li (cid:26)~2L2(µLR+2 1) − RC66(cid:27)δL,L′δML,ML′ |TLqu,ML|2 =(cid:18)EVbc(cid:19)L+1/2. (10) C (L,L′;M ) 3 L − R3 δML,ML′ (8) Consequently, the quenching cross section and rate coefficient are approximated by whereRdenotesthedistancebetweenthetwomolecules. Tfeomhreenthtdseiargceoopnlrlaiedlsiennegtlemmcoeounlpetscliunlrgeessprbaenesetdwntetehneeffetnhcotenimv-ed.iapgoTotnheanelticaeolls-- σLqu,ML = 2µ~V2Lπ+21 EcL−21 ×∆ b seuffiCmc3ie/ednRt3toiCsb6teheiisstoetrtrmhoepciocvrraiennsptohdneerdpinrWgesateoanlttshterceeoaleetfficmtcreiinecntd.t,ipToalhese-- KLqu,ML = 2µ~32VπbL+12 <EcL >×∆ (11) − dipole interaction expressed in the partial wave ba- p sis set LM V (R,θ,ϕ)L′M′ between two polar- for Ec <Vb. The QT model has the simple and intuitive ized mohleculLes| didn the el|ectricLifield direction, with advantageofshowinghowthe crosssectionsandrateco- V (R,θ,ϕ)=d2(1 3cos2θ)/(4πε R3),whered=d( ) efficientsscalewiththeheightoftheentrancecentrifugal dd 0 is the inducedelectr−icdipole moment, andθ,ϕ represeEnt barrier. For Ec Vb, it is easy to find the correspond- ≥ ing expression of the cross section in Eq. (1) by setting the relative orientation between the molecules. In the basis set of partial waves, C3 takes the form |Tqu|2 = 1. The cross section σLqu,ML will reach the unitarity limit at E V . It is also easy to find the cor- c b ≥ d2 responding expression of the rate coefficient in Eq. (2). C (L,L′;M )=α(L,L′;M ) 3 L L 4πε The QT model is general for any collision between two 0 particles provided that there is a barrier in the entrance =2 ( 1)ML √2L+1 √2L′+1 − collision channel and that chemical reactions occur with L 2 L′ L 2 L′ d2 near unit probability at short range. The only informa- . (9) (cid:18) 0 0 0 (cid:19) (cid:18)−ML 0 ML′ (cid:19) 4πε0 tion on short range chemistry is that chemical reactions occur at full and unit probability and the only informa- The large bracket symbols denote the usual 3 j coeffi- − tiononlongrangephysicsisprovidedbytheheightofthe cients. Thecoefficientαisintroducedtosimplify further entrance barrier V . The QT model describes the back- notations. The combination between repulsive and at- b ground scattering process, it does not take into account tractivetermsintheeffectivepotentials(diagonalterms) scattering resonances. Note that the model will not be ofEq.(8)generateapotentialbarrierofheightV which b appropriate in the present form for barrierless (s wave) isplottedschematicallyinFig.1. The heightofthis bar- − collisions since V = 0. For this particular type of colli- rier plays a crucial role as it can prevent the molecule b sions that do not possess a centrifugal barrier, the QDT from accessing the short range region where reactive theory can be usefully applied [22, 32, 33]. The present chemistry occurs. form of the QT model does not take into account the The quantum threshold (QT) model consists of two anisotropy of the intermolecular potential at intermedi- conditions. First, for E <V , we use the Bethe–Wigner c b aterangeand/ortheelectronicandnuclearspinstructure threshold laws [29, 30] for ultracold scattering. Second, of the molecular complex but remains suitable as far as weusetheclassicalcapturemodel(Langevinmodel)[18] theentrancecentrifugalbarriertakesplaceatlongrange. to estimate the probability of quenching for E V . A c ≥ b The QT model will have to be modified if longer range classical capture model is indiferent to collision energies interactions takes place. For example, collisions between E < V since the barrier prevents the molecules from c b N = 0 and N = 1 polar molecules can have long range coming close together. In real-life quantum scattering, interactions between hyperfine states due to dipolar and collisionsdooccurattheseenergiesduetoquantumtun- hyperfine couplings [28, 34]. However, for collisions be- neling, and they are the ones relevant to ultracold colli- tween rotationless N = 0 polar molecules, the hyperfine sions. Moreover,collisions in this energy regime are dic- couplings are weak and the