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The Efimov Effect in Ultracold Gases
Weakly Bounds Systems in Atomic and Nuclear PhysicsMarch 8 - 12, 2010
Institut für Experimentalphysik, Universität Innsbruck
Martin Berninger, Francesca Ferlaino, Alessandro Zenesini, Walter Harm, Hanns-Christoph Nägerl, Rudi Grimm
The Efimov Puzzle (an experimentalists view...)
TheoryExperiment
Efimov States in the molecules and nuclei, Rome 2009
Weakly-Bound Systems in Atomic and Nuclear Physics, Seattle 2010
Outline
• Introduction atomic few-body physics• The Efimov scenario• Experimental Efimov physics with Cs• Overview experimental Efimov physics• New results in caesium samples
(preliminary)
• Collisions in Dimer-Dimer samples• Ultracold exchange reactions
last bound level
(y r) halo dimer
kHz
2
2
maEb
2-body
CsCs
4-body
Cs2Cs2
Cs2
CsCs
Cs3 Cs
CsCsCsCs
Few-body physics
3-bodyCs2
CsCsCsCs
In general complex problem:• strong dependence on potential
k
ka
k
)(tanlim 0
0
manyvib.
levels
non-universal dimer
U(r)~1/r6
r
U(r)
schematic drawing
THz
Universal regimescattering length a>>r0
r0: range of the potential
r0 ~ lvdW ~ 100a0 for Cs
s wave scattering length:
dimers trimers tetramers
Universal connection:
?Halo
dimersEfimovtrimers
Universaltetramers
Ultracold atomic gases as a model system
nT3 1
1
T 2h2
mkBT
Quantum gas
Classical gasT ~ 1µK – 20nK
Temperature
Control knobs
Interactions
Interaction strength a
crossed-beam trap
wy ≈ w z ≈ wx
3D3D 2D2D 1D1D
Geometry optical lattice
„pancake“ trap
Mixtures
• different interactions• different mass ratios• Bosonic / Fermionic systems
State
mF=3
F=3
mF=4
F=432
MW transfer
caesium
magnetic moment of bound statediffers from the magnetic moment of the incident channel
B
a
abg
B0
F=3+F=3
F=3+F=4
F=4+F=4
(example Cs)
r
U(r)
incident channel
bound state
2
2
bg
bma
E
Tunable interaction:
Feshbach resonance
Magnetic tunability of the scattering length
energya < 0 a > 0
a/1
Two-particle picture
attractive repulsive
halodimer
s-wave resonances for Cs in F1=3 F2=3 channel
50 G 100 G
-1000
1000
2000
-2000
3000
0
magnetic field (G)
sca
tte
ring
len
gth
(a
0)
0 150 G
s-wave + d-wave resonances in Cs
bound state in open channel: EB~10kHzbackground scattering length abg~2000a0
EbB
44(6) 34(7) 34(6)
F1 F2 (F1+F2)E. Tiesinga et al.
