Studying Bose-Einstein Condensates with a linear accelerator · 1. Production and Acceleration of...

Studying Bose-Einstein Condensates with a linear accelerator

Christian Buggle, Jeremie Leonard, Wolf von Klitzing, Tobias Tiecke, Jook Walraven

Van der Waals – Zeeman InstituteFOM-Institute AMOLF

University of AmsterdamLes Houches: 14 February 2005

references

1. Production and Acceleration of two clouds• Tiecke, Kemmann, Buggle, Shvarchuck, von Klitzing and Walraven,

J. OPT. B 5 (2003) 119• Thomas, Wilson, Foot PRA 65 (2002) 063406

2. Observing the collision of two condensates• Buggle, Leonard, von Klitzing, Walraven, PRL 93 (2004) 173202• Thomas, Kjaergaard, Julienne, Wilson, PRL 93 (2004) 173201

3. Getting the elastic scattering amplitudes at any low energy• Buggle, Leonard, von Klitzing, Walraven, PRL 93 (2004) 173202

4. Interference observed in Feshbach-dissociation of ultra-cold molecules• Volz, Durr, Syassen, Rempe, van Kempen, Kokkelmans, cond-mat/0410083

Ioffe Quadrupole TrapRadial

pinch coils

offset coils

Ioffe bars

Axial

Ioffe-Pritchard Quadrupole Trap

Trap parameters:• axial level splitting 1 nK• radial level splitting 23 nK

23ωω

||

≈⊥

-1s 4772πω ×=⊥

-1|| s 212πω ×=

cm 104 -3140 ×=n

105.3 6×=N

µK 17.1 0 =T

Ioffe-Pritchard Trap||. BBµE ∝−=

rrTrapping potential:

Radial|B|

-4 -3 -2 -1 1 2 3 4

-4-2

246810

Axial (z)|B|

B0 < 0

|B|

Majorana spin flips

-1s 4772πω ×=⊥-1

|| s 212πω ×=

Two wells, but...

B0 > 0

B0

B0

Rotating the Trap Axis (TOP-field)

Add a rotating magnetic field perpendicular to Z:

“Circle of death”

Petrich, Anderson, Ensher, Cornell, PRL 74 (1995) 3352

Thomas, Wilson, Foot PRA 65 (2002) 063406

Tiecke, Kemmann, Buggle, Shvarchuck, von Klitzing and Walraven, J. OPT. B 5 (2003) 119

Producing two clouds

1) Pre-cooling in standard Ioffe-Pritchard trap

2) Apply rotating field andramp B0 to negative values

3.6 mm max.

=>

Tiecke, Kemmann, Buggle, Shvarchuck, von Klitzing and Walraven, J. OPT. B 5 (2003) 1193) Final cooling to BEC

Double BEC

Production of a Double cloud.

Thermal:Gaussian shape

~ 1 mm

Condensed:Parabolic shape(Thomas-Fermi)

Acceleration Principle

B0 > 0Rot. field OFF

axial radial

B0 < 0Rot. field ON

Z

Acceleration Principle

axial radial

PracticeB0-ramp

PrincipleB0-jump

collision of two ultracold clouds

Ec/kB=138 µK Ec/kB=1230 µK

Y00 (θ)

s-waveY2

0 (θ)d-wave

3D spatial distribution to be retrieved

Thomas, Kjaergaard, Julienne, Wilson, PRL 93 (2004) 173201

=>TOMOGRAPHY transformation

tomography

Reconstruction of 3D images from a set of 2D x-ray pictures (scanner) Nobel prize in medicine in 1979 (A. M. Cormack and G. N. Hounsgield)

AXIAL SYMMETRY: only ONE picture is neededBessel Function

1D F.F.T. z

x

Optical density

z

ρ

3D density

see e.g. : M. Born and E. Wolf, Principles of Optics,7th (expanded) Edition, Cambridge University Press, Cambridge 1999.

