Universal stereodynamics of cold atom-molecule collisions in electric fields
UUniversal stereodynamics of cold atom-molecule collisions in electric fields
Timur V. Tscherbul and Jacek K(cid:32)los , Department of Physics, University of Nevada, Reno, NV, 89557, USA Department of Physics, Joint Quantum Institute,University of Maryland College Park, College Park, Maryland, 20742, USA Department of Physics, Temple University, Philadelphia, PA 19122, USA (Dated: February 26, 2021)We use numerically exact quantum dynamics calculations to demonstrate universal stereoselec-tivity of cold collisions of Π molecules with S-state atoms in an external electric field. We showthat cold collisions of OH molecules in their low-field-seeking f -states, whose dipole moments areoriented against the field direction, are much more likely to lead to inelastic scattering than those ofmolecules oriented along the field direction, causing nearly perfect steric asymmetry in the inelasticcollision cross sections. The universal nature of this effect is due to the threshold suppression ofinelastic scattering between the degenerate ± M Stark sublevels of the high-field-seeking e -state,where M is the projection of the total angular momentum of the molecule on the field axis. Abovethe Λ-doublet threshold, the stereodynamics of inelastic atom-molecule collisions can be tuned viaelectric-field-induced resonances, which enable effective control of Ne + OH scattering over the rangeof collision energies achievable in current merged beam experiments. Modern experimental studies of ultracold moleculargases [1] have reached an extraordinary level of controlover molecular degrees of freedom using external electro-magnetic fields [2–4]. In particular, control over the ro-tational motion [5] makes it possible to address a centralquestion of chemical physics concerning the role of therelative orientation of the reactants in determining theoutcome of molecular collisions and chemical reactions[6–15]. The steric effects have been the subject of numer-ous experimental studies in crossed molecular beams [7–13, 16], external field traps [17, 18], and, more recently, inmerged molecular beams [19, 20]. The latter experimentsprobed the stereodynamics of cold HD + D collisions at1 K [19, 20] and observed a dramatic preference for theperpendicular alignment of collision products. The sin-gle partial-wave regime accessed in these experiments isoptimal for studying and controlling collision stereody-namics due to the absence of detrimental averaging overmany partial waves (or impact parameters), which tendsto obscure steric effects [6].Recent quantum scattering calculations revealed theimportant role of single scattering resonances in deter-mining the stereodynamics of cold HD( v = 1 , j = 2) + H collisions [21] and suggested the possibility of tuningshape resonances in cold HD( v = 1 , j = 2) + H col-lisions by aligning the rotational angular momentum ofHD with respect to the initial relative velocity vector [22].Additional calculations explored the stereodynamics ofcold rotationally inelastic He + HD [23] and HCl + H collisions [24] in the presence of overlapping resonancesand identified a universal trend in the sterodynamic pref-erence of state-to-state integral cross sections.Previous theoretical work on steric effects in coldmolecular collisions [21–24] has focused on molecules innondegenerate electronic states of Σ symmetry in the ab-sence of external fields. Open-shell molecular radicalssuch as OH( Π) and NO( Π) are readily controllable byexternal fields due to their quasi-degenerate Λ-doublet levels of the opposite parity [25]. The OH radical wasamong the first molecules cooled and trapped at lowtemperatures [26, 27] and its cold collisional propertieswith rare-gas atoms have been extensively studied [26–31]. External fields orient or align the molecules alonga laboratory-fixed quantization axis [3, 32–35], providingan extra spatial direction for observing novel stereody-namical effects. In addition, external fields are commonlyused to tune the scattering properties of cold atoms andmolecules via Feshbach resonances [36]. A combinationof steric and external field control may thus lead to newand powerful ways to engineer the quantum dynamics ofmolecular collisions at ultralow temperatures.Here, we explore the stereodynamics of cold atom-molecule collisions in an electric field using Ne + OH as arepresentative example (rare gas - OH collisions serve asprototype systems for studying steric effects in molecularcollisions [7–11]). Using rigorous quantum scattering cal-culations, we uncover a universal stereodynamical trend:Collisions of Π molecules initially oriented against thefield direction are much more likely to lead to inelasticscattering than those of molecules oriented along the fielddirection. We also show that the stereodynamics of coldatom-molecule collisions can be controlled by an externalelectric field, and find that such control can be exten-sive even in the multiple partial wave regime, which canbe reached experimentally in merged molecular beams[37, 38]. Our predictions can thus be verified in currentmolecular beam scattering experiments.
