Ramsey fringes in a Ramsey fringes in a BoseBose--Einstein Einstein condensate between atoms and condensate between atoms and

moleculesmolecules

Servaas KokkelmansENS, Paris

Collaboration:Theory ExperimentMurray Holland Neil ClaussenJosh Milstein Liz DonleyMarilu Chiofalo Carl Wieman

JILA, University of Colorado and NIST

AtomAtom--molecule coherencemolecule coherenceRecent 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

0 20 40 60 80 100154

155

156

157

158

159

160

161

162

163

t (µs)

B (G

auss

)

tevolve

t (µs)

B (G

auss

) tevolve

0 155 300

attractive

B (G)

-∞

a<0

Nmax=80

+∞ repulsivea>0

a (a

0)

What happens to BEC?What happens to BEC?

Expanded BEC, no B-field pulseN0 ~ 17,000

480 µm

In trap focused burst atoms(150 nK) Nburst/N0 = 25% - 40%

Cold < 3 nK BEC remnantNrem/N0 = 65% - 25%After B-field pulse,

See 2 components

Also missing atoms…………..

AtomAtom--molecule coherencemolecule coherenceTwo observed components oscillate!

RemnantBurst

10 15 20 25 30 35 400

4

8

12

16

tevolve (µs)

Num

ber (

x103 )

Looks likeRamsey-Fringes!

Molecular stateMolecular stateOscillations correspond to binding energy Feshbachmolecular state

Molecules play an important role close to resonance!

Coupled channels calculationUsed analysis from Kempen, Kokkelmans, Verhaar, Phys. Rev. Lett. 88, 093201 (2002)

Simple model

2

2

2)(

aBE

µη

=B (Gauss)

What is What is Feshbach Feshbach resonance?resonance?Coupling between open and closed channels:

Separate out bound state and treat explicitly

Resonance: short-range molecular stateRelatively long-lived moleculesScattering becomes strongly energy-dependent

closed channel

open channel

a

B

abg

Ekin

Resonance scattering: no GP equationResonance scattering: no GP equationClose to resonance, pairing field is important

Scattering length a large, na3 > 1Correlations induced by molecular stateEnergy-dependent scatteringInclude explicitly short-range molecular state in Hamiltonian

Describe two-body interaction with few parameters:Sc

atte

ring

leng

th

Detuning v

Width g

abg

Resonance HamiltonianResonance HamiltonianSplit interactions into two parts:

Direct non-resonant interaction (background process)Resonance coupling to intermediate molecular state

with

( )

( )]..)()()()(

)()()()()([

)()()()()()(

121212

12122123

13

3

cHxxxgX

xxxVxxxdxd

xxHxxxHxxdH

aam

aaaa

mmmaaa

++

+

+=

+

++

++

∫∫

ψψψ

ψψψψ

ψψψψ

mxHa 2/)( 22∇−= η 2112 xxx −=2/)( 2112 xxX +=ν+∇−= mxHm 4/)( 22η

and V(x12) and g(x12) contact interactions

Field equationsField equationsHartree-Fock-Bogoliubov approx.: Define mean-fields

Hartree-Fock-Bogoliubov approx. gives rise to coupled field equations:

,aa ψφ = ,mm ψφ = ,aaNG χχ += ,aaAG χχ=

))()(2]())0(([

)())0(|(|4)(2

)(

)]()())0(([Im2)(

))0((2

))0(())0(2|(|

2

222

**2

2

*2

rrGgGV

rGGVrGdtrdGi

rGgrGGVdtrdG

Ggdtdi

gVGGVdtdi

NmAa

ANaAA

AmAAaN

maAam

amAaNaa

δφφ

φµ

φφ

νφφφφ

φφφφφ

++++

++∇

−=

++=

++=

+++=

ηη

η

η

ηatomic condensate

molecular condensate

normal field

anomalous field

Resonance scattering equations insideResonance scattering equations insideSetting density-dependent terms to zero

Get coupled two-body scattering eqns.

Energy-dependent scattering close to resonanceContact interaction gives rise to divergence in k-space

mm

m

Pgdtdi

rgrPrVdtrdPi

νφφ

φδδµ

+=

+

+

∇−=

)0(2

)()()(2

)( 22

η

ηη

See PRA 65, 053617 (2002)

How to resolve this?Renormalize equations

Get the 2Get the 2--body physics rightbody physics rightSteps involved to get to renormalized resonance scattering theory:

Full CC scattering

Feshbach model

Analytic square-well

Renormalized scattering

Feshbach Feshbach theorytheoryShows 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

−−

−= −

)(21)( 242

22

kinm

ika

EikgikgekS bg

νπη

[ ]1)(2)( −= kSkikT ηπ

)0()(424

22

→− kgam m

bg νπ

πη

ηScattering Energy

Re[

T]-m

atrix

ν

Contact scattering Contact scattering -- renormalizationrenormalizationLimiting case: R 0

Cut-off gives renormalization!Define parameters

∑∫

∫

−

−+

−=

i ikin

K

iii

K

V

E

dpkTgcg

dpkTVcVkT

cutoff

cutoff

ν0

0

)(

)()(Solve Lippmann-Schwinger equation with contact potentials and contact coupling:

