Atom-molecule molecule collisions in spin-polarized polarized alkalis: potential energy surfaces and quantum dynamics

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1 Atom-molecule molecule collisions in spin-polarized polarized alkalis: potential energy surfaces and quantum dynamics Pavel Soldán, Marko T. Cvitaš and Jeremy M. Hutson University of Durham with Jean-Michel Launay and Pascal Honvault Université de Rennes

2 CCP6 booklet on Interactions of Cold Atoms and Molecules Edited by Pavel Soldán, Marko T. Cvitaš and Jeremy M. Hutson and Charles S. Adams University of Durham For a free copy, send to M.T.Cvitas@durham.ac.uk (with your postal address in a form easy to cut-and-paste to an address label)

3 Motivation In a Bose-Einstein condensate, ultracold molecules (alkali dimers) can be produced by photoassociation [Wynar et al., Science 287, 1016 (2000)] Feshbach resonance tuning [Donley et al, Nature 417, 529 (2002)] the molecules are produced in highly excited vibrational states: inelastic collisions will lead to trap Very little is known about atom-molecule collisions of this type Quantum dynamical collision calculations require accurate potential energy surfaces

4 Atom-diatom collisions Focus on atom-diatom collisions involving spin-polarised atoms: triplet dimers and quartet trimers Rb 2 (v) + Rb Rb 2 (v <v) + Rb Neglect nuclear spin effects (for now) First approximation: pairwise additivity V(r 12,r 23,r 31 ) = V(r 12 ) + V(r 23 ) + V(r 31 ) The triplet alkali dimer potentials are (fairly) well known, but unfortunately pairwise additivity is a poor approximation

5 What next? Ab initio calculations on trimers Standard ab initio electronic structure methods are now capable of giving reliable results (a few %) for intermolecular forces Attractive forces dominated by dispersion, so a high-level treatment of electron correlation is crucial Methods of choice are coupled-cluster methods: CCSD(T) has a good track record For high accuracy, need to morph the ab initio potentials using experimental data (spectroscopy, perhaps scattering lengths, etc.)

6 Methodology Carry out ab initio calculations on a grid of r 12,r 23,r 13 values [Choice of coordinate system is important: could use grid in Jacobi coordinates, ellipsoidal coordinates, hyperspherical coordinates, etc. Advantages of bond length coordinates are: good coverage of space with product grid easy avoidance of excluded volume] Investigate pairwise additivity for high-symmetry geometries (linear D h and equilateral D 3h ) Fit or interpolate potential for use in scattering calculations

7 Ab initio calculations Highly correlated ab initio method with medium-sized basis sets. Li 3 : RCCSD(T) / cc-pv5z basis set (all electrons) Na 3 : Higgins et al., J. Chem. Phys. 112, 5751 (2000) K 3 Rb 3 Cs 3 : RCCSD(T) / ECP10MWB + (11s 9p 5d 1f) valence basis set : RCCSD(T) / ECP28MWB + (11s 9p 5d 1f) valence basis set : RCCSD(T) / ECP46MWB + (11s 9p 5d 1f) valence basis set All the ab initio calculations were carried out using the MOLPRO package [H.-J. Werner & P. J. Knowles]. All results corrected for BSSE.

8 Beyond pairwise additivity: non-additivity and three-body potentials V trimer (r 12,r 23,r 13 ) = V 2 (r 12,r 23,r 13 ) + V 3 (r 12,r 23,r 13 ) V 2 (r 12,r 23,r 13 ) = V dimer (r 12 ) + V dimer (r 23 ) +V dimer (r 13 ) The 3-body term V 3 (r 12,r 23,r 13 ) is not small

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13 Observations Non-additive forces cause large changes in the binding energies of alkali trimers (more than a factor of 4 for Li, 1.3 to 1.5 for heavier alkalis) Non-additive forces are responsible for significant shortening of equilibrium distances of trimers (by 0.6 to 1.1 Å, compared to dimers). Most of the binding non-additive contributions exist even at the SCF level (85 110% at trimer minimum), so they arise from chemical bonding effects (not dispersion)