Theory . To explore the stereodynamics of cold atom-molecule collisions in an external electric field, we carryout rigorous quantum scattering calculations for thebenchmark collision system Ne + OH parametrized byaccurate ab initio interaction potentials (see the Supple-mental Material [39].) The stereodynamical observablesof interest are encoded in the atom-molecule scattering a r X i v : . [ phy s i c s . a t o m - ph ] F e b amplitude [40, 41] q α → α (cid:48) (ˆ k i , ˆ R ) = 2 π (cid:88) (cid:96),m (cid:96) ,(cid:96) (cid:48) ,m (cid:48) (cid:96) i (cid:96) − (cid:96) (cid:48) Y ∗ (cid:96)m (cid:96) (ˆ k i ) Y (cid:96) (cid:48) m (cid:48) (cid:96) ( ˆ R ) T α(cid:96)m (cid:96) ; α (cid:48) (cid:96) (cid:48) m (cid:48) (cid:96) (1)where k i = k i ˆ k i the incident wavevector, ˆ k i gives thedirection of the incident flux with respect to the space-fixed (SF) Z -axis defined by the direction of the externalelectric field, ˆ R specifies the direction of the scatteredflux, Y lm ( ˆ R ) are the spherical harmonics, (cid:96) and m (cid:96) arethe quantum numbers for the orbital angular momentumand its SF projection, α refers to the internal states of themolecule, and T γ(cid:96)m (cid:96) ; γ (cid:48) (cid:96) (cid:48) m (cid:48) (cid:96) are the transition T -matrixelements.We will assume that the external field is collinear withthe incident relative velocity vector ( Z (cid:107) ˆ k i ) [21–24]which allows us to set ˆ k i = 0 in Eq. (1) to yield [41] q (0) α → α (cid:48) ( ˆ R ) = √ π (cid:88) (cid:96),(cid:96) (cid:48) ,m (cid:48) (cid:96) i (cid:96) − (cid:96) (cid:48) (2 (cid:96) + 1) / Y (cid:96) (cid:48) m (cid:48) (cid:96) ( ˆ R ) T α(cid:96) α (cid:48) (cid:96) (cid:48) m (cid:48) (cid:96) (2)The integral cross section (ICS) corresponding to afixed orientation of the incident flux (ˆ k i = 0) maybe obtained by integrating the differential cross section dσ γ → γ (cid:48) ( ˆ R ) /d Ω = k − γ | q (0) γ → γ (cid:48) ( ˆ R ) | over all angles. Substi-tuting the scattering amplitude from Eq. (2) and per-forming the integration, we obtain [41] σ α → α (cid:48) = πk α (cid:20)(cid:88) (cid:96) (cid:88) (cid:96) (cid:48) m (cid:48) (cid:96) (2 (cid:96) + 1) | T α(cid:96) α (cid:48) (cid:96) (cid:48) m (cid:48) (cid:96) | + (cid:88) (cid:96) (cid:54) = (cid:96) (cid:96) (cid:48) ,m (cid:48) (cid:96) [(2 (cid:96) +1)(2 (cid:96) +1)] / i (cid:96) − (cid:96) T ∗ α(cid:96) α (cid:48) (cid:96) (cid:48) m (cid:48) (cid:96) T α(cid:96) α (cid:48) (cid:96) (cid:48) m (cid:48) (cid:96) (cid:21) . (3)The first term on the right represents the incoherentcontribution to the ICS, which does not depend on thephases of T -matrix elements. The second term is an in-terference term, which originates from fixing the direc-tion of the incident collision flux ˆ k i in Eq. (2), as requiredfor the description of molecular beam stereodynamics ex-periments [21–24]. The “steric” ICS defined by Eq. (3) isnotably different from the conventional state-to-state ICS σ α → α (cid:48) = πk − γ (cid:80) (cid:96),m (cid:96) (cid:80) (cid:96) (cid:48) ,m (cid:48) (cid:96) | T α(cid:96)m (cid:96) ; α (cid:48) (cid:96) (cid:48) m (cid:48) (cid:96) | obtained byaveraging the absolute square of the scattering amplitude(1) over ˆ k i and integrating over ˆ R [40] leading to the dis-appearance of interference terms. We note that the stericICS (3) becomes identical to the conventional ICS in the s -wave limit, where (cid:96) = (cid:96) = 0 [41] and both types ofICS obey the same threshold laws for s -wave scattering[42]. Below, we will omit the prefix “steric” when refer-ring to the ICS (3) unless necessary to avoid confusion.To explore the stereodynamics of cold atom-moleculecollisions in an external electric field, we obtain the T -matrix elements in Eqs. (2)–(3) using numerically exact quantum scattering methodology [43, 44] based on theaccurate ab initio Ne-OH PESs calculated as describedin the Supplemental Material [39]. Unlike previous theo-retical studies [7–11], our calculations explicitly accountfor the effects of external electric fields on quantum dy-namics [43, 44] as required for the proper theoretical de-scription of cold atom-molecule collisions [45, 46].
