Modeling cold collisions Atoms Molecules

Transcription

1 Modeling cold collisions Atoms Molecules E. Tiemann, H. Knöckel, A. Pashov* Institute of Quantum Optics *University Sofia, Bulgaria

2 collisional wave function for E 0 A R=0 hk r B adopted from J. Weiner et al, Rev. Mod. Phys., Vol. 71, 1 (1999)

3 Accumulated phase method energy Φ(R 0 ) Moerdijk, Verhaar, Axelsson, Phys.Rev.A51,4852 (1995) Left boundary condition for eigenstate replaced by phase at R 0 ( R ) ( D E) J( J ) Φ = Φ + Φ + Φ E e J 1 potential branch 3 phenomenological parameters R 0 and eigenstates are described by ( 0 ) e i Φ R, D C phase scattering length at asymptotic energy reliable extrapolation by fitting spectroscopic observations dispersion coefficient estimation of spectral intensities by the asymptotic wave function, the inner part keeps its form Extension to multi channel problem channel dynamics in the inner region

4 Collisions of atoms with internal structure alkali atom: s A i A f A m A > atom pair A + B with rotation l hamiltonian basis Hund s case (e) (f A,f B )f,l,f,m F > atomic hfs and mechanical rotation diagonal potential energy and spin-spin/spin-orbit non-diagonal 1 Σ g+ and 3 Σ u + manifold of channels to 3 asymptotes f A f B +1 f A f B or f A -- f B +1 f A -- f B

5 How much knowledge of molecular potentials? v, J LeRoy-Bernstein formalism photo association spectroscopy RKR IPA cold collisions A + A boundary conditions accumulated phase Verhaar et al, Phys.Rev. A48, R3429 (1993) Vogels et al, Phys.Rev. A61, (2000)

6 Test case Rb + Rb highly precise measurements and theory: 1. Raman transitions Heinzen et al (2000) and Bloch et al (2003) 2. Feshbach resonances on 87 Rb and 85 Rb from several authors 3. Theoretical analysis by Verhaar group 2002/03

7 Observed Levels by D. Heinzen et al 2000 energy (MHz) f=4 ( ) (f A,f B ) (2,2) rotational information l=2 (1,2) (1,1) MHz 87 Rb + 87 Rb 5S 1/2 + 5S 1/2 f=2 ( ) l=0 f=2 ( ) spin-spin splitting observed between F=2 and F=4 90kHz r r r F = f + l f=2 ( ) R(a 0 ) singlet-triplet mixing 3 more down to 30 GHz by I. Bloch 2003 unpublished

8 Reconstruction of scattering calculation 1. Long range behavior: dispersion parameters C 6 C 8 and C 10 and exchange energy ( C C C10 ) 6 8 γ U R = D A R exp( R asymptote ± ) 6 8 R R R 10 ex α 2. Add potential branch for X 1 Σ + g and a3 Σ + u and adjust it to get scattering lengths as reported from the analysis 3. Set up the hyperfine and rotational Hamiltonian for calculating the observed spectra and compare with reported ones 2. Fit the observed spectra while keeping C i and exchange energy, but vary accumulated phase or logarithmic derivative 3. Calculate the scattering lengths and compare with reported ones

9 Comparison of the two cases 1. Use the spectra fit σ = 0.89 property Verhaar Rempe present (au) with V./R. a t (87) 98.98(4) /98.98 a s (87) 90.4(2) /91.5 a t (85) -388(3) -376/-373 a s (85) / Use scattering lengths of 87 Rb spectral data of 87 Rb gives σ with about ten times bigger value scattering length of 85 Rb a s =2768 au and a t =-373au singlet triplet coupling not properly described!

10 Construction of the adiabatic potentials and Hamiltonian energy (cm -1 ) B V + R ( R) = A 12 R i R o V E ( s R) = ex C R ( R) = A ex C R R γ e C R B ex R ± E ex V i ( R) = aix Dm i x = R Rm R + b R m R(Å)

11 Status of Rb 2 Fourier transform spectroscopy of X 1 Σ + g C. Amiot et al 2000 accuracy 30 MHz highest level 6.5 cm GHz below the asymptote outer turning point 12.8Å Ab initio calculation of a 3 Σ + u Krauss/Stevens 1990 and Foucrault et al 1992 Two-color photo association in 85 Rb accuracy 60 MHz Raman spectroscopy in BEC of 87 Rb accuracy 12 khz pure triplet and mixed levels Heinzen et al 1997/2000 Heinzen et al 2000 Bloch et al 2003 Feshbach resonance in 85 Rb f a = 2 m fa = 2 JILA, revision in 2001 l=0 f=4 f a =3 f b =3 singlet-triplet mixed resonance <S> = Feshbach resonances in 87 Rb with high precision Rempe et al 2003 Sengstock et al 2004

