Lectures on Quantum Gases. Chapter 5. Feshbach resonances. Jook Walraven. Van der Waals Zeeman Institute University of Amsterdam

Transcription

1 Lectures on Quantum Gases Chapter 5 Feshbach resonances Jook Walraven Van der Waals Zeeman Institute University of Amsterdam 1

2 Schrödinger equation thus far: fixed potential What happens if we add internal structure? First we recapitulate: commute with and separation of variables: good for systems like helium 2

3 Interactions between alkali atoms molecule in electronic ground state - no electronic angular momentum: Λ = 0 For two ground-state alkali atoms two (not more than two) potentials Conclusion: exchange determines interatomic interaction To solve Schrödinger equation we turn to the basis: 3

4 Interactions between alkali atoms To represent exchange we construct a spin hamiltonian: Properties of operator : Hamiltonian including exchange: 4

5 Interactions between alkali atoms Let us add magnetic field: is good quantum number Hamiltonian including spin Zeeman term: good basis states: 5

6 Interactions between alkali atoms Hamiltonian including spin Zeeman term: good basis states: Solve radial wave equation for given l, S and M S : 6

7 Interactions between alkali atoms Conclusion: magnetic field lifts degeneracy of triplet potential This makes it possible to shift the triplet potential with respect to the singlet potential 7

8 Feshbach resonance Conclusion: magnetic field lifts degeneracy of triplet potential This makes it possible to shift the triplet potential with respect to the singlet potential Any weak singlet-triplet coupling induces Feshbach resonance 8

9 Interactions between alkali atoms Solve radial wave equation for given l, S and M S : Solutions for given l, S and M S : Continuum states : Bound states : 9

10 Interactions between alkali atoms Hamiltonian including spin Zeeman term: Add nuclear Zeeman terms (unlike atoms): Good basis states: Potential shift: 10

11 Interactions between alkali atoms Hamiltonian including spin Zeeman term: Add nuclear Zeeman terms (identical atoms): Good basis states: Potential shift: 11

12 Interactions between alkali atoms Add hyperfine interactions (unlike atoms): Is still a good quantum number? Answer: yes! Is S still a good quantum number? 12

13 Interactions between alkali atoms Is S still a good quantum number? identical atoms can change M S but not S. With in hamiltonian S remains a good quantum number! Analysis shows that converts singlet in triplet and vice versa 13

14 Interactions between alkali atoms Hamiltonian including spin Zeeman term: all terms conserve M F only term not singlet/triplet conserving Good basis states: Potential shift: 14

15 Interactions between alkali atoms Find all s-wave molecules with M F = 0 free atom pair with M F = 0 15

16 Interactions between alkali atoms M F = 0 0 S = 1 a ε 1 2 free atoms S = 0 ε 0 magnetic field 16

17 Interactions between alkali atoms M F = 0 M F = 0 S = 1 S = 0 17

18 Diagonalization of Hamiltonian Franck-Condon factor (only part that depends on radial wavefunction) Les Houches

19 Franck-Condon factor a. Halo states (universal regime) overlap: b. Asymptotic bound states ( ) can be calculated numerically starting from Van der Waals tail 19

20 singlet/triplet overlap Potentials reduce to two parameters: 20

21 6 Li- 40 K pairs E Z F = 3/2 2 0 S = ½, I = 1 6 Li m F +3/2 +1/2 1/2 F = 7/ S = ½, I = 4 40 K m F 7 /2 5 /2 3 /2 1/2 + 1/2 + 3/2 + 5/2 + 7/2 + 9 /2 F = 1/2-2 γ n > magnetic field 3/2 1/2 + 1/2 F = 9/2-5 γ n < magnetic field + 7/2 + 5/2 + 3/2 + 1/2 1/2 3 /2 5 /2 7 /2 9 /2 81 Gauss M F = Gauss 21

22 6 Li- 40 K loss features with M F = 3 0,0 M F = 3 Energy/h [GHz] -0,5-1,0-1,5-2, Magnetic field [G] 22

23 6 Li- 40 K loss features with M F = 3 0,0 M F = 3 Energy/h [GHz] -0,5-1,0-1,5-2, Magnetic field [G] 23

24 Include l = 1 molecular levels (p-wave) potential Conclusion: l = 0 and l = 1 levels can be related through C 6 coefficient 24

25 6 Li- 40 K loss features with M F = 3 0,0 M F = 3 Energy/h [GHz] -0,5-1,0-1,5-2, Magnetic field [G] 25

26 Relation to scattering length Feshbach range (new characteristic length) 6 Li/ 40 K 26

27 s-wave resonance near threshold 27

28 Feshbach resonance 28

29 cross section versus scattering length What happens to cross section when a diverges? (c) 9/26/2014 Les Houches

30 Magnetic field dependence 30

31 Rb-85 S.L. Cornish et al., PRL 85 (2000)

32 Li-6 a > 0 a < 0 6 Li 32

33 experimental adiabatic expansion of Gaussian trap induces evaporation: 6 40 Li K 6 Li 40 K Measure evaporation of 6 Li due to collisions with 40 K 33

34 Feshbach scattering length closed channel open channel B res B open channel (background) closed channel (resonance) 6 a/a bg 4 resonance strength parameter: 2 E (B-B 0 )/ Β zero crossing Les Houches

35 width from: elastic cross section 30 a/a bg 20 zero crossing σ/σ bg σ/σ bg k (B-B 0 )/ Β k > E (B-B 0 )/ Β on resonance ( ): unitarity limited scattering ( ) for asymmetric line shape: Fano profile Les Houches

36 characterization of a Feshbach resonance Lithium atom number (10 3 ) s 2.5s 7.5s magnetic field (G) position: B 0 = (5) G width: B = 1.5(5) G T.G. Tiecke, et al. PRL , (2010) Les Houches

37 Thanks Singapore 37

38 Feshbach coupling closed and open channels a. No coupling (unperturbed states) b. With coupling (perturbed states) 38

39 Feshbach coupling a. No coupling (unperturbed states) b. With coupling (perturbed states) Poject on unperturbed bound states: 39

40 Feshbach coupling a. No coupling (unperturbed states) b. With coupling (perturbed states) Poject on unperturbed continuum states: 40

41 Feshbach coupling a. No coupling (unperturbed states) b. With coupling (perturbed states) Approximation: with cutoff at 41

42 Feshbach coupling Approximation: with cutoff at 42

43 Two regimes a. : far from threshold b. : close to threshold 43