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Hyperfine-Changing Collisions of Cold Molecules J. Aldegunde, Piotr Żuchowski and Jeremy M. Hutson University of Durham EuroQUAM meeting Durham 18th April 2009 Contents 1. Hyperfine molecular levels (QUDIPMOL). 2. Hyperfine changing collisions (CoPoMol). Fine and hyperfine structure Hyperfine molecular levels Atomic physics: •Gross spectra: the spectra predicted by considering nonrelativistic electrons and neglecting the effect of the spin. • Fine structure: energy shifts and spectral lines splittings due to relativistic corrections (including the interaction of the electronic spin with the orbital angular momentum). • Hyperfine structure: energy shifts and splittings due to the interaction of the nuclear spin with the rest of the system. Gross structure >> Fine structure >> Hyperfine structure This classification can be extended into the molecular realm. • Stability. Molecular fine and hyperfine levels • Bose-Einstein condensate formation. Atomic hyperfine structure Hyperfine molecular levels S → Electronic spin L → Orbital angular momentum I → Nuclear spin Alkali atoms → L=0, S=1/2 F=S+I Ĥhf= A IRb ∙ SRb Ĥz= gs μB B∙SRb - gRb μN B∙IRb Hyperfine splitting ≈ GHz ≈ 10-1 K Hyperfine molecular levels 1Σ diatomic molecules 7Li133Cs 1Σ molecules 133Cs 2 ( M. Weidemüller (Freiburg)) (Hanns-Christoph Nägerl (Innsbruck)) 40K87Rb (Jun Ye, D. Jin (JILA)) S=0 (no fine structure) Two sources of angular momentum: • N → Rotational angular momentum (L in atom-atom collisions). • I1, I2 → Nuclear spins of nucleus 1 and 2. Hyperfine molecular levels Rotational angular momentum Nuclear spins H^ = H^ r + H^ hf + H^ S + H^ Z H^ r = H^ hf = 1Σ diatomic molecules N ! I 1; I 2 ! B À N 2 ¡ D À N 2 ¢N 2 X2 Vi : Q i + c4 I 1 ¢I 2 + ot her t erms i= 1 H^ S = H^ Z = ¡ ¹ ¢E X2 ¡ gi ¹ N I i ¢B ¡ gr ¹ N N ¢B i= 1 Hyperfine molecular levels Rotational angular momentum Nuclear spins H^ = H^ r + H^ hf + H^ S + H^ Z H^ r = H^ hf = 1Σ diatomic molecules N ! I 1; I 2 ! B À N 2 ¡ D À N 2 ¢N 2 X2 Vi : Q i + c4 I 1 ¢I 2 + ot her t erms i= 1 H^ S = H^ Z = ¡ ¹ ¢E X2 ¡ gi ¹ N I i ¢B ¡ gr ¹ N N ¢B i= 1 Hyperfine molecular levels Rotational angular momentum Nuclear spins H^ = H^ r + H^ hf + H^ S + H^ Z H^ r = H^ hf = 1Σ diatomic molecules N ! I 1; I 2 ! B À N 2 ¡ D À N 2 ¢N 2 X2 Vi : Q i + c4 I 1 ¢I 2 + ot her t erms i= 1 H^ S = H^ Z = ¡ ¹ ¢E X2 ¡ gi ¹ N I i ¢B ¡ gr ¹ N N ¢B i= 1 Hyperfine molecular levels Rotational angular momentum Nuclear spins H^ = H^ r + H^ hf + H^ S + H^ Z H^ r = H^ hf = B À N 2 ¡ D À N 2 ¢N 2 X2 Vi : Q i + c4 I 1 ¢I 2 + ot her t erms i= 1 1Σ diatomic molecules N ! I 1; I 2 ! H^ S = H^ Z = ¡ ¹ ¢E X2 ¡ gi ¹ N I i ¢B ¡ gr ¹ N N ¢B i= 1 Hyperfine molecular levels Rotational angular momentum Nuclear spins H^ = H^ r + H^ hf + H^ S + H^ Z H^ r = H^ hf = B À N 2 ¡ D À N 2 ¢N 2 X2 Vi : Q i + c4 I 1 ¢I 2 + ot her t erms i= 1 1Σ diatomic molecules N ! I 1; I 2 ! H^ S = H^ Z = ¡ ¹ ¢E X2 ¡ gi ¹ N I i ¢B ¡ gr ¹ N N ¢B i= 1 Hyperfine molecular levels Rotational angular momentum Nuclear spins H^ = H^ r + H^ hf + H^ S + H^ Z H^ r = H^ hf = B À N 2 ¡ D À N 2 ¢N 2 X2 Vi : Q i + c4 