QT model can be applied tated by the the Bethe–Wigner quantumthresholdlaws. without further modifications. Forquenchingcollisions,the thresholdlaws[29–31]state that Tqu 2 EL+12. For E V , a classical capture | L,ML| ∝ c c ≥ b modelwillguaranteetodeliverthemoleculepairtosmall C. Rates in zero electric field values of R, where chemical reactions are likely to occur with unit probability (see Fig. 1). Following this classical argument, we will assume that when E V , the In the absence of an electric field in Eq. (8), the long c b quenching probability reaches unitarity Tqu 2≥= 1. Us- rangepotentialreducestoadiagonalterminthebasisset | | ingthisassumptiontogetherwiththequantumthreshold of partial waves. The position and height of the barrier 4 K(2S) + K (3Σ +,v=1,N=0) temperatures when L=1, we get 2 u 10-11 L = 1 π 313µ3C3 1/4 L = 2 Kqu = 6 k T ∆. (13) L = 3 L=1,ML 8 ~10 B × 10-12 L = 4 (cid:18) (cid:19) L = 5 In Eq. (11), we used the fact that < E >= 3k T/2 in 2) c B m L=1 three dimensions. Note that to get the overall contribu- u (c10-13 tion for a given L, we have to multiply Eq. (13) by the q σ L=2 degeneracyfactor (2L+1) correspondingto all values of M . We canget similar expressionsfor any partialwave 10-14 solid: full quantum L=3 L.L dashed: QT L=4 dotted: QT (p=0.34) Totestthe validityofthemodel, wecompareinFig. 2 fit L=5 thequenchingcrosssectionsof39K(2S)+39K (3Σ+,v = 10-15 2 u 10-9 10-8 10-7 10-6 10-5 10-4 10-3 10-2 1,N = 0) collisions as a function of the collision en- E (K) c ergy for the partial waves L = 1 5: (i) calculated in − Ref. [10] with a full quantum time-independent close- FIG. 2: (Color online) Quenching cross section of 39K + coupling calculation basedon hypersphericaldemocratic 39K asafunctionofthecollision energyforthepartialwave 2 coordinates [35] and the full potential energy surface of L=1−5: (i)calculatedwithafullquantumcalculation(solid K (solid lines) (ii) using the simple QT model (dashed lines), reproduced from Ref. [10] (ii) using the QT model 3 lines) with a value of C = 9050 a.u. given in Ref. [10] (dashed lines) (iii) fitting the QT model (dotted line), using 6 (1 a.u. = 1 E a6 where E is the Hartree energy and p=0.34 in Eq.(14). h 0 h a is the Bohr radius). In this example, the QT model 0 provides an upper limit to the cross sections. This is K(2S) + K(3Σ +,v=1,N=0) due to the factthat the quenchingcrosssectiondoesnot 2 u reach a maximum value at the height of the barrier V , 100 b p=0.34 but rather at somewhat higher energy, say γ V , with b 10-1 γ > 1 (see Ref. [10]). For all partial waves,×there is a 10-2 worse agreement for collision energies in the vicinity of 10-3 the height of the barrier where the passage from the ultralow regime to the unitarity limit is smoother than for qu2|10-4 theQTmodel. ThissmootherpassageisvisibleinFig.3 |T10-5 for the full quantum L = 1 quenching probability com- 10-6 pared to the QT model, which has a sharp corner in the L=1: full quantum vicinity of V . 10-7 L=1: QT To accounbt for more flexibility in the QT model, one 10-8 L=1: QTfit Vb γ Vb (γ=2.06) canreplaceV in Eq.(10) by γ V (γ >1),and use the b b 10-9 coefficient γ as a fitting param×eter to reproduce either 10-9 10-8 10-7 10-6 10-5 10-4 10-3 10-2 E (K) fullquantumcalculationsorexperimentalobserveddata. c Alternatively,wecancorrecttheQTquenchingtunneling FIG.3: (Coloronline) Quenchingprobabilityof39K+39K probability with an overallfactor p, 2 as a function of the collision energy for thepartial wave L= 1: comparison between the full quantum calculation (solid E L+1/2 Tqu 2 = p c (14) line) and the QT model (dashed line). The fitted QT model | L,ML|fit × V appearsasadottedline. TheheightofthebarrierVb andthe (cid:18) b(cid:19) corrected height γ×Vb (γ =2.06) appear as vertical lines. with p = γ−(L+1/2). p < 1 can be interpreted as the quenching probability reached at the height of the barrier V in the QT model, rather than the rough full b are given by unit probability (p = 1). As an example, we find that γ 2.06 reproduces the quantum L = 1 partial wave R = 6µC6 1/4 cro≈ss section for 39K + 39K2 (dotted line in Fig. 2 and b ~2L(L+1) Fig.3). Thisyieldsamaximumquenchingprobabilityof (cid:18) (cid:19) p = γ−3/2 0.34 instead of 1. In other words, the QT ~2L(L+1) 3 1/2 model is on≈ly a factor of p−1 2.96 higher than the full Vb = (cid:0) 54µ3C6 (cid:1) ! . (12) quantumcalculationfor39K+≈39K2collisionsatultralow energies. WecaninsertEq.