Ramsey fringes in a Bose-Einstein condensate between atoms and molecules

1. Ramsey fringes in a Bose-Einstein condensate between atoms and molecules Servaas Kokkelmans Collaboration: Theory Experiment Murray Holland Neil Claussen Josh Milstein Liz Donley Marilu Chiofalo Carl Wieman JILA, University of Colorado and NIST

2. 163 162 161 t evolve 160 159 B (Gauss) 158 157 156 155 154 0 20 40 60 80 100 t (μs) m t ( s) + repulsive a>0 a (a0) attractive a<0 Nmax=80 - 0 155 300 B (G) Atom-molecule coherence • Recent experiment at JILA with 85Rb condensate: • Feshbach resonance causes coherent coupling • Atoms molecules Donley et al., Nature 412 295 (2002). Apply two field- pulses close to resonance tevolve B (Gauss)

3. What happens to BEC? Expanded BEC, no B-field pulse N0 ~ 17,000 Cold < 3 nK BEC remnant Nrem/N0 = 65% - 25% After B-field pulse, See 2 components In trap focused burst atoms(150 nK)Nburst/N0 = 25% - 40% 480 mm Also missing atoms…………..

4. 16 12 8 4 0 10 15 20 25 30 35 40 Atom-molecule coherence • Two observed components oscillate! Remnant Burst Looks like Ramsey- Fringes! Number (x103) tevolve (μs)

5. Molecular state • Oscillations correspond to binding energy Feshbach molecular state • Molecules play an important role close to resonance! Coupled channels calculation Used analysis from Kempen, Kokkelmans, Verhaar, Phys. Rev. Lett. 88, 093201 (2002) Simple model B (Gauss)

6. What is Feshbach resonance? • Coupling between open and closed channels: Separate out bound state and treat explicitly closed channel Ekin open channel abg a B • Resonance: short-range molecular state • Relatively long-lived molecules • Scattering becomes strongly energy- • dependent

7. Resonance scattering: no GP equation • Close to resonance, pairing field is important • Scattering length a large, na3 > 1 • Correlations induced by molecular state • Energy-dependent scattering Include explicitly short-range molecular state in Hamiltonian • Describe two-body interaction with few parameters: abg Scattering length Detuning v Width g

8. Resonance Hamiltonian • Split interactions into two parts: • Direct non-resonant interaction (background process) • Resonance coupling to intermediate molecular state • with and V(x12) and g(x12) contact interactions

9. Field equations • Hartree-Fock-Bogoliubov approx.: Define mean-fields • Hartree-Fock-Bogoliubov approx. gives rise to coupled field equations: atomic condensate molecular condensate normal field anomalous field

10. Resonance scattering equations inside • Setting density-dependent terms to zero • Get coupled two-body scattering eqns. • Energy-dependent scattering close to resonance • Contact interaction gives rise to divergence in k-space See PRA 65, 053617 (2002) How to resolve this? Renormalize equations

11. Get the 2-body physics right Steps involved to get to renormalized resonance scattering theory: Full CC scattering Feshbach model Analytic square-well Renormalized scattering

12. Interactions between alkali atoms Hamiltonian of two colliding particles: Hyperfine and Zeeman interaction (here Cs): At large internuclear distances we define two-atom hyperfine states:

13. Central and magn. dip. Interaction Central interaction (All coulomb interactions) Electronic ground state Singlet and Triplet potentials (dep. on electr. spin) Dipole interaction direct spin-spin interaction Interaction spin magnetic moments

14. Cold collisions: Only few needed Conservation of Coupled channels equations We use complete symmetrized basis of channels: and write total scattering solution as Schrödinger equation for scattering problem (coupled channels equation): with coupling matrix

15. Scattering length a for s-wave scattering Cross sections Inelastic decay, collisional freq. shifts, binding energies Scattering matrix Expand solution for large r in incoming and outgoing waves: with the scattering matrix Contains all observable collision properties Examples:

16. Re[T]-matrix Scattering Energy Feshbach theory Shows that only few parameters needed to describe full energy-dependent scattering: Coupling open en closed channels • Resonant S-matrix • T-matrix • Zero limit: • scattering length: closed channel open channel

17. a (Units of a0) B (Gauss) Can do better: 6Li Feshbach resonance • Two lowest hyperfine states (1/2,1/2)+(1/2,-1/2) • Double resonance! • Double-resonance S-matrix: • With , • And coupling strengths g1 and g2 Real background

18. Double res. needed for binding energy • Compare different models for calculation of binding energy (85Rb) B (T) Single res. Simple contact model Binding energy Coupled channels Eff. range Double res.

