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
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 lt 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
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 gt 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

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
Coupled channels equations
We use complete symmetrized basis of channels

• Cold collisions Only few needed
• Conservation of

and write total scattering solution as
Schrödinger equation for scattering problem
(coupled channels equation)
with coupling matrix
15
Scattering matrix
Expand solution for large r in incoming
and outgoing waves

• Scattering length a for s-wave scattering
• Cross sections
• Inelastic decay, collisional freq. shifts,
  binding energies

with the scattering
matrix Contains all observable collision
properties
Examples
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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
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
Double res.
Eff. range
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
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
Simulation experiment

• Solve resonance theory for experimental
  conditions

Atomic condensate fraction
t (µs)
Oscillations at binding-energy frequency!
Molecular condensate fraction
t (µs)
23
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!
Remnantburst
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)
n196.6(11) kHz
Bound state spectroscopy
29
Improving the interactomic potentials

• Ingredients
• 6 most accurate oscillation frequencies
• Position of zero crossing scattering length
  (a0) B165.75(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 gt 0
m lt 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)