(12)inEq.(11)togetanalyticalforms Giventhe factthatfullquantumcalculationsarecom- ofthequenchingcrosssectionorratecoefficient. Fortwo putationally demanding [8–14] and impossible at the indistinguishable fermionic polar molecules at ultracold present time for alkali molecule-molecule collisions, the 5 accuracy of the QT model is satisfactory and can be a KRb(1Σ,v=0,N=0) + KRb(1Σ,v=0,N=0) ; L=1 ; T=350 nK quick and powerful alternative way to estimate ordersof magnitudeforthescatteringobservables. Besides,agree- total diabatic total adiabatic ment between the QT model with experimental data or M=0: diabatic L full quantum calculations is expected to be satisfactory -1s)10-10 MML==0+:1 a adniadb -a1t:i cdiabatic ftohratcoslhliosriotnrsaningveoqlvuienngchailnkaglicosuppecliinesg,sbweiclaludsoemiitniastelikaenldy -1cule MLL=+1 and -1: adiabatic e lead to high quenching probability in the region where mol10-11 the two particles are close together [16]. Very recently, 3m total Euaqt.io(1n3o)foufltthraecQolTd cmhoemdeilcahlaqsubeenecnhianpgplriaetdefoofrctohleliseivoanls- quK (c|MMLL=|=01 10-12 of two 40K87Rb molecules in the same internal quantum state,andprovidedgoodagreementwiththe experimental data [32]. 0 0.05 0.1 0.15 0.2 d (D) D. Rates in non-zero electric field FIG.5: (Color online) Quenchingratecoefficients oftwo indistinguishable fermionic polar 40K87Rbmolecules asa function of the induced electric dipole moment for L = 1 and 1. Numerical expressions for a temperature of T = 350 nK (black curves). The rates havebeen calculated using thebarrier heights of Fig. 4. The red lines represent the L = 1,ML = 0 partial wave contri- KRb(1Σ,v=0,N=0) + KRb(1Σ,v=0,N=0) ; L=1 bution. The blue lines represent the sum of L = 1,ML = 1 10-3 and L=1,ML =−1partial wavecontributions. Thedashed linesrepresent theratescalculated with thediabaticbarriers 10-4 |ML|=1 MM|MLL==|=001:: :da didaiiabababataicttiicc wdehtialeil)t.heTshoelitdotlianl,esMwLith=t0heanaddia|MbaLt|ic=ba1rrciuerrsve(sseheavteexbtefeonr L indicated in theleft hand side. |M|=1:adiabatic 10-5 M=0 L K) L (b V 10-6 of C = 16130 a.u. [36]. We plot in Fig. 4 the heights 6 <E>=525 nK c of the diabatic (dashed lines) and adiabatic (solid lines) 10-7 barriers for the quantum numbers ML = 0 (red curves) and M =1 (blue curves). The adiabatic barriershave L | | 10-80 0.1 0.2 0.3 0.4 0.5 Ebeqe.n(8c)a.lcTuhlaeteeffdecutsionfgthfievceoupparlitniaglscwaanvebseLcle=arly1s−ee9niinn d (D) this figure by comparingdiabatic and adiabatic barriers. Especially for the M = 1 case for d 0.16 D (1 D F(sIoGli.d4l:ine(sC)obloarrroinerlinhee)igDhtiasbVatdi,ca(adsaashfeudnclitnioens)oafntdheadiniadbuacteidc = 1 Debye = 3.33|61L0−|30 C.m), couplin≈gs with higher b partial waves make the adiabatic barrier decrease as the dipole moment d for the partial waves L = 1,ML = 0 (red curves) and L=1,|ML|=1 (bluecurves). dipole increases while the diabatic barrier continues to increase. Using these heights of the barriers, we use Eq. (11) In the presence ofanelectric field in Eq.