energya < 0 a > 0
The Efimov scenario
„Efimov – states“
halodimer
×22.7
×(22.7)2
...there exists an infinite series of weakly bound trimer states for resonant
two-body interaction...V. Efimov, Phys. Lett. B 33, 563-664 (1970)
weakly bound trimer
even more weaklybound trimer
a/1
a < 0
deeply bound dimer
Trap loss
energy3
3 AA nLn
3-atomic Efimov resonance
OFF resonance
ON resonancenew decay channel Enhancement of losses
10nK
200nK
3-Atomic Efimov resonance
Kraemer et al., Nature 440, 315 (2006)
three-body recombination rate
33 AA nLn
aLm
4
1
333
2
a4
recombination length:
energy
Ultracold sample of 133Cs atomsin atomic ground state: F=3, mF=3N ~ 105 atomsT = 10/200nK
3-atomic Efimov resonance
10nK
200nK
3-Atomic Efimov resonance
Kraemer et al., Nature 440, 315 (2006)
three-body recombination rate
33 AA nLn
aLm
4
1
333
2
a4
recombination length:
energy
20
2 sinh)]/ln([sin
)2sinh(4590)(
aasaC
43 )(3 a
maCL
• for a<0, a :
C(a)=C(22.7a)
Braaten & Hammer
am
aCAD
)(
*2
*02
*
sinh)]/ln([sin
)2sinh(2.10)(
aas
aCAD
• for a>0, a :
)]/ln([(cos1.67)( 0
22 aaseaC
)1(8.16))2sinh( 4 e
Atom-Dimer relaxation rate :
s0=1.00624
Braaten-Hammer theory
aAAA=-850 a0 amin=210 a0
L3max=5.7*10-22 cm6/s
L3min=1.33*10-28 cm6/s
3-atomic Efimov resonance
energyE
20
2 sinh)]/ln([sin
)2sinh(4590)(
aasaC
43 )(3 a
maCL
• for a<0, a :
C(a)=C(22.7a)
Braaten & Hammer
am
aCAD
)(
*2
*02
*
sinh)]/ln([sin
)2sinh(2.10)(
aas
aCAD
• for a>0, a :
)]/ln([(cos1.67)( 0
22 aaseaC
)1(8.16))2sinh( 4 e
Atom-Dimer relaxation rate :
s0=1.00624
a > 0
halodimer
a/1
s-wave state
d-wavestate
# dimer: ~ 4000
# atoms: (3-6)x104
T = 30-300 nK
Separate atoms and dimers by magnetic gradient field before imaging
Measure the time-evolution & extract atom-dimer relaxation rate coefficient b
Production of 6s-molecules viaFeshbach association
Atom-dimer Efimov resonance
Atom-dimer resonance at B=25 G aAD=+400 a0
• universality a>0 and a<0 via a=0 ?• transition universal to non-universal ? (r0~100a0)• any relation to Efimov physics at different
Feshbach resonances (@800G)?
Universal relation via pole:)1'()'()( 7.2206.1/ nnn
AAAn
AD aa
for n=0, n‘=1 aAD/aAAA= 0.47
Knoop et. al., Nature Physics 5, 227 (2009)
1/a
a < 0 a > 0
Atom-dimer Efimov resonance
energya < 0 a > 0
Tetra1
Tetra2
The extended Efimov scenario
Prediction of twouniversal 4-body statestied to each Efimov trimer!
H. Hammer and L. Platter, Eur. Phys. J. A 32, 113 (2007)J. von Stecher, J. P. D’Incao, and C. H. Greene, Nature Physics 5, 417 - 421 (2009)
F. Ferlaino et. al., PRL 102, 140401 (2009)
Tetra1 Tetra2
thold=250ms thold=8ms
Four-body states - experimental results
Experiment ~ 0.47 a*T
~ 0.84 a*T
Position of theuniversal 4-body states
Theorya*Tetra1 ~ 0.43 a*T
a*Tetra2 ~ 0.9 a*T
4-bodymixed3-body
Fitting function
simple 3 body
simple 4 body
3 + 4 body
44
33 AAA nLnLn
F. Ferlaino et. al., PRL 102, 140401 (2009)
Tetra1 Tetra2
thold=250ms thold=8ms
Four-body states - experimental results
44
33 AAA nLnLn
Experiment ~ 0.47 a*T
~ 0.84 a*T
Position of theuniversal 4-body states
Theorya*Tetra1 ~ 0.43 a*T
a*Tetra2 ~ 0.9 a*T
Overview experimental Efimov physics
Barontini et al., Phys. Rev. Lett. 103, 043201 (2009)
Ottenstein et al., Phys. Rev. Lett. 101, 203202 (2008)Huckans et al., Phys. Rev. Lett. 102, 165302 (2009)Williams et al., Phys. Rev. Lett. 103, 130404 (2009)Wenz et al., Phys. Rev. A 80, 040702(R) (2009)
41K + 87Rb
6Li
Fermionic systems
Bosonic mixtures
Bosonic systems
Zaccanti et al., Nature Physics 5, (2009)
Pollack et al., Science 326 (2009)Gross et al., Phys. Rev. Lett 103, 163202 (2009)
Kraemer et al., Nature 440, 315 (2006)Knoop et. al., Nature Physics 5, 227 (2009)F. Ferlaino et. al., Phys. Rev. Lett. 102, 140401 (2009)
133Cs
39K
7LiF=1, mF=1
F=1, mF=0
Successive Efimov Features – bosonic system (39K)
Zaccanti et al., Nature Physics, Vol. 5 (2009)
Florence-Group
Comparison with universal theory:Valid only for |a|>>r0
Model for finite-range interactions?