Results.
Figure 1(a) shows the Stark energy levels ofOH( X Π) in its ground vibronic state. The ground-state J = 3 / M -components ofthe upper f -state ( | M | = 1 / | M | = 3 /
2) increase inenergy with increasing electric field. The OH moleculesresiding in the f states are oriented against the directionof the applied electric field [7]. In contrast, the energy ofthe high-field-seeking Stark sublevels of the lower e -statedecreases with increasing field as their dipole momentsare oriented along the field direction [7].In Fig. 1(b) we show the inelastic ICS σ inel i = (cid:80) k (cid:54) = i σ (0) i → k for the highest-energy low-field seeking Starkstate | i (cid:105) = | f, M = 3 / (cid:105) of OH( J = 3 /
2) as a func-tion of collision energy and electric field. We observetwo pronounced resonance peaks in the collision energy( E coll ) dependence of the ICS near E = 1 kV/cm. Theseresonances are due to the trapping of the collision part-ners behind centrifugal barriers in either the incoming oroutgoing collision channels [44]. At E coll > . − ,Ne + OH collisions occur in the multiple partial-waveregime, where destructive interference between differ-ent partial wave contributions washes out the resonancestructure in the ICS apparent at lower collision energies.Figure 1(c) shows the total ICS for the ground high-field-seeking state of OH | e, M = 3 / (cid:105) colliding withNe. At low collision energies ( E coll < ∆ E Λ , where∆ E Λ = 0 .
055 cm − is the Λ-doublet splitting energy)inelastic scattering of OH molecules oriented along theelectric field direction is strongly suppressed, leading tothe universal steric preference phenomenon consideredbelow. The origin of the suppression is that only threeinelastic channels ( | e, M (cid:48) = − / (cid:105) and | e, M (cid:48) = ± / (cid:105) )remain open at zero collision energy in the low field limit.As these channels have M (cid:48) (cid:54) = M inelastic scatteringmust be accompanied by a change in M . According tothe Krems-Dalgarno threshold laws [42] transitions withnonzero ∆ M scale with the collision energy as E ∆ M coll (foreven ∆ M ) and E ∆ M +1coll (for odd ∆ M ). Thus, transitionswith ∆ M (cid:54) = 0 will be suppressed in the s -wave limit bycentrifugal barriers in the outgoing collision channel.At higher collision energies above the Λ-doubletthreshold, inelastic channels of opposite parity and∆ M = 0 (such as | f, M (cid:48) = 3 / (cid:105) ) open up, causing a sub-stantial increase of the ICS and the appearance of near-threshold resonances. A total of 8 isolated resonancesoccur over the range of collision energies 0.06-0.2 cm − below 9 kV/cm. Remarkably, the resonances survive notonly in the few partial wave regime ( E coll < .