collision energy = 138 µK

(almost) pure s-waveExperiment:

zθ

),(),(OD θrnyx →

Tomography

cos(θ)

W(θ)

Differential cross section

)(),( θWθrn →

Optical density

Theory : 2)()(2)( θπθπθσ −+= ff (identical bosons => symetrization)

2

sin)(cos)12(22)( ∑=

+=evenl

li

llePl

kηθπθσ η η l : phase shift for the partial wave l

Pl : Legendre polynomials2

222

02 sin)1cos3(25sin8

0 ηθηπ ηη ii eek

−+≈

s-wave d-wave

(Low Energy)

| + | 2 ⇒ matter-wave interference

collision energy = 1.23 mK

cos(θ)

z

θ

Experiment:(almost) pure d-wave

Experiment:

Optical density

),(),(OD θrnyx →

Tomography

cos(θ)

W(θ)

Differential cross section

)(),( θWθrn →

)()( θσθ ∝W2

22

02 sin)1cos3(25sin8

20 ηθηπ ηη −+≈ ii eek

( ) θθθσσπ

dsin2/

0∫=

Otago results

Thomas, Kjaergaard, Julienne, Wilson, PRL 93 (2004) 173201

collision energy = 1.23 mK

cos(θ)z

θ

Experiment:

(almost) pure d-wave

)(),(),(OD θWθrnyx →→

Optical density Tomography Differential cross section

-1 0 1 2

12345

W(θ)

u

W(θ)

)1cos3( 2 −≡ θu

)()( θσθ ∝W2

22

02 sin)1cos3(25sin8

20 ηθηπ ηη −+≈ ii eek

)1( 2uBuAC ++=

parabolic fit => (A, B) => (η0,η2)

Scattered waves and Phase shifts

)()(22

rmkrH rhrψψ = ),()()( ϕθ

χψ m

ll

krrr Υ⋅=

rwith

( ) ( ) 0,)1()(, 22 =

+

−−+ rkrllrVkrk ll χχ&&Radial Schrödinger Equation :

0)( 6

6

∞→→−≈rr

CrV 0)1(2 ∞→

→+

rrllAsymptotic solution (r →∞):

)2

sin( πηχ lkr ll −+≈⇒

Obtained from the experiments

Scattering amplitude, differential cross section2

22

02 sin)1cos3(25sin8)( 20 ηθη

πθσ ηη −+≈ ii ee

kPhase Shift

Obtaining the differential cross section

at ANY (low) energy ...

Integrating the Schrödinger equation

0 1000 2000

0 1000 2000

0 1000 2000

0 1000 2000

(a0)

experimental phase shifts:

boundary conditionfor r → ∞

s-waves

d-waves

( )kη0

( )kη2

inward

Long-range part only:

66)(rCrV −=

( ) ( ) 0,)1(, 66

22 =

+

+−+ rk

rC

rllkrk ll χχ&&

Integrating the Schrödinger equation

0 20 40 60 80

(a0)

0 1000 2000

0 1000 2000

(a0)

all are in phase at rin

rinexperimental phase shifts:

boundary conditionfor r → ∞

s-waves

d-waves

( )kη0

( )kη2

inward

ResultsPh

ase

Shift

η0

Phas

e Sh

ift η

2

d-wave

s-wave

Collision Energy (µK)1 10 100 1000

-2

0

2

-π/2

π/21 10 100 1000

-2

0

2

-π/2

π/2

knowing only C6/r6

we extractONE accumulated phase

at rin = 20 a0It allows us to calculate

the phase shifts atANY (low) energy

ResultsPh

ase

Shift

η0

Phas

e Sh

ift η

2

d-wave

s-wave

Collision Energy (µK)

knowing only C6/r6




1 10 100 1000-2

0

2

-π/2

π/2

1 10 100 1000-2

0

2

-π/2

π/2

Buggle, Leonard, von Klitzing, Walraven, PRL 93 (2004) 173202

ResultsPh

ase

Shift

η0

Phas

e Sh

ift η

2

d-wave

s-wave


1 10 100 1000-2

0

2

-π/2

π/2

1 10 100 1000-2

0

2

-π/2

π/2


knowing only C6/r6




Results

atriplet= + 102(6)a0

( ) kakk

−=→ 00

limη

atriplet = 98.99(2)a0

Van Kempen, Kokkelmans, Heinzen, Verhaar PRL 88, 93201 (2002)