(8), the long- to plot in Fig. 5 the total quenching rate coefficients range interaction potential matrix is no more diagonal (blackcurves)asafunctionofdfortwoindistinguishable andcouplingsbetweendifferentvaluesofLoccur. M is fermionicpolar40K87Rb moleculesinthe samequantum L stillagoodquantumnumber. Afirstapproximation(di- stateforL=1andatatypicalexperimentaltemperature abatic approximation) consists of neglecting these cou- of T =350 nK [32]. For T =350 nK, the mean collision plings and using only the diagonal elements of the di- energy < E >= 3k T/2 = 525 nK, and the maximum c B abatic matrix directly. Then one can find numerically dipole moment for which V <525 nK (that is for which b for which R the centrifugal barriers are maximum and Eq. (11) does not apply anymore) is around d 0.24 D ≈ evaluate the height of the diabatic barriers Vd. This is (see Fig. 4). The dashed curves correspond to rates cal- b repeated for all values of the induced dipole moment d. culated with the diabatic approximation while the solid A second approximation(adiabatic approximation)is to curves correspond to rates calculated with the adiabatic diagonalize this matrix (including the non-diagonal cou- approximation. The M = 0 contribution is plotted in L pling terms) for each R and again find the maximum of red and the contribution of M = +1 and M = 1 is L L − the centrifugalbarriersto getthe heightof the adiabatic plotted in blue. The rates highly reflect the behavior of barriers Va. As an example, we compute these barrier the centrifugalbarriers in the entrance collision channel. b heights for 40K87Rb 40K87Rb collisions, using a value When the barrier increases with the dipole, it prevents − 6 KRb(1Σ,v=0,N=0) + KRb(1Σ,v=0,N=0) ; L=1 ; T=350 nK in Eq. (8). In this case, the position and height of the V barrier are given by III analytical limits (I,II,III,IV,V) ML=0: diabatic d6 3µC M=0: adiabatic 3 -1-1ules)10-10 MttooLLttaa=ll+ da1id aaibanabdta ic-ti1c: diabatic Rb = ~~2L2L(L(L++11)) 3 molec10-11 da Vb = (cid:0) 54µ3C32 (cid:1) ∝d−4. (16) 3m total quK (c|MM10LL=|-=1021 ccoonnsstt.. dr IIIVI Trehgeiopnowsihtieorne Eofq.th(1e5b)airsrsieartiisnfieEdq..T(h16is)hhaapspteonsbfeorinsutihte- ably large dipole moments, d>d where d-6 a 0 0.05 0.1 0.15 0.2 ~2L(L+1) 3 C (4πε )4 1/8 d (D) d = 6 0 . (17) a 27µ3α4 (cid:0) (cid:1) ! FIG. 6: (Color online) Same as Fig. 5 but we use the analytical expressions for the rates (see text for detail). The Thesubscriptastandsfortheattractiveinteraction. For total, ML = 0 and |ML| = 1 curves have been indicated in twoindistinguishablefermionicpolar40K87Rbmolecules, thelefthandside. Theindividualanalyticalcurveshavebeen and for L = 1 and M = 0, α(1,1;0) = 4/5 and we get L indicated in the right hand side by roman numbers. d = 0.103 D. The threshold laws for quenching colli- a sions in an electric field are the same as in the zero-field limit. Consequently, the quenching cross sections and rate coefficients behaves as in Eq. (11) except that V is themoleculesfromgettingclosetogetherandthequench- b given now by Eq. (16) and varies with d. We can insert ingratesdecreases. Whenthebarrierdecreases,thetun- Eq. (16) in Eq. (11) to get the corresponding analytical neling probability is increased allowing the molecules to expressions. ForapartialwaveL>0,thequenchingrate get close together, and the quenching rates increases. scalesasd4(L+21). Fortwoindistinguishablefermionicpo- lar molecules at ultracold temperatures when L=1 and M =0, we get L 2. Analytical expressions Kqu = L=1,ML=0 In order to have an intuitive sense of how the chemi- 3π 69µ6 1/2 d2 3 calquenchingratescaleswiththeinduceddipolemoment k T ∆. (18) (andtheelectricfield),weevaluateanalyticalexpressions 8 (cid:18)56~14(cid:19) (cid:18)4πε0(cid:19) B × of the barriers and the rates as it has been done in the Thus the L = 1,M = 0 quenching rate increases as previous section for a zero electric field. The analyti- L cal expression of the height of the diabatic barrier Vd is d6. This is a more rapid dependence on dipole moment b than for purely long-range dipolar relaxation in dipolar complicatedby the occurrenceoftwodistinctlong-range gases [4]. potentialsinthediagonalmatrixtermofEq.(8). Wecir- For negative C coefficients in Eq. (9), C /R3 is re- cumventthisdifficultybylookinginthetwolimitswhere 3 3 − pulsive in Eq. (8). For L= 1 partial waves for example, onedominatesovertheother. Forsmallelectricfields,we this occurs when M = 1, which favorsa repulsive ori- usethe zeroelectricfieldlimitdiscussedinthepreceding L ± entation of dipoles. We consider section by setting C = 0. For larger electric fields we 3 ignore the C coefficient in Eq. (8) if the electric dipole- 6 C ~2L(L+1) dipole interaction is attractive (positive C3). We ignore 3 (19) thecentrifugalterminEq.