14 Where does the extra binding come from? In triplet alkali dimers (and rare gases) effects due to bonding and antibonding orbitals almost cancel at the equilibrium geometry In alkali trimers, the ns orbitals can form bonding (a 1 ) and antibonding (e) molecular orbitals (MOs); again, the effects would be naively expected to cancel But in alkali atoms the vacant np orbitals are nearby and can also form a 1 and e combinations; level repulsion then provides bonding and antibonding orbitals that do not cancel Natural atomic orbital analysis of SCF wavefunctions gives: Dimer Trimer ns np ns np radial np tangent Li 99% 0.5% 75% 5% 20% Na 99.8% 0.1% 98.5% 0.3% 0.9% K 99.5% 0.3% 95% 1% 3% The extra binding energy thus comes mainly from sp hybridisation.

15 Fitting: best to use known analytic long-range behaviour Dimer: Dispersion energy: dipole-dipole, dipole-quadrupole, and dipole-octupole + quadrupole-quadrupole terms V dimer (r) ~ C 6 / r 6 C 8 / r 8 C 10 / r 10 Trimer non-additive part: Axilrod-Teller-Muto triple-dipole term V 3 (r 12,r 23,r 13 ) ~ C 9 [1 + 3 cos θ 1 cos θ 2 cos θ 3 ] / (r 12 r 23 r 13 ) 3

16 Quantum collision dynamics In dynamical terms, alkali + alkali dimer collisions are not simply elastic / inelastic As an atom A approaches a diatom A 2 at a T-shaped geometry, the energy decreases all the way to the equilateral equilibrium; then any one of the 3 atoms can detach this is reactive scattering Even the linear geometry is below the A + A 2 reactant / product energy Need to do quantum reactive scattering calculations at ultralow energies, taking account of all three product channels (and including boson symmetry). Collaboration with Jean-Michel Launay (Rennes), who has developed a code to do exactly this at higher collision energies

17 Reactive scattering methodology Work in hyperspherical coordinates, which treat all 3 atoms equally: internal coordinates are hyperradius ρ and hyperangles θ and χ Begin with Na + Na 2 using surface of Higgins et al. Use basis set of pseudo-hyperspherical harmonics in inner region (4 Å < ρ < 20 Å) In inner region, prediagonalise at each ρ; keep lowest 135 states and solve coupled differential equations At boundary of inner region, project onto Na 2 vibration-rotation states (0 < v < 8 and 0 < j < 50) Propagate out to 5000 Å, neglecting channel couplings (but including -C 6 /R 6 ); match to boundary conditions to extract cross sections and (complex) scattering length

18 Na + Na 2 (v=1) on pairwise-additive potential

19 Na + Na 2 (v=1) on nonadditive potential

20 Observations Factor of 10 difference between additive and nonadditive potentials Wigner threshold laws hold below 10-5 K for both potentials Quenching cross section varies as E 1/2, giving rate coefficient independent of temperature: k = 5 x cm 3 s -1 (additive) and k = 5 x cm 3 s -1 (nonadditive) These are very fast rates, which are not conducing to building up a population of vibrationally excited molecules There is no reason to expect the rates to decrease with increasing v

21 Rotational distributions Na 2 (v = 1, j = 0) has enough energy to populate v = 0 rotational levels up to j = 20 Population of higher j levels is not suppressed: for total angular momentum J = 0, outgoing channel has l = j but the barrier is in a lower-lying channel Identical particle symmetry allows only even j levels in products

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24 Conclusions Quantum dynamics calculations on alkali + alkali dimer collisions require accurate potential energy surfaces Ab initio calculations can provide good-quality surfaces Non-additive effects can be very large, e.g. a factor of 4 in well depth for Li 3 Pairwise-additive potentials are not good enough for collision calculations, even where the non-additive effects are smallest (for Na 3 ) Rates of inelastic / quenching collisions are very large, and vibrationally excited alkali dimers will be destroyed quickly under BEC conditions

25 Extensions planned Full potential surfaces for other systems Li 3 and improved Na 3 surfaces being fitted now K 3, Cs 3 and Rb 3 surfaces being calculated Collision calculations required: including nuclear spin in magnetic fields including coupling to doublet surfaces Funding: EPSRC, JILA, University of Durham