Phas

e Sh

ift η

0Ph

ase

Shift

η2

d-wave

s-wave


1 10 100 1000-2

0

2

-π/2

π/2

1 10 100 1000-2

0

2

-π/2

π/2


Phase Shifts and Elastic Cross-Section

1 10 100 10000

1x10-11

2x10-11

3x10-11

4x10-11

3

0

1

2

Elas

tic C

ross

Sec

tion

(cm

2 )

Collision Energy (µK )

( ) ∑∫=

+==evenl

llk

d ηπ

θθθσσπ

22

2/

0

sin)12(8sin

Phas

e Sh

ift η

0Ph

ase

Shift

η2 d-wave

s-wave


1 10 100 1000-2

0

2

-π/2

π/2

1 10 100 1000-2

0

2

-π/2

π/2

d-wave resonance obtained at 300(70) µK

first Ramsauer minimum at 2.1(2) mK

Comparison with the Otago results

Thomas, Kjaergaard, Julienne, Wilson,PRL 93 (2004) 173201

Phas

e Sh

ift η

0Ph

ase

Shift

η2 d-wave

s-wave


1 10 100 1000-2

0

2

-π/2

π/2

1 10 100 1000-2

0

2

-π/2

π/2

σ[1

0-12

cm2 ]

Comparison with the Garching results

‘closed’ channel

0=l

2=l

87Rb F =2 mF=2

0=l

2=l

87Rb F =1 mF=1

Volz, Durr, Syassen, Rempe, van Kempen,Kokkelmans, cond-mat/0410083

Molecular association by Feshbach tuning


BB ETk <<

87Rb F =1 mF=1


B

EB

0

Molecular dissociation by Feshbach tuning


87Rb F =1 mF=1

)}(){,()( 022

000

20 θββ δδ YeYetrgr ii −=Ψrr

β0,β2 = branching ratios β0+β2 = 1


B

EB

0

Molecular dissociation by Feshbach tuning


87Rb F =1 mF=1


)}(){,()( 022

000



dissociation versus collision

Branching ratio’s

Molecular dissociation:

)}(){,()( 022

000



Collision between free atoms (plane incident wave):

)}(sin5sin{),(')( 022

000

20 θδδ δδ YeYetrgr ii −=Ψrr

22

02

22

2 sin5sinsin5

δδδ

β+

=β0 = 1-β2 and

Comparison with the Garching results

0.1 1 100.0

0.2

0.4

0.6

0.8

1.0

Beta2 from Servaas phase shifts Beta2 from our individual data points Beta2 from our Accumulated-phase best phase shifts.

Bran

chin

g ra

tio fo

r the

d-w

ave

B-Bres [G]

0.0 0.5 1.0 1.5 2.0

B-Bres [G]

=> The Feshbach resonance shifts the d-wave shape resonance towards lower energies

Summary

• ‘New’ method to determine scattering amplitudes

+ C6/r6

1 10 100 10000

1x10-11

2x10-11

3x10-11

4x10-11

3

0

1

2

Elastic Cross Section (cm

2 )

Colision Energy ( µK )

• Requires very little a priori knowledge of the potential Only C6 (single-atom property)

• Provides fairly accurate results

• Provides good agreement with other determinations

Group Quantum Gases

Wolf von Klitzing, Paul Cleary, JTMW, Jeremie Leonard, Christian Buggle

Join us in Amsterdam!

Studying Bose-Einstein Condensates with a linear accelerator · 1. Production and Acceleration of...

Documents

Transcript of Studying Bose-Einstein Condensates with a linear accelerator · 1. Production and Acceleration of...