(8)iftheelectricdipole-dipole (cid:12)R3(cid:12)≫(cid:12) 2µR2 (cid:12) interaction is repulsive (negative C ). These two cases (cid:12) (cid:12) (cid:12) (cid:12) 3 (cid:12) (cid:12) (cid:12) (cid:12) are discussed below. In between, to accommodate the in Eq. (8). The l(cid:12)ong(cid:12)-rang(cid:12)e potential a(cid:12)gain experiences a barrier, but now it is generated by the balance between transition between the low-field and high-field limit, we the repulsive dipole potential at large R, and the attrac- will simply add the rate coefficients derived in the two tive van der Waals potential at somewhat smaller R. In limiting cases. thiscase,thepositionandheightofthisbarrieraregiven For positive C coefficients in Eq. (9), C /R3 is at- 3 − 3 by tractive in Eq. (8). For L=1 partialwavesfor example, this occurs when M = 0, which favors an attractive L 1/3 2C orientation of dipoles. We consider R = 6 b C (cid:18)| 3|(cid:19) C3 C6 (15) V = |C3|2 d4. (20) R3 ≫ R6 b 4C ∝ (cid:12) (cid:12) (cid:12) (cid:12) 6 (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) 7 Forthisapproximationtohold,thepositionofthebarrier analyticalsumI+II+V+IVisrepresentedasablacksolid in Eq. (20) has to be in the region where Eq. (19) is line. Neglecting the d−6 contributionatlargerd, the an- satisfied. This requires that d>d where alytical p wave quenching rate (taking into account the r − adiabatic barriers)is given by the simple expression ~2L(L+1) 3 C (4πε )4 1/8 6 0 dr = (cid:0) 4µ(cid:1)3α4 ! . (21) KLqu=1 = π8 p1 317~µ130C63 1/4 (cid:26) (cid:18) (cid:19) The subscript r stands for the repulsive interaction. For 29311µ6 1/2 d6 twoindistinguishablefermionicpolar40K87Rbmolecules, +1.51p k T ∆. (23) 2 56~14 (4πε )3 B × and for L = 1 and ML = 1 or ML = 1, α(1,1; 1) = (cid:18) (cid:19) 0 (cid:27) − ± 2/5andwe getdr =0.186D. We canreplace Eq.(20)in p1 (p2)isthequenchingprobabilityreachedattheheight Eq. (11) to get the correspondinganalyticalexpressions. of the barrier in the QT model for the zero (non-zero) For a partial wave L > 0, the quenching processes scale electric field regime. The QT model assumes that p = as d−4(L+12). For two indistinguishable fermionic polar p = 1 but become fitting parameters (p ,p < 1) w1hen 2 1 2 molecules at ultracold temperatures when L = 1 and comparedwithfullquantumcalculationsorexperimental |ML|=1, we get data. The limiting value da = 0.103 D (dr = 0.186 D), where the d6 (d−6) behavior begins, has also been indi- Kqu = cated with an arrow. It turns out that the total rates L=1,|ML|=1 for L = 1 calculated analytically (for both the use of 3π 50~34 C6 3/2 d2 −3 diabatic and adiabatic barriers) are very similar to the k T ∆. (22) 8 µ ! (cid:18)4πε0(cid:19) B × numericalonesofFig.5(10%differenceatmost,around d ). However, the sub-components L = 1,M = 0 and a L The L=1,|ML|=1 quenching rate decreases as d−6 as L = 1,|ML| = 1 have different behaviors. For example the electric field grows. the numerical L = 1,ML = 0 (L = 1, ML = 1) com- | | These analytical expressions use the diabatic barri- ponent startsto increase(decrease)at earlierdipole mo- ers. If we consider that at large d, the total rate is ment (typically at 0.02 D) than their analytical analogs mostly given by the M = 0 contribution (we neglect (typically after 0.06 D). The use of the simple analyt- L the M = 1 contributions at large d), one can have an ical expressions (using the diabatic or adiabatic barri- L | | analytical expression using the adiabatic barrier. If we ers) can be useful to estimate the total rate coefficients, take into account the couplings between L = 1,M = 0 while the numerical ones are prefered to estimate the L and L = 3,ML = 0, we can diagonalize analytically the L=1,ML =0 and L=1,|ML|=1 individual rates. 