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

moleculesmolecules

Servaas Kokkelmans

Collaboration:

Theory Experiment

Murray Holland Neil Claussen

Josh Milstein Liz Donley

Marilu Chiofalo Carl Wieman

JILA, University of Colorado and NIST

Atom-molecule coherenceAtom-molecule 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 (s)

B (

Ga

uss)

tevolve

t (μs)

B (

Gau

ss) tevolve

0 155 300

attractive

B (G)

-

a<0

Nmax=80

+repulsive

a>0

a (a

0)

What happens to BEC?What happens to BEC?

480 m

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

After B-field pulse,See 2 components

Cold < 3 nK BEC remnantNrem/N0 = 65% - 25%

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

Also missing atoms…………..

10 15 20 25 30 35 400

4

8

12

16

Atom-molecule coherenceAtom-molecule coherence

Burst Remnant

tevolve (μs)

Two observed components oscillate!N

umbe

r (x

103 )

Looks likeRamsey-Fringes!

Molecular stateMolecular 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

2

2

2)(

aBE

B (Gauss)

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

Separate out bound state and treat explicitly

Resonance: short-range molecular state Relatively long-lived molecules Scattering becomes strongly energy-

dependent

closed channel

open channel

a

B

abg

Ekin

Resonance scattering: no GP equationResonance 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:S

catt

erin

g le

ngth

Detuning v

Width g

abg

Resonance HamiltonianResonance Hamiltonian

Split 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

mxH a 2/)( 22

mxHm 4/)( 222112 xxx

2/)( 2112 xxX

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

Field equationsField equations

Hartree-Fock-Bogoliubov approx.: Define mean-fields

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

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

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

)(

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

))0((2

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

2

222

**2

2

*2

rrGgGV

rGGVrGdt

rdGi

rGgrGGVdt

rdG

Gg

dt

di

gVGGVdt

di

NmAa

ANaAA

AmAAaN

maAam

amAaNaa

,aa ,mm ,aaNG ,aaAG

atomic condensate

molecular condensate

normal field

anomalous field

Resonance scattering equations insideResonance 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

mm

m

Pg

dt

di

rgrPrVdt

rdPi

)0(2

)()()(2

)( 22

See PRA 65, 053617 (2002)

How to resolve this?Renormalize equations

Get the 2-body physics rightGet the 2-body physics right

Steps involved to get to renormalized resonance scattering theory:

Full CC scattering

Feshbach model

Analytic square-well

Renormalized scattering

Interactions between alkali atomsInteractions between alkali atoms

Hyperfine and Zeeman interaction (here Cs):

At large internuclear distances we definetwo-atom hyperfine states:

Hamiltonian of two colliding particles:

dipcent2

1i

inti

2

VVH2μ

pH

zzNze2hf

zhfint

B)is(1

i.sa

VVH

)1(2

,|)1(,|},{|}mf,mf{|

21 f2f1

Electronic ground stateSinglet and Tripletpotentials(dep. on electr. spin)

(All coulomb interactions)

Dipole interaction direct spin-spin interaction

Central interaction

Interaction spin magnetic moments

1s2s

Central and magn. dip. InteractionCentral and magn. dip. Interaction

TTSSc PVPVV

32121dip

r4

)s.r̂)(s.r̂(3s.sV 2

2e0

We use complete symmetrized basis of channels:

and write total scattering solution as

Schrödinger equation for scattering problem (coupled channels equation):

with coupling matrix

Coupled channels equationsCoupled channels equations

}}{m{|

}{|)r̂(Yir

Fm

}{ m

}{m

)()()(2

)1(

2 }''{''}''{ ''

}''{''}{}{22

22

rFrCrFErdr

dm

m

mmm

}''{''||}{'}''{''}{

mVVmiC dipcmm

Cold collisions: Only few needed

Conservation of )2,1,0(

totm

Expand solution for large r in incoming andoutgoing waves:

with the scattering matrix

Contains all observable collision properties

Examples:

Scattering matrixScattering matrix

ASrOrIBrOArIrF ])()([)()()(

1 ABS

]2exp[]2exp[}{00},{00 ikaiS

Scattering length a for s-wave scattering

Cross sections

Inelastic decay, collisional freq. shifts, binding energies

l

lSlk

2

,,2, )12()1(

Feshbach theoryFeshbach 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

)(

21)( 242

22

kinm

ika

Eikg

ikgekS bg

1)(2

)( kSk

ikT

)0()(4

24

22

kg

am m

bg

Scattering Energy

Re[

T]-

mat

rix

B (Gauss)

a (U

nits

of

a 0)