Res:
second order process: A+A+A D*+AaAD* losses in an atom sampledue to elastic scattering
Loss a (a0)
a<03B Max a1
- -1500
4B Max aT* -650
a>0
3B Mina1
+ 224
a2+ 5650
AD Maxa1* 30
a2* 930
Experiment with 39K atomic sampleacross Feshbach resonance, r0=64a0
atomic threshold
Usually, in the three-body process 3 particles are lost
Efimov physics in 39K: AD resonances
Thanks to M. Zaccanti & Co-Workers for the slides!
…but if AD cross section is large particle losses can be >>3!!!
Efimov physics in 39K: AD resonances
Thanks to M. Zaccanti & Co-Workers for the slides!
Successive Efimov Features – bosonic system (7Li – F=1,mF=1)
Rice-Groupatomic sample 7Li (F=1,mF=1) across Feshbach resonance, r0=33a0
Pollack et al., Science 326 (2009)
Comparison universal theoryValid only for each side, systematic discrepancy (factor 2)
Variation in the short range phase acrossthe Feshbach resonance?
Loss a (a0) a (a0)
a<0
3B Max a1- -298 a2
- -6301
4B MaxaT
1,1 -120 aT1,2 -295
aT2,1 -2950 aT
2,2 -6150
a>0
3B Mina1
+ 224
a2+ 5650
AD Maxindirect
a2* 608
DD Maxdebate
a*2,1 1470
a*2,2 3910
Res:a>0
a<0
a
Ottenstein et al., PRL 101, 203202 (2008)Huckans et al., PRL 102, 165302 (2009)Williams et al., PRL 103, 130404 (2009)Wenz et al., PRA 80, 040702(R) (2009)
Braaten et al., PRL 103, 073202 (2009)Naidon et al., PRL 103, 073203 (2009)Floerchinger et al., PRA 79, 053633 (2009)Braaten et al., PRA 81, 013605 (2010)
Jochim & O‘Hara6Li 3 componentFermi-Spin-mixture:
|3> mF= -3/2|2> mF= -1/2|1> mF= 1/2
Comparison with universal theoryUsing fit results for high field resonance (895G)reproduces low field resonances accurately: 125(3)G & 499(2)GÞ No change in the three body parameter
for B ~ 750G? for aij ~ lvdw?
Efimov features in fermionic spin mixtures (6Li)
Loss state B(G)
a<0 3B Max
n=0 127
n=0 500
n‘=1 895
Res:
Gross et al., PRL 103, 163202 (2009)
Khaykovich-Groupatomic sample 7Li (F=1,mF=0)across Feshbach resonance
Comparison with universal theory:a+/|a-| = 0.92(14) (Theory=0.96(3))
Why does 7Li agree so nicely in (F=1,mF=0)
and not in (F=1,mF=1)?
Bosonic system showing universality (7Li – F=1,mF=0)
Loss a (a0)
a<0 3B Max a- -264
a>0 3B Min a+ 1160
Results:
Barontini et al., Phys. Rev. Lett. 103, 043201 (2009)
Efimov Resonances – Heteronuclear systems (41K + 87Rb)
Florence-GroupSystem composed of distinguishable particles with different masses
Experiment with bosonic mixture of 41K and 87Rbat a interspecies Feshbach resonanceÞ Two resonantly interacting pairs are
sufficient for Efimov physicsÞ Existence of two Efimov series:
KRbRb: exp(/s0) = 131KKRb: exp(/s0) = 3.51105
Results:
KKRb-resonance
Loss a (a0)
a<03B Max KRbRb -246
3B Max KKRb -22000
a>0AD Maxindirect
a* 667
No oscillations for a>0 observed
K3
scattering length
900
K3
B(G)800
6d6
B (Gauss)
preliminary
K3
preliminary
Lifetime measurements @ high magnetic fields
Recombination rate @ 6s6 resonance ~ 800G, width ~ 90GT~200nK
Resonance!