2 2 matrix in Eq. (8). It can be shown that for each × dipolemomentd,thecouplingwithL=3,M =0lower L the diabatic barrier of L=1,M =0 by a factor of 0.76 III. PROSPECTS FOR COLLISIONS IN TWO L DIMENSIONS at the position of the barrier, to give rise to the adiabatic barrier. Inserting this correction of the barrier in Eq. (11), this yields a correction of 0.76−3/2 1.51 for In three dimensional collisions, the quenching loss is Kqu . The difference between diabatic≈and adia- largely due to incident partial waves with angular mo- L=1,ML=0 baticcalculationscanbealreadyseeninFig.4andFig.5 mentum projection ML = 0, emphasizing head-to-tail for the numerical barriers at large dipole moment. orientations of pairs of dipoles. These are the kind of In Fig. 6 the black curve corresponds to the total collisionsthatarelargelysuppressedintrapswithapan- quenching rate coefficient as a function of d for two in- cakegeometry,withthedipolepolarizationaxisorthogo- distinguishablefermionicpolar40K87Rbmoleculesinthe naltotheplaneofthepancake[37]. Ifthesecollisionscan same quantum state for L = 1 and at a temperature of be removed, then it is likely that increasing the electric T = 350 nK. The analytical expressions I, II, III, IV, field will suppress quenching collisions, making evapora- V (green thin lines) correspond respectively to Eq. (13), tivecoolingpossible. Ifweassumeanidealpancaketrap 2 Eq. (13), Eq. (18), 2 Eq. (22), 1.51 Eq. (18). thatconfinestheparticlestomovestrictlyonaplane,one × × × The curvesIII andIVareforthe diabaticbarriers,while can apply the present model to estimate the behavior of curve V is to account for the adiabatic barrier. The red the quenching processes. We assume that the molecules dashed line (I+III) represents the L=1,M =0 partial are polarized along the electric field axis, perpendicular L wavecontributionforthediabaticbarrierswhiletheblue tothetwodimensionalplane. Inthiscase,thelongrange dashedline(II+IV)representsthesumofL=1,M =1 potential is given by L and L = 1,ML = −1 partial wave contributions for the ~2(M 2 1/4) C6 d2 diabatic barriers. The analytical sum I+II+III+IV is V(ρ)= | | − + (24) 2µρ2 − ρ6 4πε ρ3 represented as a black dashed line. To account for the 0 adiabatic barriers we assume that the correction for the where ρ stands for the distance between 2 particles in total rate comes only from the L = 1,M = 0 partial a two dimensional plane, M stands for the angular mo- L wave, and we replace I+III by I+V (red solid line). The mentum projection on the electric field axis. The last 8 KRb(1Σ,v=0,N=0) + KRb(1Σ,v=0,N=0) ; |M|=1 trifugal term in Eq. (8) leads to the threshold laws in 10-2 Eq. (10), the replacement L(L+1) M 2 1/4 (that → | | − analytical d4 is L M 1/2) in Eq. (10) leads to numerical →| |− 10-3 E |M| Tqu 2 = c (25) | M| V (cid:18) b(cid:19) K) (b10-4 where Vb denotes the height of the centrifugal barrier in V two dimensions. This resultrequires that the centrifugal potential is repulsive, i.e., that M > 0. For M = 0 | | 10-5 the threshold law exhibits instead a logarithmic diver- const. gence [31]. In two dimensions, quenching cross sections and rate 10-6 coefficients have respectivelly units of length and length 0 0.1 0.2 0.3 0.4 0.5 d (D) squared per unit of time, and are given by [39] ~ FIG. 7: (Color online) Barrier heights as a function of the σqu = Tqu 2 ∆ induceddipolemomentdforthepartialwaves|M|=1intwo M √2µEc | M| × dimensions. The green thin curves represent the analytical ~ Eq.(28)(constant)andEq.(29)(d4). Thedashedblackcurve KMqu = µ |TMqu|2×∆. (26) isthesumofthem. Thesolid blackcurveistheheightofthe barrier in Eq. (24). Within this model, it follows that the quenching cross section and rate coefficient for M >0 are given by | | ~ KRb(1Σ,v=0,N=0) + KRb(1Σ,v=0,N=0) ; |M|=1 ; T=350 nK σqu = E|M|−1/2 ∆ M √2µV|M| c × b analytical limits ~ total analytical Kqu = <E|M| > ∆. (27) -1s)10-5 total numerical M µVb|M| c × -1ecule const. d2D VI TfohuendeninerRgyef.d[e4p0e].ndInenEceq.i(s27in),agreement with the one ol10-6 m 2m (~2(M 2 1/4))3 1/2 c V = | | − (28) quK ( b (cid:18) 54µ3C6 (cid:19) 10-7 d-4 for the zero-electric field regime and VII 1 d2 2 0 0.05 0.1 0.15 0.2 V = (29) d (D) b 4C 4πε 6 (cid:18) 0(cid:19) forthenon-zeroelectricfieldregime. Theheightofthese FIG.8: (Coloronline)Twodimensionalquenchingratecoef- ficient(blackcurves)oftwoindistinguishablefermionicpolar barriers has been reported in Fig. 7 (green thin lines). 