Can do better:Can do better: 6 6Li Feshbach resonanceLi 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

Realbackground

!31 0aabg

211222

21

1222

212

)(

)(21)(

ggik

ggikekS bgika

mkinE24

11 )( mkinE24

22 )(

0.0155 0.0156 0.0157 0.0158 0.0159 0.0161

-600

-500

-400

-300

-200

-100

Double res. needed for binding energyDouble res. needed for binding energy

Compare different models for calculation of binding energy (85Rb)

Single res.

2

2

maE

Simple contactmodel

Coupled channels

Double res.Eff. range

Bin

ding

ene

rgy

B (T)

0.0155 0.016 0.0165 0.017 0.0175 0.018

-30000

-20000

-10000

10000

20000

30000

More interesting structure arisesMore interesting structure arises

Double resonance model shows also quasi-bound state:

Also virtual states arise:

Work in progress!

Bin

ding

ene

rgy

B (T)

Simple model to describe Feshbach resonance

Coupled square well

Range R 0: Contact potentials

Double square wellDouble square well

-V1

-V2

Potential range R

ε

Ekin0

Detuning

Simple wave-Functions:Molecular and “free”

Coupling:Boundarycondition

uP(r), uQ(r)

u1(r), u2(r)

)(

)(

cossin

sincos

)(

)(

2

1

Ru

Ru

Ru

Ru

Q

P

Contact scattering - renormalizationContact scattering - renormalization

Limiting case: R 0

Cut-off gives renormalization!

Define parameters

i ikin

K

iii

K

V

E

dpkTgcg

dpkTVcVkT

cutoff

cutoff

0

0

)(

)()(

Solve Lippmann-Schwinger

equation with contact potentials

and contact coupling:

U

1

1

222

cutoffmK

bgam

U24

UU

11g g

1111 gg

Relation between “real”

and cut-off parameters:

(for single resonance)

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

Simulation experimentSimulation experiment

Solve resonance theory for experimental conditions :

t (μs)

Atomic condensatefraction

Molecular condensatefraction

Oscillations at binding-energy frequency!

t (μs)

2|| a

2|| m

0 20 40 60 80 100154

155

156

157

158

159

160

161

162

163

t (s)

B (

Ga

uss

)

tevolve

Binding energyBinding energy

Oscillation 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)

EB (

kHz)

Oscillations

Coupled channels

Simulation experiment (2)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

10 15 20 25 30 35 400

2000

4000

6000

8000

10000

12000

14000

16000

18000

t (s)

Num

ber

Ramsey FringesRamsey 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 statetevolve (μs)

Num

ber

Atomic condensate

Non-condensate

Change pulse shapeChange pulse shape

Longer fall time

155 G

10 s 160 s

tevolve (s)

2 4 6 8 10 12 14 16

Num

be

r

0

4000

8000

12000

16000

tevolve (s)

10 15 20 25 30 35 40

Num

ber

(x10

-3)

0

4

8

12

16

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

short fall time

long fall time

Remnant+burst

Remnant

Burst

More molecules!More molecules!

Precision binding energy measurementPrecision binding energy measurement

tevolve

(s)10 20 30 40

Re

mn

an

t N

um

be

r (

x10-3

)

9

10

11

12

13

14

=196.6(11) kHz

oscillation freq. +

B-field (pulsed NMR)

Bound statespectroscopy

Improving the interactomic potentialsImproving the interactomic potentials

B (G)156 157 158 159 160 161 162

Fre

quen

cy (

kHz)

10

100

1000

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

How to detect the moleculesHow to detect the molecules

25 50 75 100-500

-250

0

250

12840

12880

12920

12960

ener

gy (c

m-1

)

internuclear distance (a0)

Og

- (5P3/2

)

ground state (5S1/2

)

v = 0 - 10

molecule

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

Look for bound-bound transitions.

Short laser pulse

- Scott Papp, Sarah Thompson, Carl Wieman

Stern-Gerlach separatorStern-Gerlach separator

dimer 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> 0 < 0

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

Molecules remain

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

ConclusionsConclusions

PRL 89, 180401 (2002), PRL 87, 120406 (2002), PRA 65, 053617 (2002)