I 1 ¢I 2 + ot her t erms i= 1 1Σ diatomic molecules N ! I 1; I 2 ! H^ S = H^ Z = ¡ ¹ ¢E X2 ¡ gi ¹ N I i ¢B ¡ gr ¹ N N ¢B i= 1 Hyperfine molecular levels 1Σ(N=0) diatomic molecules. Zero field splittings. Zero field splittings dominated by the scalar spin-spin interaction (c4I1·I2). c4(40K87Rb) ≈ -2 kHz c4(133Cs2) ≈ 13 kHz Hyperfine splitting ≈ tens to hundreds of kHz ≈ 1 to 10 μK The ratio |c4/(eQq)| ratio determines the zero field splitting partner: • Large |c4/(eQq)| values → the splitting is determined by the scalar spinspin Interaction and coincides with that for N=0. • Small |c4/(eQq)| values → the splitting is determined by the electric quadrupole interaction. 85Rb 2 (N=1) Hyperfine splitting ≈ hundreds to thousands of kHZ ≈ 10 to 100 μK 1Σ (N≠0) diatomic molecules. Zero field splittings. Hyperfine molecular levels eQq(85Rb2) ≈ 2 MHz Hyperfine molecular levels (2I+1) components (N=0). Each level splits into 1Σ diatomic molecules. Zeeman splitting. (2F+1) components (N≠0). The slope of the energy levels and the corresponding splittings are determined by the nuclear g-factors. 1Σ diatomic molecules. Zeeman splitting. Hyperfine molecular levels • Energy levels with the same value of MI display avoided crossings (the red lines correspond to MI =-3) • I remains a good quantum number for values of the magnetic field below those for which the avoided crossings appear. • For large values of the magnetic field the individual projections of the nuclear spins become good quantum numbers. 1Σ diatomic molecules. Zeeman splitting. Hyperfine molecular levels • Energy levels with the same value of MI display avoided crossings (the red lines correspond to MI =-3) • I remains a good quantum number for values of the magnetic field below those for which the avoided crossings appear. • For large values of the magnetic field the individual projections of the nuclear spins become good quantum numbers. 1Σ diatomic molecules. Stark splitting. Hyperfine molecular levels • Mixing between rotational levels is very important and increases with the electric field. • The number of rotational levels required for convergence becomes larger with field. • For the levels correlating with N=0, the Stark effect is quadratic at low fields and becomes linear at high fields. Hyperfine molecular levels 1Σ diatomic molecules. Stark splitting. Energy levels correlating with N=0 referred to their field-dependent average value: • Each level splits into I+1 components labelled by |MI|. • At large fields the splitting approach a limiting value and the individual projections of the nuclear spins become well defined. Hyperfine changing collisions Rb + OH(2Π3/2) Rb + OH(2Π3/2) collisions M.Lara et al studied these collisions (Phys. Rev. A 75, 012704 (2007)). • Rb, OH or both of them undergo fast collisions into high-field-seeking states. • Sympathetic cooling is not going to work unless both species are trapped in their absolute ground states. Cs + Cs collisions Hyperfine changing collisions Cs( 2 S)+ Cs( 2 S) Int eract t hrough a singlet and a t riplet pot ent ial 2 ¹ h ^ H = 2¹ " ¡ R ¡ 1 2 ^2 # d L ^1 + h ^2 + V ^ c (R) + V ^ d (R) R + + h dR 2 R2 cent ral dipolar ^ 1; h ^ 2 ! Monomer Hamiltonians h ^ c (R) = V0 (R) P^ (0) + V1(R) P^ (1) V singlet (isotropic) t riplet ^ d (R) ! Small anisotropic spin-dependent interactions V Hyperfine changing collisions 2 Cs + Cs collisions ¹ h ^ H = 2¹ " ¡ R ¡ 1 2 ^2 # d L ^1 + h ^2 + V ^ c (R) + V ^ d (R) R + + h dR 2 R2 cent ral dipolar ^ c (R) drives spin-exchange collisions: M S1 ! M S1 § 1 V0 (R) 6 = V1 (R) ! V M S2 ! M S2 ¨ 1 V0 (R) = V1 (R) ! ^ c (R) diagonal in S; M S ; M S1 and M S2 . It s mat rix V represent at ion is proport ional t o t he unit mat rix. ^ d (R) Inelasticity due to V Slow inelastic collisions. Hyperfine changing collisions Rb + CO(1Σ) collisions Rb and 12C16O interact through a doublet potential I ( 12 C) = I ( 16 O) = 0 ! no hyper¯ne structure " 2 ¹ h ^ H = 2¹ # ¡ R ¡ 1 d2 L^ 2 ^ Rb + h ^ CO + V ^ (R; µ) R + + h 2 2 dR R ^ CO = B À N 2 ¡ gr ¹ N B ¢N h ^ Rb = AI Rb ¢SRb + gs¹ B B ¢SRb ¡ gN ¹ Rb B ¢I Rb h No spin-relaxation collisions will take place. 2 ¹ h ^ H = 2¹ " ¡ R ¡ 1 2 ^2 # d L ^ CO + V ^ Rb ^ (R; µ) + h R + + h dR 2 R2 jLM L N M N i jSM S I M I i Rb + CO(1Σ) collisions Hyperfine changing collisions 1. T he dipolar int eract ion bet ween t he Rb elect ronic magnet ic moment and t he rot at ional magnet ic moment of t he CO molecule is given by ^ = ¡ ¹ 0 gs ¹ B gr ¹ N (S ¢N ¡ 3(S ¢R^ )(N ¢R^ )) U 4¼ R3 ^ \ connect s" t he jL M L N M N i , jSM S I M I i spaces and drives t ransit ions U ^ where M S , M N and M L change by up t o one. M at rix element s of U bet ween L = 0 or N = 0 st at es are zero. 2. T he at omic hyper¯ne coupling const ant (A ) is R dependent . T his is equivalent t o include and ext ra-t erm in t he Hamilt onian 2 h ¹ H^ = 2¹ · ¸ L^ 2 ¡ 1 d ^CO + h ^Rb ^ (R; µ) + (A (R; µ) ¡ A (1 ))I R b ¢S R b + h ¡ R R+ 2 +V 2 dR R t hat causes t ransit ions M S = 1=2 $ M S = ¡ 1=2 MI $ MI + 1 M S = ¡ 1=2 $ M S = 1=2 MI $ MI ¡ 1 or whenever t he elect ric ¯eld is di®erent from zero. Hyperfine changing collisions Rb + CO(1Σ) collisions Bot h mechanisms will drive slow Rb inelast ic collisions. ^ / gsgr where gs À gr . 1. U 2. T he t erm (A(R; µ) ¡ A(1 ))I Rb ¢SR b is short -range. (A(R; µ) ¡ A(1 )) ¡¡R ¡ ¡!¡ ¡ 1 ¡! 0 Collisions of alkali at oms wit h singlet molecules will not cause fast at omic inelast icity Rb + ND3 collisions Hyperfine changing collisions The ND3 molecule in low-¯eld-seeking st at es correlat ing wit h t he jJ; K i = j1; 1ui st at e can be St ark-decelerat ed. Cross sect ions for t he j1; 1ui ! j1; 1ui (| ) j1; 1ui ! j1; 1li (| ) processes in t he absence of ¯eld ND3 molecules in low-¯eld-seeking st at es can probably not be cooled t o sub-mK t emperat ures by collisions wit h ult racold Rb at oms Hyperfine changing collisions Rb + ND3 collisions What about sympathetic cooling of ND3 high-¯eld-seeking states? T he same analysis applied t o Rb+ CO collisions suggest s t hat sympat het ic colling of ND 3 (or NH 3 ) molecules in high-¯eld-seeking st at es by magnet ically t rapped Rb at oms is likely t o be feasible. jSM SI M I i (Rb monomer) Rb and ND3 int eract t hrough a doublet pot ential jLM L i jJK I N I D i 1. Dipole int eract ions between magnet ic moment s of t he monomers. 