40K87Rbmoleculesasafunctionoftheinducedelectricdipole These results imply that for M =1 the quenching pro- | | moment for the M = 1 and M = −1 components at a tem- cesseswithinthismodelwillbeindependentofthedipole perature of T =350 nK. The dashed lines represent the rate moment in the zero electric field regime, where usinganalyticalexpressionswhilethesolidlinerepresentsthe rateusingthenumerical expression (seetextfordetail). The 27µC 1/2 individual analytical curves have been indicated in the right K|qMu|=1 = ~4 6 kBT ×∆ (30) hand side by roman numbers. (cid:18) (cid:19) andwillscaleasd−4 inthenon-zeroelectricfieldregime, where term comes from the repulsive dipole-dipole interaction Kqu = 4~C6 d2 −2 k T ∆. (31) whenthe dipolesarepointingalongthe electricfieldand |M|=1 µ 4πε B × (cid:18) 0(cid:19) approacheachotherside byside. The heightofthis bar- We use the fact that < E >= k T in two dimensions. rier has been plotted as a function of d in Fig. 7 (black c B The non-zero electric field regime is reached when d > solid line). At ultralow energy and large molecular sep- d where aration, the Bethe–Wigner laws for quenching processes 2D d1/epRe2nd[31o,nl3y8o].ntThheelornepg-urlasnivgeecreenpturlisfiuvgeacletnetrrmifusgaarletedrimf- d = ~2(|M|2−1/4) 3 C6(4πε0)4 1/8 (32) ferent in Eq. (8) and Eq. (24). As the repulsive cen- 2D (cid:0) 4µ3(cid:1) ! 9 For two indistinguishable fermionic polar 40K87Rb determine analyticalexpressionsofthe chemicalquench- molecules, and for M =1, we get d =0.081 D. ing cross section and rate coefficient as a function of 2D | | The behavior of the quenching rate (black lines) is the collision energy or the temperature. We also pro- shownin Fig. 8 for two indistinguishable fermionic polar vide an estimate as a function of the induced electric 40K87Rb molecules as a function of the induced electric dipole moment d in the presence of an electric field. We dipole momentfor M =1 and M = 1 componentsat a found that the quenching rates of two ultracold indis- − temperature of T = 350 nK. The dashed line represents tinguishablefermionicpolarmoleculesgrowsasthesixth the analytical rate which is the sum of the analytical powerofd. Forweakerelectricfield,quenchingprocesses expression VI corresponding to 2 Eq. (30) and analyt- are independent of the induced electric dipole moment. × ical expression VII corresponding to 2 Eq. (31). The Prospects for two dimensional collisions have been dis- × solidlinerepresentstherateusingthegeneralexpression cussed using this model and we predict that the quench- Eq. (27) and the numerical height of the barrier calcu- ing rate will decrease as the inverse of the fourth power lated in Eq. (24). The limiting value d = 0.081 D, of d. This fact may be useful for efficient evaporative 2D where the d−4 behavior for the quenching rate begins, cooling of polar molecules. This model provides a gen- has also been indicated with an arrow. The difference eral and comprehensive picture of ultracold collisions in betweenthe numericalcalculationand the analyticalex- electric fields. Preliminary data suggest that this model pression reflects the difference in the calculation of the gives good agreement with experimental chemical rates height of the barrier, already seen in Fig. 7. The nu- for three dimensional collisions in an electric field [42]. merical calculation is more exact, while the other is analytical. However, at large d, the numerical rate tends to the analytical d−4 behavior. The quenching rate decreases rapidly as the dipole moment increases and this may be promising for efficient evaporative cooling of polar molecules since the elastic rate is expected to grow V. ACKNOWLEDGEMENTS with increasing dipole moment [41]. We acknowledge the financial support of NIST, the IV. CONCLUSION NSF, and an AFOSR MURI grant. We thank K.