05 cm − ) (b)(a)
10 20
Electric field (kV/cm) -0.100.10.2 E n e r gy ( c m - ) f f e e (c)UR FIG. 1. Stark energy levels of OH( Π) in the J = 3 / a ) for Ne + OH plotted vs. collision energy and electric fieldfor the | f, M = 3 / (cid:105) (b) and | e, M = 3 / (cid:105) (c) initial states ofOH. The zero-field Λ-doublet energy is marked by the verticaldashed line and the dependence ∆ E Λ ( E ) is marked by theslanted line. The area labeled “UR” indicates the universalthreshold regime, where the inelastic ICS σ inel f,M is suppressed. but also at higher collision energies. This is due to thelimited number of inelastic channels available for the | e, M = 3 / (cid:105) initial state to decay into, resulting in amore pronounced S -matrix pole structure compared tothe | f, M = 3 / (cid:105) initial state [47].The resonance structure shown in Fig. 1(c) displays aninteresting pattern, shifting to higher collision energieswith increasing field. This occurs due to the field-inducedrepulsion between the opposite parity e and f states. Asa result of widening Λ-doublet energy gap, the minimumcollision energy required to access the inelastic channelsin the f -manifold increases with the E -field, shifting theonset of the resonance pattern to higher collision energies.Additionally, as seen in Fig. 1(c), increasing the electricfield suppresses inelastic scattering from the | e, M = 3 / (cid:105) state of OH: the resonance maxima of the ICS becomeless pronounced at higher electric fields. This is causedby the energy gap between the | M | = 1 / | M | = 3 / e -manifold growing linearly with increas-ing field [see Fig. 1(a)] until the | M | = 1 / | e, M = 3 / (cid:105) → | e, M (cid:48) = − / (cid:105) scales as σ inel (cid:39) E ∆ M +1coll = E [42] and is thus stronglysuppressed at ultralow collision energies. As shown be-low, the suppression of the inelastic ICS is a universaltrend , which manifests itself in ultracold collisions of Πmolecules in the | e, M = 3 / (cid:105) initial state with spheri-cally symmetric atoms. In contrast, no such trend existsfor the | f, M = 3 / (cid:105) initial state.To gain additional insight into cold Ne + OH colli-sion stereodynamics, we calculate the steric asymmetryparameter [7–11] S inel = σ inel e,M − σ inel f,M σ inel e,M + σ inel f,M (4)where σ inel i = (cid:80) k σ inel i → k is the total inelastic ICS for theinitial states of OH | i (cid:105) aligned along and against the fieldaxis (see above), M = 3 /
2, and the k sum runs over allenergetically accessible final channels. The steric asym-metry measures the difference between the collisionalproperties of OH molecules oriented along vs. againstthe field direction (with coincides with the incident atom-molecule velocity vector). A value of S inel close to -1 (+1)indicates a strong stereodynamic preference for Ne + OHcollisions with OH oriented against (along) the field axis.Experimental measurements and theoretical calculationsof the steric asymmetry have provided a wealth of valu-able information about stereodynamic effects in collisionsof OH and NO molecules with rare-gas atoms at collisionenergies of 300 K and above [7–11].Figure 2(a) is a two-dimensional map of the stericasymmetry for inelastic Ne + OH collisions plotted asa function of collision energy and electric field. Twodistinct regions may be observed in the map, whichwe will refer to as the universal region and the reso-nance region. The universal region corresponds to theregime E coll < ∆ E Λ , where the upper components of theΛ-doublet are closed, and inelastic scattering from thelowest low-field-seeking state | e, J = 3 / , M = 3 / (cid:105) ofOH is strongly suppressed by the threshold laws for M -changing collisions as noted above. Thus, in the universalregion, σ inel f,M (cid:29) σ inel e,M and S inel (cid:39) −
1. This remarkabletrend in only apparent in the inelastic ICSs [39].Significantly, the universal suppression of inelasticscattering persists at nonzero electric fields because the M = 3 / M = − / | e, M = 3 / (cid:105) →| e, M = − / (cid:105) transition. The degeneracy can be liftedby an external magnetic field, which is expected to breakthe universal behavior of the steric asymmetry, leadingto a rapid increase of S inel as the s -wave threshold scalingof the M -changing ICS changes from E to E − / .In the resonance regime defined by the condition E coll > ∆ E Λ excitation transitions occur from the ini-tial e -state, such as | e, M = 3 / → | e, M = ± / (cid:105) and | e, M = 3 / → | f, M = ± / (cid:105) . While these are also M -changing transitions, they are not so strongly suppressed UR (b)(a) UR FIG. 2. (a) Steric asymmetry (4) for inelastic Ne + OH col-lisions as a function of collision energy and electric field com-puted using the present Ne-OH PESs [39] (a) and the PESsfrom Ref. [48] (b). The Λ-doublet splitting energy of OH ismarked by the vertical dashed line ( E = 0) and by the slopeddashed line (as a function of E ). compared to the | e, M = 3 / (cid:105) → | e, M = − / (cid:105) tran-sition due to their smaller ∆ M . As s result, the back-ground value of the inelastic ICS increases and so doesthe steric asymmetry. In addition, scattering resonancesbegin to appear near and above the excitation thresh-olds leading to distinct spikes in σ inel e,M [see Fig. 1(c)]. Asnoted above and seen in Fig. 1(b), σ inel f,M does not varystrongly with either collision energy or electric field in theresonant regime. Taken together, these factors cause theappearance of the resonance peaks in the steric asymme-try in Fig. 2(a). As the details of the resonance structureare sensitive to the underlying PESs (a well-documentedphenomenon in cold molecular collisions [47, 49, 50]), theresonance regime can also be regarded as nonuniversal.To illustrate the distinction between the universal vs.nonuniversal regimes, we plot in Fig. 2(b) the stericasymmetry calculated using a different set of Ne-OH in-teraction PESs [48]. While the differences between thePESs are small [39], they have a dramatic effect on theresonance structure at E coll ≥ ∆ E Λ , as expected in thenonuniversal regime [47, 49, 50].Remarkably, by comparing Fig. 2(a) and Fig. 2(b) weobserve that the steric asymmetries calculated using thedifferent PESs are in nearly perfect agreement with eachother at E coll < ∆ E Λ ( S inel (cid:39) − C r o ss s ec ti on ( a t o m i c un it s ) -4 -2 Electric field (kV/cm) e f e -1/2 f -1/2 e e -3/2 e f e -3/2 e f -1/2 e e f e -1/2 f f e (a)(b) (c)(d) FIG. 3. (a) Electric field dependence of state-to-state inelasticICSs σ f,M =3 / → k [panels (a), (b)] and σ e,M =3 / → k [panels (c),(d)] for Ne + OH collisions. The collision energy is 0.02 cm − [panels (a), (c)] and 0.2 cm − [panels (b), (d)]. The total ICSssummed over all final k are shown by the top traces. scattering of OH( | e, M = 3 / (cid:105) ) is strongly suppresseddue to the threshold effects [42] (see above).In Figs. 2(a) and 2(b) we observe small deviations fromthe universal behavior at E coll (cid:39) ∆ E Λ . To understandthe origin of these deviations, we plot in Fig. 3 the state-to-state ICSs σ f,M =3 / → k and σ e,M =3 / → k , which definethe steric asymmetry (4). Below the Λ-doublet thresholdthe electric field dependence of σ f,M =3 / → k is determinedby isolated shape resonances with the dominant contribu-tion due to the | e, M = 3 / (cid:105) final state. At higher electricfields and/or collision energies, the resonances broadenand begin to overlap, leading to the disappearance ofdistinct peaks in the total ICS [44]. We note that theresonance structure in the state-to-state ICS can surviveat collision energies as high as 0.2 cm − , as illustrated inFig. 3(b) for the final state | e, M = 1 / (cid:105) .As shown in Figs. 3(c) and 3(d), the ICSs σ e,M =3 / → k increase by 2-4 orders of magnitude with increasing col-lision energy by a factor of 10, which is consistent withtheir E ∆ M coll threshold scaling discussed above [42]. Incontrast, the ICSs σ f,M =3 / → k decrease due to their dif-ferent threshold scaling (cid:39) E − / . We verified that thedeviations from the perfect universal scaling ( S inel = − (cid:96) ≥ σ e,M =3 / → k at E coll < ∆ E Λ [seeFig. 3(c)]. At lower collision energies, these contributionsfreeze out as E and the universal relation S inel = − Π molecular radicalswith S -state atoms in an external electric field. Usingrigorous quantum scattering calculations based on highlyaccurate ab initio interaction potentials, we show that thesteric anisotropy of Ne + OH collisions approaches − M -changing transitions from the | e, M = 3 / (cid:105) initialstate, in which the dipole moment of OH is oriented alongthe field direction. The suppression occurs universally inthe s -wave threshold regime, where the M -changing crosssections vanish [42], and it persists at collision energiesbelow the Λ-doublet energy ∆ E Λ regardless of the mag-nitude of the applied electric field [see Fig. 2]. Abovethe Λ-doublet energy, nearly perfect stereoselectivity is lost and scattering occurs in the nonuniversal resonantregime, where extensive control is possible over collisionstereodynamics via electric field-induced resonances. 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