12 Potentials potential for X 1 Σ + g with 30 parameters construction of a potential for a 3 Σ + u with 9 parameters common long range behavior of X 1 Σ + g and a 3 Σ + u

13 Results for 87 Rb spectroscopic data described in the full range of observations resonance positions modeled by accumulated phase method < 0.5% with potentials about 0.1% or σ = 0.37G reliable potentials for future modeling: cold collisions and cold molecules property (au) Verhaar 2002 Rempe 2003 present with V./R. present new C i a t (87) 98.98(4) /98.89 a s (87) 90.4(2) /91.4 a t (85) -388(3) -376/-376 a s (85) /4800 C same C i assumed as by C Verhaar and Rempe resonances calculated

14 Scaling to 85 Rb? Using the potentials derived from 87 Rb: 85 Rb : spectra show systematic deviation of 45MHz and large scatter Feshbach resonance f a = 2 m fa = -2 calculated at G observed at 154.9(4) G 85 Rb separate evaluation with C i and exchange force from theoretical estimates spectra described within error limits Feshbach resonance at G isotope a s (from 85) a t (from 85) a s (from 87) a t (from 87) 85 Rb(au) Rb(au) Isotope effect beyond mass scaling? Correction to Born-Oppenheimer approximation?

15 Advantage by full potentials scientifically complete system within definition of potentials systematic analysis of existing data set and internal consistency connecting experimental regimes check for isotope effects in the cold collision regime transformation of atom pairs molecules data available for Na 2, Rb 2, NaRb, NaCs KRb and LiCs are processed presently

16 Potential scheme of mixed alkalis example NaRb Na fine structure Rb fine structure entrance channel high resolution Fourier spectroscopy spectroscopy entrance

17 Example NaRb 0.10 B 1 Π - X 1 Σ + system 0.08 x30 Intensity [B 1 Π, c 3 Σ +, b 3 Π ] - a 3 Σ + system Frequency [cm -1 ]

18 Progression to state a 3 Σ +

19 Overlap of X 1 Σ + and a 3 Σ X a 2.0 intensity asymptote f Na =1 + f Rb =2 J = energy (cm -1 ) shape resonance compare spacing exchange energy

20 Full potentials for states X 1 Σ + and a 3 Σ + of NaRb Prediction of collision properties in the energy range up to 10mK and Feshbach resonances up to 1000G approximations: atomic hyperfine interaction neglected second order spin-orbit interaction Born-Oppenheimer approximation

21 Collisions at different asymptotes Scattering length and resonances 23 Na 87 Rb M= ,0 Na 2,2 Rb 1,1 Na 2,1 Rb ag 0.00 f=2, ,2 Na 1,0 Rb hb f=2,3 2/1 f=2, 1/1 f=2,3 2/1 f=2, 1/1 energy [cm -1 ] B [Gauss] resonance spectroscopy 2,1 Na 1,1 Rb ga 1,1 Na 1,1 Rb aa

22 Summary and Conclusions on polar molecules Full potentials at ground state asymptote available with sufficient accuracy NaRb, NaCs, (KRb), LiCs Description with atomic hyperfine parameters very accurate Feshbach resonance structure very rich Electronic structure of excited states very complex Doorways to cold molecules with good predictions of transition moments from theory

23 channel 1,1 Na 1,1 Rb lowest asymptote 400 scattering length [a 0 ] B [Gauss]

24 channel 1,1 Na 2,1 Rb scattering length [a 0 ] B [Gauss]

25 mass scaling of scattering properties phase integral asymptote isotope independent? R 0 R 1 R 2 µ h R 2 ( v + ) = E U ( R)dR R 1 v π ( A) µ ( A) 1 ( + ) = 0 U ( R)dR v 2 2µ h R 0 v = v max + v d v d scattering length isotope relation ( B) µ ( B) v + 1 / 2 v + 1/ 2 = scattering length(a) v d (A) v max (A) knowledge of full potential v (B) v max (B) and v d (B) scattering length(b)

26 Feshbach resonances compared to magnetic resonance spectroscopy (NMR) E simplified diagram shown but hyperfine interaction important a 3 Σ + u scattering continuum v v s v-1 +1 Feshbach resonance v-2 v s -1 0 perturbation X 1 Σ + g M S = -1 B

27 Extrapolating from Feshbach resonances typical magnetic resonance spectroscopy Rb levels below asymptote f a =1 f b =1 m f = 2 and l = 0 E (cm -1 ) ref.: center of gravity B (Gauss)