2. R dependence of t he Rb hyper¯ne coupling const ant . Rb + ND3 collisions Hyperfine changing collisions ² Int eract ion between t he elect ronic magnet ic moment of Rb and t he rot at ional magnet ic moment of ND 3 . ^ / gsgr U Dipole int eract ions ² Int eract ion between t he elect ronic magnet ic moment of Rb and t he nuclear magnet ic moment s of ND 3 . ^ / gsgN ; H ; D U gs À gH > gD > gN > gr ² T he int eract ion involving t he ammonia nuclear spins predominat es. Nevert heless bot h int eract ions are much smaller t han t he elect ron-elect ron dipole int eract ion. ² ND 3 may be a bet t er candidat e t o sympat het ic cooling t han NH3 . Conclusions • The rotational levels of 1Σ alkali metal dimers split into many hyperfine components. • For nonrotating states, the zero-field splitting is due to the scalar spin-spin interaction and amounts to a few μK. • For N≠1 dimers, the zero-field splitting is dominated by the electric quadrupole interaction and amounts to a few tens of μK. • External fields cause additional splittings and can produce avoided crossings. • For molecules in closed shell single states colliding with alkali atoms, the atomic spin degrees of freedom are almost independent of the molecular degrees of freedom and the collisions will not change the atomic state even if the potential is highly anisotropic. • Prospects for sympathetic cooling of ND3/NH3 molecules with cold Rb atoms: 1. Poor for ND3/NH3 low-field-seeking states. 2. Good for ND3/NH3 high-field-seeking states. ND3 better than NH3. • Quantitative calculations are necessary. Hyperfine changing collisions Rb + OH(2Π3/2) collisions There are ¯ve pot ent ial energy surfaces corresponding t o t he Rb( 2 S) and OH( 2 ¦ 3=2 ) int eract ion. ^ operat or. The monomers Hilbert spaces are coupled by t he V At omic inelast ic collisions will be fast . M. Lara et al, Phys. Rev. A 75, 012704 (2007) Rb + OH(2Π3/2) collisions Hyperfine changing collisions ² Prospect s for sympat het ic cooling from t he low-¯eldseeking t hresholds correlat ing wit h C1 are bleak. ² At omic inelast ic collisions will be fast . Conclusions • Molecular energy levels split into many fine and hyperfine components. • 1Σ alkali dimers only display hyperfine splittings. • For nonrotating states, the zero-field splitting is due to the scalar spin-spin interaction and amounts to a few μK. • For N≠1 dimers, the zero-field splitting is dominated by the electric quadrupole interaction and amounts to a few tens of μK. • Except for short range terms, the system Hamiltonian for collisions between 2s atoms and singlet molecules can be factorised. The collisions will not cause fast atomic inelasticity. • This factorization will not be possible when the 2s atoms collides with doublet or triplet molecules. In this case, the potential operator will drive fast atomic Inelastic collisions. • Prospects for sympathetic cooling of ND3/NH3 molecules with cold Rb atoms: 1. Poor for ND3/NH3 low-field-seeking states. 2. Good for ND3/NH3 high-field-seeking states. ND3 better than NH3. • Quantitative calculations are necessary.