23
5max3 )(
844
TkmL
B
Unitarity limit:
Another piece to the puzzle!
L 3L 3
f l mf
nD= -L2 nD 2Measuring relaxation rate L2:
Ferlaino et al., PRL 101, 023201 (2008)
Experimental results: dimer-dimer collisions
s-wave state
d-wavestate
2 atoms in F=3, mF=3microwaveSample of universal dimers in 6s-state:
crossed dipole trap (1060nm) ND ~ 4000 T ~ 40 – 350 nK kBT << EB ~ h50kHz << EvdW ~ h2.7MHz
105 ultracold 133Cs atoms (40nK) Feshbach associationÞ Removal of atoms with microwaveÞ Sample of ultracold dimers
0 200 400 600 800 1000
0,1
1
120 nK
re
laxa
tio
n r
ate
(10-1
0 c
m3/s
)
scattering lenght (a0)
0 200 400 600 800 1000
0,1
1
80 nk
re
laxa
tio
n r
ate
(10-1
0 c
m3/s
)
scattering lenght (a0)
0 200 400 600 800 1000
0,1
1
40 nK
re
laxa
tio
n r
ate
(10-1
0 c
m3/s
)
scattering lenght (a0)scattering length (a0)
.~L,1
~1
~ 22/1constv
Tv inin
2-body reaction cross section (Wigner 1948)
energya < 0 a > 0
Tetra1
Tetra2
?
Exchange reactions with distinguishable particles
B + A2
F=4, mF=2, 3 or 4 Feshbach molecule / halo dimer2x (F=3, mF=3)
mF=3
F=3
F=4 2
mF=43
MW transfer
A + A2
F=3, mF=3
?
total lossexchange
T=50 nK
b: atom-dimer loss rate coefficient
Exchange reactions loss rates
Knoop et al., Phys. Rev. Lett. 104, 053201 (2010) Theory: Jose D’Incao & Brett Esry
B
E
A+A+B
A2+B
A+ABDE
new decay channel
mF=4
mF=3
mF=2
resonance @ 35 G:opening exchange channel
B
E
A+B
A+A
AB
A2
A2(v=-1)+B → A+AB(v=-1)
Closer look around 35 G
appearance of trapped atoms in state A!
Ultracold exchange reactioncontrolled by magnetic fieldT=100 nK, thold=22ms
mF=4
mF=3mF=2
Role of the large scattering length
A2(v=-1)+B
A+AB(v’<v)
A+AB(v=-1)
A2(v’<v)+B
A+A
A+B(mF=2)
A+B(mF=3)A+B(mF=4)
(y r)
2
2
maEb
2/ar
TheoryExperiment
Experimentalists wish list for Theory
Is there any relation for Efimov physics at different Feshbach resonances (133Cs low fields and Feshbach resonance @ 800G)?
Model for finite-range interactions, transition universal to non-universal (39K & 133Cs)?
Variation in the short range phase across the Feshbach resonance (7Li) – Factor 2?
Why does 7Li agree so nicely in (F=1,mF=0) and not in (F=1,mF=1)?
Why there is no change in the three body parameter in 6Li spin mixture for B ~ 750G and/or for aij ~ lvdw?
Coming soon:
Cs data for 800G resonance
Any connection of Efimov physics from a>0 to a<0 via a=0 (133Cs)? – Factor 1/2?
Temperature dependence in 133Cs halo molecules?
a << a
The Caesium-Efimov-Team
M.B.
Rudi Grimm
FrancescaFerlaino
AlessandroZenessini
Hanns-Christoph
Nägerl
WalterHarm