Uα−=Γ

11

222 ηπα

cutoffmK=

bgamU

24 ηπ=

UΓ=U

11g gΓ=

1111 ggανν +=

Relation between “real” and cut-off parameters:

(for single resonance)

Simulation experimentSimulation experimentSolve resonance theory for experimental conditions :

0 20 40 60 80 1000

0.005

0.01

0.015

0 20 40 60 80 1000.4

0.5

0.6

0.7

0.8

0.9

1

t (µs)

Atomic condensatefraction

Oscillations at binding-energy frequency!

t (µs)

2|| aφ

2|| mφ

Molecular condensatefraction

Binding energyBinding energyOscillation frequency agrees with molecular binding energy:

158 158.5 159 159.5 160 160.5 1610

50

100

150

200

250

300

350

400

450

500

B (Gauss)

E B(k

Hz)

Oscillations

Coupled channels

Simulation experiment (2)Simulation experiment (2)Crucial aspect:

Growth of non-condensate component!Oscillates almost out of phase with atomic condensateNot a usual thermal gas: coherent because of rise pairing field GA

GN(r) is correlation functionCan determine temperature of these atoms:Is consistent with experiment

Ramsey FringesRamsey Fringes

10 15 20 25 30 35 400

2000

4000

6000

8000

10000

12000

14000

16000

18000

t (µs)

Num

ber

Simulate experiment for different tevolve:Correct visibility, mean valueCorrect oscillation frequencySame (small) phase-shift as in experiment

Identify different fractions as:

Remnant atomic condensate

Burst atoms coherent non-condensate

Missing atoms atoms in molecular state

tevolve (µs)

Num

ber

Atomic condensate

Non-condensate

160 µs

Change pulse shapeChange pulse shape

Longer fall time

10 µs

155 G

tevolve (µs)

2 4 6 8 10 12 14 16

Num

ber

0

4000

8000

12000

16000

tevolve (µs)10 15 20 25 30 35 40

Num

ber (

x10-3

)

0

4

8

12

16

short fall time

long fall time

Remnant+burst

Remnant

Burst

More molecules!More molecules!

Phase shift smaller, so much bigger oscillationsin total number of observed atoms.

Precision binding energy measurementPrecision binding energy measurement

oscillation freq.+

B-field (pulsed NMR)

tevolve (µs)10 20 30 40

Rem

nant

Num

ber (

x10-3

)

9

10

11

12

13

14

ν=196.6(11) kHz

Bound statespectroscopy

Improving the Improving the interactomic interactomic potentialspotentials

B (G)156 157 158 159 160 161 162

Freq

uenc

y (k

Hz)

10

100

1000

Ingredients:6 most accurate oscillation frequenciesPosition of zero crossing scattering length (a=0): B’=165.75(1)

Very close to thresholdIn agreement with previous 87Rb-85Rb determinationUncertainty in B0 reduced by factor 10

abg = -450.5 +- 4 a0

B0= 154.95 +- 0.03 G

25 50 75 100-500-250

0250

12840

12880

12920

12960

ener

gy (c

m-1)

internuclear distance (a0)

Og- (5P3/2)

ground state (5S1/2)

v = 0 - 10

νmolecule

- Scott Papp, Sarah Thompson, Carl Wieman

How to detect the moleculesHow to detect the molecules

Short laser pulse

Minimize photoassociationof BEC (B-field, laser freq).

Look for bound-bound transitions.

SternStern--GerlachGerlach separatorseparator

µdimer 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

µ > 0 µ < 0

Other possibility: Large detuning (for optical trap), blow away atomsMolecules remain

B

ConclusionsConclusionsExplain observed coherent oscillations atoms-molecules in 85Rbcondensate

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)

Double square wellDouble square wellSimple model to describe Feshbach resonance

Coupled square well

Range R 0: Contact potentials

-V1-V2

Potential range R

ε

Ekin0

Detuning

Simple wave-Functions:Molecular and “free”

Coupling:Boundarycondition

uP(r), uQ(r)

u1(r), u2(r)

−

=

)()(

cossinsincos

)()(

2

1

RuRu

RuRu

Q

P

θθθθ

Can do better:Can do better: 66Li Li FeshbachFeshbach resonanceresonance

B (Gauss)

a (U

nits

of a

0)

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

Realbackground

!31 0aabg =

∆∆−∆+∆

∆+∆−= −

211222

21

1222

212

)()(21)(

ggikggikekS bgika

mkinE24

11 )( ηπν −=∆ mkinE24

22 )( ηπν −=∆

Double res. needed for binding energyDouble res. needed for binding energyCompare different models for calculation of binding energy (85Rb)

0.0155 0.0156 0.0157 0.0158 0.0159 0.0161

-600

-500

-400

-300

-200

-100 Single res.

2

2

maE η=

Simple contactmodel

Coupled channels

Double res.Eff. range

Bin

ding

ene

rgy

B (T)

More interesting structure arisesMore interesting structure arises

0.0155 0.016 0.0165 0.017 0.0175 0.018

-30000

-20000

-10000

10000

20000

30000

Double resonance model shows also quasi-bound state:

Also virtual states arise:Work in progress!

Bin

ding

ene

rgy

B (T)