19. More interesting structure arises • Double resonance model shows also quasi-bound state: • Also virtual states arise: Work in progress! Binding energy B (T)

20. Detuning Ekin -V1 -V2 Potential range R Double square well • Simple model to describe Feshbach resonance • Coupled square well • Range R 0: Contact potentials uP(r), uQ(r) u1(r), u2(r) Simple wave- Functions: Molecular and “free” Coupling: Boundary condition

21. Contact scattering - renormalization • Limiting case: R 0 • Cut-off gives renormalization! • Define parameters Solve Lippmann-Schwinger equation with contact potentials and contact coupling: Relation between “real” and cut-off parameters: (for single resonance)

22. 1 0.9 0.8 0.7 0.6 0.5 0.4 0 20 40 60 80 100 0.015 0.01 0.005 0 0 20 40 60 80 100 Simulation experiment • Solve resonance theory for experimental conditions : Atomic condensate fraction Oscillations at binding-energy frequency! t (μs) Molecular condensate fraction t (μs)

23. 500 450 400 350 300 250 200 150 100 50 0 158 158.5 159 159.5 160 160.5 161 Binding energy • Oscillation frequency agrees with molecular binding energy: Oscillations EB (kHz) Coupled channels B (Gauss)

24. Simulation experiment (2) • Crucial aspect: • Growth of non-condensate component! • Oscillates almost out of phase with atomic condensate • Not a usual thermal gas: coherent because of rise pairing field GA • GN(r) is correlation function • Can determine temperature of these atoms: • Is consistent with experiment

25. Ramsey Fringes • Simulate experiment for different tevolve: • Correct visibility, mean value • Correct oscillation frequency • Same (small) phase-shift as in experiment • Identify different fractions as: • Remnant • atomic condensate • Burst atoms • coherent non-condensate • Missing atoms • atoms in molecular state Number Atomic condensate Non-condensate tevolve (μs)

26. Change pulse shape Longer fall time 10 ms 160 ms 155 G

27. More molecules! Remnant+burst Remnant Burst long fall time Phase shift smaller, so much bigger oscillations in total number of observed atoms. short fall time

28. Precision binding energy measurement oscillation freq. + B-field (pulsed NMR) n=196.6(11) kHz Bound state spectroscopy

29. Improving the interactomic potentials • Ingredients: • 6 most accurate oscillation frequencies • Position of zero crossing scattering length (a=0): B’=165.75(1) • Very close to threshold • In agreement with previous 87Rb-85Rb determination • Uncertainty in B0 reduced by factor 10 abg = -450.5 +- 4 a0 B0= 154.95 +- 0.03 G

30. How to detect the molecules Short laser pulse Minimize photoassociation of BEC (B-field, laser freq). Look for bound-bound transitions. - Scott Papp, Sarah Thompson, Carl Wieman

31. Stern-Gerlach separator • mdimer strong function of B near resonance • Choose Bevolve where dimers are untrapped • After pulse #1, wait for dimers to fall • Apply 2nd pulse, look for position shift of atoms E B m > 0 m < 0 Other possibility: Large detuning (for optical trap), blow away atoms Molecules remain

32. Conclusions • Explain observed coherent oscillations atoms-molecules in 85Rb condensate • Pairing field plays crucial role, gives rise to coherent non-condensate atoms • Non-condensate larger than molecular condensate • Agreement also for different densities • Based on formulation of resonance pairing model by separating out highest bound states • Resonance scattering built-in in many-body theory: coupled channels with contact potentials • High precision bound state spectroscopy improves potentials • Previously used for description of resonance superfluidity PRL 89, 180401 (2002), PRL 87, 120406 (2002), PRA 65, 053617 (2002)