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
we extractONE accumulated phase
at rin = 20 a0It allows us to calculate
the phase shifts atANY (low) energy
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
Collision Energy (µK)
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
knowing only C6/r6
we extractONE accumulated phase
at rin = 20 a0It allows us to calculate
the phase shifts atANY (low) energy
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
Collision Energy (µK)
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
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
Collision Energy (µK)
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
Collision Energy (µK)
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
‘closed’ channel
BB ETk <<
87Rb F =1 mF=1
Volz, Durr, Syassen, Rempe, van Kempen,Kokkelmans, cond-mat/0410083
B
EB
0
Molecular dissociation by Feshbach tuning
‘closed’ channel
87Rb F =1 mF=1
)}(){,()( 022
000
20 θββ δδ YeYetrgr ii −=Ψrr
β0,β2 = branching ratios β0+β2 = 1
Volz, Durr, Syassen, Rempe, van Kempen,Kokkelmans, cond-mat/0410083
B
EB
0
Molecular dissociation by Feshbach tuning
‘closed’ channel
87Rb F =1 mF=1
Volz, Durr, Syassen, Rempe, van Kempen,Kokkelmans, cond-mat/0410083
)}(){,()( 022
000
20 θββ δδ YeYetrgr ii −=Ψrr
β0,β2 = branching ratios β0+β2 = 1
dissociation versus collision
Branching ratio’s
Molecular dissociation:
)}(){,()( 022
000
20 θββ δδ YeYetrgr ii −=Ψrr
β0,β2 = branching ratios β0+β2 = 1
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
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