-K. Ni, S. Ospelkaus, D. Wang, M. H. G. de Miranda, B. We have proposed a simple model which combines Neyenhuis, P. S. Julienne, J. Ye, and D. S. Jin for help- quantumthresholdlawsandaclassicalcapturemodelto ful discussions. [1] J. M. Sage, S. Sainis, T. Bergeman, and D. DeMille, J.-M. Launay,Phys. Rev.Lett. 94, 200402 (2005). Phys.Rev.Lett. 94, 203001 (2005). [13] M. Lara, J. L. Bohn, D. Potter, P. Soldán and J. M. [2] K.-K. Ni, S. Ospelkaus, M. H. G. de Miranda, A. Pe’er, Hutson, Phys.Rev. Lett.97, 183201 (2006). B.Neyenhuis,J.J.Zirbel,S.Kotochigova,P.S.Julienne, [14] G.Quéméner,J.-M.Launay,andP.Honvault,Phys.Rev. D.S. Jin, and J. Ye,Science 322, 231 (2008). A 75, 050701(R) (2007). [3] J. Deiglmayr, A. Grochola, M. Repp, K. Mörtlbauer, C. [15] J.M.HutsonandP.Soldán,Int.Rev.Phys.chem.26,1 Glu¨ck, J. Lange, O. Dulieu, R. Wester, and M. Wei- (2007). demu¨ller, Phys.Rev. Lett.101, 133004 (2008). [16] G. Quéméner, N. Balakrishnan, and A. Dalgarno, in [4] S.Hensler et al., Appl.Phys.B 77, 765 (2003). Cold Molecules: Theory, Experiment, Applications, R. [5] J.L.Bohn,M.Cavagnero,andC.Ticknor,NewJ.Phys. V.Krems,W.C.Stwalley,andB.Friedrich(Eds.),CRC 11, 055039 (2009). Press (2009). [6] A. V. Avdeenkov and J. L. Bohn, Phys. Rev. A 66, [17] G. Quéméner, N. Balakrishnan, and B. K. Kendrick, 052718 (2002). Phys. Rev.A 79, 022703 (2009). [7] N. Balakrishnan and A. Dalgarno, Chem. Phys. Lett. [18] P. Langevin, Ann.Chim. Phys.5, 245 (1905). 341, 652 (2001). [19] P. Julienne and F. H. Mies, J. Opt. Soc. Am. B 6, 2257 [8] P.Soldán,M.T.Cvitaˇs,J.M.Hutson,P.Honvault,and (1989). J.-M. Launay,Phys. Rev.Lett. 89, 153201 (2002). [20] J. P. Burke, Jr., C. H. Greene, and J. L. Bohn, Phys. [9] G.Quéméner,P.Honvault,andJ.-M.Launay,Eur.Phys. Rev. Lett.81, 3355 (1998). J. D 30, 201 (2004). [21] F. H. Mies and M. Raoult, Phys. Rev. A 62, 012708 [10] G. Quéméner, P. Honvault, J.-M. Launay, P. Soldán, D. (2000). E. Potter, and J. M. Hutson, Phys. Rev. A 71, 032722 [22] P. S. Julienne, Faraday Discuss. 142, 361 (2009). (2005). [23] C.Orzel,M.Walhout,U.Sterr,P.S.Julienne,andS.L. [11] M.T.Cvitaˇs,P.Soldán,J.M.Hutson,P.Honvault,and Rolston, Phys.Rev. A 59, 1926 (1999). J.-M. Launay,Phys. Rev.Lett. 94, 033201 (2005). [24] E. R. Hudson, N. B. Gilfoy, S. Kotochigova, J. M. Sage, [12] M.T.Cvitaˇs,P.Soldán,J.M.Hutson,P.Honvault,and and D.DeMille, Phys.Rev.Lett. 100, 203201 (2008). 10 [25] E. R.Hudson et al., Phys.Rev.A 73, 063404 (2006). ted (2009). [26] T. V. Tscherbul and R. V. Krems, J. Chem. Phys. 129, [34] J. Aldegunde, B. A. Rivington, P. S. Z˙uchowski, and J. 034112 (2008). M. Hutson,Phys. Rev.A 78, 033434 (2008). [27] J. P. Burke, Jr., PhD Thesis, Univer- [35] J.-M. Launay and M. Le Dourneuf, Chem. Phys. Lett. sity of Colorado (1999), available online at: 163, 178 (1989). http://jilawww.colorado.edu/pubs/thesis/burke. [36] S. Kotochigova, privatecommunication (2009). [28] S. Ospelkaus, K.-K. Ni, G. Quéméner, B. Neyenhuis, D. [37] A.Micheli,G.Pupillo,H.P.Bu¨chler,andP.Zoller,Phys. Wang, M. H. G. de Miranda, J. L. Bohn, J. Ye, and D. Rev. A 76, 043604 (2007). S.Jin, Phys.Rev.Lett. 104, 030402 (2010). [38] A.R.P.Rau,Comments At.Mol. Phys.14, 285(1984). [29] H.A. Bethe, Phys.Rev. 47, 747 (1935). [39] P. Naidon and P. S. Julienne, Phys. Rev. A 74, 062713 [30] E. P. Wigner, Phys. Rev.73, 1002 (1948). (2006). [31] H.R.Sadeghpouretal.,J.Phys.B:At.Mol.Opt.Phys. [40] Z. Li, S. V. Alyabyshev, and R. V. Krems, Phys. Rev. 33, R93 (2000). Lett. 100, 073202 (2008). [32] S.Ospelkaus,K.-K.Ni,D.Wang, M. H.G.deMiranda, [41] C. Ticknor, Phys. Rev.A 80, 052702 (2009). B. Neyenhuis, G. Quéméner, P. S. Julienne, J. L. Bohn, [42] K.-K. Ni, S. Ospelkaus, D. Wang, G. Quéméner, B. D. S. Jin, and J. Ye, arXiv:0912.3854, Science, in press Neyenhuis,M. H.G.deMiranda, J.L.Bohn, J.Ye,and (2010). D. S. Jin, arXiv:1001.2809, submitted (2010). [33] Z.IdziaszekandP.S.Julienne,arXiv:0912.0370,submit-